Nuclear power stations have a relatively high capital cost and low fuel cost, so that there is a tendency to use them for base load. On the other hand, it will be difficult to use
very big power stations only for base load. In general all power stations tend to be used for part load as they get older and more efficient plants become available.

At present the ability of following load variation, at least between 100% and 40% is a general condition posed by all customers. Sometimes it is even required that the station should be able to restart at any moment after shut-down (full Xe override).

Enough excess reactivity must be invested in the control rods in order to allow for Xe override. A typical curve of reactivity as a function of time for a 100%-40%-100% power variation is given in Fig. 12.8. The value Ak = 0 corresponds to the equilibrium Xe concentration for 100% power.

One can see that there is a decrease in reactivity about 5 h after a power reduction and an increase about 3 h after return to full power. The minimum of Ak has to be compensated by extraction of control rods which are normally inserted in the reactor in the equilibrium condition. This results in a neutron loss and in an increase of the fuel-cycle costs. It is therefore necessary to weight the increased plant flexibility against the resulting additional costs.

The excess reactivity required for Xe override is dependent on the core composition, and fuel rating. A softer spectrum tends to increase the effective Xe absorption cross-section, while a high rating tends to increase the number of Xe atoms relative to the number of fuel atoms.

The problem of the Xe override has different aspects depending on the fuel- management scheme. In reactors with batch or discontinuous loading excess reactivity is present for a great part of the fuel life. In this case, if one can tolerate limitations in Xe override toward the end of the refuelling period, the additional excess reactivity can be reduced, using for Xe override the excess reactivity available at the beginning of a cycle for burn-up compensation (e. g. full override for some part of the fuel life only, and reduction down to 40% at any moment). In reactors with continuous reloading no excess reactivity is available in the equilibrium condition for burn-up compensation.

The control-rod requirement for Xe override may vary during the running-in period and it is necessary to check the Xe overriding capability as a function of time during this

100

phase. Theoretically instead of extracting control rods it could be possible also to insert fissionable material in the reactor, but the movement of active and heat-producing parts gives rise to great difficulties. This solution is not considered practicable at present.

In the same period of time a series of experiments has been performed by General Atomic. The models to calculate thermalization in graphite have been tested measuring the neutron spectrum in a block of graphite with the time-of-flight technique’6’ with various temperatures and poisoning. The Peach Bottom HTGR critical experiment which ran from 1959 to 1962 included measurements of temperature coefficient, flux distributions, control-rod worth, Th resonance integral and reactivity coefficients of various materials in a rather simple and clean system.<7) The initial criticality of the Peach Bottom reactor in 1966 provided also a good test of the calculational methods.

Criticality was achieved when 682 fuel elements were loaded in the core, the predicted value being 689 ±20 elements.<8)

AVR

In Germany the commissioning of the pebble-bed AVR reactor gave also the opportunity of testing calculations of criticality, temperature coefficient and control rod worth.<U) The agreement between experiments and calculations was good, but one must notice that pebble-bed systems are not very suited to an accurate test of calculational methods, especially because of the difficulty of exactly locating the position of the fuel elements in the core.

From the time of these older experiments the calculational methods have evolved somewhat, besides, in the case of the HTRs with prismatic fuel the core geometry changed considerably so that it was felt that new experiments were needed.

(a) Accident analysis

The reactor dynamics calculations for HTRs are often performed with a point model, at least as far as neutron kinetics are concerned. This sort of calculation is often considered sufficient for accident analysis as also the cause of the accident (e. g. the amount of water entering the core) can only be determined with difficulty.

Comparisons performed between space-dependent and point model calculations show that even for the maximum reactivity transients possible for HTRs, there is very little discrepancy in the behaviour of the total reactor power with time. This means that if the flux shape is not altered during the transient (e. g. control-rod movement) the results of the point model are sufficiently accurate as far as neutron kinetics are concerned. This is not valid for the heat-transfer calculation because the shape of the axial temperature profile may change strongly during the power excursion. An example is given in Fig. 12.3 which refers to a hypothetical ejection of a central control rod in the 300-MWe THTR pebble-bed reactor. This distortion in the axial temperature profile gives rise to a considerable discrepancy between the zero and the one-dimensional calculations. This is shown in Fig. 12.4 where the results of the zero-dimensional DYN code’111 are compared with those of a one-dimensional COSTANZA calculation.’81

In the case of big reactivity excursions where the reactor is nearly prompt critical, point model calculations can only give an accurate time behaviour of the reactor power if the proper prompt neutron lifetime is used. The effect of / on point model calculations of power excursion is given in Fig. 12.5. A space-dependent calculation, limited to a few time steps, may be necessary in order to obtain the proper / (see § 12.8).

Still for the case of the central control rod ejection, Fig. 12.6 shows the time behaviour of the flux shape. In order to visualize better the change in shape, all distributions have been renormalized to the initial power level.

One can see that after about 0.05 sec the flux has settled to the new fundamental mode and all higher modes have vanished. As the thermal capacity of THTR fuel elements is about 2 (MW sec/m3°C) even in a pessimistic assumption of no heat release to the coolant (adiabatic assumption in the thermodynamic meaning of this word) one needs an average power density of 400 MW/m3 to have a temperature increase of 10°C in 0.05 sec. In the case of a 0.64% reactivity step the overall power increases in 0.05 sec only by less than a factor of 2, and even in the central region does not increase by more than a factor of 13. As the HTR power density is always in the range 6-8 MW/m3 it does not appear to be possible with foreseeable accidents to reach a power density of the order of 400 MW/m3 in 0.05 sec. This means that the contributions of the higher modes to core temperatures is negligible in HTRs.<12>

Various codes for zero-dimensional or space-dependent reactor dynamics are described in refs. 13 to 18.

The reactor control is based on physical quantities which are measured with various accuracy, delay and reliability. The quantities of interest are usually: neutron flux, temperature of fuel, coolant and structures, coolant mass flow, coolant pressure, coolant activity.

Core-temperature measurements are rather unreliable in HTRs so that they can be used for providing information on the operating condition of the reactor, but cannot be relied upon for control and safety purposes.

The coolant outlet temperature measurement is sufficiently reliable and provides information about core temperatures, but with considerable delay.

This means that this measurement can be used for the regulation of slow transients, but cannot be used for counteracting fast excursions. The neutron flux measurement is very fast. The proportionality factor between the flux in the neutron detectors and the average core power may depend on the control-rod configurations so that it may be necessary to build a mean value over the signal of various detectors or place the detectors rather far away from the core.

Because of the great range of variation of the neutron flux between approach to criticality and full reactor power, it is necessary to have different sets of instruments each of which is used for a limited range. Usually three different sets are used. In order to follow the neutron flux variation during start-up it is necessary to have a neutron source in the core so that a neutron flux can be measured also in undercritical conditions. Particular problems may be posed to the instrumentation of HTRs in case of spatial instability. Informations on the spatial variations of flux and temperatures must in this case be obtained from measurements made from outside the core.

In the period 1966-9 a programme of HTGR critical experiments covered in a first part measurements of the reactivity worth of special elements in a “clean” essentially homogeneous graphite and fully enriched U core. A second part covered lattice measurements (Doppler coefficients, control-rod worth, reactivity worth of burnable poison rods, flux plots) on a central part of a critical facility which simulated the HTR block fuel geometry.

In the first part the C/235U ratio was varied between 432 and 5000. The analysis of the reactivity coefficient measurements indicated that the reaction rates of core materials can be predicted within the following accuracy:01)

Boron ± 2% 238U + 2 to + 5%

235U ± 2% 236U + 3 to + 5%

233U ± 4% 237Np + 2 to — 10%

232Th + 2 to + 5%

At the time of evaluation the data available were the version I of ENDF/B. Newer cross-section sets such as the versions II and III of ENDF/B do not involve significant changes in the data concerning thermal reactors.

The analysis of these experiments has been repeated at CEA with the APOLLO code.<40)

The control-rod worths measured as part of the lattice measurements have shown discrepancies of the order of 5-10% (the calculated rod worth was in general overestimated).<12)

Many space-dependent dynamics codes<8,9) have a version for slow transient calcula­tions in which Xe and Sm equations are treated and delayed neutrons are neglected.

The compensation of the changes in Xe poisoning requires a movement of the control rods. If the rods used for this purpose are not uniformly distributed over the core, the resulting changes in flux distribution can only be properly treated with space-dependent codes. A typical example is given by the pebble-bed THTR reactor where a possible strategy of operation foresees that only reflector control rods are used for Xe override. Even in cases in which the flux distortion due to rod movement is kept to a minimum, there are discrepancies between zero and space-dependent calculations because the maximum change in Xe concentration occurs in the regions of maximum flux and hence of maximum importance.

An accurate value of the excess reactivity needed for Xe override can in many cases only be obtained with a two-dimensional calculation.19’

Various abnormal operating conditions are possible, some of which can rapidly damage the reactor, some only if the abnormal operation is protracted over a very long time. It is very important to ensure both a safe and reliable operation. In order to ensure plant reliability it is important to avoid unnecessary scrams while on the other hand it is necessary to protect the reactor from dangerous conditions. Scram signals are released when limiting values are exceeded in various quantities, of which the most important are: too high flux, too short period and too high gas outlet temperature.

Before these limits are reached it is possible to have alarms and power set back. A scram signal causes the control rods to be inserted in the core with their maximum speed.

The maximum temperature reached after an accident depends on the delay between the beginning of the abnormal condition and the introduction of the required negative reactivity to shut down the reactor.

After a scram the coolant mass flow must be reduced in order to avoid too rapid a cooling down of the reactor structures. The mass flow must be adjusted to the requirement posed by the decay heat.

In the period 1970-1 a series of measurements at very high temperature (up to 1000°C) have been performed at the Battelle Northwest Laboratory (BNWL) in the High Temperature Lattice Test Reactor (HTLTR). The method of measurement in the HTLTR determines the excess neutron production in a test lattice from the ratio of the reactivity coefficients of a test lattice sample and a normalizing neutron absorber, usually copper.<15) The measurements have been repeated at several temperatures to obtain the temperature coefficient. Several correction factors, both analytical and experimental, must be applied to each measurement, and this complicates the analysis. With this technique only a small region of the exact lattice under study is required, and this fact made possible the study of lattices with high 233U or Pu concentrations.

Five different lattices have been measured with various 235U, 233U and Th loads. Lattice 4 was fuelled with Pu and Th, and lattice 5 was designed to simulate the Molten Salt Breeder Reactor (233U-Th). At high temperature the measured temperature coefficient is less negative than calculated. The measurements on Pu lattice showed a very good agreement with the calculations.

Supposing that the control system keeps constant the total reactor power output, it is still possible to have local power variations in which the power output is increasing in some part of the reactor while decreasing in other parts. There are two possible forms of spatial instability: temperature and Xe instability.

Temperature instability can occur for sufficiently large values of a positive moderator temperature coefficient. To a local flux increase a temperature increase corresponds which through the positive temperature coefficient tends to enhance further the flux

disturbance. This effect must compete with the negative fuel coefficient and the increase in leakage corresponding to the distortion of the flux distribution. Depending on the reactor size and the fuel temperature coefficient there is a threshold value for the moderator coefficient above which spatial temperature instability occurs. While the temperature instability has a monotone divergent character, the Xe instability gives rise

to oscillations which can have divergent amplitude. A local flux increase causes a local reduction of Xe concentration because of neutron absorption in Xe. This Xe reduction will, in turn, produce a further flux increase, and at the same time increases the 135I production from fission, so that after a few hours the decay of 135I will overcompensate the 135Xe loss and the flux will start to decrease, reversing the trend.

A positive moderator temperature coefficient will emphasize this phenomenon, while a damping effect comes from the increase in leakage due to the flux distortion.

The threshold beyond which the oscillations diverge is determined by the reactor size, equilibrium Xe concentration (power level) and temperature coefficient.

Power-flattening usually tends to give a lower stability.

The high migration length of HTRs is a stabilizing factor in comparison to LWRs, but the lower power density of HTRs requires bigger dimensions for a given power level, so that these two reactor types have, for the same power, approximately similar stability problems. On the other hand, HTRs are more stable than other graphite reactors (e. g. Magnox). Because of their very long period these spatial instabilities do not usually pose a safety problem, but rather a control problem.

Xenon and temperature instabilities can be counteracted by providing different regions with independent control systems (sector controllers) which respond to the temperature of each considered region and not to the overall reactor temperature or flux.

A problem is posed by the measurement of the flux and temperature distribution which is, in HTRs, more difficult than in other reactor types. Because of the high temperature, in-core instrumentation is either impossible or unreliable, so that one must rely on informations obtained from measurements made outside the core (e. g. flux in outer reflector).

Xenon and temperature instabilities can be numerically studied with space dependent dynamics codes for slow transients of the type described in the preceding paragraphs.

A first investigation on the instability of a reactor can be performed with simplified analytical methods.

The system is described by the Xe and I equations (12.24) together with the one group diffusion equation in the form (§8.11)

V2(/> + В 2ф — 0.

Simplified solutions can be obtained assuming only small perturbations about the steady state119’ so that [see eqn. (8.41)]

Фіг, t) = ф'(г) + (f (r, f),

Br, t) = B2r) + ^2= B2r) + jpATm(r, t) + j^-2ATt(r, t) + fpx(r, t),

Xe(r, f) = Xe'(r) + x(r, f),
l(r, t) = l'(r)+i(r, t),

where Ф*(г), B2′(r), Xe*(r) and I*(r) are steady-state values, ax is a Xe reactivity coefficient, ©/ and 0m the fuel and moderator temperature coefficients, AT/ and ДTm the fuel and moderator temperature perturbations. (In ref. 19 the Xe and I concentra­
tions are expressed as relative values to the saturation Xe concentration which is reached at very high flux levels.)

The perturbations are then assumed to have a time dependence of the form

<p(r, t) = <pt(r) e“",

x(r, t) = Xi(r) e“”, (12.39)

і (r, t) = 11 (r) e“

and the space dependence is expanded as sum of a truncated infinite series of the orthonormal eigenfunctions f of the boundary-value problem that has the unperturbed flux shape as its fundamental solution

<Pi(r) = 2 a‘f‘

x,(r) = 2V< (12.40)

fi(r) = 2 cf,

(V2+B2*)/i + p2f =0 (12.41)

with this modal expansion it is possible to linearize the system of eqns. (12.24) and (12.26) and the associated heat-transfer equations, neglecting the product of the perturbations. If coefficient like 0m, 0/, ax. . . can be considered as space — and time-independent, it is possible to obtain a characteristic equation in w, for each mode i, using the orthogonality property:

ffdV = 8„.

The mode і is stable if all roots of its characteristic equation have a negative real part.

A further simplification consists in neglecting the heat-transfer equations, defining a temperature coefficient aT expressed in terms of reactivity per unit flux

0mATm(r, t) + QfATf(r, t) = aT(p(r, t). (12.43)

With these assumptions091 it is possible to obtain the following stability criterion:
The threshold for oscillations in the jth mode of the flux distribution occurs when

1

M2B2 д aT , _ 1 + Ах! сгхф ^x ax 1 ax + (Xx + Хі)Ісгхф

where Л, is the difference between the eigenvalue of the jth mode of the flux distribution and the eigenvalue of the fundamental mode, divided by the geometrical buckling.

В2 = is the geometrical buckling,

A„ Ax = decay constant of l35I and 135Xe,

yx = fraction of l35Xe formed directly from fission rather than through 135I, crx — l35Xe cross-section,

a*=is the reactivity held by saturated 135Xe (corresponding to the limit concentration Xesal for very high flux) «’,= a, Xe“t,

M2 = migration area.

In this way it is possible to assess the stability of each mode independently.

The smallest Л occurs usually for the first azimuthal mode, which is then the first mode to become unstable.

The simplest eigenfunction shapes are the least stable because they have fewer nodal lines, which determine the increase in leakage relative to the fundamental mode.

The form of some of the radial-azimuthal and axial eigenfunctions is shown schematically in Fig. 12.7.<20) The form of the imperturbed flux has an important influence on stability, so that it is important to repeat the investigation with different control-rod configurations (which in the simplified treatment appear through the term B2′(r)). The solutions are usually studied for a range of values of the moderator temperature coefficient and of power levels. As the temperature coefficient gets positive and bigger the period of the Xe oscillation increases and the oscillatory Xe instability tends toward the monotone temperature instability. With this simplified treatment it is possible to check the stability margin for each mode. The stability margin is usually defined as the difference between the actual moderator temperature coefficient and the value of the coefficient for which instability occurs. Independent sector controllers are required not only in case of instability, but also in case of too-small a stability margin when the natural damping of external perturbations is insufficient.

The number and position of the sector controllers is dictated by the form of the modes which need stabilizing. The reactor operation should be possible also in case of failure of one controller.

It is possible to include the feedback of the reactor controller in simplified analytical methods, but usually only numerical codes allow an accurate investigation of the effect

of power distribution and control-rod position on stability and the optimization of the sector control system.

Two-dimensional (г, в) codes are necessary to study azimuthal and radial instability, while axial instability may be studied with one-dimensional codes.

Codes to study Xe spatial oscillations are described in refs. 21, 22 and 23. About the optimization of the control strategy see ref. 24. Investigations carried out for HTRs<20> show that axial instability is not likely to occur for core heights < 10 m (the present values are around 5 m). Also radial modes appear to be very stable. The most unstable mode is the first azimuthal and its natural damping might become critically small already for reactor sizes of the order of 1000 MWe, depending from core composition, power-flattening and height-to-diameter ratio. The control-rod requirement necessary to damp Xe oscillations is, in any case, very small.

13.1. Control-rod requirement

The control system of a reactor must be able to shut down the reactor and keep it in undercritical condition at any moment of its life, compensating also the reactivity introduced by the worst possible accident.

Beside that, enough excess reactivity must be provided to regulate the system and to allow for Xe override and burn-up compensation.

The above-mentioned requirements must be met also if one or more control rods fail to enter the core.

In this way we get then the following list of requirements:

1. Temperature effect between cold and operating condition.

2. Decay of fission products after shutdown (135Xe^> 135Cs).

3. Decay of 233Pa into 233U in Th cycles, of 239Np in 239Pu in 238U cycles (this effect is considerably smaller than that of 233Pa)

4. Compensation of worst accident.

Points 1 to 4 give the reactivity which has to be compensated by the control rods after a shut-down. To this the so-called excess reactivity must be added. This reactivity is invested in control rods which are initially inserted in the critical reactor in order to be able to compensate criticality losses due to:

5. Fine regulation.

6. Xe override.

7. Burn-up compensation in case of batch or discontinuous loading.

8. Damping of spatial oscillation.

The excess reactivity should be reduced to the minimum compatible with the operational requirements. The neutrons absorbed in control rods are lost for conversion and increase the fuel cycle cost.

The increase in number of control rods due to excess reactivity means a considerable increase in plant cost.

The absorption in control rods tends also to disturb the power distribution, and the uniformity of the burn-up, so that hot spots can result.

The requirement of point 5 is very small.

Point 6 depends on the operation requirement of the power station.

Point 7 can be strongly reduced by frequent or continuous reloading, and by the use of burnable poisons. With proper lumping, the rate of destruction of the burnable poison can be made similar to the change in reactivity due to the burn-up.

Point 8 is only important for very big systems.

The build-up of 149Sm after shut-down could be considered to reduce the control-rod requirement, but this contribution is rather small, and not always available: if the reactor has run at reduced power in the last days before shut-down, also the 149Sm build-up is reduced.

The control-rod requirement is dependent on the reactor composition and therefore has to be calculated as a function of burn-up. In reactors with continuous reloading the control requirement has to be calculated during the running-in period and in the equilibrium condition.

The reactivity requirement of the control system Akr is normally obtained as a difference of two calculations, one corresponding to the highest reactivity (cold reactor without Xe and Sm, at the beginning of a refuelling cycle, after decay of 233Pa and 239Np) and one to the lowest (hot reactor with maximum Xe concentration, equilibrium 233Pa and 239Np, at the end of a refuelling cycle). To this value is added the reactivity introduced for the requirement of regulation and by the worst accident.