Special tests will not be required to determine whether a system is stable. Unstable behavior will be unpleasantly apparent without testing, but measurements will be required to assess the stability reserve of stable systems. Furthermore, prudent operators of reactor systems should prize quantitative information on changes in the stability reserve with changing system characteristics. Examples of system characteristics that will change and that will have an influence on the stability reserve are:

1. Fuel composition—this will change because of burn-up, fission product accumulation, and refueling.

2. Moderator composition—This will change in some pressurized-water reactors because of changes in dissolved poison concentration.

3. Control-element concentration and location—This includes burnable poisons, removable but stationary poison, and movable control elements.

4. Core heat-transfer characteristics—These may change because of radiation damage and partial fouling of heat-transfer surfaces.

5. Heat exchanger heat-transfer characteristics—These may change because of partial fouling of heat-transfer surfaces.

6. Controller characteristics—Changes made in controller settings may be made to optimize the controller and to adapt it to changing system character­istics.

Numerous experimentally observable stability reserve measures are used in dynamics studies. These include (1) resonance peak, (2) gain margin, (3) phase margin, (4) decay ratio, and (5) peak overshoot. The first three may be deduced from frequency response measurements as well as from calcula­tions using a theoretical model. The next two are based on the response of the system to a step input and may be obtained from a test or from calcula­tions using a theoretical model.

The resonance-peak approach is based on the observation that decreasing stability reserve often manifests itself as a growing resonance peak in the closed-loop frequency response amplitude results. The interpretation involves the extrapolation of resonance amplitudes at several different operating conditions to the condition that gives infinite amplitude. For ease of extrapolation, the usual procedure is to plot the reciprocal of the resonance amplitude versus the selected operating condition and to extra­polate to zero.

The Nyquist method involving the open-loop frequency response (see Section 2.4) is useful for experimental evaluation of stability margins. In most cases, it is not possible to measure the open-loop frequency response directly, but it can be obtained from the measured closed-loop frequency response. The relation between the closed-loop frequency response G, the forward-loop frequency response G1, and the feedback frequency response H is

G = GJ{ 1 + G, H) (6.1.1)

The desired open-loop frequency response is GiH. If separate measurements of G and G1 are made, then the open-loop frequency response may be obtained as follows:

GiH = (GJG) — 1. (6.1.2)

In reactor systems, a measurement of the zero-power frequency response (power/reactivity) gives Gt, and the at-power frequency response (power/ reactivity) gives G. Also, it is often possible to use theoretical predictions for G[, since it is usually known with adequate accuracy.

A good example of the use of phase-margin measurements is the work done on the experimental boiling-water reactor (EBWR) (1). Frequency response measurements (using the oscillator method) were made at several different power levels. The phase margin was determined at each power level and an extrapolation was made to predict the power level at which an instability would occur (the phase margin would go to zero). The results are shown in Fig. 6.1. These indicate that the EBWR would have become unstable at a power level of 66 MW.

The decay ratio is defined as the ratio of the magnitude of the second overshoot to the magnitude of the first overshoot resulting from a step input. This is shown in Fig. 6.2. This performance measure is used extensively by the General Electric Company in experimental assessments of boiling — water reactor dynamics (2). Clearly the decay ratio must be less than unity for a stable system. Furthermore, generally accepted performance standards specify a decay ratio of less than 0.25. General Electric has set a value of

0.25 as the maximum allowable decay ratio for some of the process variables and a value of 0.5 for others. This is based on the requirement that certain key parameters must remain near design values while others can be allowed a greater range.

The peak overshoot is closely related to the decay ratio. It is defined as follows:

Square waves are included because they are useful in tests in which a sinu­soidal input is not feasible and it is desirable to get the best results at a single frequency in the shortest time using binary input signals. The power spectrum of a square wave is given by:

Pk = 0.8M2//c2 for к odd

= 0 for к even

Since this spectrum is identical with the maximum (see Section 2.12), the square wave gives the greatest possible energy at harmonic frequencies. The first harmonic contains 81 % of the total signal energy, and the next nonzero harmonic (к = 3) contains only 9 %.

(a) Magnetic Jack

This CRDM is used widely in pressurized-water reactors. The magnetic jack has the advantage that it does not require mechanical or electrical penetrations of the reactor pressure boundary. Electrical coils around the CRDM housing are capable of supplying magnetic forces that can engage appropriate latches and can lift or drop the control rod between stops. The motion of a rod is accomplished by a series of lifts or drops with appropriate latching operations between moves. A typical step size is about in. and the minimum time between moves is 1 sec. The move time is virtually instantaneous compared to the time constants of interest in most reactor dynamics tests. Adequate reactivity worth can usually be obtained by applying the electrical signal for rod motion simultaneously to several CRDMS. This was demonstrated on the magnetic jack CRDMs used in the dynamics tests in the Halden reactor and several pressurized-light-water reactors. For a minimum time between moves of 1 sec, the highest frequency that one would expect to be able to measure is 0.5 Hz. Since this is within the range of important frequencies for PWRs, we conclude that the speed of the magnetic jack is satisfactory. Also, since the transition is so rapid, the position indication is no problem. The rod may be assumed to move instantly between two known positions at the time of the move command.

Even though this book deals with experimental measurements, it is neces­sary to consider some mathematical questions. First, it is essential to examine briefly the methods used for theoretical analysis of dynamic systems so that the significance of test results will be appreciated. It is also necessary to study Fourier analysis and related topics as a background for later discus­sions on data analysis. The treatments in this chapter are not complete mathematical developments, but are condensations that should be useful as a review for the reader.

Noise will contaminate the input and output signals in a test. The severity of this problem will vary widely from one case to another, but achieving an adequate signal-to-noise ratio is a general problem. The signal available for analysis, Z(t), will be the sum of the desired signal x(t) and the noise N(t):

Z(t) = x(t) + N(t)

The computed Fourier integral for a periodic signal will be

where Ck is the computed Fourier integral, Ck the true Fourier integral, n the number of periods of data analyzed, and T the period. The integral in Eq.

(4.9.2) is the error due to noise contamination. It has been shown (10) that this error is inversely proportional to the amplitude of the periodic signal and inversely proportional to the square root of the length (time duration) of the data record:

1 /(flvA) (4.9.3)

where a is the amplitude of the periodic input signal.

The experiences with frequency response testing on heavy-water reactors have been at the NPD and NRU reactors in Canada, the Agesta reactor in Sweden, and the Halden reactor in Norway. Each of these countries has devoted considerable effort to development of practical testing procedures. The experiences with all heavy-water reactors have been combined in this section, but the reactor characteristics and testing procedures were quite different.

The Canadian reactors NPD and NRU use fluid level in the core to control reactivity. In NPD, the level of D20 in the reactor calandria is controlled. In NRU, reactivity is controlled by adjusting the level of light water in special control compartments. In Agesta and Halden, conventional control rods are used for reactivity control.

Measurements on Canadian reactors have used the oscillator technique and the PRBS technique. Results were obtained at frequencies up to about 1 Hz in NRU and NPD. These tests provided valuable experience in on-line data analysis, since a PDP-8 computer with an FFT analysis algorithm was

t See the literature (85-93).

f See the literature (93-109).

used. Tests at NPD uncovered a need for changing the settings in the control system.

The measurements at Agesta were made to gain experience with testing methods and to study system dynamic behavior as a function of fuel burn-up. Standard control rods were used to introduce PRBS and MFBS reactivity inputs.

The Halden tests were also extensive and innovative. The testing program included PRBS and MFBS input signals. The reactivity perturbations were introduced using magnetic jack control rods. A significant aspect of these tests was that up to ten control rods were moved simultaneously by the introduction of a single signal from the control panel. This indicates that increased input amplitude can be obtained in systems with magnetic jack control rod drives simply by using enough rods. There is no need to wait until sufficient reactivity can be introduced by multiple steps of a single rod.

As will be shown in later chapters, two-level (binary) signals are well suited for tests on reactor systems. In this section, the upper bound on the energy in any harmonic of a binary signal will be derived. We will see in Chapter 3 that the harmonic amplitude of several important signals is less than this upper bound by a constant factor at all harmonics. Also, the upper bound indicates the limits on spectral shape that can be achieved in any binary signal.

Consider a periodic signal u(t) consisting of Z equal-amplitude positive and negative pulses of length At seconds. Each pulse in the chain is called a bit. It should be noted that it is possible to have strings of bits of the same sign, giving a pulse that is several bits long. This is called a run. The Fourier coefficient Ck of the fcth harmonic is

where cok = 2knjT and T = ZAt. Equation (2.12.1) can be written as

where и,- is the polarity of u(t) over the interval (і — 1) At < t < і At and A is the amplitude of the signal. Performing the integration and rearranging

f See Buckner (5).

Since

we obtain

This gives the upper bound on the power spectrum of a binary pulse chain.

The actual spectrum for a particular pulse chain depends on the actual set of polarities щ in the signal.

The bandwidth of a signal (the frequency range over which the signal power is at least half as large as its greatest value) is useful in estimating the maxi­mum frequency at which results can be obtained. Since the spectra for several important signals have the same shape but a smaller magnitude than the maximum (Eq. 2.12.7), bandwidth relations derived for the spectral shape of the maximum will have general utility. The shape function is

sin(o> At/2)

_ to At/2

The bandwidth obtained by setting

sin(o> At/2)

_ со At/2

is to At/2 = 1.39 or to = 2.78/Дг. Since to is a harmonic frequency for a periodic signal, we may write

2kn/T = 2.78/ Дг

or

T

к = 0.44— = 0.44Z (2.12.10)

At

That is, in a signal with Z bits, all harmonics out to harmonic number 0.44Z have at least half as much power as the largest harmonic.

The term system identification has been used to describe a number of things, including the measurement of the system frequency response. However, we will use it here to describe procedures for extracting useful information on system models or parameters from experimental data. Different levels of system identification may be applied to test data. The different levels vary in complexity and in the value of the resulting informa­tion. Three system identification procedures that are useful in nuclear reactor applications are described in this section.

Several nonperiodic signals such as single pulses or steps are very easy to input into a system. However, we shall see that the total available energy is limited, and these signals have difficulty in providing a high enough signal — to-noise ratio to give accurate frequency response results. Nevertheless, they may be used to give approximate results very easily. These tests can comple­ment more accurate methods involving periodic inputs. One important use would be for frequent checks for major changes in system behavior. More accurate methods would be used on a less frequent schedule or when the tests with nonperiodic signals indicate a need for the more accurate tests.

(a) Pulse Tests t

Frequency response tests using a single pulse as the input have been used in several nuclear reactor tests and in numerous tests on chemical process equipment. The simplest pulse shape is a square pulse. The square pulse and its energy density appear in Fig. 3.15. Note that the signal energy is given in terms of an energy density (energy per unit frequency), since the signal distri­butes a finite amount of energy over a continuous frequency range. The energy density e for a square pulse of duration T is given by

_ T2T2rsin(coT/2)

Є 2n (x>T/2

t See Hougen and Walsh (66) and Hougen (67).

It is important to note that the energy density in a nonperiodic signal is simply the power density times the duration of the input (PT). In contrast, the energy Ek in the kth harmonic of a periodic signal is given by

Ek = PknT (3.8.2)

where n is the number of cycles and T the period. Thus the signal energy can be made as large as desired by increasing n for a periodic signal, but the energy for a nonperiodic signal with a fixed amplitude and duration is fixed. The only way to improve the results from tests with a given nonperiodic signal is to average the results from multiple tests. This procedure may be difficult to use, since it may be difficult to achieve the same initial conditions for each test.

Since the signal energy relative to the noise energy determines the accuracy of the test results, it is useful to use the energy density results and determine the total energy in a specified frequency range for a square pulse.

The energy in frequency range cu, — w2 is given by

Eicot — to2) = A2T

Since A2 T represents the total signal energy, the term in brackets represents the fraction of the total signal energy in the range coj — co2. The integral

1 f« (sm(a}T/2)2 M ^

2nJaT (oT/2 I d{(° ] (3.8.5)

has been evaluated and is shown in Fig. 3.16 (50). This allows the evaluation of E(tol — co2) as follows :

E(tol — a>2) = £(«! — oo) — E(co2 — oo) (3.8.6)

This type of information is useful for comparing different signals.

For an example, let us determine the signal energy between 0.1 and 0.2 rad/sec in a pulse with an amplitude of 2 and a duration of 10 sec. From Fig. 3.16, we obtain

and

£(0.2 — oo) = 40(0.43) = 17.2

Then

£(0.1 — 0.2) = 27.6 — 17.2 = 10.4 energy units

(b) Step Tests t

The frequency reponse can also be obtained from tests using a step change in the input. A step input may be used if the output settles to a constant value (including zero) after the step input. Stated differently, the frequency response gain at zero frequency must be finite.

The Fourier transform is not defined for a function that does not return to its original value. The output of a system with a finite, nonzero gain at zero frequency does not return to its initial value in a step test. However, the derivative of the signal does return to its initial value. The ratio of the Fourier transforms of the derivatives of the output and input gives the system frequency response:

It is convenient to express the input and output signals relative to their final values:

30* = 30(oo) — 30 31* = 3I(oo) — 31

Thus

d 30*/dt = — d 30/dt and d 31*/dt = — d 31 dt

and we obtain

OUTPUT

INPUT

Fig. 3.17. A typical step test.

t See Nyquist, Schindler, and Gilbert (68).

This may be integrated by parts to give

The energy in a true step input signal would be infinite. In an actual test that is terminated after the system settles out, the useful energy is finite. The total energy in a step test that is terminated at time T is half the useful energy in a square pulse of duration T. As an example, consider the step test shown in Fig. 3.17 and the pulse test shown in Fig. 3.18. The pulse test con­tains the same useful energy as two step tests.

T 2T

Fig. 3.18. A typical pulse test.

The roller nut CRDM, like the magnetic jack, requires no mechanical or electrical penetrations. It is used in PWRs and fast reactors. Coils are mounted outside the CRDM housing. Inside the housing is a threaded nut that is constrained to a fixed axial position. The magnetic field causes a torque that causes this nut to rotate. Generally the torque is applied in steps so that the nut rotates a fixed angle at each step. The control rod is connected to a threaded shaft that passes through the nut. The rotary motion of the nut causes axial motion of the threaded shaft. Typical rod motions are yj in. per step and the minimum time between steps is ygsec. The motion is probably fast enough to permit one to neglect the move time. A roller nut CRDM with a minimum time between moves of r^sec will permit measure­ment to frequencies up to about 8 Hz if the worth per step is large enough. However, because of the small displacement per step, it may take multiple steps to obtain adequate reactivity even when several rods are moved simultaneously. For example, if a motion of in. were required, the rod would have to take 16 steps and would take 1 sec. This would reduce the useful maximum frequency to 0.5 Hz. The position may be determined by counting the number and timing of move steps.