Optimisation of Marine Boilers using Model-based Multivariable ...

32 3. SUMMARY OF CONTRIBUTIONS
pa
˙mr
pp
Figure 3.1: Diagram of feed water system. Water pumped from the feed water tank is injected into the boiler
in the forward path, and in the return path the water is led back to the feed water tank.
boiler pressure was found as
�
˙mfw = g(zfw,kr,ps) = kff(zfw)
zfw
˙mfw
where ∆pp is given as the solution to a quadratic equation:
˙m f
fw
FT
ps
∆pp(zfw,kr,ps) + pa − ps
∆pp = −a1(zfw,kr,ps) − � a1(zfw,kr,ps) 2 − 4a2(zfw,kr)a0(zfw,ps)
2a2(zfw,kr)
(3.1)
(3.2)
kf and kr are the valve gains of the forward control valve and the return valve respectively.
a2, a1 and a0 can be found in [Solberg et al., 2008d].
The function f(·) relates the valve stroke, zfw, to the flow:
f(zfw) = 1
R (Rzfw − e −R0zfw ) (3.3)
The function indicates that this is an exponential valve. Further, the last term in the
parenthesis involving R0 is not standard, but is included to describe the valve through
the whole operating range. The feed water flow dynamics and the dynamics of the
pneumatic valve positioning are fast compared to the flow sensor dynamics which is
why this model was constructed purely static. The sensor dynamics were described as:
˙m f
fw (s) = Gfw(s) ˙mfw(s) =
1
τfws + 1 ˙mfw(s) (3.4)
This model provides a good fit to measuring data, and the sensor time constant is about
τfw = 4s. The only parameters in this model that cannot be found from datasheets are
the positioning of the return valve, kr, and the sensor time constant. It is obvious that
the system is very nonlinear, which is illustrated in Figure 3.2.
Note that the feed water flow is dependent on the boiler steam pressure, such that
˙mfw ∈ [0, ˙mfw(ps)]. Further, note that at nominal boiler pressure, ps = 8bar, the
small gain varies up to a factor of 22.