Page 307 - Dynamics and Control of Nuclear Reactors

APPENDIX F State variable models and transient analysis 309

The sensitivity for parameter, P, is given by Δ(x (t))/ΔP; that is, change in the
response x(t) for a change in the parameter from a nominal value. The quantity,
P, may appear in more than one matrix element. The sensitivity is a time-dependent
vector quantity that gives the change in calculated response per unit change in the
parameter of interest.
Sensitivity information enables the analyst to evaluate the consequences of
uncertainties in design parameters and to determine changes needed in specified
design parameters in order to obtain a desired change in system response. Sensitiv-
ities may be obtained by “brute force”. That is, change a parameter in the model,
repeat the simulation and see what happens. This means that the whole analysis must
start over with a new coefficient matrix. Efficient sensitivity analysis is possible for
linear model simulations using the matrix exponential approach. It avoids the need to
re-analyze by using the same C 1 and C 2 matrices that were used to obtain the solution
for x.
The sensitivity to a matrix element, a jk, is dx/da jk . An equation for sensitivities is
obtained by differentiating Eq. (F.7a) with respect to a jk as follows:

d dx dx dA
¼ A + x (F.40)
dt da jk da jk da jk
Define the sensitivity vector as.

dx
S ¼ (F.41)
jk
da jk
The differential equation of the sensitivity vector becomes.
dS dA
jk
¼ AS + x (F.42)
dt jk da jk
Note that (dA/da jk ) is a matrix with 1 in the j-th row and the k-th column and zeroes in
all other matrix locations. Also, note that Eq. (F.42) has exactly the same form as
Eq. (F.7a). Therefore, the solution of Eq. (F.42) for the sensitivity vector at time step,
(i+1), is

i
ð
S ð i+ 1Þ ¼ C 1 S ðÞ +C 2 dA=da jk xi + 1Þ (F.43)
jk
jk
Note that (dA/da jk ) is a simple matrix with one non-zero element, and C 1 and C 2 are
the same as used in the solution for X; and X(i+1) is known from the solution for X.
Therefore, the solution for the sensitivity equation only involves matrix multiplica-
tions of known quantities.
It is common for a design parameter, P, to appear in several matrix elements. The
sensitivity to P is obtained from.

dx X dx da jk
¼ (F.44)
dP da jk dP
j,k
Sensitivity analysis can be tedious, requiring inspection of large sets of time-
dependent sensitivity results, but providing important and useful results for the reac-
tor designer. If the system description is nonlinear, involving several parameters