Using the CET cladding durability estimation method, an analysis of the cladding (stress relieved zircaloy) durability estimation sensitivity to the WWER-1000 main regime and design initial data uncertainty, under variable loading conditions, has been done. The WWER-1000 main regime and design parameters have been devided into two groups: the parameters that influence the cladding failure conditions slightly and the parameters that determine the cladding failure conditions. The second group includes such initial parameters that any one of them gives a change of t0 estimation near 2 % (or greater) if the initial parameter has been specified at the value assignment interval of 3 %. This group consists of outer cladding diameter, pellet diameter, pellet hole diameter, cladding thickness, pellet 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 dhde = 0.140 cm, 0.112 cm and 0.168 cm, is shown in Fig. 4.

Fig. 4. Dependence of SDE on N for dhole: 0.140 cm (1); 0.112 cm (2); 0.168 cm (3).

Dependence of cladding equivalent stress (r’max (T and yield stress a’0nax (T, for the cladding point having the maximum temperature, on the number of effective days N, for dhole = 0.112 cm and 0.168 cm, is shown in Fig. 5.

Fig. 5. Dependence of cladding yield stress (1) and equivalent stress (2; 3) on N for dhoie: 0.112 cm (2); 0.168 cm (3). Determination of t0 for dhoie = 0.112 cm.

Using the value of T0 and the calculated dependence of SDE on N, the value of A0 is found — see Fig. 6.

Fig. 6. Calculation of A0.

For the combined variable load cycle, dependence of cladding SDE on the number of effective days N for a medium-loading FE of UTVS, TVS-А and TVS-W, is shown in Fig. 7.

Fig. 7. Dependence of SDE on N for UTVS, TVS-A and TVS-W.

For the combined cycle, the maximum SDE value was obtained for a medium-loading FE of the FA produced by WESTINGHOUSE, which has no pellet centre hole (see Table 1). The same result was obtained for the stationary regime of WWER-1000 (Maksimov and Pelykh,

2010) .

It has been found that cladding running time, expressed in cycles, for the WWER-1000 combined load cycle decreases from 1925 to 1351 cycles, when FE maximum LHR ql/max increases from 248 W/cm to 298 W/cm (Maksimov and Pelykh, 2010). Having done estimation of cladding material failure parameter со after 1576 ef. days, it was found that the WWER-1000 combined load cycle has an advantage in comparison with stationary operation at 100 % power level when ql/max < 273 W / cm — see Table 4.

According to FEM, a FE length is divided into n equal length AS. In the first publications devoted to the CET-method it was supposed that the central AS is most strained and shortest-lived. However, this assumption does not consider that segments differ in LHR jump value. In addition, it was assumed that a FA stays in the same place over the whole fuel operating period (Maksimov and Pelykh, 2009).

		FE maximum LHR, W/ cm
248	258 263	273	298
		Average fast neutron flux density, cm-2 s	1
	11014	1.04 1014 1.06 1014	1.11014	1.2 1014
		Stationary loading
T0, ef. d.	2211	2078 2016	1904	1631
A0, MJ/m3	33.37	35.66 36.87	39.74	47.64
о, %	60	65 68	74	94
Combined variable loading
T0, ef. d.	2246	2102 2032	1903	1576
A0, MJ/m3	27.36	29.14 30.05	32.10	37.69
о, %	57	64 67	74	100

Table 4. Cladding damage parameter for stationary loading and the combined variable loading of WWER-1000.

At last, influence of cladding corrosion rate on cladding durability at variable loading was not taken into account. Thus it is necessary to estimate influence of varying duty on all AS, to take account of a real FA transposition algorithm as well as to consider influence of cladding corrosion rate on its durability.

3. Method to determine the most strained cladding axial segment

The amplitude of LHR jumps in AS occurring when the NR thermal power capacity N increases from 80% to 100% level, was estimated by the instrumentality of the RS code, which is a verified tool of the WWER-1000 calculation modelling (Philimonov and MaMichev, 1998). Using the RS code, the WWER-1000 core neutron-physical calculation numerical algorithms are based on consideration of simultaneous two-group diffusion equations, which are solved for a three-dimensional object (the reactor core) composed of a limited number of meshes.

The amplitude of LHR jumps was calculated for the following daily power maneuvering method: lowering of N from Ni=100% to N2=90% by injection of boric acid solution within 0.5 h — further lowering of N to N3=80% due to reactor poisoning within 2.5 h — operation at N3=80% within 4 h — rising of N to the nominal capacity level N1=100% within 2 h (Maksimov et al., 2009). According to this maneuvering method, the inlet coolant temperature is kept constant while the NR capacity changes in the range N=100-80%, and the initial steam pressure of the secondary coolant circuit changes within the standard range of 58-60 bar. It was supposed that the only group of regulating units being used at NR power maneuvering was the tenth one, while the control rods of all the other groups of regulating units were completely removed from the active core. The next assumption was that the Advanced power control algorithm (A-algorithm) was used. The WWER-1000 core contains ten groups of regulating units in case of the A-algorithm — see Fig. 8.

Fig. 8. Disposition of the WWER-1000 regulating units in case of the A-algorithm: (upper figure) the FA number; (middle figure) the lowest control rod axial coordinate (at 100% NR power level) measured from the core bottom, %; (lower figure) the regulating unit group number.

The lowest control rod axial coordinates for Ni=100% and N3=80% were designated Hi=90% and H3=84%, respectively. That is when N changes from N1=100% to N3=80%, the lowest control rod axial coordinate measured from the core bottom changes from H1=90% to H3=84%.

It has been found using the RS code that the WWER-1000 fuel assemblies can be classified into three groups by the FA power growth amplitude occurring when the NR capacity increases from 80% to 100% level — see Table 5 (Pelykh et al., 2010).

FA group	The number of fuel assemblies	FA power growth, %	FA numbers (according to the core cartogram )
1	6	28	31, 52, 58, 106, 112, 133
2	37	26	20, 42, 43, 46, 51, 53.. .57, 66.. .71, 80.. .84, 93.. .98, 107…111, 113, 118, 121, 122, 144
3	120	< 25	all other fuel assemblies

Table 5. Three groups of the WWER-1000 fuel assemblies.

When the eighth, ninth and tenth regulating groups are simultaneously used, the central FA (No. 82) as well as fresh fuel assemblies are regulated by control rods. But when using the A-algorithm, the tenth regulating group is used only. In this case, such a four-year FA transposition algorithm can be considered as an example: a FA stays in the 55-th FA (FE maximum LHR q, max = 236.8 W/ cm, FA group 2) position for the first year — then the FA stays in the 31-st FA (q;max = 250.3 W/cm, group 1) position for the second year — further the FA stays in the 69-th FA (q, max = 171.9 W/cm, group 2) position for the third year — at last, the FA stays in the central 82-d FA (q, max = 119.6 W/cm, group 2) position for the fourth year (the algorithm 55-31-69-82).

The average LHR for i-segment and j-FA is denoted as < ql r j >. For all segments (n = 8) of the 55-th, 31-st, 69-th and 82-nd fuel assemblies, the values of < ql r j > have been calculated at power levels of N3=80% and N1=100% using the RS code. The < ql r j > (100%)/ < ql r j > (80%) ratio values are listed in Table 6.

FA number
AS	55	31	69	82
8	1.341	1.517	1.328	1.340
7	1.308	1.426	1.297	1.309
6	1.250	1.241	1.263	1.268
5	1.229	1.213	1.238	1.250
4	1.224	1.217	1.232	1.242
3	1.241	1.229	1.243	1.259
2	1.255	1.251	1.271	1.270
1	1.278	1.275	1.288	1.302
Table 6. The < ql r	j > (100%)/ < q,,,	О A	ratio values for fuel assemblies 55, 31, 69, 82.

Though the Nb-containing zirconium alloy E-110 (Zr + 1% Nb) has been used for many years in FE of WWER-1000, there is no public data on E-110 cladding corrosion and creep rates for all possible loading conditions of WWER-1000. In order to apply the cladding durability estimation method based on the corrosion and creep models developed for Zircaloy-4 to another cladding alloy used in WWER-1000, it is enough to prove that using these models under the WWER-1000 active core conditions ensures conservatism of the E — 110 cladding durability estimation. Nevertheless, the main results of the present analysis will not be changed by including models developed for another cladding alloy.

The modified cladding failure criterion at NR variable loading is given as (Pelykh and Maksimov, 2011):

T

air) = A(T) / A0 = 1; A(r) = Jamax (r) pemax (r) dr; A0 at amax (r0) = n ОТ* (T), (15)

0

where i(r) is cladding material failure parameter; т is time, s; A(r) is SDE, J/m3; A0 is SDE at the moment r0 of cladding material failure beginning, when аЄпах(т0) = n a(5nax(T)); C^sx(t) and pm^ (r) are equivalent stress (Pa) and rate of equivalent creep strain (s-1) for the cladding point of an AS having the maximum temperature, respectively; сг™3* (T is yield stress for the cladding point of an AS having the maximum temperature, Pa; n is some factor, n < 1.

Assuming the 55-31-69-82 four-year FA transposition algorithm and n = 0.6, the i(r) values have been calculated by Eq. (15) using the following procedure: calculating aemax(r), pemax(r) and оП^Г) by the instrumentality of FEMAXI-V code (Suzuki, 2000); calculating A(t) ; determining the moment t0 according to the condition

a-max^) = n оП’Ч’Т)); determining A0 = A(t0) ; calculating i(r) — see Table 7 (Pelykh and

Maksimov, 2011).

т, days		AS
4	5	6	7
360	0.063	0.151	0.190	0.175
720	0.598	0.645	0.647	0.547
1080	0.733	0.783	0.790	0.707
1440	0.788	0.838	0.848	0.779

Table 7. Cladding failure parameters i(r) for the axial segments 4-7.

For the other axial segments No. 1-3 and 8, on condition that a FA was transposed in concordance with the 55-31-69-82 four-year algorithm, the i(r) value was less than 1.0, i. e. there was no cladding collapse up to т = 2495 days. For т > 2495 days calculations were not carried out. For all the axial segments, on condition that a FA was transposed in concordance with the 55-31-69-82 four-year algorithm, it has been found that there was no cladding collapse up to т = 2495 days with i(r) = 1. At the same time, for all the axial segments, on condition that a FA stayed in the 55-th FA position for all fuel operation period, as well as on condition that a FA stayed in the 55-th FA position for the first year, then it stayed in the 31-st FA position for the remaining fuel operation period, the а(т) value reached 1.0 and the cladding collapse was predicted at т < 2495 days with n = 1.

The prediction shown in Table 7 that the largest value of (o(t) exists at the fifth (central) axial segment and above it the value drops in the sixth segment situated between the axial coordinates z = 2.19 and 2.63 m reflects the fact that the most considerable LHR jumps take place at the core upper region (see Table 6). Thus, taking account of the 55-31-69-82 four — year FA transposition algorithm as well as considering the regulating unit disposition, on condition that the FE length is divided into eight equal-length axial segments, the sixth (counting from the core bottom) AS cladding durability limits the WWER-1000 operation time at daily cycle power maneuvering.

Growth of the water-side oxide layer of cladding can cause overshoot of permissible limits for the layer outer surface temperature prior to the cladding collapse moment. The corrosion models of EPRI (MATPRO-09, 1976) and MATPRO-A (SCDAP/RELAP5/MOD2, 1990) have been used for zircaloy cladding corrosion rate estimation. According to the EPRI model, the cladding corrosion rate for a bubble flow is estimated as

dS / dt = (A / S2) exp(-Q1 / R Тъ )(1 + COR), (16)

where dS / dt is the oxide growth rate, pm/day; A = 6.3*109 pm3/day; S is the oxide layer thickness, pm; Ql=32289 cal/mol; R=1.987 cal/(mol K); Тъ is the temperature at the oxide layer-metal phase boundary, K; COR is an adjusting factor which is added in the FEMAXI code (Suzuki, 2010).

According to the MATPRO-A model, the oxide layer thickness for a nucleate boiling flow is estimated as

о о 1/3

S = (4.976x 10-3 A t exp(-15660 / Тъ) + S03) (1 + COR), (17)

where S is the oxide layer thickness, m; A = 1.5 (PWR); t is time, days; Тъ is the temperature at the oxide layer-metal phase boundary, K; S0 is the initial oxide layer thickness, m.

The cladding failure parameter values listed in Table 7 have been obtained using the MATPRO-A corrosion model at COR = 1. If COR is the same in both the models, the MATPRO-model estimation of cladding corrosion rate is more conservative than the EPRI — model estimation, under the WWER-1000 conditions. Regardless of the model we use, the factor COR must be determined so that the calculated oxide layer thickness fits to experimental data. The oxide layer thickness calculation has been carried out for the described method of daily power maneuvering, assuming that a FA was transposed in concordance with the 55-31-69-82 four-year algorithm. The calculations assumed that, the Piling-Bedworth ratio was 1.56, the initial oxide layer thickness was 0.1 pm, the maximum oxide layer thickness was restricted by 100 pm, the radial portion of cladding corrosion volume expansion ratio was 80%. It has been found that the calculated cladding oxide layer thickness, for the WWER-1000 conditions and burnup Bu = 52.5 MW day / kg, conforms to the generalized experimental data obtained for PWR in-pile conditions (Bull, 2005), when using the EPRI model at COR = — 0.431 — see Fig. 9.

Fig. 9. Cladding oxide layer thickness S subject to height h: (■) calculated using the EPRI model at COR = — 0.431; In accordance with (Bull, 2005): (1) zircaloy-4; (2) improved zircaloy-4; (3) ZIRLO.

The EPRI model at COR = — 0.431 also gives the calculated cladding oxide layer thickness values which were in compliance with the generalized experimental data for zircaloy-4 (Kesterson and Yueh, 2006). For the segments 5-8, assuming that a FA was transposed in concordance with the 55-31-69-82 four-year algorithm, the maximum oxide layer outer surface temperature T^^Ut during the four-year fuel life-time has been calculated (EPRI, COR = — 0.431) — see Table 8. Also, for the segments 5-8, the calculated oxide layer thickness S and oxide layer outer surface temperature Tox Out subject to time т are listed in Table 8.

The maximum oxide layer outer surface temperature during the four-year fuel life-time does not exceed the permissible limit temperature T0XmOut =352 °C (Shmelev et al., 2004).

i	T max °c ox ,out		S ^m ( Tox	out, °C )
		360 days	720 days	1080 days	1440 days
5	345.1	11.3 (342.3)	40.6 (344.8)	58.1 (328.2)	69.8 (316.7)
6	349.6	16.1 (347.6)	49.8 (349.4)	69.3 (332.6)	82.5 (320.1)
7	351.2	18.1 (350.0)	52.7 (351.0)	74.1 (336.1)	88.5 (323.0)
8	348.0	14.2 (347.9)	38.3 (346.9)	58.0 (335.6)	71.2 (323.3)

Table 8. The maximum oxide layer outer surface temperature.

The same result has been obtained for the EPRI model at COR = 0; 1; 2 as well as for the MATPRO-A model at COR = — 0.431; 0; 1; 2. Hence the oxide layer outer surface temperature should not be considered as the limiting factor prior to the cladding collapse moment determined in accordance with the criterion (15). Though influence of the outer oxide layer thickness on the inner cladding surface temperature must be studied.

Having calculated the SDE by the instrumentality of FEMAXI (Suzuki, 2010), assuming that a FA was transposed in concordance with the 55-31-69-82 four-year algorithm, it has been found for the sixth axial segment that the number of calendar daily cycles prior to the beginning of the rapid creep stage was essentially different at COR = — 0.431; 0; 1; and 2. As a result, the rapid creep stage is degenerated for both the corrosion models at COR = — 0.431 (Fig. 10).

Let us introduce a dimensionless parameter I

10-6 T

1 = °С ■ day J Tdad’in ‘dt, (18)

where Tdad in is the cladding inner surface temperature for an axial segment, °С; and t is time, days.

Having analysed the described method of daily power maneuvering, the maximum cladding oxide layer outer surface temperature during the period of 2400 days, as well as I(2400

days) and the 2400 days period averaged cladding inner surface temperature < Tdad in > have been calculated for the sixth segment, using the EPRI corrosion model — see Table 9.

COR	T max °c *0X	I(2400 days)	О о л "чГ h? V
2	349.2	0.951	396.2
1	349.5	0.947	394.5
0	349.6	0.938	390.7
-0.431	349.6	0.916	381.8

Table 9. Cladding temperatures subject to COR for the sixth segment, the EPRI model corrosion.

This shows that the effect of cladding outer surface corrosion rate (with COR) on the cladding SDE increase rate (see Fig. 10) is induced by the thermal resistance of oxide thickness and the increase in Tciadrin (see Table 9).

It should be noticed that the metal wall thickness decrease due to oxidation is considered in the calculation of the SDE, as effect of the cladding waterside corrosion on heat transfer and mechanical behavior of the cladding is taken into account in the FEMAXI code. Since
temperature and deformation distributions physically depend on each other, simultaneous equations of thermal conduction and mechanical deformation are solved (Suzuki, 2000).

It is obvious that the cladding temperature at the central point of an AS increases when the outer oxide layer thickness increases. At the same time, according to the creep model (MATPRO-09, 1976) used in the code, the rate of equivalent creep strain p max(T) for the central point of an axial segment increases when the corresponding cladding temperature increases. Hence the waterside corrosion of cladding is associated with the evaluation of SDE through the creep rate depending on the thickness of metal wall (Pelykh and Maksimov, 2011).

It should be noted, that neutron irradiation has a great influence on the zircaloy corrosion behavior. Power maneuvering will alter neutron flux to give a feedback to the corrosion behavior, either positive or negative. But in this paper, the EPRI model and MATPRO code are used in the corrosion model, where irradiation term is not evidently shown. Although either temperature or reactivity coefficient is introduced in applying the model, it does not fully represent such situation.

For the studied conditions, the maximum cladding hoop stress, plastic strain and oxide layer outer surface temperature do not limit cladding durability according to the known restrictions a’m’ax < 250 MPa, ^ < 0.5% (Novikov et al., 2005) and T^, < 352 °C (Shmelev et al., 2004), respectively. A similar result has been obtained for the corrosion model MATPRO-A.

Setting COR = 0 and COR = 1 (MATPRO-А), the SDE values for the algorithms 55-31-55-55 and 55-31-69-82 have been calculated. Then the numbers of calendar daily cycles prior to the beginning of rapid creep stage for Zircaloy-4 (Pelykh and Maksimov, 2011) and rapid w(t) stage for E-110 alloy (Novikov et al., 2005) have been compared under WWER-1000 conditions — see Fig. 11.

Fig. 11. Cladding damage parameter (E-110) and SDE (Zircaloy-4) as functions of time: (1) co(t) according to equation (2); (2.1, 2.2) A(t) at COR = 0 for the algorithms 55-31-55-55 and 55-31-69-82, respectively; (3.1, 3.2) A(t) at COR = 1 for the algorithms 55-31-55-55 and 55-31­69-82, respectively.

It is necessary to notice that line 1 in Fig. 11 was calculated using separate consideration of steady-state operation and varying duty. When using equation (2), the fatigue component has an overwhelming size in comparison with the static one (Novikov et al., 2005).

Use of the MATPRO-A corrosion model under the WWER-1000 core conditions ensures conservatism of the E-110 cladding durability estimation (see Fig. 11). Growth rate of A(x) depends significantly on the FA transposition algorithm. The number of daily cycles prior to the beginning of rapid creep stage decreases significantly when COR (cladding outer surface corrosion rate) increases.

Setting the WWER-1000 regime and FA constructional parameters, a calculation study of Zircaloy-4 cladding fatigue factor at variable load frequency v << 1 Hz, under variable loading, was carried out. The investigated WWER-1000 fuel cladding had an outer diameter and thickness of 9.1 mm and 0.69 mm, respectively. The microstructure of Zircaloy-4 was a stress-relieved state. Using the cladding corrosion model EPRI (Suzuki, 2000), AS 6 of a medium-load FE in FA 55 (maximum LHR q;max =229.2 W/cm at N=100 %) has been analysed (COR = 1, inlet coolant temperature Tin=const=287 °C). The variable loading cycle 100-80-100 % was studied for Ax =11; 5; 2 h (reactor capacity factor CF=0.9): N lowering from 100 to 80 % for 1 h ^ exploitation at N = 80 % for Ax h ^ N rising to Nnom=100 % for 1 h ^ exploitation at N = 100 % for Ax h, corresponding to v =1; 2; 4 cycle/day, respectively( v << 1 Hz).

Calculation of the cladding failure beginning moment x0 depending on v showed that if v << 1 Hz and CF=idem, then there was no decrease of t0 after v had increased 4 times, in comparison with the case v =1 cycle/day, taking into account the estimated error < 0.4 % (n=0.4, AS 6). At the same time, when N=100 % =const (CF=1), the calculated t0 decreases significantly — see Table 10.

Hence, the WWER-1000 FE cladding durability estimation based on the CET model corresponds to the experimental results (Kim et al., 2007) in principle.

CF		0.9		1
v, cycle/day	1	2	4	—
T0, day	547.6	547.0	549.0	436.6

Table 10. Change of cladding failure time depending on v and CF.

In the creep model used in the FEMAXI code (Suzuki, 2000), irradiation creep effects are taken into consideration and cladding creep strain rate pe (t) is expressed with a function of fast neutron flux, cladding temperature and hoop stress (MATPRO-09, 1976). Thus creep strain increases as fast neutron flux, irradiation time, cladding temperature and stress increase. Fast neutron flux is predominant in cladding creep rate, whereas thermal neutron distribution is a determining factor for reactivity and thermal power (temperature of cladding) in core. It can be seen that both types of neutron flux are important for the cladding life.

One of main tasks at power maneuvering is non-admission of axial power flux xenon waves in the active core. Therefore, for a power-cycling WWER-1000 nuclear unit, it is interesting to consider a cladding rupture life control method on the basis of stabilization of neutron flux axial distribution. The well-known WWER-1000 power control method based on keeping the average coolant temperature constant has such advantages as most favorable conditions for the primary coolant circuit equipment operation, as well as possibility of stable NR power regulation due to the temperature coefficient of reactivity. However, this method has such defect as an essential raise of the secondary circuit steam pressure at power lowering, which requires designing of steam generators able to work at an increased pressure.

Following from this, it is an actual task to develop advanced power maneuvering methods for the ENERGOATOM WWER-1000 units which have such features as neutron field axial distribution stability, favorable operation conditions for the primary circuit equipment, especially for FE claddings, as well as avoidance of a high pressure steam generator design. The described daily power maneuvering method with a constant inlet coolant temperature allows to keep the secondary circuit initial steam pressure within the standard range of 58­60 bar (N=100-80%).

The nonstationary reactor poisoning adds a positive feedback to any neutron flux deviation. Therefore, as influence of the coolant temperature coefficient of reactivity is a fast effect, while poisoning is a slow effect having the same sign as the neutron flux deviation due to this reactivity effect, and strengthening it due to the positive feedback, it can be expected that a correct selection of the coolant temperature regime ensures the neutron flux density axial distribution stability at power maneuvering. The neutron flux axial stability is characterized by AO (Philipchuk et al., 1981):

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

The variables AO, Nu, Ni, N are represented as

where A°o, Nu0, Nl,0- N0 are the stationary values of АО, NU, Nt, N, respectively; 8АО, SNU, 8Nl, 8N are the sufficiently small deviations from АО0, Nu0 , Nl0, N0, respectively.

The small deviations of NU and Nt caused by the relevant average coolant temperature deviations 8 < TU > and 8 < T, > are expressed as

where 8NU and 8Nt are the small deviations of NU and Nt, respectively; 8 < T > is the average coolant temperature small deviation for the whole core; 8 < TU > and 8 < Tt > are the
average coolant temperature small deviations for the upper half-core and for the lower half­core, respectively.

The term SN / S < T > is expressed as

SN _Sp/S< T > ^ fcL

S< T > Sp/SN kN’ (

where p is reactivity; kT and kN are the coolant temperature coefficient of reactivity and the power coefficient of reactivity, respectively.

Having substituted equations (20)-(22) in (19), the following equation for a small deviation of AO caused by a small deviation of N is derived after linearization:

SO _ jL. n-1 ■ [(1 — АО0) S< Tu > -(1 + АО0) S < T, >] (23)

kN

In case of the assumption

АО0 << 1

equation (23) is simplified:

SAO _. N-1. [S< Tu >-S< T, >] (25)

kN

The criterion of AO stabilization due to the coolant temperature coefficient of reactivity (the coolant temperature regime effectiveness criterion) is obtained from (25):

where i is the power step number; m is the total number of power steps in some direction at reactor power maneuvering.

Use of the criterion (26) allows us to select a coolant temperature regime giving the maximum LHR axial distribution stability at power maneuvering. Let us study the following three WWER-1000 power maneuvering methods: M-1 is the method with a constant inlet coolant temperature Tin=const; M-2 is the method with a constant average coolant temperature <T>=const; and M-3 is the intermediate method having Tin increased by 1 °C only, when N lowers from 100% to 80%. Comparison of these power maneuvering methods has been made using the RS code. Distribution of long-lived and stable fission products causing reactor slagging was specified for the KhNPP Unit 2 fifth campaign start, thus the first core state having an equilibrium xenon distribution was calculated at this moment. The non-equilibrium xenon and samarium distributions were calculated for subsequent states taking into account the fuel burnup. The coolant inlet pressure and coolant flow rate were specified constant and equal to 16 MPa and 84 TO3 m3/h, respectively. When using M-1, the coolant inlet temperature was specified at Tin=287 °C. When using M-

2, the coolant inlet temperature was specified according to Table 11 (Tout is the coolant outlet temperature).

N, %	o о	Toub °С	<T>, °С
100	287	317	302
90	288	316	302
80	290	314	302

Table 11. Change of the coolant temperature at <T>=const in the M-2 method.

Denoting change of the lowest control rod axial coordinate (%) measured from the core bottom during a power maneuvering as AH, the first (М-2-а) and second (М-2-b) variants of М-2 had the regulating group movement amplitudes AH2a =4% and AH2b =6%, respectively. The reactor power change subject to time was set according to the same time profile for all the methods (Fig. 12).

Fig. 12. Change of the reactor power subject to time.

For all the methods, N lowered from Ni=100 % to N2=90 % within 0.5 h, under the linear law dN1_2/dr=-2%/6 min, at the expense of boric acid entering. Also for all the methods, N

lowered from N2=90% to N3=80% within 2.5 h, under the law dN2-3/dx= -0.4%/6 min, at the expense of reactor poisoning. The coolant concentration of boric acid was the criticality parameter when N stayed constant during 4 h. The NR power increased from N3=80% to N1=100% within 2 h, under the law dN3-1/dx=1.0%/ 6 min, at the expense of pure distillate water entering and synchronous return of the regulating group to the scheduled position. When N increased from N3=80% to N1=100%, change of the regulating group position H subject to time was set under the linear law (Fig. 13).

Fig. 13. Change of the regulating group position subject to time: (1) the methods М-1, М-2-а and М-3; (2) the method М-2-b.

Thus, modelling of the non-equilibrium WWER-1000 control was made by assignment of the following control parameters: criticality parameter; T^ci ; dTn/dN; N1; N2; N3; H0; AH;dN/dx. Setting the WWER-1000 operation parameters in accordance with the Shmelev’s method (Shmelev et al., 2004), for the methods М-1, М-2-а, М-2-b and М-3, when N changed from 100% to 80%, the change of core average LHR distribution was calculated by the RS code. Let us enter the simplifying representation

Using the obtained LHR distribution, by the FEMAXI code (Suzuki, 2010), the average coolant temperatures of the upper < Tu > and lower < T > half-cores were calculated for M-

6

was

i=1

found for М-1, М-2-а and М-3 having the same AH (Table 12).

Method	t, h	N, %	< Tu >	< T >	S< Tu >	S<T >	AST	6 Z AST i=1
M-1; M-2-a; M-3	0.1	100	318.3	296.825	0	0	0
	0.6	90	317.975	296.575	-0.325	-0.25	-0.075
	1.1	88	316.375	296	-1.6	-0.575	-1.025
M-1	1.6	86	315.725	295.725	-0.65	-0.275	-0.375	2.65
2.1	84	315.1	295.525	-0.625	-0.2	-0.425
	2.6	82	314.5	295.3	-0.6	-0.225	-0.375
	3.1	80	313.9	295.075	-0.6	-0.225	-0.375
	0.6	90	319.25	298.025	0.95	1.2	-0.25
	1.1	88	317.875	297.575	-1.375	-0.45	-0.925
M-2-a	1.6	86	317.45	297.575	-0.425	0	-0.425
2.1	84	316.925	297.575	-0.525	0	-0.525	2.85
	2.6	82	316.65	297.625	-0.275	0.05	-0.325
	3.1	80	316.35	297.725	-0.3	0.1	-0.4
	0.6	90	318.4	297.075	0.1	0.25	-0.15
	1.1	88	316.9	296.55	-1.5	-0.525	-0.975
M-3	1.6	86	316.35	296.4	-0.55	-0.15	-0.4
2.1	84	315.7	296.2	-0.65	-0.2	-0.45
	2.6	82	315.225	296.125	-0.475	-0.075	-0.4
	3.1	80	314.775	296	-0.45	-0.125	-0.325

Table 12. Change of the average coolant temperatures for М-1, М-2-а, М-3.

Having used the criterion (26), the conclusion follows that the coolant temperature regime М-1 ensures the most stable АО, while the regime М-2-а is least favorable — see Table 12. In order to check this conclusion, it is useful to compare АО stabilization for the discussed methods, calculating the divergence AAO between the instant and equilibrium axial offsets (Philimonov and MaMichev, 1998) — see Fig. 14.

Fig. 14. Equilibrium and instant axial offsets (1, 2a, 2b, 3) subject to time for М-1, М-2-а, М-2- b and М-3, respectively: (lower line) the equilibrium АО; (upper line) the instant АО.

The regulating group movement amplitude is the same (4%) for M-1, М-2-а and М-3, but the maximum divergence between the instant and equilibrium offsets are AAOmax ~ 1.9% (M-1), AAO2maax ~ 3% (M-2-a) and AAO^ax ~ 2.3% (M-3). This result confirms the conclusion made on the basis of the criterion (26). If the regulating group movement amplitude, at power maneuvering according to the method with a constant average coolant temperature, is increased from 4 to 6%, then the maximum AO divergence lowers from 3% to 1.9% (see Fig. 14). Therefore, when using the method with < T > =const, a greater regulating group movement amplitude is needed to guarantee the LHR axial stability, than when using the method with Tin = const, on the assumption that all other conditions for both the methods are identical.

Having used the RS code, the core average LHR axial distribution change has been calculated for the methods М-1, М-2-а, М-2-b and М-3, for the following daily power maneuvering cycle: lowering of N from N1=100% to N2=90% during 0.5 h by injection of boric acid solution — further lowering of N to N3=80% during 2.5 h due to reactor poisoning — operation at N3=80% during 4 h — rising of N to the nominal capacity level N1=100% during 2 h at the expense of pure distillate water entering and synchronous return of the regulating group to the scheduled position — operation at N1=100% during 15 h.

When using the criterion (15) for comparative analysis of cladding durability subject to the FA transposition algorithm, the position of an axial segment and the power maneuvering method, the value of n should be set taking into account the necessity of determining the moment t0, when the condition cr^To) = П C^To) is satisfied. In addition, as the maximum number of power history points is limited by npim =10,000 in the FEMAXI code, the choice of n depends on the analysed time period Tmax and the complexity of a power maneuvering method, because a greater time period as well as a more complicated power maneuvering method are described by a greater number of history points np. Therefore, the value of n should be specified on the basis of simultaneous conditions CmaxT0) = П c0max(T0); np < npim; t0 < Tmax. Though the cladding failure parameter values listed in Table 6 were obtained assuming n =0.6 (the MATPRO-A corrosion model, COR = 1), comparison of cladding failure parameters for different power maneuvering methods can be made using the cladding collapse criterion (15), for instance, at n =0.4. Assuming n =0.4, on the basis of the obtained LHR distributions, the cladding failure parameters have been calculated by the instrumentality of the FEMAXI code (Suzuki, 2000) for the methods М-1, М-2-а, М-2-b and М-3, for the axial segments six and seven (the MATPRO-A corrosion model, COR = 1) — see Table 13.

	Method	М-1	М-2-а	М-2-b	M-3
		t0 , days	504.4	497.4	496.0	501.4
	6	A0, MJ/m3	1.061	1.094	1.080	1.068
Axial		ю(500 days)	0.957	1.027 (+7.3%)	1.040 (+ 8.7%)	0.988 (+3.2%)
Segment		t0 , days	530.0	519.4	519.0	525.0
	7	A0, MJ/m3	1.044	1.055	1.019	1.043
		ю(500 days)	0.766	0.848 (+10.7%)	0.848 (+10.7%)	0.804 (+5.0%)

Table 13. Cladding failure parameters for the methods М-1, М-2-а, М-2-b and М-3.

Among the regimes with the regulating group movement amplitude AH =4%, the coolant temperature regime М-1 ensuring the most stable АО is also characterized by the least calculated cladding failure parameter r»(500 days), while the regime М-2-a having the least stable АО is also characterized by the greatest r»(500 days) — see Table 12, Fig. 14 and Table 13. The intermediate method M-3 having Tin increased by 1 °С only, when N lowers from 100% to 80%, is also characterized by the intermediate values of AO stability and r»(500 days).

In addition, the second variant of М-2 (М-2-b) having the regulating group movement amplitude AH2b =6% is characterized by a more stable AO in comparison with the method М-2-a (see Fig. 14) and, for the most strained axial segment six, by a greater value of a)(500 days) — see Table 13.

It should be stressed that the proposed cladding rupture life control methods are not limited only in WWER-1000. Using the FEMAXI code, these methods can be extended into other reactor types (like PWR or BWR). At the same time, taking into account a real disposition of regulating units, a real coolant temperature regime as well as a real FA transposition algorithm, in order to estimate the amplitude of LHR jumps at FE axial segments occurring when the NR (PWR or BWR) capacity periodically increases, it is necessary to use another code instead of the RS code, which was developed for the WWER-1000 reactors.

The FA transposition algorithm 55-31-69-82 is characterized by a lower fuel cladding equivalent creep strain than the algorithm 55-31-55-55. At the same time, it has a lower fuel burnup than the algorithm 55-31-55-55 (see Table 14).

FA transposition algorithm	55-31-69-82	55-31-55-55
COR	0	1	0	1
Bu, MW-day/kg	57.4	71.4
CTemax, MPa (% of 00)	69.9 (33)	127.4 (61)	107.2 (51)	146.7 (70)
pe, %	4.22	11.22	9.36	16.02

Table 14. Fuel burnup and cladding equivalent creep strain for AS 6 (after 1500 d).

Thus, an optimal FA transposition algorithm must be set on the basis of cladding durability — fuel burnup compromise.