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: