6. Linearized model
In order to compare linear and nonlinear model analysis and control tech- niques, the linearized version of the nonlinear model – state equations are in Equations (4.78)–(4.79) – is presented here:
˙
x=Ax+Bu (4.84)
where A=
∂f
∂x
x=0
=
0 −àmax(K2XS020+S(K20S+K02−K1)21)
−(K2Sà02max+S0S+K0 1)Y
àmaxX0(K2S02−K1) (K2S20+S0+K1)2 −FV0
(4.85)
B=g(0) = −XV0
SF−S0
V
(4.86) The steady-state point (X0, S0, F0) has been used in the linearized version of the model. The system matrices at the optimal operating point are:
A=
0 0.4011
−1.6045 1.2033
, B=
−1.227 2.4453
(4.87)
4.6 Case Study: Modeling a Gas Turbine
Gas turbines are important and widely used prime movers in transportation systems such as aircraft and cars. They are also found in power systems, where they are the main power generators, and in process plants as well.
The investigation of steady-state behavior and static characteristics of gas turbines is a traditional area in engineering. This kind of model is based upon the characteristics of the component parts of the engine. The static characteristics can be given in the form of polynomials reflecting the results of the preliminary calculations or the measurements.
This section is devoted to the model building of a low-power gas turbine [1], which will be used later on in the book as a case study for nonlinear analysis and control.
4.6.1 System Description
The main parts of a gas turbine include the compressor, the combustion chamber and the turbine. For a jet engine, there is also the inlet duct and the nozzle. The interactions between these components are fixed by the phys- ical structure of the engine. The operation of these two types of gas turbine is basically the same. The air is drawn into the engine by the compressor, which compresses it and then delivers it to the combustion chamber. Within the combustion chamber the air is mixed with fuel and the mixture is ignited,
producing a rise in temperature and hence an expansion of the gases. These gases are exhausted through the engine nozzle or the engine gas-deflector, but first pass through the turbine, which is designed to extract sufficient energy from them to keep the compressor rotating so that the engine is self- sustaining. The main parts of the gas turbine are shown schematically in Figure 4.4. In this section we analyze a low-power gas turbine, which is in-
1 2 3 4
Compressor Combustion chamber Turbine
Mload
Figure 4.4.The main parts of a gas turbine
stalled on a test-stand in the Technical University of Budapest, Department of Aircraft and Ships.
Engineering intuition suggests that the gas turbine is inherently stable because of its special dynamics. This is also confirmed by our measurements and simulation results. The most important control aim could then be to keep the number of revolutions constant, unaffected by the load and the ambient conditions (pressure and temperature). The temperatures and the number of revolution has to be limited, their values are bounded from above by their maximum values.
4.6.2 Modeling Assumptions
In order to get a low order dynamic model suitable for control purposes, some simplifying modeling assumptions should be made.
General assumptions that apply in every section of the gas turbine 1. Constant physico-chemical properties are assumed. These include the specific heats at constant pressure and at constant volume, the specific gas constant and adiabatic exponent.
2. Perfectly stirred balance volumes (lumps) are assumed in each main part of the gas turbine. This means that a finite dimensional concentrated parameter model is developed and the values of the variables within a balance volume are equal to those at the outlet.
4.6 Case Study: Modeling a Gas Turbine 67
Other assumptions
3. Efficiency of the combustion is constant.
4. In the compressor and in the turbine the mass flow rates are constant:
νCin=νCout=νC and νT in=νT out=νT
4.6.3 Conservation Balances
The nonlinear state equations are derived from first engineering principles.
Dynamic conservation balance equations are constructed for the overall mass m and internal energy U for each of the three main parts of the turbine system [32]. The notation list is given separately in Table 4.2.
Table 4.2.The variables and parameters of the gas turbine model
Variables Indices
m mass comb combustion chamber
U internal energy f uel refers to the fuel
T temperature C compressor
p pressure T turbine
n number of revolution in inlet c specific heat out outlet
i enthalpy p refers to constant pressure
M moment v refers to constant volume
R specific gas constant comb refers to combustion
η efficiency mech mechanical
Θ inertial moment load loading ν mass flow rate air refers to air
gas refers to gas
The development of the model equations is performed in the following steps:
1. Conservation balance for total mass (applies to each section of the gas turbine):
dm
dt =νin−νout (4.88)
2. Conservation balance for total energy in each section of the gas turbine, where the heat energy flows and work terms are also taken into account:
dU
dt =νiniin−νoutiout+Q+W (4.89)
We can transform the above energy conservation equation by considering the dependence of the internal energy on the measurable temperature:
dU dt =cv
d
dt(T m) =cvTdm
dt +cvmdT
dt (4.90)
From the two equations above we get a state equation for the tempera- ture:
dT
dt = νiniin−νoutiout+Q+W −cvT(νin−νout)
cvm (4.91)
3. The ideal gas equation is used as a constitutive equation together with two balance equations above to develop an alternative state equation for the pressure:
dp dt = RT
V (νin−νout) +p
T
νiniin−νoutiout+Q+W−cvT(νin−νout) cvm
(4.92) Note that both the extensive and the intensive forms of the model equations are used later for model analysis.
4.6.4 Conservation Balances in Extensive Variable Form
The state equations in extensive variable form include the dynamic mass con- servation balance for the combustion chamber, the internal energy balances for all of the three main parts of the turbine and an overall energy balance for the system originating from the mechanical part. The indices in the balance equations and variables therein refer to the main parts of the turbine: to the compressor (i= 2), to the combustion chamber (i = 3) and to the turbine (i= 4), while the inlet variables are indexed byi= 1.
Thus five independent balance equations can be constructed, therefore the gas turbine can be described by only five state variables.
Total mass balance dmComb
dt =νC+νf uel−νT (4.93)
Total energy balance dU2
dt =νCcpair(T1−T2) +νTcpgas(T3−T4)ηmech−2Π 3
50nMload (4.94) dU3
dt =νCcpairT2−νTcpgasT3+Qfηcombνf uel (4.95) dU4
dt =νTcpgas(T3−T4)−νCcpair(T2−T1)
ηmech − 2Π
ηmech
3
50nMload (4.96) Mechanical dynamic equation
dPn
dt =νTcpgas(T3−T4)ηmech−νCcpair(T2−T1)−2Π 3
50nMload (4.97)
4.6 Case Study: Modeling a Gas Turbine 69
4.6.5 Model Equations in Intensive Variable Form
There are several alternatives for the model equations in intensive variable form. We choose the set that includes the dynamic mass balance for the com- bustion chamber (that is, Equation (4.93)), the pressure form of the state equations derived from the energy balances and the intensive form of the overall mechanical energy balance, expressed in terms of the number of rev- olutionsn.
dp2
dt = Rair
VCcvair
(νCcpair(T1− p2VC
mCRair
) +νTcpgas( p3VComb
mCombRmed
− p4VT
mTRgas)ηmech−2Π 3
50nMload) (4.98)
dp3
dt = p3
mComb
(νC+νf uel−νT) + + Rmed
VCombcvmed
(νCcpair
p2VC
mCRair −νTcpgas
p3VComb
mCombRmed
+Qfηcombνf uel−cvmed
p3VComb
mCombRmed(νC+νf uel−νT)) (4.99) dp4
dt = Rgas
VTcvgas
(νTcpgas( p3VComb
mCombRmed
− p4VT
mTRgas
)−νCcpair(mpC2RVCair −T1)
ηmech − 2Π
ηmech
3
50nMload) (4.100) dn
dt = 1
4Π2Θn(νTcpgas( p3VComb
mCombRmed− p4VT
mTRgas
)ηmech
−νCcpair( p2VC
mCRair −T1)−2Π 3
50nMload) (4.101)
4.6.6 Constitutive Equations
Two types of constitutive equations are needed to complete the nonlinear gas turbine model. The first is the ideal gas equation
T = pV mR
which has already been used before, and has been substituted into the state equations to get alternative intensive forms.
The second type of constitutive equations describes the mass flow rate in the compressor and in the turbine.
νC= const(1)q(λ1) p1
√T (4.102)
νT = const(2)q(λ3) p3
√T3
(4.103) In these equationsq(λ1) andq(λ3) can be calculated as follows:
q(λ1) =f( n
√T1
,p2
p1
) (4.104)
q(λ3) =f(const(3) n q p3VComb
mCombRmed
,p3
p4
) (4.105)
The parameters and constants of these functions can be determined using measured data and the compressor and turbine characteristics.
4.6.7 Operation Domain and System Variables
Themeasurable intensive set of state variables for the gas turbine test stand at the Technical University of Budapest is:
x= [mComb p2 p3 p4 n]T (4.106)
Experimental values of these variables are constrained to the following do- main:
0.003≤mComb≤0.0067 [kg] 180000≤p∗2≤280000 [Pa]
170000≤p∗3≤270000 [Pa] 100000≤p∗4≤140000 [Pa]
39000≤n≤51000 [1/min]
The value of the onlyinput variable νf uelis also constrained by:
0.009≤νf uel≤0.017 [kg/sec]
The set of possibledisturbances includes:
d= [T1 p1 Mterh ]T (4.107)
where the domain of its elements is:
273≤T1≤310 [K] 97000≤p1≤103000 [Pa] 0≤Mterh≤180 [Nm]
Finally we construct theset of output variables by noticing that all the pres- suresp∗i and the number of revolutionsn in the state vector above can be measured, but the massmComb cannot:
y= [p2 p3 p4 n] (4.108)
4.8 Questions and Application Exercises 71