INTRODUCTION
A modulation of the nuclear power plants must be able to respond to the demand
on the network. The pressurized water nuclear reactor has to yield correctly
a load set point (SiFodil et al., 2000; Khan
et al., 2011). Referring to the principle of operating a nuclear
power plant, securing the sufficient of its cooling source and preventing the
damage of turbine blades must be done by stabilizing water level in the SG of
nuclear power plant. The water level control problem of SG has been a main cause
of unexpected shutdowns of nuclear power plants. Stabilizing the water level
around the predetermined level is very important for the safety and efficient
operation of nuclear power plants. It becomes more difficult to maintain water
level at low power since the nonminimum phase effects, known as ‘swell
and shrink’, become greater. This reverse dynamic effect prevents using
the automated controller and forces manual operation. Even for experts in the
field, however, it is not so easy to react effectively to the reverse dynamics
(Kim et al., 2004).
Consequently, many researchers deploy many efforts to apply various control
techniques to the water level control of steam generators (Parlos
and Rais, 2000; Strohmayor, 1982) . One of them is
using PID control of UTSG water level, based on model predictive technique (based
on standard Irving’s model) to automatically tune the PID gains. More of
them, like it is shown in the literature, use the fuzzy control technique which
receives many attentions among such of advanced control techniques, due to its
resemblance to humanlike characteristics.
It was demonstrated through several previous works that fuzzy control technique
is an interesting approach which can be successfully applied to control water
level of steam generator of nuclear reactors (Choi, 1987;
Munasinghe et al., 2005; Na,
2001). This last process is highly complex, nonlinear and describable only
by a (not analytical) numerical model.
However, performance of fuzzy controllers strongly depends on the initial assignment
of membership parameters and rule structure and so on. In order to tune such
parameters, it is necessary that the designer use trialanderror procedure.

Fig. 1: 
Simplest schematic of a nuclear power plant with the utube
steam generator 
This one consists of repeating two steps: the first step is to apply current
controller to the target system and the second step is to tune the controller
observing the performance of the controller on the target system. To get rid
of this necessity, there has been an approach to apply genetic algorithm to
tune the fuzzy controller (Cho and No, 1996).
In this study, a SelfTuning Controller (STC) is proposed. The stochastic autoregressive moving average model, ARMAX, is derived. Then the adaptive MVC is implemented to regulate UTSG water level in pressurized water nuclear reactor. The adverse phenomena which are caused due to parameters uncertainties, modelled dynamics, errors of flow rate measurements and process noise is cured by using the proposed stochastic adaptive control law.
Utube steam generator model and problem statement: Utube steam generator
is the major component in nuclear power industry, where the steam is generated.
Fig. 1 shows a simplest schematic of the overall nuclear power
plant. Indeed, the heat generated at the nuclear reactor is taken away by forcedcirculated
water in the primary circuit. Due to the water contamination by radioactive
particles, the primary circuit is therefore, isolated from the rest of the system.
Referring to the figure, the primary circuit has an inverted utube bundle submerged
in the water column of the steam generator, where the heat transfer takes place
from primary circuit to secondary circuit that makes secondary circuit water
reach the state of bulkboiling. The generated steam of the secondary circuit
is sent to the turbine which is coupled to an armature to generate electricity
to be injected in the grid.
The philosophy of the control is that the water level control of the UTSG should
be maintained within its lower and upper limits, like it is shown in Fig.
1. If unfortunately, the controller is not able to maintain water level
at the desired level, it would lead to the following serious consequences including
unintended plant shutdowns and also system damage (Parlos
and Rais, 2000).
In the case, where low water level exposes the utubes, the heat transfer from the primary circuit to the secondary circuit will not take place efficiently. Consequently, primary circuit builds up heat within itself, which causes the reactor to trip off.
In either case, where the water level rises too high, the steam will contain more moisture (dryness). And the wet steam may damage the turbine blades; therefore, turbine trips off.
Therefore, it is extremely important that the water level of the UTSG should
be regulated within its limits. Significant percentage of plant shutdowns and
system unavailability at present, are reportedly due to failures in UTSG water
level control. The last one is a very difficult problem as its dynamics shows
high nonlinearity and nonminimum phase behavior. It can be approximated by
the following linearized model taking account a given power level (Kothare
et al., 2000):
Where:
p 
= 
Power level (%) 
u 
= 
Feed water flow (kg sec^{1}) 
v 
= 
Steam flow (kg sec^{1}) 
y 
= 
Water level (mm) 
r 
= 
Reference water level (mm) 
e^{l} 
= 
ry=Level error (mm) 
e^{l} 
= 
vu p=Flow error (kg sec^{1}) 
v_{p} 
= 
Rated steam flow at power (kg sec^{1}) 
Table 1: 
UTSG model parameters 

The four terms on the Right Hand Side (RHS) are, respectively in that order
mass capacity effect, nonminimum phase effect of feedwater, nonminimum phase
effects of steam and the effect of mechanical oscillation. For an ideal plant,
Table 1 gives the model parameters of Eq. 1,
i.e., {G_{1}, G_{2}, G_{3}, G_{4}, τ_{1},
τ_{2}, τ_{3}, T} originally published by Irving (Munasinghe
et al., 2005). Figure 2 graphically illustrates
UTSG dynamics given in Eq. 1, in the case, when the plant
operates at 50% of its rated power.
Figure 2c and d show the reverse dynamics
due to feedwater change and steam flow change, respectively. We propose some
assumptions which are helpful to simplify UTSG modeling, however, at an expense
of loosing credibility to represent actual plants; the two reverse dynamics
have been assumed identical (except for sign) in Na (2001).
In Na (2001) and Zheng (2004),
the mechanical oscillation effect (Fig. 2e) has been neglected.
However, in this study, we consider mechanical oscillation effect as well as nonidentical reverse dynamics for steam and feedwater, therefore, to make the model more accurate in representing actual UTSG plants.
We consider that the UTSG water level control is inherently, a very difficult
problem due to the following two particular reasons (Parlos
and Rais, 2000).
The nonminimum phase dynamics (known as “swell” and “shrink”
in UTSG literature): Indeed, a temporary increase in water level in response
to a reduction of liquid water mass in the steam generator is referred to “Swelling”
behaviour. It is momentarily observed when steam flow rate undergoes a sudden
increment (v→v+δv) (Fig. 2a, d)
or feedwater flow rate undergoes a sudden drop (u→u+δu). On the other
hand, Fig. 2a and c illustrate the “shrinking”
behaviour which is the exact opposite of “swelling”. It refers to
a temporary decrease in the water level, against an increase of the liquid water
mass in the steam generator.

Fig. 2(af): 
UTSG dynamics at 50% of the rated power. All graphs show the
deviation form its steady state value at the specified power level. Plant
excitations in 1 u = 1 v = 6.6 (kg sec^{1}) is 1% of the plant
flow rates at 50% rated power 
In fact, these behaviours, though they last momentarily, are in exact opposition
of the response, one would expect upon the nature of steam or feedwater flow
changes introduced to the system. We conclude that these reverse behaviours
make it very difficult to regulate the water level of the UTSG in pressurized
water nuclear reactor.
Errors of flow rate measurements: Steam flow rate and feedwater flow
rate are considered the most critical and widely used feedback signals. They
are not accurate enough during startup transients in the more often cases and
at low power operations. Under these conditions, feedback signals as flow rates
are small in magnitude and corrupt the process noise beyond the limit of being
useful.
Minimum variance controller: Since the characteristics of UTSG vary with time, in this study, the SelfTuning Controller (STC) is used to design the adaptive controller. The selftuning controller is suitable for controlling the timevarying plant and unknown plant.
The block diagram of UTSG control system based on STC is shown in Fig. 3. The UTSG controller is composed of the selftuning controller and the parameter identifier. y(k) is water level, y_{m}(k) is setpoint and μ(k) is control input.
The controller and the identifier are designed assuming the following ARMAX
model (Zheng, 2004; Huang et
al., 2005):
where e(k) is white noise, q^{d} u(k) denotes control delay (d≥1) and q^{1} is backward shift operator with one sampling period, viz., q^{1}.y(k) = y(k1), A(q^{1}) = 1+a_{1}.q^{1}+...+a_{n}.q^{n}, a_{1},...a_{n} are parameters of the Autoregressive (AR) part, is the AR order, B(q^{1}) = b_{0}+b_{1}q^{1}+...+b_{m}.q^{m}, b_{o} ... b_{m} are the parameters of the exogenous (X ) input part, m is the input order, c(q^{1}) = 1+c_{1}.q^{1} +...+ c_{n}.q^{n} c_{1}...c_{n} are parameters of the moving average (MA) part and n is the MA order.
The adaptive controller using STC is designed with the generalized MVC law.
The control input μ(k) based on the generalized MVC law is selected to
minimize the following J(k) cost function (Clarke, 1984;
Grimble, 1981):
where, E[] is the expectation operator, λ_{o} is the control weighting that is introduced to control nonminimum phase systems.
The control input μ(k) to minimize the cost function J(k) is obtained by the following manipulations. The following Diophantine’s equation uniquely determines the polynomial F(q^{1}) and G(q^{1}) if the polynomial is given by Bezout identity:
Equation 4 is rewritten multiplying y(k) to both side in
Eq. 4 and using Eq. 2 as:
Equation 5 represents the nonminimum phase realization of
Eq. 2.

Fig. 3: 
General view of the proposed control system 
The control input u(k) to J(k) minimize is obtained, taking account ∂y(k+d)/∂u(k) = b_{0} of by using. ∂J(k)/∂u(k) = 0:
Equation 6 determines the control input u(k) based on the generalized MVC law.
Recursive least square estimation: The controller must know the UTSG
parameters to calculate the control input u(k). There are many parameter identification
algorithms and approaches for realizing adaptive control (Jha
et al., 2011; Naeimi et al., 2009;
Halbaoui et al., 2009). This study adopts the
recursively least square method for parameter identification for the sake of
simplification of the parameter identification algorithm.
The parameter identification based on the recursively least square method for
unknown plant is to find the parameter vector to minimize the following cost
function J(k) (Landau, 1993):
where,
The identification algorithm of the recursively least square method is finally obtained by calculating ∂J/∂θ = 0. The identified parameter vector θ is obtained by:

Fig. 4(ad): 
Simulation under MVC control. During first 31000 sec, the
operator changes randomly the reference level within 80120 mm. Random disturbances
are introduced in the steam flow at all times. The time interval between
consecutive disturbances is 200 sec and between consecutive reference changes
is 300 sec. The magnitude and sign of disturbances and reference changes
are also randomly determined 
Equation 8, 9 and 10
are the identification algorithm of the recursively least square method.
Application to the steam generator water level control: Using the MVC proposed algorithm, a prolonged simulation of the UTSG plant was carried and the results are shown in Fig. 4. Starting from the beginning of the simulation, the reference water level was intentionally changed in 5 mm steps in every 300 sec intervals, over the entire range of 100, 120, 80 and 100 mm, which takes 10000 sec in total. Then, another 2 h was given for random reference changes.
The total duration of one simulation epoch is, therefore, 31000 sec.
CONCLUSION
In this study, an adaptive controller was developed to control the water level of nuclear steam generators. The developed controller was applied to the linear ARMAX models for nuclear steam generators. The steam generator water level controller was designed to effectively cope with water level deviation and steam flow disturbance and especially, computer simulations were conducted to investigate the output tracking performance.