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

36 3. SUMMARY OF CONTRIBUTIONS
The model was based on first principles using mass and energy balances for describing
the temperature and pressure dynamics. As flow through and efficiency of the turbocharger
was required in these balance equations, approximations of the turbocharger
turbine and compressor maps partly based on physical insight were made. The oxygen
was modelled by describing mole balances for the oxygen in the gas generator and furnace
volumes. The resulting model was of sixth order and presented in descriptor form
as:
F(x) dx
= h(x,u,d)
dt
(3.7a)
y = g(x,u,d) (3.7b)
where u = [ ˙mfu1, ˙mfu2] T is the vector of inputs being the fuel mass flow to the gas
generator and the furnace. d = [Ta,Tfu,Tm] T is the vector of disturbances being
temperatures of the inlet air, of the fuel and of the metal separating the hot flue gas
and the water/steam in the boiler. x = [pgg,Tgg,ω,Tfn,xgg,O2 ,xfn,O2 ]T is the state
vector being pressure in the gas generator, temperature in the gas generator, shaft speed,
temperature in the furnace, and oxygen fraction in the gas generator and the furnace.
Finally, y = [ ˙mfu, ˙ Q,xfn,O2 ]T is the output vector with ˙mfu = ˙mfu1 + ˙mfu2 and
˙Q is the energy transferred to the metal walls. Expanding, (3.7a) reveals the model
structure as:
⎡
⎢
⎣
f11 f12 0 0 0 0
f21 f22 0 0 0 0
0 0 f33 0 0 0
0 0 0 f44 0 0
0 0 0 0 f55 0
0 0 0 0 0 f66
⎤⎡
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎦⎢
⎣
dpgg
dt
dTgg
dt
dω
dt
dTfn
dt
dxgg,O2 dt
dxfn,O2 dt
⎤
⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ = ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
h1
h2
h3
h4
h5
h6
⎤
⎥
⎦
(3.8)
where the elements fij and hi were found in the model derivation – see [Solberg et al.,
2008c]. F is never singular, hence has a well defined inverse, so (3.7a) can be written
as an ordinary differential equation: ˙x = f(x,u,d) = F −1 (x)h(x,u,d).
Validation against measurements collected from the test setup shows good agreement
between model and measuring data in terms of capturing the dynamical behaviour.
However, in terms of stationary values these are not represented well by the model
for other outputs than the oxygen level, see Figure 3.5. This was accepted for now
as the main focus was on oxygen control, and constraints on internal states were not
considered.
The performance requirements for the controller of the burner were to deliver the requested
fuel flow while keeping the oxygen percentage above 3% and maximise efficiency.
No requirement regarding off-set free tracking of the fuel flow reference was
introduced meaning that differences between requested and actual fuel flow should be
handled by e.g. including integral action in the outer controller.