**1. Introduction**

174 Nuclear Reactors

in the downstream region is significantly dependent on the lift force caused by a strongly inhomogeneous bubble diameter distribution and that it is necessary to adequately evaluate the influence of void fraction on bubble diameter in order to avoid strongly inhomogeneous

The present study includes the result of "Research of simulation technology for estimation of quake-proof strength of nuclear power plant" conducted by the University of Tokyo as Core Research for Evolutional Science and Technology (CREST). This research was

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**5. Acknowledgment** 

**6. References** 

38, No. 12, pp.1074-1080

*Systems*, Vol.2, No. 1, pp.262-270

*(SNA+MC2010)*, Tokyo, Japan

*journal*, Vol.32, No.5

According to complexity of nuclear reactor technology, applying a highly developed simulation is necessary for controlling the nuclear reactor control rods, so in this proposal the processes of a controlling model for nuclear reactors have been developed and simulated by the SIMULINK tool kit of MATLAB software and all responses, including oscillation and transient responses, have been analyzed.

In this work an arbitrary value of Keff as a comparable value is purposed and attributed to input block (*H*) of diagram and then this value with the received feedback value from block diagram is compared. Since the stability of the cited simulation depends on either velocity or delay time values, therefore according to this simulation the best response and operation which a reactor can have from stability aspect, have been derived. Meantime by viewing the results, the best ranges of velocity and delay time of control rod movement (in unit per second and millisecond respectively) for stability a nuclear reactor has been deduced.

Though the highlights of this proposal are respectively the following:


In view of the great advancing the nuclear reactors technology, the phenomenal and significant changes in evolution of made nuclear reactors is observed. Since the make of the first nuclear reactor on 1948 until modern reactors, too changes are obvious. The major of these changes to: the kind of reactor design, the percent of fuel enrichment, the kind of coolant and neutron moderator, more safety and the dimensions of core are referred.

The power control system is a key control system for a nuclear reactor, which directly affects the safe operation of a nuclear reactor. Much attention has been spent to the power control system performance of nuclear reactor in engineering (Zhao et al., 2003).

The Theoretical Simulation of a Model by SIMULINK for

sink accident) and LOFA (loss of flow accident).

reactor sets in the normal status.

loops are not done.

accident might occur.

transfer's function is determined.

5. External events such as earthquake, enemy attack and etc.

through breaking in primary loop of reactor either hot leg or cold leg.

demineralizer tank) or other existent valves in the secondary loop.

steam generator and sets after condenser pump.

such as:

2. Control rod ejection. 3. Spent fuel handling. 4. Stem line break.

Surveying the Work and Dynamical Stability of Nuclear Reactors Cores 177

Each mentioned issues are divided to other sub issues. Over power accident is due factors

1. Control rod withdrawal including uncontrolled rod withdrawal at sub critical power

In each mentioned issues the positive reactivity to nuclear reactor core can be injected. But there is another important accident that is: under cooling accident. Under cooling accident is classified to three sub accident among: LOCA (loss of coolant accident), LOHA (loss of heat

LOCA accident from loosing coolant is derived. This event in PWR reactor can occur

In this state the existent water in the primary loop along with steam are strongly leaked that blow down event occurs. But when the lost water of primary loop through RHRS (Residual and Heat Removal System) including HPIS (High Pressure Injection System), LPIS (Low Pressure Injection System) and Accumulator (passive system) are filled this process is entitled Refill. When the primary is filled and all the lost water in it is compensated then the

In the blow down status the raised steam is due loss of pressure in primary loop and

LOHA accident from loosing heat sink in reactor is derived. Heat sink is as steam generator in nuclear reactor. This event when occurs heat exchanging between primary and secondary

This event might occur through lacking water circulation in the secondary loop of reactor. Not circulating the water might occur through closing either block valve (which sets after

LOFA accident from disabling and loosing the pumps in either primary loop or secondary is derived. In case either primary loop's pump or secondary encounter with problems LOFA

In the secondary loop the main feed water pump has duty of circulation of feed water to

In view of dynamically stability of a nuclear reactor, there is a stable system so that an excess reactivity is injected to it and it able to be stabled again in shortest time. The stability of linear systems in the field of complex numbers by defining the polarity of closed loop

In case all the polarities are in the left side of imagine page then system will be stabled. In the time field the stability definition means system's response to each input will be

definitive. In the matrix form all the Eigen values of system have real negative part.

moving the situation of reactor's primary loop from single phase to two phase flow.

and uncontrolled rod withdrawal at power that will cause power excursion.

High reliability is one of the main objectives of the design and operation of control systems in nuclear power plants (Basu and Zemdegs, 1978; Stark, 1976).

Prototyping a control-rod driving mechanism (CRDM), which is a crucial safety system in the Taiwan Research Reactor (TRR-II) has been implemented, by iterative parallel procedures. Hence to ensure the mechanical integrity and substance of the prototype, a series of performance testing and design improvement have been interactively executed. Functional testing results show that the overall performance of the CRDM meets the specification requirements (Chyou and Cheng, 2004).

Also the SCK.CEN/ININ joint project, which deals with the design and application of modern/expert control and real-time simulation techniques for the secure operation of a TRIGA Mark III research nuclear reactor, has been undertaken (Dong et al., 2009).

This project has been proposed as the first of its kind under a general collaboration agreement between the Belgian Nuclear Research Centre (SCK·CEN) and the National Nuclear Research Institute (ININ) of Mexico (Benítez et al., 2005). In addition to the fuzzy proportional-integral-derivative (fuzzy-PID) control strategy has been applied recently as a nuclear reactor power control system. In the fuzzy-PID control strategy, the fuzzy logic controller (FLC) is exploited to extend the finite sets of PID gains to the possible combinations of PID gains in stable region and the genetic algorithm (Cheng et al., 2009; Park and Cho, 1992).

Until now, manual controlling systems have been used for controlling and tuning the control rods in the core of Gen II and some Gen III reactors (Tachibana et al., 2004).

But by application of this simulation that is the subject of this proposal the best response for operating and the best velocity and delay time of control rod movement in which can be caused to stability and critical state of a nuclear reactor, have been derived.

The safe situation is state in which the reactor stabilizes in the critical situation, meaning that the period is infinite and the Keff is 1 (Lamarsh, 1975).

#### **1.1 Accidents**

In which two states the positive reactivity overcomes the temperature coefficient of reactivity (αT):

Increasing the power and temperature of reactor core might decrease concentration of boric acid. Accordingly this event might cause to inject positive reactivity.

In addition for the reactors which apply fuels including Pu, because of having a resonance for Pu in thermal neutrons range so through increasing core's temperature the related resonance is broadened and absorbs more neutrons and because of Pu is fissionable therefore fissionable absorption occurs and is caused excess reactivity.

Also either accident or unfavorable issues as a feedback can be considered. Accidents of a nuclear reactor are totally classified based on following:


Each mentioned issues are divided to other sub issues. Over power accident is due factors such as:


176 Nuclear Reactors

High reliability is one of the main objectives of the design and operation of control systems

Prototyping a control-rod driving mechanism (CRDM), which is a crucial safety system in the Taiwan Research Reactor (TRR-II) has been implemented, by iterative parallel procedures. Hence to ensure the mechanical integrity and substance of the prototype, a series of performance testing and design improvement have been interactively executed. Functional testing results show that the overall performance of the CRDM meets the

Also the SCK.CEN/ININ joint project, which deals with the design and application of modern/expert control and real-time simulation techniques for the secure operation of a

This project has been proposed as the first of its kind under a general collaboration agreement between the Belgian Nuclear Research Centre (SCK·CEN) and the National Nuclear Research Institute (ININ) of Mexico (Benítez et al., 2005). In addition to the fuzzy proportional-integral-derivative (fuzzy-PID) control strategy has been applied recently as a nuclear reactor power control system. In the fuzzy-PID control strategy, the fuzzy logic controller (FLC) is exploited to extend the finite sets of PID gains to the possible combinations of PID gains in stable region and the genetic algorithm (Cheng et al., 2009;

Until now, manual controlling systems have been used for controlling and tuning the

But by application of this simulation that is the subject of this proposal the best response for operating and the best velocity and delay time of control rod movement in which can be

The safe situation is state in which the reactor stabilizes in the critical situation, meaning

In which two states the positive reactivity overcomes the temperature coefficient of

Increasing the power and temperature of reactor core might decrease concentration of boric

In addition for the reactors which apply fuels including Pu, because of having a resonance for Pu in thermal neutrons range so through increasing core's temperature the related resonance is broadened and absorbs more neutrons and because of Pu is fissionable

Also either accident or unfavorable issues as a feedback can be considered. Accidents of a

control rods in the core of Gen II and some Gen III reactors (Tachibana et al., 2004).

caused to stability and critical state of a nuclear reactor, have been derived.

that the period is infinite and the Keff is 1 (Lamarsh, 1975).

acid. Accordingly this event might cause to inject positive reactivity.

therefore fissionable absorption occurs and is caused excess reactivity.

nuclear reactor are totally classified based on following:

TRIGA Mark III research nuclear reactor, has been undertaken (Dong et al., 2009).

in nuclear power plants (Basu and Zemdegs, 1978; Stark, 1976).

specification requirements (Chyou and Cheng, 2004).

Park and Cho, 1992).

**1.1 Accidents** 

reactivity (αT):

1. Over power accident. 2. Under cooling accident. 5. External events such as earthquake, enemy attack and etc.

In each mentioned issues the positive reactivity to nuclear reactor core can be injected. But there is another important accident that is: under cooling accident. Under cooling accident is classified to three sub accident among: LOCA (loss of coolant accident), LOHA (loss of heat sink accident) and LOFA (loss of flow accident).

LOCA accident from loosing coolant is derived. This event in PWR reactor can occur through breaking in primary loop of reactor either hot leg or cold leg.

In this state the existent water in the primary loop along with steam are strongly leaked that blow down event occurs. But when the lost water of primary loop through RHRS (Residual and Heat Removal System) including HPIS (High Pressure Injection System), LPIS (Low Pressure Injection System) and Accumulator (passive system) are filled this process is entitled Refill. When the primary is filled and all the lost water in it is compensated then the reactor sets in the normal status.

In the blow down status the raised steam is due loss of pressure in primary loop and moving the situation of reactor's primary loop from single phase to two phase flow.

LOHA accident from loosing heat sink in reactor is derived. Heat sink is as steam generator in nuclear reactor. This event when occurs heat exchanging between primary and secondary loops are not done.

This event might occur through lacking water circulation in the secondary loop of reactor. Not circulating the water might occur through closing either block valve (which sets after demineralizer tank) or other existent valves in the secondary loop.

LOFA accident from disabling and loosing the pumps in either primary loop or secondary is derived. In case either primary loop's pump or secondary encounter with problems LOFA accident might occur.

In the secondary loop the main feed water pump has duty of circulation of feed water to steam generator and sets after condenser pump.

In view of dynamically stability of a nuclear reactor, there is a stable system so that an excess reactivity is injected to it and it able to be stabled again in shortest time. The stability of linear systems in the field of complex numbers by defining the polarity of closed loop transfer's function is determined.

In case all the polarities are in the left side of imagine page then system will be stabled. In the time field the stability definition means system's response to each input will be definitive. In the matrix form all the Eigen values of system have real negative part.

The Theoretical Simulation of a Model by SIMULINK for

input reactivity to transfer function as shown in Fig.2 :

Fig. 2. The closed loop for several feedback reactivities

There are several controlling factors in nuclear reactors such as:

**1.3 System's variables of nuclear reactor** 

1. Coolant flow rate.

4. Reaction rate. 5. Error function. 6. Temperature of core. 7. Power of reactor. 8. Core expansion.

2. Movement of control rods. 3. Concentration of boric acid.

dynamical system.

Surveying the Work and Dynamical Stability of Nuclear Reactors Cores 179

In order to survey the stability of a closed loop system the term of [1 ( ). ( )] + *Gs Hs* must be set zero and by solving this equation, all the roots that are as zero and pole for closes loop system, will be defined. The stability condition of a closed loop system is lack of positive

Reactivity feedback causes the steady operation of nuclear reactor and equilibrium of its

A transfer function can be either linear or not. Each system variable can be affected as an

real part of poles. It means all the poles must be the left side of real-imagine graph.

#### **1.2 Dynamics of nuclear reactor**

There are several methods for investigation of nuclear reactors dynamics.

One of the most important methods to study reactor dynamics and the stability of nuclear reactor is define of transfer's functions and application of it to analyze the closed loop function.

According to following figure a closed loop system including transfer function, feedback and related applied reactivity are shown:

Fig. 1. The closed loop conversion function based on reactivity

Where:

( ) *<sup>i</sup>* ρ *s* is: input reactivity in frequency field, ( ) *<sup>f</sup>* ρ *s* is: reactivity due to feedback in frequency field, ( ) *<sup>e</sup>* ρ *s* is: error reactivity in frequency field that is as input reactivity to transfer function, *G s*( ) is: transfer function, *H s*( ) is: feedback function and *n s*( )is: output of closed loop conversion function that means the density of neutrons.

There is also:

$$
\rho\_e(\mathbf{s}) = \rho\_i(\mathbf{s}) - \rho\_f(\mathbf{s}) \tag{1}
$$

According to Fig.1 for both transfer function and feedback function existing in closed loop can write:

$$G(s) = \frac{n(s)}{\rho\_\varepsilon(s)},\tag{2}$$

$$H(\mathbf{s}) = \frac{\rho\_f(\mathbf{s})}{n(\mathbf{s})} \tag{3}$$

and it can also be written:

$$\text{Conversion Function:} \qquad \qquad T(\mathbf{s}) = \frac{n(\mathbf{s})}{\rho\_i(\mathbf{s})} = \frac{\mathbf{G}(\mathbf{s})}{\mathbf{1} + \mathbf{G}(\mathbf{s}) \cdot H(\mathbf{s})} \tag{4}$$

One of the most important methods to study reactor dynamics and the stability of nuclear reactor is define of transfer's functions and application of it to analyze the closed loop

According to following figure a closed loop system including transfer function, feedback

ρ

 ρρ

According to Fig.1 for both transfer function and feedback function existing in closed loop

( ) ( ) ( ) *<sup>e</sup> n s G s* ρ

> ( ) ( ) ( ) *f s*

*n s* ρ

*H s*

*ns Gs T s* ρ

 *s* is: error reactivity in frequency field that is as input reactivity to transfer function, *G s*( ) is: transfer function, *H s*( ) is: feedback function and *n s*( )is: output of closed

*s* is: reactivity due to feedback in frequency

*ss s* = − (1)

*<sup>s</sup>* <sup>=</sup> , (2)

= (3)

*s Gs Hs* = = <sup>+</sup> (4)

There are several methods for investigation of nuclear reactors dynamics.

Fig. 1. The closed loop conversion function based on reactivity

loop conversion function that means the density of neutrons.

() () () *eif*

Conversion Function: () () ( ) ( ) 1 ( ). ( ) *<sup>i</sup>*

ρ

*s* is: input reactivity in frequency field, ( ) *<sup>f</sup>*

**1.2 Dynamics of nuclear reactor** 

and related applied reactivity are shown:

function.

Where:

( ) *<sup>i</sup>* ρ

field, ( ) *<sup>e</sup>* ρ

There is also:

can write:

and it can also be written:

In order to survey the stability of a closed loop system the term of [1 ( ). ( )] + *Gs Hs* must be set zero and by solving this equation, all the roots that are as zero and pole for closes loop system, will be defined. The stability condition of a closed loop system is lack of positive real part of poles. It means all the poles must be the left side of real-imagine graph.

Reactivity feedback causes the steady operation of nuclear reactor and equilibrium of its dynamical system.

A transfer function can be either linear or not. Each system variable can be affected as an input reactivity to transfer function as shown in Fig.2 :

Fig. 2. The closed loop for several feedback reactivities
