Corresponding author: Sergey B. Vygovsky ( vigovskii@mail.ru ) Academic editor: Georgy Tikhomirov
© 2020 Rashdan Talal Al Malkawi, Sergey B. Vygovsky, Osama Wasef Batayneh.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Al Malkawi RT, Vygovsky SB, Batayneh OW (2020) Investigation of the impact of steadystate VVER1000 (1200) core characteristics on the reactor stability with respect to xenon oscillations. Nuclear Energy and Technology 6(4): 289294. https://doi.org/10.3897/nucet.6.60464

The article presents a method for obtaining an analytical expression for the criterion of stability of a VVER1000 (1200) reactor with respect to xenon oscillations of the local power in the core, containing an explicit dependence of the criterion ratio coefficients on the arbitrary axial neutron field distribution in steady states of the core. Based on the data of numerical experiments using a fullscale model of the Kalinin NPP power units, the authors present the results of checking the validity of this expression for the reactor stability criterion with respect to xenon oscillations for different NPPs with VVER1000 (1200) reactors.
Reactor stability, xenon oscillations, axial offset, VVER1000 (1200), flexible (load tracing) operating modes
For the first time, analytical expressions were obtained for the criterion of stability of a VVER1000 (1200) reactor with respect to xenon oscillations of the local power in the core, containing an explicit dependence of the criterion ratio coefficients on the arbitrary axial neutron field distribution in steady states of the core.
Under the conditions of maneuvering modes with a change in power during the day, nonstationary poisoning of the core with xenon occurs, which can lead to the occurrence of xenon oscillations of the local power throughout the VVER1000 (1200) reactor core (
For a symmetric neutron field distribution, it is not difficult to obtain an analytical expression for the reactor stability criterion. Similar expressions are found in many works (
In (
The analytical expressions obtained by us confirm the above findings and indicate explicitly the dependence of the temperature coefficient of reactivity on the coefficient of axial nonuniformity of energy releases (or the value of the axial neutron power offset), and thus allow us to quantitatively evaluate the effect of this dependence on the reactor stability to xenon oscillations.
In (
The axial offset in practice for the second fuel loads of VVER1000 (1200) always has a noticeable positive value, since the neutron distribution shifts to the top of the core due to significantly less the burnout of fuel in the upper part of the core as compared to the lower one for the first fuel load. Fuel burnup in the upper part of the core occurs to a lesser extent as compared to the lower part for the first fuel load due to a shift in the axial neutron field distribution for the first load to the lower part of the core. For every second load, it is useful to have an analytical assessment of the reactor stability, taking into account the influence of the axial distribution on it. This is what the practice of applying the obtained analytical expressions for the operation of NPP power units with VVER1000 (1200) can be. This is especially important for making a decision on testing maneuvering modes for the second fuel load of NvNPP7 and LNPPII2.
Undoubtedly, the obtained criterion is useful for clarifying the idea of the nature of the occurrence of xenon oscillations in thermal neutron reactors. It is advisable to use the information on the obtained criterion in the process of training future specialists for the nuclear industry and for retraining personnel of nuclear power plants. Moreover, these materials are already used in the educational process at the Departments of Automation of NRNU MEPhI and Equipment and Operation of Nuclear Power Plants of OINPE NRNU MEPhI.
Xenon transients in the reactor core are caused by a violation of the equilibrium state, i.e., the dynamic equilibrium between the neutron flux density and the concentration of ^{135}Хе and ^{135}I nuclei. Xenon stability is understood as the ability of the core to restore the equilibrium state of spatial xenon distribution and spatial local power distribution throughout the core.
The changes in reactivity caused by xenon processes (integral xenon processes) as well as changes in the spatial distribution of energy releases in the core (spatial xenon processes) are of practical importance for the reactor operation.
The reactor stability to xenon oscillations is characterized by the stability index (α) and the oscillation period (Т_{Xe}), which are determined in the analysis of free xenon oscillations obtained experimentally or by means of computational modeling. In this case, the time variation of a certain scalar quantity characterizing the energy release distribution in the core is considered. Such a scalar value is the axial offset (АО), i.e., the percentage ratio of the difference between the powers of the upper and lower halves of the core to the total power. Free spatial xenon oscillations have a sinusoidal character with periodic alternation of the ascending and descending phases, corresponding to an increase and decrease in the values of the parameter characterizing xenon oscillations. The deviation of the АО value from its equilibrium value (АО^{*}), corresponding to the equilibrium distribution of xenon, is represented as (
A (t) = A (t_{0})exp(αt)cos(ωτ), (1)
where A = AO – AO^{*}; ω = 2π/T; τ = t – t_{0} (t_{0} is the moment of reaching the first extremum); T is the period of free xenon oscillations; α = T^{–1}×ln(A_{2}/A_{1}), A_{1} is the amplitude of the first maximum, A_{2} is the amplitude of the second maximum.
The behavior of АО in time controls the stability of the reactor. The operational efficiency of a power unit with a VVER1000 (1200) reactor is determined by minimizing the deviation of АО from its stationary values (
A characteristic of the stability of stationary states is such a parameter as the reactor stability index (α) with respect to free xenon oscillations of the local power in the core (
At α < 0, the reactor is stable, and the oscillations damp; at α ³ 0, the reactor is unstable, and the oscillations do not damp.
To obtain an expression that determines the criterion of reactor stability to xenon oscillations, the twopoint approximation of the neutron kinetics model is used. This approximation is described by a system of ordinary nonlinear differential equations for the balance of xenon and iodine, supplemented by an algebraic neutron balance equation in onegroup in neutron energy and onedimensional in geometry approximations, and simplified expressions for feedbacks in reactivity with small deviations of the average power. To analyze the reactor stability to xenon oscillations, the original system of ordinary nonlinear differential equations is reduced to a consolidated system of linearized equations. Further, the Laplace transform is applied to this system. Using this transform, we can obtain the final form of the third order characteristic equation. Based on the Hurwitz criterion, a criterion of the reactor stability with respect to xenon oscillations is derived in the case of an asymmetric axial neutron field distribution:
where b is the conversion factor from MW to fis/(s×cm^{3}); γ = γ_{I} + γ_{Xe}, γ_{I} ≈ γ_{Te}; γ_{I} is the total fraction of the yield of ^{135}I per fission of the heavy isotope (^{235}U, ^{239}Pu, ^{241}Pu); γ_{Xe} is the total fraction of the yield of ^{135}Xe per fission of the heavy isotope (^{235}U, ^{239}Pu, ^{241}Pu); γ_{Te} is the total fraction of the yield of ^{135}Te per fission of the heavy isotope (^{235}U, ^{239}Pu, ^{241}Pu); D_{1} is the diffusion coefficient throughout the lower part of the core; D_{2} is the diffusion coefficient throughout the upper part of the core; aN_{0} isthe coolant heating, deg;〈Σ_{f}〉 is the average macroscopic uranium fission cross section; σ_{Xe} is the average microscopic cross section of thermal neutron absorption by ^{135}Xe; H is the height of the reactor core; V_{0} is the volume of the reactor core; λ_{Xe} is the xenon radioactive decay constant; λ_{I} is the iodine radioactive decay constant; k_{z} is the axial core nonuniformity factor, k_{z} = N_{10}/(0,5N_{0}); N_{0} is the average number of fissions (or average power), N_{0} = N_{10} + N_{20}; N_{10} is the average number of fissions in the lower part of the core (or average power); N_{20} is the average number of fissions in the upper part of the core (or average power); dρ/dN is the power coefficient of reactivity; dρ/dT_{1} is the temperature coefficient of reactivity in the lower part of the core; dρ/dT_{2} is the temperature coefficient of reactivity in the upper part of the core;. ν s the number of secondary neutrons.
Note that the coolant temperature in the lower part of the core is lower than in the upper part. It is known that the temperature coefficient of reactivity consists of two parts. One of them is determined by the density effect of reactivity and depends on the derivative of the water density with respect to its temperature, and this part of the coefficient is directly proportional to the value of this derivative. The value of the derivative of the water density with respect to temperature (modulo) decreases as the water temperature decreases, which follows from the thermodynamic properties of water and steam, and this leads to a decrease (modulo) in the temperature coefficient of reactivity. For this reason, the temperature coefficient of reactivity in the lower half of the core is smaller modulo than in the upper one.
The left side of (2) can be represented as a function F of the parameters k_{z} and σ_{Xe}N_{0}/(λ_{Xe}〈Σ_{f}〉V_{0}): F (k_{z}, σ_{Xe}N_{0}/(λ_{Xe}〈Σ_{f}〉V_{0})). Provided that σ_{Xe}N_{0}/(λ_{Xe}〈Σ_{f}〉V_{0}) > 1, the derivative F with respect to the parameter k_{z} turns out to be positive, i.e., δF/δk_{z} > 0. For VVER1000 and VVER1200 reactors at rated power, the parameter σ_{Xe}N_{0}/(λ_{Xe}〈Σ_{f}〉V_{0}) is noticeably greater than unity, which suggests that an increase in the parameter k_{z} leads to an increase in the value of the left side of inequality (2) and contributes to an increase in the ractor stability to xenon vibrations. Otherwise, with a decrease in k_{z} and a shift in the axial neutron field distribution to the upper part of the core, the reactor stability to xenon oscillations will decrease.
To check the validity of the expression given above, we carried out computational studies of the VVER1000 xenon stability using the PROSTOR software package (
This software package has been verified on many experimental and calculated NPP data based on acceptance test programs for the abovementioned fullscale simulators and analyzers (
Stability index and period of axial xenon oscillations for the first campaign of Kalinin NPP3,4 at 0 eff. days.
Oscillation parameters  Stability index α, h^{–1}  Oscillation period Т, h 

Calculation for Kalinin NPP3  –34×10^{3}  27.7 
Experiment at Kalinin NPP3  –(33.4±0.7)×10^{–3}  27.9±0.5 
Calculation for Kalinin NPP4  –1.91×10^{–3}  28.0 
Experiment at Kalinin NPP4  –(1.93±0.04)×10^{–3}  27.7±0.5 
Stability index and period of axial xenon oscillations depending on power for the 22^{nd} campaign of Kalinin NPP2 at 305 eff. days.
Reactor power, %  α, h^{–1}  Т_{Xe}, h  Н_{к}, %  АО^{*}  σ_{Xe,} barn  〈Σ_{f}〉, cm^{–1}  ν  δρ/δN×10^{3}, %/MW 

95  –0.030  30.08  70  2.07  150393  0.00944  2.58  –0.4285 
85  –0.040  31.27  70  3.66  150922  0.00946  2.58  –0.4413 
75  –0.045  32.56  70  5.15  151596  0.00949  2.58  –0.4593 
65  –0.050  34.05  70  6.57  152210  0.00952  2.58  –0.4816 
55  –0.053  35.74  70  8.00  152798  0.00954  2.58  –0.5036 
45  –0.056  37.42  70  9.48  153221  0.00956  2.58  –0.5095 
Oscillations were excited by immersion of the working (tenth) group of control elements of the reactor control and protection system (CPS) from the initial position of 90% extraction along the core height at the initial steady state of the reactor. Throughout the entire process, the critical state of the reactor was maintained at a constant power level by changing the concentration of boric acid in the coolant.
Tables
Stability index and period of axial xenon oscillations depending on power for the 22^{nd} campaign of Kalinin NPP2 at 305 eff. days.
Reactor power, %  α, h^{–1}  Т_{Xe},h  Н_{к}, %  АО^{*}  σ_{Xe,} barn  〈Σ_{f}〉, cm^{–1}  ν  δρ/δN×10^{3}, %/MW 

95  0.027  27.7  80  0.74  173338  0.00899  2.66  –0.6761 
85  0.012  28.44  80  4.15  174073  0.00902  2.66  –0.6787 
75  –0.002  29.32  80  7.33  174791  0.00905  2.66  –0.6940 
65  –0.012  30.36  80  10.40  175481  0.00907  2.66  –0.7015 
55  –0.022  31.57  80  13.57  176137  0.00910  2.66  –0.7082 
45  –0.030  33.15  80  16.90  176757  0.00913  2.66  –0.7130 
It follows from the analysis of these tables that the change in the stability index occurs in accordance with the changes in the parameters that determine the condition for the occurrence of xenon oscillations of the axial offset in the core. These parameters are the reactor power, the average macroscopic neutron fission cross section in the fuel, the number of secondary neutrons per fission, and the total power coefficient of reactivity.
Tables
Stability index and period of axial xenon oscillations in the core and in the beginning of the first campaign of Kalinin NPP3.
Reactor power, %  α, h^{–1}  Т_{Xe},h  Н_{к}, %  k_{z}  σ_{Xe,} barn  〈Σ_{f}〉, cm^{–1}  ν  δρ/δN×10^{3}, %/MW 
105  –0.027  27.09  70  1.072  209455  0.00888  2.55  –0.3059 
95  –0.033  27.98  70  1.064  209964  0.00891  2.55  –0.3245 
85  –0.039  29.18  70  1.056  210436  0.00894  2.55  –0.3455 
75  –0.045  30.55  70  1.049  210868  0.00897  2.55  –0.3692 
65  –0.049  32.49  70  1.043  211240  0.00900  2.55  –0.3978 
55  –0.054  34.82  70  1.036  211543  0.00902  2.55  –0.4262 
45  –0.058  37.80  70  1.031  211744  0.00902  2.55  –0.4462 
Stability index and period of axial xenon oscillations in the core and in the beginning of the second campaign of Kalinin NPP3.
Reactor power, %  α, h^{–1}  Т_{Xe},h  Н_{к}, %  k_{z}  σ_{Xe,} barn  〈Σ_{f}〉, cm^{–1}  ν  δρ/δN×10^{3}, %/MW 

105  0.009  27.38  70  0.983  187955  0.00908  2.56  –0.3618 
95  –0.001  27.94  70  0.970  188697  0.00911  2.56  –0.3736 
85  –0.012  28.88  70  0.957  189416  0.00913  2.56  –0.3882 
75  –0.023  29.82  70  0.946  190113  0.00916  2.56  –0.3930 
65  –0.032  31.15  70  0.935  190776  0.00918  2.56  –0.4042 
55  –0.039  32.56  70  0.924  191392  0.00921  2.56  –0.4510 
45  –0.044  34.59  70  0.912  191957  0.00923  2.56  –0.4573 
Despite the fact that the fuel enrichment and the power coefficient of reactivity increase, and the microscopic crosssection of xenon poisoning decreases for the second fuel load, the first load at the beginning of the campaign turns out to be more stable than the second load at the same power values. This occurs due to the change in the axial profile of the neutron power for the second campaign as compared to the initial load. An analysis of expression (2) proves this seemingly paradoxical result. As we can see, the values of k_{z} (k_{z} = N_{10}/(0,5N_{0})), indicated in Tab.
An analytical form of the criterion of stability of VVER1000 (1200) with respect to xenon processes in the core is obtained, taking into account the arbitrary axial distribution of the neutron field in the reactor stationary states. The quality of this criterion was checked on the basis of the results of numerical experiments using the PROSTOR software package based on the data obtained from a number of power units of Russian NPPs with VVER1000 reactors.