**1. Introduction**

196 Nuclear Reactors

Munoz, J and Verdu, G., 1991. Application of Hopf bifurcation theory and variational

Munoz, J., Verdu, G., Pereira, C., 1992. Dynamic reconstruction and Lyapunov exponents

Nahla, A., 2009. An analytical solution for the point reactor kinetics equations with one

Pandey, M., 1996. *Nonlinear Reactivity Interactions in Fission Reactor Dynamical Systems*, Ph.D.

Pankaj, W and Vivek, K., 2011. Nonlinear stability analysis of a reduced order model of

Park, G., Cho, N., 1992. Design of a nonlinear model-based controller with adaptive PI gains

Shirazi, S., Aghanajafi, C., Sadoughi, S and Sharifloo, N., 2010. Design, construction and

Smets, H and Giftopoulos, E., 1959. The application of topological methods to the kinetics of homogeneous reactors. *Nuclear Science and Engineering*, Vol. 6, 341–349.

Tachibana, Y., Sawahata, H., Iyoku, T and Nakazawa, T., 2004. Reactivity control system of

Tewari, A., 2002. *Modern Control Design with MATLAB and SIMULINK*. John Wiley, Chapter

Ward, M and Lee, J., 1987. Singular perturbation analysis of relaxation oscillations in reactor

Zhao, F., Cheung, K and Yeung, R., 2003. Optimal power control system of a research

*Nuclear Engineering and Design*, Vol. 241, No.1, (January), 134-143.

Stark, K., 1976. *Modal control of a nuclear power reactor*. Automatica 12, 613-618.

systems. *Nuclear Science and Engineering*, Vol. 95, 47–59.

*Energy*, Vol. 18, No.5, 269–302.

*Energy*, Vol. 51, No.1, 124–128.

233, No.1-3, (October 2004), 89-101.

Thesis, Indian Institute of Technology-Kanpur.

No.4, 223–235.

49.

1665.

2, 3.

252.

methods to the study of limit cycles in boiling water reactors. *Annals of Nuclear* 

from time series data in boiling water reactors. *Annals of Nuclear Energy*, Vol. 19,

group of delayed neutrons and the adiabatic feedback model. *Progress in Nuclear* 

nuclear reactors: A parametric study relevant to the advanced heavy water reactor.

for robust control of a nuclear reactor. *Progress in Nuclear Energy,* Vol. 27, No.1, 37-

simulation of a multipurpose system for precision movement of control rods in nuclear reactors. *Annals of Nuclear Energy,* Vol. 37, No.12, (December 2010)*,* 1659-

the high temperature engineering test reactor. *Nuclear Engineering and Design,* Vol.

nuclear reactor. *Nuclear Engineering and Design,* Vol. 219, No.3, (February 2003), 247-

The problem of fuel life control at nuclear power plants (NPP) with WWER-type light-water reactors (PWR) will be discussed for design (normal) loading conditions only. That is, emergency nuclear reactor (NR) operation leading to cladding material plastic deformation is not studied here, therefore the hot plasticity (stress softening) arising at the expense of yield stress decrease under emergency cladding temperature rise, will not be considered here.

Analysing the current Ukrainian energetics status it is necessary to state that on-peak regulating powers constitute 8 % of the total consolidated power system (CPS), while a stable CPS must have 15 % of on-peak regulating powers at least. More than 95 % of all thermal plants have passed their design life and the Ukrainian thermal power engineering averaged remaining life equals to about 5 years. As known, the nuclear energetics part in Ukraine is near 50 %. Hence, operation of nuclear power units of Ukraine in the variable part of electric loading schedule (variable loading mode) has become actual recently, that means there are repeated cyclic NR capacity changes during NR normal operation.

Control of fuel resource at WWER nuclear units is a complex problem consisting of a few subproblems. First of all, a physically based fuel cladding failure model, fit for all possible regimes of normal NR operation including variable loading and burnups above 50 MW·d/kg, must be worked out. This model must use a certified code developed for fuel element (FE) behaviour analysis, which was verified on available experimental data on cladding destruction.

The next condition for implementation of nuclear fuel resource control is availability of a verified code estimating distribution of power flux in the active core for any reactor normal operation mode including variable loading.

It should be noticed that calculation of nuclear fuel remaining life requires estimating change of the state of a fuel assembly (FA) rack. For instance, the state of a rack can change considerably at core disassembling (after a design accident) or at spent fuel handling. Generally speaking, the total fuel handling time period must be considered including the duration of dry/wet storage. Before designing a nuclear fuel resource control system, using

Theory of Fuel Life Control Methods at Nuclear

active core operational reliability analysis.

multiple cyclic and long-term static loads.

optimization of the alloy fabrication technique.

сladding strength criteria is defined as

operation and varying duty are considered separately:

ω τ

ω

considered.

where

ω τ

describing the material state (

Power Plants (NPP) with Water-Water Energetic Reactor (WWER) 199

long influence of high-level temperature-power stressing leading to uncontrollable cladding material creep processes causing, after a while, its destruction, and fission products enter the circuit in the quantities exceeding both operational limits and limits of safe operation. In this connection, estimation of cladding integrity time for a NR variable loading mode, taking into account some appointed criteria, becomes one of key problems of FE designing and

In accordance with the experience, there are following main characteristic cladding destruction mechanisms for the WWER-1000 varying loading mode (Suzuki, 2010): pelletcladding mechanical interaction (PCMI), especially at low burnups and stress corrosion cracking (SCC); corrosion at high burnups (>50 MWd/kg-U); cladding failure caused by

It is supposed that influence of low-burnup PCMI is eliminated by implementation of the WWER-1000 maximum linear heat rate (LHR) regulation conditions. Non-admission of cladding mechanical damage caused by SCC is ensured by control of linear heat power permissible values and jumps also. The high-burnup corrosion influence is eliminated by

As all power history affects fuel cladding, it is incorrect to transfer experimental stationary and emergency operation cladding material creep data onto the FE cladding working at variable loading. Emergency NR operation leading to cladding material plastic deformation is not studied here, therefore hot plasticity (stress softening) arising at the expense of yield stress decrease under emergency cladding temperature rise, is not

To solve this problem, we are to define main operating conditions affecting FE cladding durability and to study this influence mechanism. The normative safety factor *K*norm for

max *K RR* = / , (1)

= +< , (2)

ω

= 1, for the damaged

norm

where max *R* is the limit value of a parameter; *R* is the estimated value of a parameter.

The groupe of WWER-1000 сladding strength criteria includes the criteria SC1…SC5 – see Table 2 (Novikov et al., 2005). According to SC4, the WWER-1000 FE cladding total damage parameter is usually estimated by the relative service life of cladding, when steady-state

( ) 1 *<sup>i</sup>*

of *i*-type power-cycles and the allowable number of *i*-type power-cycles, respectively; *t* is

The cladding material damage parameter can be considered as a structure parameter

*i i*

time; max *t* is the creep-rupture life under steady-state operation conditions.

max max 0

( ) is the cladding material damage parameter; *NCi* and max *NCi* are the number

= 0, for the intact material and

τ

*NC dt NC t*

probability theory and physically based FA failure criteria, the failure probability for all FA must be estimated. Having satisfied the listed conditions, a computer-based system for control of nuclear fuel remaining life can be worked out.

The FEMAXI code has been used to calculate the cladding stress/strain development for such its quality as simultaneous solution of the FE heat conduction and mechanical deformation equations using the finite element method (FEM) allowing consideration of variable loading (Suzuki, 2000). Sintered uranium dioxide was assumed to be the material of pellets while stress relieved Zircaloy-4 was assumed to be the material of cladding (Suzuki, 2010). Cladding material properties in the FEMAXI code are designated in compliance with (MATPRO-09, 1976). But the manufacturing process and the zircaloy alloy used are not specified here.

FE behaviour for UTVS (the serial FA of WWER-1000, V-320 project), ТVS-А (the serial FA of WWER-1000 produced by OKBM named after I.I. Aphrikantov) and ТVS-W (the serial FA produced by WESTINGHOUSE) has been analysed.

The full list of input parameters used when analyzing the PWR fuel cladding durability can be seen in (Suzuki, 2000). The NR regime and FA constructional parameters were set in compliance with Shmelev's method (Shmelev et al., 2004). The main input parameters of FE and FA used when analyzing the WWER-1000 fuel cladding durability are listed in Table 1.


Таble 1. Different parameters of UТVS, ТVS-А and ТVS-W.

FE cladding rupture life control for a power-cycling nuclear unit having the WWER-1000 NR is a key task in terms of rod design and reliability. Operation of a FE is characterized by

probability theory and physically based FA failure criteria, the failure probability for all FA must be estimated. Having satisfied the listed conditions, a computer-based system for

The FEMAXI code has been used to calculate the cladding stress/strain development for such its quality as simultaneous solution of the FE heat conduction and mechanical deformation equations using the finite element method (FEM) allowing consideration of variable loading (Suzuki, 2000). Sintered uranium dioxide was assumed to be the material of pellets while stress relieved Zircaloy-4 was assumed to be the material of cladding (Suzuki, 2010). Cladding material properties in the FEMAXI code are designated in compliance with (MATPRO-09, 1976). But the manufacturing process and the zircaloy alloy

FE behaviour for UTVS (the serial FA of WWER-1000, V-320 project), ТVS-А (the serial FA of WWER-1000 produced by OKBM named after I.I. Aphrikantov) and ТVS-W (the serial FA

The full list of input parameters used when analyzing the PWR fuel cladding durability can be seen in (Suzuki, 2000). The NR regime and FA constructional parameters were set in compliance with Shmelev's method (Shmelev et al., 2004). The main input parameters of FE and FA used when analyzing the WWER-1000 fuel cladding durability are listed in Table 1.

> Cladding outer diameter, сm 0.910 0.910 0.914 Cladding inner diameter, сm 0.773 0.773 0.800 Cladding thickness, сm 0.069 0.069 0.057 Pellet diameter, сm 0.757 0.757 0.784

Pellet centre hole diameter, сm 0.24 0.14 —

Total fuel weight for a FE, kg 1.385 1.487 1.554

Equivalent coolant hydraulic diameter, сm 1.06 1.06 1.05

FE cladding rupture life control for a power-cycling nuclear unit having the WWER-1000 NR is a key task in terms of rod design and reliability. Operation of a FE is characterized by

Pellet dish — — each side

ТVS

<sup>U</sup>ТVS ТVS-<sup>А</sup> <sup>Т</sup>VS-W

control of nuclear fuel remaining life can be worked out.

produced by WESTINGHOUSE) has been analysed.

Parameter

Таble 1. Different parameters of UТVS, ТVS-А and ТVS-W.

used are not specified here.

long influence of high-level temperature-power stressing leading to uncontrollable cladding material creep processes causing, after a while, its destruction, and fission products enter the circuit in the quantities exceeding both operational limits and limits of safe operation. In this connection, estimation of cladding integrity time for a NR variable loading mode, taking into account some appointed criteria, becomes one of key problems of FE designing and active core operational reliability analysis.

In accordance with the experience, there are following main characteristic cladding destruction mechanisms for the WWER-1000 varying loading mode (Suzuki, 2010): pelletcladding mechanical interaction (PCMI), especially at low burnups and stress corrosion cracking (SCC); corrosion at high burnups (>50 MWd/kg-U); cladding failure caused by multiple cyclic and long-term static loads.

It is supposed that influence of low-burnup PCMI is eliminated by implementation of the WWER-1000 maximum linear heat rate (LHR) regulation conditions. Non-admission of cladding mechanical damage caused by SCC is ensured by control of linear heat power permissible values and jumps also. The high-burnup corrosion influence is eliminated by optimization of the alloy fabrication technique.

As all power history affects fuel cladding, it is incorrect to transfer experimental stationary and emergency operation cladding material creep data onto the FE cladding working at variable loading. Emergency NR operation leading to cladding material plastic deformation is not studied here, therefore hot plasticity (stress softening) arising at the expense of yield stress decrease under emergency cladding temperature rise, is not considered.

To solve this problem, we are to define main operating conditions affecting FE cladding durability and to study this influence mechanism. The normative safety factor *K*norm for сladding strength criteria is defined as

$$K\_{\text{norm}} = \mathbb{R}^{\text{max}} \;/\; R \;/\; \tag{1}$$

where max *R* is the limit value of a parameter; *R* is the estimated value of a parameter.

The groupe of WWER-1000 сladding strength criteria includes the criteria SC1…SC5 – see Table 2 (Novikov et al., 2005). According to SC4, the WWER-1000 FE cladding total damage parameter is usually estimated by the relative service life of cladding, when steady-state operation and varying duty are considered separately:

$$\alpha(\mathbf{r}) = \sum\_{i} \frac{\text{NC}\_{i}}{\text{NC}\_{i}^{\text{max}}} + \int\_{0}^{\mathbf{r}} \frac{dt}{t^{\text{max}}} < \mathbf{1} \,, \tag{2}$$

where ω τ( ) is the cladding material damage parameter; *NCi* and max *NCi* are the number of *i*-type power-cycles and the allowable number of *i*-type power-cycles, respectively; *t* is time; max *t* is the creep-rupture life under steady-state operation conditions.

The cladding material damage parameter can be considered as a structure parameter describing the material state (ω = 0, for the intact material and ω= 1, for the damaged

Theory of Fuel Life Control Methods at Nuclear

where *<sup>i</sup>* Δτ

equation:

stage.

CF = 0.860.

maximum NR capacity (100 %).

verified codes available through an international data bank.

Power Plants (NPP) with Water-Water Energetic Reactor (WWER) 201

describing the maximum number of cycles prior to the cladding failure, still there stays the problem of disagreement between experimental conditions and real operating environment (e.g. fluence; neutron spectrum; rod internal pressure; coolant temperature conditions; cladding water-side corrosion rate; radiation growth; cladding defect distribution; algorithm of fuel pick-and-place operations; reactor control system regulating unit movement amplitude and end effects; loading cycle parameters, etc.). In connection with this problem, to ensure a satisfactory accuracy of the cladding state estimation at variable loading conditions, it is necessary to develop physically based FE cladding durability analysis methods, on the basis of

As is known, when repair time is not considered, reactor capacity factor CF is obtained as

*n*

CF <sup>1</sup>

*i*

=

<sup>=</sup> <sup>⋅</sup> τ

Using (3), the number of daily cycles *Ne,0* that the cladding can withstand prior to the beginning of the rapid creep stage, expressed in effective days, is defined from the following

*Ne,0* = *N0*  CF, where *N0* – the number of calendar daily cycles prior to the beginning of the rapid creep

It should be stressed that CF is a summary number taking into account only the real NR

2. The NR works at 100 % capacity level within 5 days, then the reactor is transferred to 50 % capacity level within 1 hour. Further the NR works at the capacity level of 50 % within 46 hours, then comes back to 100 % capacity level within 1 hour. Such NR operating mode will be designated as the (5 d – 100 %, 46 h – 50 %) weekly load cycle,

3. The NR works at 100 % capacity level within 16 hours, then the reactor is transferred to 75 % capacity level within 1 hour. Further the NR works at 75 % capacity level within 6 hours, then comes back to 100 % capacity level within 1 hour. Such NR operating mode

will be designated as the (16 h – 100 %, 6 h – 75 %) daily load cycle, CF *=* 0.927. 4. The NR works at 100 % capacity level within 16 hours, then the reactor is transferred to 75 % capacity level within 1 hour. Further the NR works at 75 % capacity level within 6 hours, then comes back to 100 % capacity level within 1 hour. But the NR capacity decreases to 50 % level within last hour of every fifth day of a week. Further the reactor works during 47 hours at 50 % capacity level and, at last, within last hour of every seventh day the NR capacity rises to the level of 100 %. Such NR operating mode will be designated as the (5 d – 100 % + 75 %, 2 d – 50 %) combined load cycle, CF = 0.805.

loading history. For instance, the following NR loading modes can be considered:

1. Stationary operation at 100 % NR capacity level, CF = 1.

Δ

*T P*

( )

*i i*

⋅

*P*

– NR operating time at the capacity of *Pi* ; *T* – total NR operating time; *P* –

, (3)

material). The second possible approach is considering ω τ( ) as a characteristic of discontinuity flaw. That is when ω = 0, there are no submicrocracks in the cladding material. But if ω = 1, it is supposed that the submicrocracks have integrated into a macrocrack situated in some cross-section of the cladding


Таble 2. Сladding strength criteria.

An experimental study of Zircaloy-4 cladding deformation behavior under cyclic pressurization (at 350 °С) was carried out in (Kim et al., 2007). The investigated cladding had an outer diameter and thickness of 9.5 mm and 0.57 mm, respectively. The microstructure of Zircaloy-4 was a stress-relieved state. A sawtooth pressure waveform was applied at different rates of pressurization and depressurization, where the maximum hoop stress was varied from 310 MPa to 470 MPa, while the minimum hoop stress was held constant at 78 MPa. Using the cladding stress-life diagram and analyzing the metal structure and fatigue striation appearance, it was found that when loading frequency ν < 1 Hz, creep was the main mechanism of thin cladding deformation, while the fatigue component of strain was negligibly small.

Taking into account the experimental results (Kim et al., 2007), it can be concluded that estimation of ω τ( ) by separate consideration of NR steady-state operation and varying duty (2) has the following disadvantages: the physical mechanism (creep) of cladding damage accumulation and real stress history are not taken into account; uncertainty of the cladding durability estimate forces us into unreasonably assumption *K*norm = 10; there is no public data on max *Ni* and max *<sup>t</sup>* for all possible loading conditions.

Now the WWER-1000 fuel cladding safety and durability requirements have not been clearly defined (Semishkin et al., 2009). As strength of fuel elements under multiple cyclic power changes is of great importance when performing validation of a NR project, a tendency to indepth studies of this problem is observed. The well-known cladding fatigue failure criterion based on the relationship between the maximum circumferential stress amplitude max σθ and the allowable number of power-cycles max *NC* is most popular at present (Kim et al., 2007). Nevertheless, in case of satisfactory fit between the experimental and calculated data

Criterion Definition *K*norm

φ

( ) 1 *<sup>i</sup>*

*NC dt NC t*

0

τ

SC3 *P*<sup>c</sup> <sup>≤</sup> max *<sup>P</sup>*<sup>c</sup> , where *P*c is coolant pressure, Pа. 1.5

*i i*

An experimental study of Zircaloy-4 cladding deformation behavior under cyclic pressurization (at 350 °С) was carried out in (Kim et al., 2007). The investigated cladding had an outer diameter and thickness of 9.5 mm and 0.57 mm, respectively. The microstructure of Zircaloy-4 was a stress-relieved state. A sawtooth pressure waveform was applied at different rates of pressurization and depressurization, where the maximum hoop stress was varied from 310 MPa to 470 MPa, while the minimum hoop stress was held constant at 78 MPa. Using the cladding stress-life diagram and analyzing the metal structure and fatigue striation appearance, it was found that when loading frequency ν < 1 Hz, creep was the main mechanism of thin cladding deformation, while the fatigue component of

Taking into account the experimental results (Kim et al., 2007), it can be concluded that

(2) has the following disadvantages: the physical mechanism (creep) of cladding damage accumulation and real stress history are not taken into account; uncertainty of the cladding durability estimate forces us into unreasonably assumption *K*norm = 10; there is no public

Now the WWER-1000 fuel cladding safety and durability requirements have not been clearly defined (Semishkin et al., 2009). As strength of fuel elements under multiple cyclic power changes is of great importance when performing validation of a NR project, a tendency to indepth studies of this problem is observed. The well-known cladding fatigue failure criterion based on the relationship between the maximum circumferential stress amplitude max

the allowable number of power-cycles max *NC* is most popular at present (Kim et al., 2007). Nevertheless, in case of satisfactory fit between the experimental and calculated data

*t* for all possible loading conditions.

( ) by separate consideration of NR steady-state operation and varying duty

ω τ

= 1, it is supposed that the submicrocracks have integrated into a

*<sup>e</sup>* is maximum equivalent stress, Pа;

= 0, there are no submicrocracks in the cladding

is maximum circumferential stress. 1.2

= +< . 10

is cladding limit circumferential plastic strain –

is neutron fluence, cm-2s-1. –

( ) as a characteristic of

σ<sup>0</sup> is

> σθand

material). The second possible approach is considering

macrocrack situated in some cross-section of the cladding

≤ 250 МPа, where max

φ

yield stress, Pа; *Т* is temperature, К;

SC4 max max

≤ 0.5 % , where max

ω τ

> θ ,*pl* ε

ω

σθ

 , where max σ

discontinuity flaw. That is when

max σ

Таble 2. Сladding strength criteria.

 σ*<sup>e</sup>* < <sup>0</sup> ( ) *Т*,

ω

material. But if

SC2

SC1 max σθ

SC5 max θ ,*pl* ε

strain was negligibly small.

ω τ

data on max *Ni* and max

estimation of

describing the maximum number of cycles prior to the cladding failure, still there stays the problem of disagreement between experimental conditions and real operating environment (e.g. fluence; neutron spectrum; rod internal pressure; coolant temperature conditions; cladding water-side corrosion rate; radiation growth; cladding defect distribution; algorithm of fuel pick-and-place operations; reactor control system regulating unit movement amplitude and end effects; loading cycle parameters, etc.). In connection with this problem, to ensure a satisfactory accuracy of the cladding state estimation at variable loading conditions, it is necessary to develop physically based FE cladding durability analysis methods, on the basis of verified codes available through an international data bank.

As is known, when repair time is not considered, reactor capacity factor CF is obtained as

$$\text{CF} = \frac{\sum\_{i=1}^{n} (\Delta \tau\_i \cdot P\_i)}{T \cdot P} \tag{3}$$

where *<sup>i</sup>* Δτ – NR operating time at the capacity of *Pi* ; *T* – total NR operating time; *P* – maximum NR capacity (100 %).

Using (3), the number of daily cycles *Ne,0* that the cladding can withstand prior to the beginning of the rapid creep stage, expressed in effective days, is defined from the following equation:

$$N\_{e,0} = N\_0 \cdot \mathbf{CF}\_{\prime}$$

where *N0* – the number of calendar daily cycles prior to the beginning of the rapid creep stage.

It should be stressed that CF is a summary number taking into account only the real NR loading history. For instance, the following NR loading modes can be considered:


Theory of Fuel Life Control Methods at Nuclear

2007) as

ω τ

experimentally proved matter.

condition max max

 τ

situations (Semishkin, 2007).

0 00 () () στστ

by any of two ways:

stress max 0 σ

segments).

Power Plants (NPP) with Water-Water Energetic Reactor (WWER) 203

τ

ω τ

0

( ) *e e <sup>p</sup> A* σ

where *ki* are material parameters defined from experiments with micromodels cut out

According to (Semishkin, 2007), for LOCA-accidents only, using the failure condition

In contrast to the experimental technique for determining *A*0 developed in (Semishkin, 2007), the calculation method proposed in (Pelykh et al., 2008) means that *A*0 can be found

1. As the SDE value at the moment τ0 of cladding stability loss, which is determined by

2. As the SDE value at the rapid creep start moment for the cladding point having the maximum temperature. This way is the most conservative approach, and it is not

The equivalent stress σ*e* and the rate of equivalent creep strain *<sup>e</sup> p* are calculated by the LWR fuel analysis code FEMAXI (Suzuki, 2000). Though cladding creep test data must have been used to develop and validate the constitutive models used in the finite element code FEMAXI to calculate the equivalent creep strains under cyclic loading, difficulty of this problem is explained by the fact that cladding material creep modeling under the conditions corresponding to real operational variable load modes is inconvenient or impossible as such tests can last for years. As a rule, the real FE operational conditions can be simulated in such tests very approximately only, not taking into account all the variety of possible exploitation

The code FEMAXI analyzes changes in the thermal, mechanical and chemical state of a single fuel rod and interaction of its components in a given NR power history and coolant

obvious that such level of conservatism is really necessary when estimating *A*<sup>0</sup> .

(according to the calculation model, a fuel rod is divided into axial and radial

*<sup>e</sup>* = , when equivalent stress max( )

() 1 = , the SDE value *A0* accumulated by the moment of cladding failure and supposed to be temperature-dependent only, is determined from the equations (9)-(10). At the same time, the assumption that the value of *A*0 at high-temperature creep and cladding failure analysis is loading history independent, is accepted for LOCA-accidents as an

<sup>⋅</sup> = ⋅= (8)

<sup>⋅</sup> <sup>=</sup> , (10)

σ

( ) for the point of the cladding having the maximum temperature

 τ

*<sup>e</sup>* becomes equal to yield

( ) ), (9)

0 0 ( ) <sup>1</sup> *e e <sup>p</sup> <sup>d</sup> A*

The CET-method of light-water reactor (LWR) FE cladding operation life estimation can be considered as advancement of the method developed for FE cladding failure moment estimation at loss-of-coolant severe accidents (LOCA) (Semishkin, 2007). The equations of creep and cladding damage accumulation for zirconium alloys are given in (Semishkin,

τ σ

*<sup>e</sup> p* = *f(ki , T*, σe ,

ω τ

along the FE cladding orthotropy directions; *T* is absolute temperature, К.

ω τ

#### **2. The CET-method of fuel cladding durability estimation at variable loading**

The new cladding durability analysis method, which is based on the creep energy theory (CET) and permits us to integrate all known cladding strength criteria within a single calculation model, is fit for any normal WWER/PWR operating conditions (Pelykh et al., 2008). The CET-model of cladding behaviour makes it possible to work out cladding rupture life control methods for a power-cycling WWER-1000 nuclear unit. As the WWER-1000 Khmelnitskiy nuclear power plant (KhNPP) is a base station for study of varying duty cycles in the National Nuclear Energy Generating Company ENERGOATOM (Ukraine), the second power unit of KhNPP will be considered.

According to CET, to estimate FE cladding running time under multiple cyclic NR power changes, it is enough to calculate the energy *A*0 accumulated during the creep process, by the moment of cladding failure and spent for cladding material destruction (Sosnin and Gorev, 1986). The energy spent for FE cladding material destruction is called as specific dispersion energy (SDE) *A*(*τ*). The proposed method of FE cladding running time analysis is based on the following assumptions of CET: creep and destruction processes proceed in common and influence against each other; at any moment τ creep process intensity is estimated by specific dispersion power (SDP) *W* (*τ*), while intensity of failure is estimated by *A*(*τ*) accumulated during the creep process by the moment τ

$$A(\boldsymbol{\pi}) = \bigcap\_{0}^{\boldsymbol{\pi}} \mathcal{W}(\boldsymbol{\pi}) \cdot d\boldsymbol{\pi} \; \; \; \; \tag{4}$$

where SDP standing in (4) is defined by the following equation (Nemirovsky, 2001):

$$\mathcal{W}(\mathfrak{r}) = \sigma\_e \cdot \dot{p}\_e \tag{5}$$

where σ *<sup>e</sup>* is equivalent stress, Pa; *<sup>e</sup> p* is rate of equivalent creep strain, s-1. Equivalent stress σ*<sup>e</sup>* is expressed as

$$
\sigma\_{\varepsilon} = \sqrt{\frac{1}{2} \left[ \left( \sigma\_{\theta} - \sigma\_{z} \right)^{2} + \sigma\_{\theta}^{2} + \sigma\_{z}^{2} \right]},
\tag{6}
$$

where and σ σ θ*<sup>z</sup>* are circumferential stress and axial stress, respectively.

The cladding material failure parameter ω τ( ) is entered into the analysis:

0 ωτ τ( ) ( )/ = *A A* , (7)

where *A0* is SDE at the moment of cladding material failure beginning, known for the given material either from experiment, or from calculation, J/m3 (Sosnin and Gorev, 1986); ω = 0 – for intact material, ω= 1 – for damaged material.

The proposed method enables us to carry out quantitative assessment of accumulated ω τ( ) for different NR loading modes, taking into account a real NR load history (Pelykh et al., 2008). The condition of cladding material failure is derived from (4), (5) and (7):

**2. The CET-method of fuel cladding durability estimation at variable loading**  The new cladding durability analysis method, which is based on the creep energy theory (CET) and permits us to integrate all known cladding strength criteria within a single calculation model, is fit for any normal WWER/PWR operating conditions (Pelykh et al., 2008). The CET-model of cladding behaviour makes it possible to work out cladding rupture life control methods for a power-cycling WWER-1000 nuclear unit. As the WWER-1000 Khmelnitskiy nuclear power plant (KhNPP) is a base station for study of varying duty cycles in the National Nuclear Energy Generating Company ENERGOATOM (Ukraine), the

According to CET, to estimate FE cladding running time under multiple cyclic NR power changes, it is enough to calculate the energy *A*0 accumulated during the creep process, by the moment of cladding failure and spent for cladding material destruction (Sosnin and Gorev, 1986). The energy spent for FE cladding material destruction is called as specific dispersion energy (SDE) *A*(*τ*). The proposed method of FE cladding running time analysis is based on the following assumptions of CET: creep and destruction processes proceed in

estimated by specific dispersion power (SDP) *W* (*τ*), while intensity of failure is estimated by

0 *A Wd* () () τ

> ( ) *W p e e* τ σ

( ) <sup>1</sup> <sup>2</sup> 2 2

 τ

where *A0* is SDE at the moment of cladding material failure beginning, known for the given material either from experiment, or from calculation, J/m3 (Sosnin and Gorev, 1986);

for different NR loading modes, taking into account a real NR load history (Pelykh et al.,

= 1 – for damaged material.

The proposed method enables us to carry out quantitative assessment of accumulated

 *e zz* θ

*<sup>z</sup>* are circumferential stress and axial stress, respectively.

ω τ

ωτ

2008). The condition of cladding material failure is derived from (4), (5) and (7):

 σ

 θ

0

 σ

( ) is entered into the analysis:

= − ++ , (6)

( ) ( )/ = *A A* , (7)

 ττ

τ

where SDP standing in (4) is defined by the following equation (Nemirovsky, 2001):

*<sup>e</sup>* is equivalent stress, Pa; *<sup>e</sup> p* is rate of equivalent creep strain, s-1.

2

 σσ τ

= ⋅ , (4)

= ⋅ , (5)

τ creep process intensity is

ω τ( )

second power unit of KhNPP will be considered.

common and influence against each other; at any moment

*A*(*τ*) accumulated during the creep process by the moment

*<sup>e</sup>* is expressed as

σ

ω

where

ω

σ

σ

 σ

= 0 – for intact material,

The cladding material failure parameter

Equivalent stress

where and σ

θ

$$\rho\alpha(\tau) = \int\_0^{\tau} \frac{\sigma\_e \cdot \dot{p}\_e}{A\_0} \cdot d\tau = 1 \tag{8}$$

The CET-method of light-water reactor (LWR) FE cladding operation life estimation can be considered as advancement of the method developed for FE cladding failure moment estimation at loss-of-coolant severe accidents (LOCA) (Semishkin, 2007). The equations of creep and cladding damage accumulation for zirconium alloys are given in (Semishkin, 2007) as

$$
\dot{p}\_e = f(k\_i, T, \sigma\_e, \,\,\alpha(\tau) \,),
\tag{9}
$$

$$
\dot{\phi}(\mathbf{r}) = \frac{\sigma\_e \cdot \dot{p}\_e}{A\_0} \,\tag{10}
$$

where *ki* are material parameters defined from experiments with micromodels cut out along the FE cladding orthotropy directions; *T* is absolute temperature, К.

According to (Semishkin, 2007), for LOCA-accidents only, using the failure condition ω τ() 1 = , the SDE value *A0* accumulated by the moment of cladding failure and supposed to be temperature-dependent only, is determined from the equations (9)-(10). At the same time, the assumption that the value of *A*0 at high-temperature creep and cladding failure analysis is loading history independent, is accepted for LOCA-accidents as an experimentally proved matter.

In contrast to the experimental technique for determining *A*0 developed in (Semishkin, 2007), the calculation method proposed in (Pelykh et al., 2008) means that *A*0 can be found by any of two ways:


The equivalent stress σ*e* and the rate of equivalent creep strain *<sup>e</sup> p* are calculated by the LWR fuel analysis code FEMAXI (Suzuki, 2000). Though cladding creep test data must have been used to develop and validate the constitutive models used in the finite element code FEMAXI to calculate the equivalent creep strains under cyclic loading, difficulty of this problem is explained by the fact that cladding material creep modeling under the conditions corresponding to real operational variable load modes is inconvenient or impossible as such tests can last for years. As a rule, the real FE operational conditions can be simulated in such tests very approximately only, not taking into account all the variety of possible exploitation situations (Semishkin, 2007).

The code FEMAXI analyzes changes in the thermal, mechanical and chemical state of a single fuel rod and interaction of its components in a given NR power history and coolant

Theory of Fuel Life Control Methods at Nuclear

Fig. 1. Analysis model.

AS.

having four degrees of freedom, as is shown in Fig. 2.

Fig. 2. Quadrangular model element with four degrees of freedom.

Power Plants (NPP) with Water-Water Energetic Reactor (WWER) 205

In this system, stress/strain analysis is performed using FEM with quadrangular elements

Fig. 3 shows relationship between mesh division and degree of freedom for each node in an

conditions. The code analytical scope covers normal operation conditions and transient conditions such as load-following and rapid power increase in a high-burnup region of over 50 MWd/kg-U.

In the creep model used in the code, irradiation creep effects are taken into consideration and rate of equivalent cladding creep strain *<sup>e</sup> p* is expressed with a function of cladding stress, temperature and fast neutron flux (MATPRO-09, 1976):

$$\dot{p}\_c = \mathbf{K} \cdot \Phi \left(\sigma\_\theta + \mathbf{B} \cdot \exp\left(\mathbf{C} \cdot \sigma\_\theta\right)\right) \exp\left(-\mathbf{Q} / \mathbf{R} \cdot T\right) \mathbf{\tau}^{-0.5},\tag{11}$$

where *<sup>e</sup> p* is biaxial creep strain rate, s-1 ; K, B, C are known constants characterizing the cladding material properties; Φ is fast neutron flux (E > 1.0 МeV), 1/m2s; σθ is circumferential stress, Pa;Q = <sup>4</sup> 10 J/mol; R = 1.987 cal/mol⋅K; *T* is cladding temperature, K; τis time, s.

According to (11), creep strain increases as fast neutron flux, cladding temperature, stress and irradiation time increase.

For creep under uniaxial stress, cladding and pellet creep equations can be represented as (Suzuki, 2010):

$$\dot{p}\_e = f\left(\sigma\_{e'}\mathcal{E}^H, T, \Phi, \dot{F}\right)\_{\prime\prime} \tag{12}$$

where *<sup>e</sup> p* is equivalent creep strain rate, с-1; σ *<sup>e</sup>* is equivalent stress, Pa; *<sup>H</sup>*ε is creep hardening parameter; *F* is fission rate, 1/m3s.

When equation (12) is generalized for a multi-axial stress state, the creep strain rate vector { *p* } is expressed as a vector function { β} of stress and creep hardening parameter:

$$\begin{Bmatrix} \not\supset \dot{\jmath} \end{Bmatrix} = \left\{ \begin{Bmatrix} \{\sigma\} \ \varepsilon^H \end{Bmatrix} \right\} \,\, \, \tag{13}$$

where *T*, Φ and *F* are omitted because they can be dealt with as known parameters.

When a calculation at time *tn* is finished and a calculation in the next time increment *<sup>n</sup>* <sup>1</sup> *t* Δ <sup>+</sup> is being performed, the creep strain increment vector is represented as

$$\mathbb{E}\left\{\Delta p\_{n+1}\right\} = \Delta t\_{n+1} \left\{\dot{p}\_{n+\theta}\right\} = \left\{\,\,\beta\{\sigma\_{n+\theta}\}, \,\varepsilon\_{n+\theta}^{H}\right\},\tag{14}$$

where {σ θ σ θσ *n nn* + + θ} = − ⋅ +⋅ ( ) 1 { } { <sup>1</sup>} ; ( ) <sup>1</sup> 1 *<sup>H</sup> H H n* θ *n n* ε θ ε θε+ + = − ⋅ +⋅ ; 0 ≤ θ ≤ 1.

In order to stress importance of numerical solution stability, θ = 1 is set.

Then, when the (*i*+1)-th iteration by the Newton-Raphson method is being performed after completion of the (*i*)-th iteration, the creep strain rate vector is expressed (Suzuki, 2010).

As shown in Fig. 1, the analysis model includes a 2-dimensional axisymmetrical system in which the entire length of a fuel rod is divided into AS, and each AS is further divided into concentric ring elements in the radial direction.

Fig. 1. Analysis model.

conditions. The code analytical scope covers normal operation conditions and transient conditions such as load-following and rapid power increase in a high-burnup region of

In the creep model used in the code, irradiation creep effects are taken into consideration and rate of equivalent cladding creep strain *<sup>e</sup> p* is expressed with a function of cladding

K B C QR ( ) ( ) ( ) 0.5 exp exp / , *<sup>e</sup> p T*

where *<sup>e</sup> p* is biaxial creep strain rate, s-1 ; K, B, C are known constants characterizing the

circumferential stress, Pa;Q = <sup>4</sup> 10 J/mol; R = 1.987 cal/mol⋅K; *T* is cladding

According to (11), creep strain increases as fast neutron flux, cladding temperature, stress

For creep under uniaxial stress, cladding and pellet creep equations can be represented as

When equation (12) is generalized for a multi-axial stress state, the creep strain rate vector

{ } { ( ) { }, } *<sup>H</sup> <sup>p</sup>* <sup>=</sup> βσε

When a calculation at time *tn* is finished and a calculation in the next time increment *<sup>n</sup>* <sup>1</sup> *t* Δ <sup>+</sup>

{ 1 1 } { } { { }, } *<sup>H</sup> n nn n n p tp* θ

Then, when the (*i*+1)-th iteration by the Newton-Raphson method is being performed after completion of the (*i*)-th iteration, the creep strain rate vector is expressed (Suzuki, 2010).

As shown in Fig. 1, the analysis model includes a 2-dimensional axisymmetrical system in which the entire length of a fuel rod is divided into AS, and each AS is further divided into

} = − ⋅ +⋅ ( ) 1 { } { <sup>1</sup>} ; ( ) <sup>1</sup> 1 *<sup>H</sup> H H n* θ

ε

β

( ) , ,, , *<sup>H</sup> e e p* = Φ *f TF* σ ε

σ

are omitted because they can be dealt with as known parameters.

βσ

 θθ

*n n*

 θ ε θε+ + = − ⋅ +⋅ ; 0 ≤ θ ≤ 1.

 εΔ =Δ = + ++ + + , (14)

 θ  τ

, (12)

} of stress and creep hardening parameter:

*<sup>e</sup>* is equivalent stress, Pa; *<sup>H</sup>*

, (13)

σθis

is creep

ε

<sup>−</sup> = ⋅Φ + ⋅ ⋅ − ⋅ (11)

σσ

cladding material properties; Φ is fast neutron flux (E > 1.0 МeV), 1/m2s;

stress, temperature and fast neutron flux (MATPRO-09, 1976):

θ

is fission rate, 1/m3s.

is being performed, the creep strain increment vector is represented as

In order to stress importance of numerical solution stability, θ = 1 is set.

 θσ

over 50 MWd/kg-U.

temperature, K;

(Suzuki, 2010):

τ

and irradiation time increase.

hardening parameter; *F*

where *T*, Φ and *F*

σ

θ

where {

is time, s.

where *<sup>e</sup> p* is equivalent creep strain rate, с-1;

{ *p* } is expressed as a vector function {

 θ σ

*n nn* + +

concentric ring elements in the radial direction.

In this system, stress/strain analysis is performed using FEM with quadrangular elements having four degrees of freedom, as is shown in Fig. 2.

Fig. 2. Quadrangular model element with four degrees of freedom.

Fig. 3 shows relationship between mesh division and degree of freedom for each node in an AS.

Theory of Fuel Life Control Methods at Nuclear

efficiency (Maksimov et al., 2009):

power irregularity, AO is calculated as

within 2 h.

within 2 h.

respectively.

Power Plants (NPP) with Water-Water Energetic Reactor (WWER) 207

unwanted xenon oscillations are suppressed by the NR control group movement. At first, boric acid solution is injected so that the NR capacity decreases to 90 %, while the NR inlet coolant temperature is maintained constant at the expense of the Main Steam Line (MSL) pressure rise. To guarantee suppression of xenon oscillations, the optimal instantaneous Axial Offset (АО) is maintained due to the NR control group movement. Further the NR power is lowered at the expense of poisoning. The NR capacity will reach the 80% level in 2–3 h and the capacity will be stabilized by intake of the "pure distillate". The NR capacity will be partly restored at the expense of depoisoning starting after the maximal iodine poisoning. To restore the nominal NR power level, the "pure distillate" is injected into the NR circuit and the MSL pressure is lowered, while the NR coolant inlet temperature is maintained constant. The optimal instantaneous AO to be maintained, the control rod group is extracted from the active core. The automatic controller maintains the capacity and xenon oscillations are suppressed by

The proposed algorithm advantages: lowering of switching number; lowering of "pure distillate" and boric acid solution rate; lowering of unbalanced water flow; improvement of fuel operation conditions. Also, the proposed NR capacity program meaning the NR inlet coolant temperature stability, while the MSL pressure lies within the limits of 5.8–6.0 MPa and the NR capacity changes within the limits of 100–80 %, has the advantages of the well known capacity program with the first circuit coolant average temperature constancy.

The capacity program with the first circuit coolant average temperature constancy is widely used at Russian nuclear power units with WWER-reactors due to the main advantage of this program consisting of the possibility to change the unit power level when the reactor control rods stay at almost constant position. At the same time, as the MSL pressure lies within the procedural limits, the proposed algorithm is free of the constant first circuit temperature program main disadvantage consisting of the wide range of MSL pressure change. Two WWER-1000 daily maneuver algorithms were compared in the interests of

1. The algorithm tested at KhNPP ("Tested") on April 18, 2006: power lowering to 80 % within 1 h – operation at the 80 % power level within 7 h – power rising to 100 %

2. The proposed algorithm ("Proposed"): power lowering to 90 % by boric acid solution injection within 0.5 h – further power lowering tо 80 % at the expense of NR poisoning within 2.5 h – operation at the 80 % power level within 4 h – power rising to 100 %

Comparison of the above mentioned daily maneuver algorithms was done with the help of the "Reactor Simulator" (RS) code (Philimonov and Mamichev, 1998). To determine axial

> АО *N N N* <sup>−</sup> <sup>=</sup> *u l* ,

where *Nu*, *Nl*, *N* are the core upper half power, lower half power and whole power,

The instantaneous АО corresponds to the current xenon distribution, while the equilibrium АО corresponds to the equilibrium xenon distribution. Having used the proposed method

the control group movement after the NR has reached the nominal power level.

Fig. 3. Mesh division of FEM (for one AS).

In Fig. 3, the number of mesh divisions in the radial direction of pellet and cladding is fixed at 10 and 4, respectively. The inner two meshes of a cladding (11, 12) are metal phase, and the outer two meshes (13, 14) are oxide layer (ZrO2). The model used in the code takes into account that the oxide layer mesh and metal mesh are re-meshed and change their thickness with the progress of corrosion.

The fuel temperature calculation was carried out with the difference between the numerical solution and analytical solution not exceeding 0.1 %. The numerical error arising in the form of residue from iterative creep calculation on each time step, was not estimated as in most cases this error is exceeded by other uncertainties, first of all by thermal conductivity model error (Suzuki, 2010).

Denoting the number of daily load NR power cycles as *N,* using the CET-model, the dependence *A* (*N*), as well as the borders of characteristic creep stages (unsteady, steady and rapid creep) for zircaloy cladding were obtained for the WWER daily load cycle (16 h − 100 %; 6 h − *k*100 %), where *k* = 1; 0.75; 0.5; 0.25. Hence the number of daily cycles *Ne,0*  that the cladding can withstand prior to the rapid creep stage beginning could be calculated. The conclusion was made that the calculated value of *A0* is not constant for a given material and depends on the operating mode of multiple cyclic power changes (Pelykh, 2008).

It was found, that the calculated equivalent creep strain *<sup>e</sup> p* for zircaloy cladding, for all daily load modes, gradually increases and a hysteresis decrease of *<sup>e</sup> p* can be seen at the last creep stage beginning. Then, after the hysteresis decrease, *<sup>e</sup> p* starts to grow fast and achieves considerable values from cladding reliability point of view. At the rapid creep beginning, the equivalent stress σ *<sup>e</sup>* decrease trend changes into the σ *<sup>e</sup>* increase trend, at the same time *<sup>e</sup> p* decreases a little, that is there is a "hysteresis loop", when the *<sup>e</sup> p* increase has got a phase delay in comparison with the σ *<sup>e</sup>* increase. It should be noted, that the cause of the *<sup>e</sup> p* hysteresis decrease effect must be additionally studied as *<sup>e</sup> p* is expected to continuously increase unless the cladding is subjected to significant compressive creep stresses during the cycle and that this had been properly included in the creep material model.

The following new NR power daily maneuver algorithm was proposed in (Maksimov et al., 2009). It is considered that a nuclear unit is working at the nominal power level (100 %),

In Fig. 3, the number of mesh divisions in the radial direction of pellet and cladding is fixed at 10 and 4, respectively. The inner two meshes of a cladding (11, 12) are metal phase, and the outer two meshes (13, 14) are oxide layer (ZrO2). The model used in the code takes into account that the oxide layer mesh and metal mesh are re-meshed and change their thickness

The fuel temperature calculation was carried out with the difference between the numerical solution and analytical solution not exceeding 0.1 %. The numerical error arising in the form of residue from iterative creep calculation on each time step, was not estimated as in most cases this error is exceeded by other uncertainties, first of all by thermal conductivity model

Denoting the number of daily load NR power cycles as *N,* using the CET-model, the dependence *A* (*N*), as well as the borders of characteristic creep stages (unsteady, steady and rapid creep) for zircaloy cladding were obtained for the WWER daily load cycle (16 h − 100 %; 6 h − *k*100 %), where *k* = 1; 0.75; 0.5; 0.25. Hence the number of daily cycles *Ne,0*  that the cladding can withstand prior to the rapid creep stage beginning could be calculated. The conclusion was made that the calculated value of *A0* is not constant for a given material and depends on the operating mode of multiple cyclic power changes (Pelykh, 2008).

It was found, that the calculated equivalent creep strain *<sup>e</sup> p* for zircaloy cladding, for all daily load modes, gradually increases and a hysteresis decrease of *<sup>e</sup> p* can be seen at the last creep stage beginning. Then, after the hysteresis decrease, *<sup>e</sup> p* starts to grow fast and achieves considerable values from cladding reliability point of view. At the rapid creep beginning, the

decreases a little, that is there is a "hysteresis loop", when the *<sup>e</sup> p* increase has got a phase

hysteresis decrease effect must be additionally studied as *<sup>e</sup> p* is expected to continuously increase unless the cladding is subjected to significant compressive creep stresses during the

The following new NR power daily maneuver algorithm was proposed in (Maksimov et al., 2009). It is considered that a nuclear unit is working at the nominal power level (100 %),

σ

*<sup>e</sup>* increase. It should be noted, that the cause of the *<sup>e</sup> p*

*<sup>e</sup>* increase trend, at the same time *<sup>e</sup> p*

*<sup>e</sup>* decrease trend changes into the

cycle and that this had been properly included in the creep material model.

σ

Fig. 3. Mesh division of FEM (for one AS).

with the progress of corrosion.

error (Suzuki, 2010).

equivalent stress

σ

delay in comparison with the

unwanted xenon oscillations are suppressed by the NR control group movement. At first, boric acid solution is injected so that the NR capacity decreases to 90 %, while the NR inlet coolant temperature is maintained constant at the expense of the Main Steam Line (MSL) pressure rise. To guarantee suppression of xenon oscillations, the optimal instantaneous Axial Offset (АО) is maintained due to the NR control group movement. Further the NR power is lowered at the expense of poisoning. The NR capacity will reach the 80% level in 2–3 h and the capacity will be stabilized by intake of the "pure distillate". The NR capacity will be partly restored at the expense of depoisoning starting after the maximal iodine poisoning. To restore the nominal NR power level, the "pure distillate" is injected into the NR circuit and the MSL pressure is lowered, while the NR coolant inlet temperature is maintained constant. The optimal instantaneous AO to be maintained, the control rod group is extracted from the active core. The automatic controller maintains the capacity and xenon oscillations are suppressed by the control group movement after the NR has reached the nominal power level.

The proposed algorithm advantages: lowering of switching number; lowering of "pure distillate" and boric acid solution rate; lowering of unbalanced water flow; improvement of fuel operation conditions. Also, the proposed NR capacity program meaning the NR inlet coolant temperature stability, while the MSL pressure lies within the limits of 5.8–6.0 MPa and the NR capacity changes within the limits of 100–80 %, has the advantages of the well known capacity program with the first circuit coolant average temperature constancy.

The capacity program with the first circuit coolant average temperature constancy is widely used at Russian nuclear power units with WWER-reactors due to the main advantage of this program consisting of the possibility to change the unit power level when the reactor control rods stay at almost constant position. At the same time, as the MSL pressure lies within the procedural limits, the proposed algorithm is free of the constant first circuit temperature program main disadvantage consisting of the wide range of MSL pressure change. Two WWER-1000 daily maneuver algorithms were compared in the interests of efficiency (Maksimov et al., 2009):


Comparison of the above mentioned daily maneuver algorithms was done with the help of the "Reactor Simulator" (RS) code (Philimonov and Mamichev, 1998). To determine axial power irregularity, AO is calculated as

$$\text{AO} = \frac{N\_u - N\_l}{N} \,\prime$$

where *Nu*, *Nl*, *N* are the core upper half power, lower half power and whole power, respectively.

The instantaneous АО corresponds to the current xenon distribution, while the equilibrium АО corresponds to the equilibrium xenon distribution. Having used the proposed method

Theory of Fuel Life Control Methods at Nuclear

Power Plants (NPP) with Water-Water Energetic Reactor (WWER) 209

effective density, maximum FE linear heat rate, coolant inlet temperature, coolant inlet pressure, coolant velocity, initial He pressure, FE grid spacing, etc. (Maksimov and Pelykh, 2009). For example, dependence of cladding SDE on the number of effective days *N*, for pellet

centre hole diameter *hole d* = 0.140 сm, 0.112 сm and 0.168 сm, is shown in Fig. 4.

Fig. 4. Dependence of SDE on *N* for *hole d* : 0.140 сm (1); 0.112 сm (2); 0.168 сm (3).

σ

cladding point having the maximum temperature, on the number of effective days *N*, for

Fig. 5. Dependence of cladding yield stress (1) and equivalent stress (2; 3) on *N* for *dhole*: 0.112

 τ

*<sup>e</sup>* and yield stress max

0 σ

 τ

( ) , for the

Dependence of cladding equivalent stress max( )

сm (2); 0.168 сm (3). Determination of τ0 for *dhole* = 0.112 сm.

*hole d* = 0.112 сm and 0.168 сm, is shown in Fig. 5.

of cladding failure estimation for zircaloy cladding and WWER-type NR, dependence of the irreversible creep deformation accumulated energy from the number of daily load cycles is calculated for the "Tested" and "Proposed" algorithms, and efficiency comparison is fulfilled – see Table 3.


Тable 3. Efficiency comparison for two daily maneuvering algorithms.

For the "Proposed" algorithm, taking into account the lower switching number necessary to enter "pure distillate" and boric acid solution during the maneuver, slight divergency of the instantaneous and equilibrium АО diagrams, the lower amplitude of АО change during the maneuver, the higher turbo-generator efficiency corresponding to the higher CF, as well as in consideration of practically equal cladding operation times for both the algorithms, it was concluded that the "Proposed" algorithm was preferable (Maksimov et al., 2009).

Using this approach, the сomplex criterion of power maneuvering algorithm efficiency for WWER-1000 operating in the mode of variable loading, taking into account FE cladding damage level, active core power stability, NR capacity factor, as well as control system reliability, has been worked out (Pelykh et al., 2009). Also the Compromise-combined WWER–1000 power control method capable of maximum variable loading operation efficiency, has been proposed and grounded (Maksimov and Pelykh, 2010).
