Citation: Tian, D and Deng, J and Zio, E and Maio, F and Liao, F (2018) Failure modes detection of nuclear systems using machine learning. In: 2018 5th International Conference on Depend- able Systems and Their Applications (DSA). IEEE, pp. 35-43. ISBN 9781538692660 DOI: https://doi.org/10.1109/DSA.2018.00017 Link to Leeds Beckett Repository record: https://eprints.leedsbeckett.ac.uk/id/eprint/7341/ Document Version: Book Section (Accepted Version) c 2018 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. The aim of the Leeds Beckett Repository is to provide open access to our research, as required by funder policies and permitted by publishers and copyright law. The Leeds Beckett repository holds a wide range of publications, each of which has been checked for copyright and the relevant embargo period has been applied by the Research Services team. We operate on a standard take-down policy. If you are the author or publisher of an output and you would like it removed from the repository, please contact us and we will investigate on a case-by-case basis. Each thesis in the repository has been cleared where necessary by the author for third party copyright. If you would like a thesis to be removed from the repository or believe there is an issue with copyright, please contact us on [email protected]and we will investigate on a case-by-case basis.
Transcript
Citation:Tian, D and Deng, J and Zio, E and Maio, F and Liao, F (2018) Failure modes detection ofnuclear systems using machine learning. In: 2018 5th International Conference on Depend-able Systems and Their Applications (DSA). IEEE, pp. 35-43. ISBN 9781538692660 DOI:https://doi.org/10.1109/DSA.2018.00017
Link to Leeds Beckett Repository record:https://eprints.leedsbeckett.ac.uk/id/eprint/7341/
The aim of the Leeds Beckett Repository is to provide open access to our research, as required byfunder policies and permitted by publishers and copyright law.
The Leeds Beckett repository holds a wide range of publications, each of which has beenchecked for copyright and the relevant embargo period has been applied by the Research Servicesteam.
We operate on a standard take-down policy. If you are the author or publisher of an outputand you would like it removed from the repository, please contact us and we will investigate on acase-by-case basis.
Each thesis in the repository has been cleared where necessary by the author for third partycopyright. If you would like a thesis to be removed from the repository or believe there is an issuewith copyright, please contact us on [email protected] and we will investigate on acase-by-case basis.
Robust-Optimal Integrated Control DesignTechnique for a Pressurized Water-type Nuclear
Power PlantVineet Vajpayee, Victor Becerra, Nils Bausch, Jiamei Deng, S. R. Shimjith, A. John Arul
Abstract—A control design scheme is formulated for a pressur-ized water type nuclear power plant by integrating the optimallinear quadratic Gaussian (LQG) control with the robust integralsliding mode (ISM) technique. A novel robust-optimal hybridcontrol scheme is further proposed by integrating the LQG-ISMtechnique with the loop transfer recovery approach to enhancethe effectiveness and robustness capability. The control archi-tecture offers robust performance with minimum control effortsand tracks the reference set-point effectively in the presence ofdisturbances. The multi-input-multi-output nuclear power plantmodel adopted in this work is characterized by 40 state variables.The nonlinear plant model is linearised around steady-stateoperating conditions to obtain a linear model for controllerdesign. The efficacy of proposed controllers is demonstratedby simulations on different subsystems of the nuclear powerplant. The control performance of the proposed technique iscompared with other classical control design schemes. Statisticalmeasures are employed to quantitatively analyse and comparethe performance of the different controllers that are studied inthe work.
Index Terms—Optimal Control, Robust Control, PressurizedWater Reactor, Nuclear Power Plant.
I. INTRODUCTION
Nuclear power plants (NPPs) are complex non-linear sys-tems. Control of NPPs pose challenges due to parameter vari-ations caused by fuel burn-up, internal reactivity feedbacks,modelling uncertainties, and unknown disturbances. Systemparameters associated with reactor core, thermal-hydraulics,reactivity feedbacks, etc., differ significantly with operatingconditions. The routine load cycles over a broad range ofpower variations can significantly affect plant performance.Uncertainties in the actuator signals and noisy sensor measure-ments add further complexities to the control design problem.Consequently, traditional controllers often fail to deliver de-sirable performance. The plant control systems must be ableto respond promptly and safely to fast variations in demandin an uncertain environment. Thus, it is essential to develop
Vineet Vajpayee ([email protected]), Victor Becerra ([email protected]), and Nils Bausch ([email protected]) are withSchool of Energy and Electronic Engineering, University of Portsmouth,Portsmouth, PO1 3DJ, United Kingdom.
Jiamei Deng ([email protected]) is with School of Built Environ-ment, Engineering, and Computing, Leeds Beckett University, Leeds, LS63QS, United Kingdom.
S. R. Shimjith ([email protected]) is with Reactor Control System DesignSection, Bhabha Atomic Research Centre, Mumbai, 400 085, India and HomiBhabha National Institute, Mumbai, 400 094, India.
A. John Arul ([email protected]) is with Probabilistic Safety, ReactorShielding and Nuclear Data Section, Indira Gandhi Centre for AtomicResearch, Kalpakkam, 603 102, India.
improved control techniques which can provide closed-loopstability and enhance the safety and operability of NPPs.
A considerable amount of research has been done onthe application of robust control techniques in nuclear re-actors, especially for the core power control. In the lasttwo decades, various control design techniques such as statefeedback assisted control (SFAC) [1]–[5], H∞ control [6]–[9], model predictive control (MPC) [10]–[14], sliding modecontrol (SMC) [15]–[21], and soft-computing based controls[22]–[27] have been proposed to deal with disturbances anduncertainties in NPPs. Edwards et al. [1] proposed the idea ofSFAC to enhance the stability of the classical control loop byintegrating a state-feedback compensating loop. Loop transferrecovery (LTR) technique has been combined with linearquadratic Gaussian (LQG) control in an SFAC framework forthe improvement of reactor temperature and power controlsduring variation in reactor parameters [2]–[5]. H∞-basedcontrol schemes have been designed for reactor power controland to obtain enhanced robustness over the classical LQGcontrol scheme [6]–[9]. To deal with system design constraintsin an uncertain NPP system, receding horizon-based robustMPC approaches, which solve an optimization problem ateach sampling instant, have been proposed [10]–[14]. SMCis another robust control design technique applied to study theload-following problem of nuclear reactors. SMC guaranteesrobustness to the uncertainties entering through the inputchannel, once the system reaches the sliding manifold [15]–[21]. However, its implementation is sensitive to uncertaintiesduring the reaching phase. To avoid this, an integral slidingmode approach has been proposed in the literature whichenforces the system trajectories to lie on the sliding manifoldfrom the very beginning thereby avoiding the reaching phase[28]. To deal with modelling uncertainties and disturbances,soft-computing techniques have been further incorporated inthe controller design. Robust PID controller [22], fractional-order PID controllers [23], neural network controllers [24],emotional controllers [25], fuzzy logic controllers [26], andgenetic algorithms optimized controllers [27] have been pro-posed to enhance the capabilities of the classical controllers.
Generally, a robust controller often has to spend high controlenergy to achieve satisfactory performance in an uncertainenvironment, which may sometimes lead to saturation ofactuators. Practically, a robust control strategy which spendsless control energy is desired. This stimulus leads to theadvancement of hybrid control strategies by integrating robustcontrol with optimal control techniques [20], [21], [29], [30].
2
SMC has been combined with optimal control to deisgn corepower control under the assumption that the complete stateinformation is available for control design [20], [21]. However,for instance, the concentration of delayed neutron precursorsis not directly measurable in a reactor. Thus, a state estimatoris required to estimate the unmeasurable states and to designthe state feedback control strategy [17], [19].
In this paper, a new control strategy for a PWR-type NPPis proposed by combining the optimal LQG control andthe robust integral sliding mode (ISM) design approaches.The proposed LQG-ISM technique consists of the combinedactions of a nominal controller and a discontinuous controller.The nominal controller uses the LQG approach, which in-volves a linear quadratic tracker (LQT) for state feedbackcontrol and a Kalman filter for states estimation. On theother hand, the discontinuous control employs the ISM ap-proach, which allows the system motion to be invariant todisturbances throughout the entire system response. This paperfurther proposes a robust-optimal hybrid control strategy byintegrating the LQG-ISM control with the LTR technique.The overall architecture thus offers enhanced robustness withimproved system performance in the presence of uncertaintiesand disturbances.
In the NPP control design literature, the coupling effectsamong the reactor-core, steam generator, pressurizer, turbine-governor, and different piping and plenum are most oftenignored while designing the individual controllers [1]–[27].The dynamics of actuators and sensors are also frequentlyomitted. Pragmatically, it is meaningful to develop controlmethods for the whole NPP system. However, there are veryfew results for controlling an entire NPP [31], [32]. In thisregard, the proposed work designs state feedback control tech-niques using estimated states for the integrated NPP model.Both proposed techniques are applied to different subsystemsof the PWR-type NPP. In particular, the paper addresses thefollowing problems: reactor power and temperature controls,steam generator pressure control, pressurizer pressure andlevel controls, and turbine speed control. The efficacy ofthe proposed work has been tested using simulations in theMATLAB/Simulink environment. The proposed techniqueshave been further compared with other classical techniques.The main contributions of the paper are listed below:
• Robust-optimal hybrid control techniques are proposed toimprove system performance and robustness with mini-mum control efforts in the presence of disturbances.
• Design, validation, and testing of the control technique isperformed for various control loops of a PWR-type NPP.
• Reactor power and temperature controls, steam generatorpressure control, pressurizer pressure and level controls,and turbine speed control problems are studied.
The reminder of the paper is organized as follows: SectionII presents the dynamic non-linear model of a PWR-type NPP.Section III formulates the control design problem. Section IVpresents the proposed control scheme. Section V implementsthe proposed technique on the PWR-type NPP model and dis-cusses its effectiveness through simulation results. Conclusionsare drawn in section VI indicating main contributions.
II. DYNAMIC MODEL OF A PWR NUCLEAR POWER PLANT
The key variables of the model equations given below aredescribed near their first occurrence, while the constant modelparameters are all described, along with their units, in thenomenclature section. Typical parameter values are given inTable A.1.
A. Reactor Core Model
The core-neutronics model consisting of normalized power(Pn) and precursor concentration of six group of delayedneutrons (Cin) is given by,
dPndt
=
ρt −6∑i=1
βi
ΛPn +
6∑i=1
βiΛCin, (1)
dCindt
= λiPn − λiCin, i = 1, 2, . . . 6. (2)
The neutronic power in a reactor can be monitored using ex-core detectors, placed outside the core, and their associatedamplifiers. The ex-core detector current (ilo) [33] is sensed as
τ1τ2d2ilodt2
+ (τ1 + τ2)dilodt
+ ilo = Klolog10 (κloPn) . (3)
The total reactivity (ρt) consists of reactivity due to rodmovement (ρrod), and feedbacks due to variations in fuel andcoolant temperatures and primary coolant pressure as
ρt = ρrod + αfTf + αcTc1 + αcTc2 + αppp, (4)
wheredρroddt
= Gzrod. (5)
B. Thermal-Hydraulics Model
The thermal-hydraulics model is governed by the Mann’smodel which relates the core power to the temperature dropfrom fuel (Tf ) to coolant nodes (Tc1 and Tc2),
dTfdt
= HfPn −1
τf(Tf − Tc1) (6)
dTc1dt
= HcPn +1
τc(Tf − Tc1)− 2
τr(Tc1 − Tcin) (7)
dTc2dt
= HcPn +1
τc(Tf − Tc1)− 2
τr(Tc2 − Tc1) . (8)
Resistance temperature detectors (RTDs) are used to sense thecoolant temperature and its transmitter at the inlet (Trtd1) andoutlet (Trtd2). The dynamics of these sensed temperatures canbe described by [33]:
dTrtd1
dt=
1
τrtd(−Trtd1 + 2Tc1 − Trxi) (9)
dTrtd2
dt=
1
τrtd(−Trtd2 + 2Tc2 − Trxu) (10)
A current signal (irtd) can be obtained from the sensed RTDsignals as
irtd = Krtd(((Trtd1 + Trtd2)/2)− Trxi0)
(Trxu0 − Trxi0)+ 4 (11)
3
C. Piping & Plenum Model
Hot-leg (Thot) and cold-leg (Tcold) pipings, reactor lower(Trxi) and upper (Trxu) plenums, steam generator inlet (Tsgi)and outlet (Tsgu) plenums can be represented by first orderordinary differential equations as [34],
dTrxudt
=1
τrxu(Tc2 − Trxu) , (12)
dThotdt
=1
τhot(Trxu − Thot) , (13)
dTsgidt
=1
τsgi(Thot − Tsgi) , (14)
dTsgudt
=1
τsgu(Tp2 − Tsgu) , (15)
dTcolddt
=1
τcold(Tsgu − Tcold) , (16)
dTrxidt
=1
τrxi(Tcold − Trxi) . (17)
D. Steam Generator Model
A U-tube type SG can be represented by five nodes inwhich, the primary coolant lump (PCL) (Tp1 and Tp2) andmetal tube lump (MTL) (Tm1 and Tm2) are represented bytwo nodes each [35],
dTp1dt
=1
τp1(Tsgi − Tp1)− 1
τpm1(Tp1 − Tm1) (18)
dTp2dt
=1
τp2(Tp1 − Tp2)− 1
τpm2(Tp2 − Tm2) (19)
dTm1
dt=
1
τmp1(Tp1 − Tm1)− 1
τms1(Tm1 − Ts) (20)
dTm2
dt=
1
τmp2(Tp2 − Tm2)− 1
τms2(Tm2 − Ts) . (21)
The secondary coolant lump (SCL) represent steam pressure(ps) by balancing mass, volume, and heat as,
dpsdt
=1
Ks[Ums1Sms1 (Tm1 − Ts) + Ums2Sms2 (Tm2 − Ts)
− (msohss − mfwcpfwTfw)] . (22)
where,
Ks = mws∂hws∂ps
+mss∂hss∂ps
−mws
(hws − hssνws − νss
)∂νss∂ps
(23)
E. Pressurizer Model
The water level (lw) in the pressurizer can be obtained byapplying mass balance equation on water and steam phase as,
dlwdt
=1
dsAp
(Ap (l − lw)K2p −
C2p
C1p
)dppdt
+1
C2p1
(C2p
dppdt− msur − mspr
)+msur
C1p(24)
The pressure (pp) can be obtained by applying volume andenergy balances of water and steam mixture with steamcompressibility as [36],
dppdt
=
Qheat + msur
(ppνsJpC1p
+ hw
C1p
)+
mspr
(hspr − hw + hw
C1p+
ppνwJpC1p
)mw
(K3p +
K4pppJp
)+
msK4pppJp
−Vw
Jp+
C2p
C1p
(hw +
ppνsJp
) (25)
where the intermediate variables are defined as,
C1p =dwds− 1 (26)
C2p = Ap (l − lw)dwdsK2p +AplwK1p (27)
K1p =∂dw∂pp
;K2p =∂ds∂pp
;K3p =∂hw∂pp
;K4p =∂νs∂pp
. (28)
The surge rate (msur) can be represented using coolanttemperatures at different nodes as
msur =
N∑j=1
VjϑjdTjdt
(29)
F. Turbine Model
The dynamical model of a turbine consisting of the high-pressure, intermediate-pressure, and low-pressure turbines isgiven by [37],
d2Php
dt +(Orv+τipτhpτip
)dPhp
dt +(
Orv
τhpτip
)Php =
(OrvFhp
τhpτip
)¯mso
+(
(1+κhp)Fhp
τhp
)d ¯mso
dt
d2Pip
dt +(Orvτhp+τipτhpτip
)dPip
dt +(
Orv
τhpτip
)Pip =
(OrvFip
τhpτip
)¯mso
d3Plp
dt +(Orvτhp+τipτhpτip
+ 1τlp
)d2Plp
dt +(Orv(τlp+τhp)+τip
τhpτipτlp
)dPlp
dt +(
Orv
τhpτipτlp
)Plp = OrvFlp ¯mso
(30)where the steam flow is ¯mso = mso/msor, msor is the ratedsteam mass flow rate. The steam flow rate can be modifiedusing the turbine-governor control valve coefficient (Ctg) as
mso = Ctgps (31)d2Ctgdt2
+ 2ζtgωtgdCtgdt
+ ω2tgCtg = ω2
tgKtgutg (32)
where utg is the input signal to the valve. The total mechanicaloutput of turbine (Ptur) is computed as,
Ptur = Php + Pip + Plp. (33)
where Php, Pip, and Plp are mechanical power outputs ofhigh-pressure, intermediate-pressure, low-pressure turbine, re-spectively.
The turbine-generator model also consists of a turbine speedsystem which produces the rate of change in turbine speed(ztur) in accordance with the difference in generator demandpower (Pdem) and turbine output as
dzturdt
=Ptur − Pdem
(2π)2JturzturItg
. (34)
4
III. PROBLEM FORMULATION
Consider a linear time invariant system given by
x(t) = Ax(t) +Bu(t) +Bξ(t) + ω(t)
y(t) = Cx(t) + υ(t) (35)
where x(t) ∈ Rn, u(t) ∈ Rm, y(t) ∈ Rl, and ξ(t) ∈ Rmrespectively represent state vector, control input, system out-put, and uncertainty. ω(t) and υ(t) are process noise andmeasurement noise with zero mean and covariance matricesE(ω(t)ω(t)>
)= Ξ and E
(υ(t)υ(t)>
)= Θ, respectively.
A ∈ Rn×n, B ∈ Rn×m, and C ∈ Rl×n are system matrices.It is assumed that (A,B) is controllable and that the systemuncertainties are unknown and bounded, so that
‖ξ(t)‖ ≤ ξ∗, ξ∗ > 0. (36)
The control aim is to force the system output y(t) to followthe desired reference r(t) with minimal control effort in thepresence of uncertainty ξ(t). To achieve this objective thecontrol scheme, depicted in Fig. 1, is proposed in this work,where the robust control reduces the effect of uncertaintiesand the optimal control guarantees minimum control effort.The control law u(t) is formed of two parts, i.e.,
u(t) = un(t) + ud(t) (37)
where the nominal control (un(t)) is produced using LQGto obtain nominal system performance in an optimal waywhereas the discontinuous control (ud(t)) is generated by ISMto compensate for uncertainties. Thus, (35) can be written as
x(t) = Ax(t) +B(un(t) + ud(t) + ξ(t)) + ω(t)
y(t) = Cx(t) + υ(t) (38)
IV. PROPOSED HYBRID CONTROL DESIGN SCHEME
A. Nominal Control Design
The nominal control uses the LQG approach, which involvestwo steps, state estimation using a Kalman filter and optimalstate feedback control using the LQT.
1) Kalman Filter: The Kalman filter estimation problem isto find an optimal state estimate x(t) such that the followingerror covariance is minimized:
J1 = limt→∞
E{
(x− x) (x− x)>}
(39)
The Kalman filtering problem is estimated by computing theKalman gain Kf given by
Kf = PfC>Θ−1 (40)
where Pf is a symmetric positive semidefinite matrix and canbe computed using a solution of following Algebraic RiccatiEquation (ARE) as
APf + PfA> + ΓΞΓ> − PfC>Θ−1CPf = 0 (41)
where Γ ∈ Rn×m is disturbance input matrix. Thus, theestimated states x(t) for the nominal system are given by,
˙x(t) = Ax(t) +Bu(t) +Kf (y(t)− Cx(t)). (42)
Complete Nonlinear PWR
NPP Model
x
r(t)
Process Noise
Measurement Noise
y(t)
Kalman Filter
u(t)
Integral Sliding Mode Control
LQTun(t)
ud(t)
Fig. 1: Block diagram representation of the proposed hybridcontrol strategy.
2) Linear Quadratic Tracker: The classical linear quadraticregulator design is modified to track the reference signal. TheLQT design computes an optimal control input by minimizingthe cost function [38]:
J2 =
∞∫0
((Cx− r)>Q (Cx− r) + u>nRun
)dτ (43)
where Q and R are positive semidefinite and positive defi-nite weighing matrices, respectively. Thus, the state feedbackcontrol law is given by,
un(t) = −Kcx(t) +Kvs(t) (44)
where the optimal feedback gain Kc is computed by findinga solution of the following ARE
A>Pc + PcA+ C>QC − PcBR−1B>Pc = 0 (45)
It is given byKc = R−1B>Pc (46)
where Pc is a symmetric positive semidefinite matrix. Thefeed-forward gain Kv is computed as
Kv = R−1B (47)
and s(t) is given by the solution of
−s(t) = (A−BKc)>s(t) + C>Qr (t) , s (∞) = 0 (48)
Thus, the optimal state feedback nominal control law is thenimplemented using the estimated states as
un(t) = −R−1B>Pcx(t) +R−1B>s(t) (49)
B. Loop Transfer Recovery
Due to the incorporation of a Kalman filter for state es-timation, the robustness and stability margin of the nominalcontrol are weakened [39]. To resolve this, either the gain ofKalman filter or the gain of tracker can be modified using the
5
LTR approach. The gains are shaped so that the resultant filtertransfer function has guaranteed stability margins. The open-loop system with the LQG return ratio at the input is givenby
G(s) = Kc (sI −A+BKc +KfC)−1KfC(sI −A)−1B
(50)The LTR at the input can be designed as follows: First, theLQT is designed by suitably selecting Q and R. Then, Γ = B,Ξ = qΞ and Θ = I are selected. The idea of LTR design is touse a fictitious gain coefficient q and then gradually increaseq →∞, such that the final loop-transfer function approximatesto the state-feedback loop transfer function designed by theLQT as
limq→∞
G(s) = Kc (sI −A)−1B (51)
The proposed LQG/LTR-ISM scheme first designs the nominalcontrol using LQG and enhances the stability using the LTRtechnique and then combines with the ISM approach. Thus,the hybrid approach possess strong robustness capability withenhanced performance.
C. Discontinuous Control Design
The ISM works by designing first an integral sliding mani-fold followed by the design of a discontinuous control law. Thesliding motion occurs on the integral sliding manifold whereasthe control law drives the system trajectories onto the slidingmanifold and maintained on it. An integral sliding manifoldφ(t) ∈ Rm =
[φ1(t) φ2(t) · · · φm(t)
]>can be designed as
[28],
φ(t) = G
[x(t)− x(0)−
∫ >0
˙xn(τ) dτ
](52)
where G ∈ Rm×n is the design freedom. For simplicity, it isselected as left-pseudo inverse of input distribution matrix Bgiven as
G = (B>B)−1B> (53)
The term −x(0) assures that the system starts from the slidingmanifold by eliminating the reaching phase and enforcingφ(0) = 0. Thus, the closed-loop system turns out to be robusttowards matched uncertainties from the initial time instant.Here, the discontinuous control ud(t) is formulated based onthe reachability condition [40]
ud(t) = −µ sign(φ(t)) (54)
where µ is an appropriately designed positive constant andsign(.) is the standard signum function.
D. Stability Analysis
A Lyapunov function V (t) is selected as [41]
V (t) =1
2φ>(t)φ(t) (55)
Differentiating V (t) with respect to time gives
V (t) = φ>(t)φ(t) = φ>(t)G(
˙x(t)− ˙xn(t))
(56)
During sliding mode, the system trajectories follow the nomi-nal system trajectories i.e., x(t) = xn(t). Thus, (56) becomes
where δ is a small positive constant.It is apparent from (58) that the trajectories of uncertain
system will be maintained on the sliding manifold φ(t) = 0and drive towards the specified equilibrium point despite theuncertainties in finite time. The boundary layer approachcan be used to restrain the effect of chattering due to thepresence of signum function [41]. The signum function canbe approximated as,
sign (φi (t)) =φi (t)
|φi (t)|+ εi = 1, 2, · · ·m (59)
where ε is a small positive constant. The boundary layertechnique results in loss of invariance property and steadystate error proportional to boundary layer thickness. Thus, forgood performance the value of ε should be selected as smallas possible [40]. A more prominent approach to reduce theeffect of chattering is higher order sliding mode control.
V. RESULT AND DISCUSSION
The simulation results test the performance of the designedcontrollers under various conditions. The controllers are testedon the nonlinear PWR-type NPP model under disturbances ofsinusoidal and chirp nature. The following important controlloops are considered: reactor power and temperature loop,temperature loop, steam generator loop, pressurizer loop, andturbine speed loop. In each case, the results of the proposedcontrol schemes are compared with other classical state feed-back techniques such as LQG and LQG/LTR schemes. Thedefinition of input and output vector for every single-input-single-output control loop is given in Table I. The value oftuning parameters of the controllers for different loops is givenin Table I.
A. Reactor Power Loop
The performance of the designed controllers is tested fortypical load-following transients of a PWR-type NPP in thepresence of disturbances. The disturbances added to the systemare given by
Fig. 2: Variation of reactor power signals during load-following mode of operation.
1) Case I: Initially, the NPP is assumed to be operating at100% full power (FP). A load-following transient is consideredto study typical power variations in which the reference poweris varied at 6.6%/min in a ramp manner. It is given as follows:
P refn =
1, 0 ≤ t ≤ 500.0011(t− 50) + 1, 50 < t ≤ 2501.22, 250 < t ≤ 600−0.0011(t− 700) + 1.22, 600 < t ≤ 10000.78, 1000 < t ≤ 13500.0011(t− 1900) + 0.78, 1350 < t ≤ 15501, 1550 < t ≤ 2000
(62)
The performance of the controllers, in terms of measured ex-core detector current corresponding to output power is shownin Fig. 2a. The variations of control signal and its incrementalchange are shown in Figs. 2b and 2c, respectively. The designof sliding manifolds using LQG-ISM and LQG/LTR-ISMis shown in Fig. 2d. It can be noticed that the LQG andLQG/LTR controllers are unable to reject the disturbanceswhereas the LQG-ISM and LQG/LTR-ISM controllers cantrack the variations smoothly as envisaged. The LQG-ISM andLQG/LTR-ISM schemes take similar control efforts which arelower than that taken by the LQG and LQG/LTR techniques.
2) Case II: Another load-following transient is consideredto validate the performance during an emergency operation.The reference power value is brought down from 100% to20% FP in a step manner and then it is slowly brought backto its initial steady-state value power at 5%/min ramp. Thetransient is given as follows:
P refn =
1.0, 0 ≤ t ≤ 500.2, 50 < t ≤ 5000.05(t− 500)/60 + 0.2, 500 < t ≤ 14601.0, 1460 < t ≤ 2000
(63)
The performance of the controllers in terms of measured ex-core detector current are shown in Fig. 3a. The variationsof control signal and its incremental change are shown inFigs. 3b and 3c, respectively. The design of sliding manifoldsusing LQG-ISM and LQG/LTR-ISM is shown in Fig. 3d. TheLQG/LTR controller gives better performance than the LQGcontroller where the latter gives large overshoot however, bothof them are unable to handle the disturbances. The LQG-ISMand LQG/LTR-ISM are able to track the sudden 80% loadrejection transient without any overshoot and are able to rejectthe disturbances present in the system. The LQG-ISM andLQG/LTR-ISM are found to give better control performanceover the LQG and LQG/LTR approaches.
7
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Time (s)
17.5
18
18.5
19
19.5
20E
x-co
re d
etec
tor
curr
ent (
mA
)Demand LQG LQG/LTR LQG-ISM LQG/LTR-ISM
100 200 300 400 50018.24
18.28
18.32
(a) Ex-core detector current.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Time (s)
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
Con
trol
rod
spe
ed
LQG LQG/LTR LQG-ISM LQG/LTR-ISM
200 400 600 800 1000 1200
Time (s)
-6
-4
-2
0
2
4
610-3
(b) Control rod speed.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Time (s)
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Incr
emen
tal c
hang
e of
con
trol
sig
nal
LQG LQG/LTR LQG-ISM LQG/LTR-ISM
0 200 400 600 800 1000 1200 1400 1600 1800 2000
-2
-1
0
1
2
310-7
(c) Incremental change of control rod speed.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Time (s)
-5
0
5
10
15
20
Slid
ing
man
ifold
10-5
LQG-ISM LQG/LTR-ISM
(d) Sliding manifold.
Fig. 3: Variation of reactor power signals during load-following mode of operation.
B. Temperature Control Loop
The NPP power can also be controlled using the coolanttemperature. To analyse the performance of temperature con-trol, in the presence of disturbances similar to V-A, a load-following transient is considered as follows:
P refn =
1, 0 ≤ t ≤ 50−0.001(t− 50) + 1, 50 < t ≤ 2500.80, 250 < t ≤ 7000.001(t− 700) + 0.80, 700 < t ≤ 9001, 900 < t ≤ 12500.001(t− 1250) + 1, 1250 < t ≤ 14501.20, 1450 < t ≤ 1900−0.001(t− 1900) + 1.20, 1900 < t ≤ 21001, 2100 < t ≤ 2500
(64)
The performance of the controllers, in terms of measured RTDcurrent corresponding to the output power is shown in Fig. 4a.The variations of control signal and its incremental change areshown in Figs. 4b and 4c, respectively. The design of slidingmanifolds using LQG-ISM and LQG/LTR-ISM is shown inFig. 4d. The LQG-ISM and LQG/LTR-ISM controllers cantrack the variation smoothly in the presence disturbanceshowever, the LQG and LQG/LTR controllers are unable to doso. The LQG-ISM and LQG/LTR-ISM approaches take lower
control efforts than the other approaches.
C. Steam Generator Loop
The performance of the designed controllers is tested for aset-point change in steam generator pressure in the presence ofdisturbances. The disturbances added to the system are givenby
where the disturbance ω1(t) is added to the valve coefficientin (31) and the disturbance ω2(t) is added to the input signalto the turbine governor valve in (32). A set-point change inthe secondary pressure is applied as follows:
prefs =
7.285, 0 ≤ t ≤ 10010−3(t− 100) + 7.285, 100 < t ≤ 1507.335, 150 < t ≤ 300−10−3(t− 300) + 7.335, 300 < t ≤ 3507.285, 350 < t ≤ 500
(67)
The performance of the controllers is shown in Fig. 5a. Thevariation of control signal and its incremental change areshown in Figs. 5b and 5c, respectively. The design of sliding
8
0 500 1000 1500 2000 2500Time (s)
5
10
15
20
25R
TD
det
ecto
r cu
rren
t (m
A)
Demand LQG LQG/LTR LQG-ISM LQG/LTR-ISM
200 300 400 500 600 7006.5
7
7.5
(a) RTD current.
0 500 1000 1500 2000 2500
Time (s)
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
Con
trol
rod
spe
ed
LQG LQG/LTR LQG-ISM LQG/LTR-ISM
0 500 1000 1500 2000 2500
-6
-4
-2
0
2
4
610-3
(b) Control rod speed movement.
0 500 1000 1500 2000 2500-12
-10
-8
-6
-4
-2
0
2
4
Incr
emen
tal c
hang
e of
con
trol
sig
nal
10-4
LQG LQG/LTR LQG-ISM LQG/LTR-ISM
0 500 1000 1500 2000 2500
-3
-2
-1
0
1
2
3
10-7
(c) Incremental change of control rod speed.
0 500 1000 1500 2000 2500
Time (s)
-3
-2
-1
0
1
2
3
Slid
ing
man
ifold
10-5
LQG-ISM LQG/LTR-ISM
(d) Sliding manifold.
Fig. 4: Variation of reactor temperature signals during load-following mode of operation.
manifolds using LQG-ISM and LQG/LTR-ISM is shown inFig. 5d. It can be observed that the LQG-ISM and LQG/LTR-ISM are able to track the variation smoothly in the presenceof disturbances. On the contrary, the LQG and LQG/LTRcontrollers are unable to reject the disturbances. The controlefforts taken by LQG-ISM and LQG/LTR-ISM are found tobe significantly lower than that of the other approaches.
D. Pressurizer Loop
The pressurizer pressure control is usually achieved byactuating bank of heaters and by adjusting the spray flowrate. The performance of the designed controllers is tested fora set-point change in pressurizer pressure in the presence ofdisturbances. The disturbances added to the system are givenby
ω1(t) = 0.2 sin(0.1t) + 0.4 sin(0.2t) + sin(0.5t)
ω2(t) = 10−2 sin(2π × 10−4t+ 1.998π × 10−4t2
)(68)
where the disturbances ω1(t) and ω2(t) are added to the inputsignal and to the surge flow in (25), respectively.
1) Pressure Control by Heater: A set-point change in thepressurizer pressure is applied as follows:
prefp =
15.41, 0 ≤ t ≤ 5010−4(t− 50) + 15.41, 50 < t ≤ 10015.415, 100 < t ≤ 200
(69)
The increment in reference pressure actuates the heater system.The performance of the controllers is shown in Fig. 6a. Itcan be observed that the all the controllers able to track theset-point however, the LQG and LQG/LTR controllers areunable to reject the disturbances whereas the LQG-ISM andLQG/LTR-ISM are able to effectively handle the disturbances.The variation of control signal and its incremental change areshown in Figs. 6b and 6c, respectively. The design of slidingmanifolds using LQG-ISM and LQG/LTR-ISM is shown inFig. 6d. All the control schemes are found to take similarcontrol efforts out of which the LQG-ISM and LQG/LTR-ISMtakes the minimum efforts.
2) Pressure Control by Spray: A set-point change in thepressurizer pressure is applied as follows:
prefp =
15.41, 0 ≤ t ≤ 50−10−4(t− 100) + 15.41, 50 < t ≤ 10015.405, 100 < t ≤ 200
(70)
9
0 50 100 150 200 250 300 350 400 450 500Time (s)
7.24
7.26
7.28
7.3
7.32
7.34
7.36
7.38P
ress
ure
(MP
a)Setpoint LQG LQG/LTR LQG-ISM LQG/LTR-ISM
(a) Steam generator secondary pressure.
0 50 100 150 200 250 300 350 400 450 500
Time (s)
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Con
trol
sig
nal t
o tu
rbin
e go
vern
or v
alve
LQG LQG/LTR LQG-ISM LQG/LTR-ISM
(b) Control signal to turbine governor valve.
0 20 40 60 80 100 120 140 160 180 200
Time (s)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Incr
emen
tal c
hang
e of
con
trol
sig
nal
LQG LQG/LTR LQG-ISM LQG/LTR-ISM
20 40 60 80 100 120 140 160 180 200-4
-2
0
2
4
610-4
(c) Incremental change of control signal.
0 50 100 150 200 250 300 350 400 450 500Time (s)
-1
0
1
Slid
ing
man
ifold
10-4
LQG-ISM LQG/LTR-ISM
(d) Sliding manifold.
Fig. 5: Variation of steam generator during a set-point change in secondary pressure.
The decrement in reference pressure actuates the spray flowsystem. The performance of the proposed controller is shownin Fig. 7a. It can be observed that LQG and LQG/LTRcontrollers are able to track the set-point with superimposeddisturbances. The LQG-ISM and LQG/LTR-ISM are ableto reject the disturbances and smoothly track the set-pointvariation. The variation of control signal and its incrementalchange are shown in Figs. 7b and 7c, respectively. The designof sliding manifolds using LQG-ISM and LQG/LTR-ISM isshown in Fig. 7d. The control efforts taken by LQG-ISM andLQG/LTR-ISM are found to be lower than that of the otherapproaches.
3) Level Control: The pressurizer level control systemmaintains the water level for the reactor core coolant system.A set-point change in the level is applied as follows:
lrefw =
28.06, 0 ≤ t ≤ 50−0.01(t− 50) + 28.06, 50 < t ≤ 10027.56, 100 < t ≤ 2500.01(t− 250) + 27.56, 250 < t ≤ 30028.06, 300 < t ≤ 400
(71)
The performance of the controllers is shown in Fig. 8a. TheLQG-ISM and LQG/LTR-ISM are able to track the set-pointvariation smoothly in the presence of disturbances whereas
the LQG and LQG/LTR controllers are unable to reject thedisturbances. The control signal variation and the incrementalchange of control signal are shown in Figs. 8b and 8c,respectively. The design of sliding manifolds using LQG-ISMand LQG/LTR-ISM is shown in Fig. 8d. The LQG-ISM andLQG/LTR-ISM are found to spend lower control energies thanthat of the other approaches.
E. Turbine Speed Loop
The turbine speed control system regulates the shaft speedby controlling the steam flow to the turbine through the turbinegovernor valve. The performance of the proposed technique istested in regulating the demand power using turbine speed inthe presence of disturbances. The disturbances added to thesystem are given by
where ω1(t) is added to valve coefficient in (31) and ω2(t) isadded to the input signal to the turbine governor valve in (32).
10
0 20 40 60 80 100 120 140 160 180 200
Time (s)
15.408
15.409
15.41
15.411
15.412
15.413
15.414
15.415
15.416P
ress
ure
(MP
a)
Setpoint LQG LQG/LTR LQG-ISM LQG/LTR-ISM
10 20 30 4015.4097817
15.4097819
15.4097821
(a) Pressurizer pressure.
0 20 40 60 80 100 120 140 160 180 200
Time (s)
-200
0
200
400
600
800
1000
1200
1400
Rat
e of
hea
t add
ition
(W
)
LQG LQG/LTR LQG-ISM LQG/LTR-ISM
60 80 1001267
1268
1269
1270
(b) Rate of heat addition.
0 20 40 60 80 100 120 140 160 180 200
Time (s)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Incr
emen
tal c
hang
e of
con
trol
sig
nal
LQG LQG/LTR LQG-ISM LQG/LTR-ISM
20 40 60 80 100 120 140 160 180 200-4
-2
0
2
4
610-4
(c) Incremental change of control signal.
0 20 40 60 80 100 120 140 160 180 200Time (s)
-3
-2
-1
0
1
2
3
Slid
ing
man
ifold
10-3
LQG-ISM LQG/LTR-ISM
(d) Sliding manifold.
Fig. 6: Variation of pressurizer heater signals during set-point change in pressure.
The demand power from the generator is changed as follows:
P refdem =
1, 0 ≤ t ≤ 500.002(t− 50) + 1, 50 < t ≤ 1000.90, 100 < t ≤ 1500
(74)
The performance of the proposed controllers for trackingthe set-point change in demand power is shown in Fig. 9a.The LQG and LQG/LTR controllers track the variation withdisturbances superimposed. The LQG-ISM and LQG/LTR-ISM are able to track the set-point variation smoothly in thepresence of disturbances. The control signal variation and theincremental change of control signal are shown in Figs. 9b and9c, respectively. The design of sliding manifolds using LQG-ISM and LQG/LTR-ISM is shown in Fig. 9d. The LQG-ISMand LQG/LTR-ISM are found to spend lower control energiesthan that of the other approaches.
F. Performance Assessment
The performance of different controllers is dependent on thetuning parameters. In the case of LQG, the Q and R matricesregulate the penalties on the states variables and control input,respectively. If Q is a diagonal matrix, large diagonal elementsresults in the poles of the closed-loop system far from theorigin and the state tracks the reference rapidly. On the
contrary, if R is a diagonal matrix, large diagonal elementsresults in the poles of the closed-loop system close to theorigin and the state tracks the reference slowly. Thus, thevalues of Q and R are tuned such that the set-point can betracked quickly without any overshoot. In the case of LTR, therecovery gain q is selected based on the frequency response ofthe target feedback loop. The value of q is selected such thatthe loop transfer function approaches the ideal return ratiogiven by the target feedback loop. The tuning parameter ofISM is selected to ensure that the discontinuous control signaldoes not contain high-freuqnecy noise. The values of differenttuning parameters adopted during simulations for each schemeare given in Table I.
The control performance can be statistically analysed basedon the following measures. Firstly, the percentage root meansquared error (PRMSE) is calculated on the basis of trackingerror. Secondly, the effect of control action on input is analysedby computing the total variation of input (TVI) and the L2-norm of input (L2NI). These are given by,
PRMSE =
√√√√ 1
N
N∑i=1
(yi − ri)2 × 100%, (75)
11
0 20 40 60 80 100 120 140 160 180 200Time (s)
15.404
15.405
15.406
15.407
15.408
15.409
15.41P
ress
ure
(MP
a)
Setpoint LQG LQG/LTR LQG-ISM LQG/LTR-ISM
100 120 140 160 180 20015.40476
15.40478
15.4048
(a) Pressurizer pressure.
0 20 40 60 80 100 120 140 160 180 200Time (s)
-5
0
5
10
15
20
25
30
35
Spr
ay fl
ow r
ate
(kg/
s)
LQG LQG/LTR LQG-ISM LQG/LTR-ISM
60 65 70 75 80 85 90 95 10032
33
34
35
(b) Rate of spray flow.
0 20 40 60 80 100 120 140 160 180 200Time (s)
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
Incr
emen
tal c
hang
e of
con
trol
sig
nal
LQG LQG/LTR LQG-ISM LQG/LTR-ISM
0 20 40 60 80 100 120 140 160 180 200-1
-0.5
0
0.5
110-3
(c) Incremental change of control signal.
0 20 40 60 80 100 120 140 160 180 200Time (s)
-3
-2
-1
0
1
2
3
Slid
ing
man
ifold
10-3
LQG-ISM LQG/LTR-ISM
(d) Sliding manifold.
Fig. 7: Variation of pressurizer spray flow signals during set-point change in pressure.
TABLE I: Tuning parameters for different control approaches
where N denotes the total number of samples, which isequal to simulation time divided by sampling interval wherethe sampling interval is taken as 1 ms. Table II comparesthe control performance of LQG, LQG/LTR, LQG-ISM, andLQG/LTR-ISM approaches. It is found that the values of
PRMSE for the LQG/LTR-ISM approach is lower than thoseof the other approaches in all the cases. The value of TVI andthe L2NI are also found to be lower for the LQG-ISM andLQG/LTR-ISM approaches. The performance of LQG/LTR isslightly better than the LQG, however both of the techniquesare unable to provide the desired response in the presenceof disturbances. It can be concluded that the proposed LQG-ISM and LQG/LTR-ISM controllers provide better set-pointtracking over other control approaches with minimum controlefforts.
12
0 50 100 150 200 250 300 350 400Time (s)
27.4
27.5
27.6
27.7
27.8
27.9
28
28.1
28.2Le
vel (
m)
Setpoint LQG LQG/LTR LQG-ISM LQG/LTR-ISM
100 150 200 250
27.56
27.561
(a) Pressurizer level.
0 50 100 150 200 250 300 350 400Time (s)
-20
-15
-10
-5
0
5
10
15
20
Con
trol
sig
nal t
o C
VC
S (
kg/s
)
LQG LQG/LTR LQG-ISM LQG/LTR-ISM
250 260 270 280 290 30014
15
50 60 70 80 90 100
-15
-14
(b) Control signal to CVCS system.
0 50 100 150 200 250 300 350 400
Time (s)
-0.04
-0.02
0
0.02
Incr
emen
tal c
hang
e of
con
trol
sig
nal
LQG LQG/LTR LQG-ISM LQG/LTR-ISM
0 50 100 150 200 250 300 350 400-3
-2
-1
0
1
2
10-4
(c) Incremental change of control signal.
0 50 100 150 200 250 300 350 400Time (s)
-5
-4
-3
-2
-1
0
1
2
3
Slid
ing
man
ifold
10-3
LQG-ISM LQG/LTR-ISM
(d) Sliding manifold.
Fig. 8: Variation of pressurizer level signals during set-point change in level.
VI. CONCLUSIONS
This work proposes state feedback-based hybrid controldesign techniques by integrating robust-optimal approachesfor the control of a pressurized water-type nuclear powerplant. The robust-optimal hybrid control technique combinesthe LQG-ISM approach with the loop transfer recovery tech-nique. The control architecture thus offers enhanced robustnesswith improved performance and tracks the reference set-pointsmoothly in the presence disturbances. The effectiveness of thetechniques has been validated using simulations on differentsubsystems of the PWR-type NPP. The control performancesof the proposed approaches have been quantitatively comparedwith LQG and LQG/LTR control approaches using differentstatistical measures for reactor power and temperature con-trols, steam generator pressure control, pressurizer pressureand level controls, and turbine speed control. The proposedcontrollers can handle disturbances in the system and they havebeen found giving better performance over other controllers.
VII. ACKNOWLEDGEMENT
The work presented in this paper has been financially sup-ported under grants EP/R021961/1 and EP/R022062/1 fromthe Engineering and Physical Sciences Research Council.
APPENDIX
The value of different parameters used in the model aregiven in Table A.1 [34]–[37].
NOMENCLATURE
Ap Cross-sectional area of pressurizer (m2)Cin Normalized delayed neutron precursor concen-
tration (per unit)Ctg Turbine governor valve coefficientG Reactivity worth (mK)H Rate of rise of temperature (0C/J)I Moment of inertia (kg.m2)J Conversion factorK GainPn Normalized power (per unit)Qheat Rate of heat addition (kW )S Effective heat transfer area (m2)T Average temperature (0C)U Heat transfer coefficient (W/m2.0C)V Volume (m3)cp Specific heat (J/kg.0C)d Density (kg/m3)h Enthalpy (J/kg)i Current (mA)
13
0 500 1000 1500
Time (s)
0.88
0.9
0.92
0.94
0.96
0.98
1
1.02N
orm
aliz
ed m
echa
nica
l pow
erDemand LQG LQG/LTR LQG-ISM LQG/LTR-ISM
(a) Normalized mechanical power.
0 500 1000 1500Time (s)
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
Con
trol
sig
nal t
o tu
rbin
e go
vern
or v
alve
LQG LQG/LTR LQG-ISM LQG/LTR-ISM
(b) Control signal to turbine governor valve.
0 500 1000 1500
Time (s)
-2
-1
0
1
2
3
4
5
6
Incr
emen
tal c
hang
e of
con
trol
sig
nal
10-4
LQG LQG/LTR LQG-ISM LQG/LTR-ISM
200 250 300 350 400-5
0
510-5
(c) Incremental change of control signal.
0 500 1000 1500
Time (s)
-5
-4
-3
-2
-1
0
1
2
3
4
5
Slid
ing
man
ifold
10-3
LQG-ISM LQG/LTR-ISM
(d) Sliding manifold.
Fig. 9: Variation of turbine speed signals during a set-point change in demand power from generator.
l Pressurizer length (m)m Mass (kg)m Mass flow rate (kg/s)p Pressure (MPa)q Torque (J/rad)z Speed (m/s)Λ Neutron generation time (s)α Temperature coefficient of reactivity (0C−1)β Fraction of delayed neutronsκ Constantλ Decay constant (s−1)ρ Reactivity (mK)ζ Damping ratioτ Time constant (s)ν Specific volume (m3/kg)ω Natural frequency of oscillation (rad/s)Subscriptsc1, c2, cin Coolant at node 1, 2 and inletdem Demandf Fuelfw Feed-waterhot, cold Hot and cold leghp, ip, lp, High, intermediate, and low pressure steami ith group of delayed neutron precursor
lo, lr Logarithmic and Log rate amplifierm1, m2 MTL 1 and 2mp1, mp2 Transfer from MTL 1 and 2 to PCL 1 and 2ms1, ms2 Transfer from MTL 1 and 2 to SCLp Pressurizerp1, p2 PCL 1 and 2pm1, pm2 Transfer from PCL 1 and 2 to MTL 1 and 2rod Regulating rodrxi, rxu Reactor lower and upper plenums Steamss Steam in secondary lumpsg, sgi, sguSteam generator, inlet, and outlet plenumspr, sur Spray and surgertd1, rtd2 RTD 1 and 2tg Turbine-Governortur Turbinew Waterws Water in secondary lump
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14
TABLE II: Performance comparison of different control ap-proaches
Case Technique PRMSE TVI L2NI
A.1LQG 2.214× 100 8.730× 10−2 2.595× 100
LQG/LTR 3.387× 100 8.610× 10−2 2.474× 100
LQG-ISM 9.811× 10−1 9.070× 10−2 3.132× 100
LQG/LTR-ISM 9.806× 10−1 8.216× 10−2 3.130× 100
A.2LQG 1.084× 101 2.626× 10−1 1.959× 101
LQG/LTR 1.343× 101 5.966× 10−1 9.341× 100
LQG-ISM 8.487× 100 2.598× 10−1 7.590× 100
LQG/LTR-ISM 8.484× 100 8.940× 10−1 9.350× 100
BLQG 1.456× 102 2.005× 10−1 3.675× 100
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LQG-ISM 1.371× 102 1.563× 10−1 3.470× 100
LQG/LTR-ISM 1.369× 102 1.688× 10−1 3.455× 100
CLQG 1.540× 100 7.815× 100 2.771× 101
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LQG-ISM 1.804× 10−1 6.850× 10−2 9.857× 100
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LQG-ISM 3.673× 100 2.215× 10−1 2.071× 102
LQG/LTR-ISM 3.669× 100 2.263× 10−1 2.068× 102
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15
TABLE A.1: Typical Parameters of a PWR Nuclear Power Plant
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