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Dynamic behaviour of material states in steam power plant control
Automatica, Vol. 16, pp. 45 52 Pergamon Press Ltd. 1980. Printed in Great Britain © International Federation of Automatic Control
0005-1098,,80/0101-0045$02.00/0
Dynamic Behaviour of Material States in Steam Power Plant Control* A. D U R M A Z t
Mathematical models, developed for the investigation of the dynamic behaviour of material state variables (temperature, strain, stress), indicate that the maximum allowable values of these variables should be approached and maintained in nonstationary operation of steam power plants. Key Words--Control system analysis; computer-aided designs; stress control; material states dynamics; multivariable control system; steam generator; nonstationary operation control; identification; modelling; frequency response; simulation.
Almtract.--The enormous price increase in petroleum and the limited reserves of other fossil fuels have resulted in the increased use of nuclear power plants for the basic load of electricity supply. Consequently, conventional steam power plants have been used to meet the variable and peak loads in electricity generation. The control of fast and large load changes is becoming more and more important for conventional power plant operations. An economical control of the nonstationary operation (start-up, shut-down and large load changes) is possible only by approaching and maintaining the m a x i m u m allowable values of the material state variables (temperature 0, strain ~ and stress a) at the critical points of the power plant components. To achieve this primary control goal, the dynamic behaviour of material states must be well understood. A direct measurement of the changes in material states especially due to thermal stresses, caused by temperature gradients, is not possible. The material states can be calculated, however, by means of mathematical models describing the nonstationary temperature distribution. The working fluid temperature, pressure and mass flow rate are the input variables for these mathematical models. This paper presents a linear mathematical model for the investigation of the dynamic behaviour of material states. Another mathematical model is also developed for calculating the thermal stresses for large temperature and load changes. Thus, the steam generator models so far describing only the states of the working fluid can be extended to the material states. The basic idea is demonstrated for an insulated thick wall tube containing a working fluid. The simulation of the mathematical models is performed. a E F Mn Mo ~/E P PD r ri r,
q,. t cto # v oaD 0 a
heat loss at the outside wall time heat transfer coefficient (steam-tube wall) Poisson's ratio linear thermal coefficient of expansion steam temperature material temperature (tube wall temperature) material strain material stress
Indices a B D i
outside fuel steam inside
m p r v
mean pressure radial resultant
w thermal z axial q~ peripheral a stress
1. I N T R O D U C T I O N
load change in steam power plants is limited by material states, especially due to thermal stresses occurring in the thick wall elements during the transient state of steam generator and turbine. The material temperature distribution 0 and thus the thermal strain ew and thermal stresses a w in the critical parts are dependent mainly on the steam temperature and the heat transfer coefficient, i.e. the steam mass flow rate. The steam generator models that are used in a number of publications, such as the ones for optimization of controller settings (Durmaz, 1973) and driving an optimal control policy for load changes (Dettinger, Wolfonder and Herbrik, 1974) can be extended as shown in Fig. 1 to take care of the limitations imposed by the material state variables such as temperatures 0, strains and stresses a. For achieving this goal, the dynamic behaviour of material states, which depends on the dynamics of the working fluid, must be known. The extension of the dynamics of working fluid variables, such as the steam temperature 0o, steam mass flow rate )f/o and the steam pressure PD, to the dynamics of material states in represented in Fig. 2. The mathematical model used for investigating THE
NOMENCLATURE thermal diffusivity (a=2/pc) modulus of elasticity transfer function fuel mass flow rate steam mass flow rate spray water mass flow rate electric power steam pressure spatial variable in radial direction tube inside radius tube outside radius
*Received 19 October 1978; revised 25 June 1979. The original version of this paper was presented at the 7th IFAC Congress on A Link Between Science and Application of Automatic Control which was held in Helsinki, Finland during June 1978. The published Proceedings of this IFAC Meeting may be ordered from: Pergamon Press Ltd, Headington Hill Hall, Oxford OX3 0BW. U.K. This paper was recommended for publication in revised form by associate editor B. Wollenberg. tMiddle East Technical University, Department of Electrical Engineering, Ankara, Turkey. 45
RATE o f
46
A. DURMAZ turers, plant operators and control system designers is required. The mathematical models derived in this paper can also be utilized for the investigation of the dynamic behaviour of the existing measurement methods for thermal stresses or for the development of the new measurement methods, which are appropriate for on-line or off-line stress calculations (Durmaz, 1974).
Po
I/I w
u=!
Steam
t____ ~ 1 ofDynamics materiat Dynamics states Generator
t Controtter
I-I¢i. 1. Extension
t]
of steam generator control by considering material states.
the material state dynamics is coupled to the working fluid side and the controller side by the operator blocks KG1 and KG2, respectively. K G I gives the heat transfer coefficient c~D and KG2 gives the resultant stress at, Strains and stresses are caused by thermal effects and the other effects due to working fluid pressure and external loads. The material strains and stresses due to the two types of effects are superimposed as c = ~:w+ %, a = c%,+ crp, where the subscripts w and p refer to the thermal effects and internal pressure effects, respectively. In this study, these model extensions are explained by considering a thick wall tube that is completely insulated from the surroundings. This is admissible idealization for the parts which are generally agreed to be the most critical ones, namely the steam headers which operate especially in the high steam temperature range. The material stresses due to the steam pressure are of proportional-behaviour. In the following sections the dynamic behaviour, particularly of material states due to the thermal stresses is investigated by means of the developed mathematical models. These models are useful for the design of control systems and for the economic guidance of nonstationary power plant operations. One of the main problems of stress control system design and its effective utilization is the determination of the critical points and the change of critical sections in the whole power plant. This is particular to every power plant and for this purpose a close cooperation between power plant manufac-
",.hDff)
.
Ctu(tl
2. BASIC EQUATIONS FOR INVESTIGATING THE DYNAMIC BEHAVIOUR OF MATERIAL STATES
2.1. Basic equations for material temperatures The general expression for heat transfer in an isotropic and homogeneous body is givetn by 90
~
c(O)p(O)~t-4(O)dlv(gradO)=O.
(1)
The dimensions of the tube wall geometry as well as the steam mass flow rate and the physical properties of the working fluid are assumed to be constant over the length of the tube element. The axial heat conduction in the wall can be neglected, since the steam temperature is nearly constant along the header. Then, the temperature changes of the tube wall in the radial direction are given by
~O(r,t)
,,~,[t'~ZO(r,t) +l t?O(r,t)]
(2)
with the boundary conditions -2(0)
gO(r, , ~ t) ] =%['%(t)-O(r,t) r=r
i
(2a) r=r
on the inside surface of the tube, and
), 0 ?O(r. t) -~() ~ I =0 ('F
12b)
r-r
on the outside surface (0,.,=0 since outside wall is completely isolated). 2.2. Basic equation ,/'or thermal stresses The thermal stresses in the principal directions
I
e I,.,L
,,
~'(r.t}
qv~ FIG. 2. Block diagram for investigating the dynamic behaviour of material states of a thick wall tube with the coupling elements KG1 and KG2 on the steam side and controller side.
Dynamic behaviour of material states in steam power plant control for a tube wall are given as follows (Parkus, 1959):
47
3. INVESTIGATION OF THE DYNAMIC BEHAVIOUR OF THERMAL STRESSES FOR SMALL TEMPERATURE AND LOAD CHANGES
in the axial direction
3.1. Derivation of the transfer functions for ther-
mal stresses
vE
~Ar, t)=i-s~[O.,(t)-O(r,t)]
Linearization of equations (2)-(7) gives the linear equations which describe the dynamic behaviour of the material temperature and thermal stresses for small variations of steam temperature and mass flow rate. The block diagram in Fig. 3 shows how the dynamics of thermal stresses are coupled with the steam generator dynamics and the main controllers. The transfer functions, which represent various parts of the mathematical model, are derived from these linear equations through Laplace transformation. The transfer functions for investigation of the dynamic behaviour of unsteady state temperature fields in cylindrical structures are treated in the literature (Schittke and Durmaz, 1974). The transfer functions for the significant temperatures O(r, t), Ore(t), O,,~(t) of Fig. 3 are given in Table 1 where I 0, It are Bessel functions and K0, K1 are Neuman functions. The remaining transfer functions of Fig. 3 are of proportional type with the following gains:
(3)
in the peripheral directions
vE
r~
+(1
r~
l
OIr
I4,
and in the radial direction vE
~,Ar, t)=~(i~_#) 1-7~ [oAt)-omAt)] (5) where Ore(t) and Om,(t) represent mean temperatures over the tube cross-section, and are given by
Ore(t ) - 2 2 r~ if" O(r,t)rdr r a -ri 2
O(r, Ordr.
(6) (7)
F(r, s)=F(r, s)=F(r, s)= The changes of material properties 2, p, c, a, E, #, v of boiler materials are functions of temperature (Pich, 1964).
Workin 9
f I.uid
dynamics
F(r,s)=F(r,s)=F(r,s) Orelr~qjw
Thermot
stress
Q{r)
_r7~
1~- o.I
om I
Omr~arw
vE 2(1-#)\
dynamics
_ _ _
Oo
Om~r w
~ .
,.
= ~ -1oo- ~,.I
O'zw I t |
0'~ow(r)
.-
vE
2
FIG. 3. Block diagram showing the coupling of the steam generator dynamics with the thermal stress dynamics for small temperature and load changes.
(1
(8)
r~. r2j (9)
48
A. DURMAZ TABLE l.
TRANSFER I~UNCIIONSOF ]'HE SIGNIFICANF FEMPERATURES FOR CALCULATINGTHE IHERMAL STRESSES
F'r "'= lo(flr )K , (fir~)~ l ~ (~r~)Ko(flr ! :}," }~J C, [ l , (flr, )K , ( f l r . ) - l , (flr~)K , (flr~ )] + lo(flr,)K , {flr.) + I i {flr,i )Ko{flr,) C2 [1, (fir,)K, (fir.) - l ~(fir.)K 1 (flrl)]
fi(:],) = c f)i(/~rlK;il~r;,)Zll(fir.-iK;(l~r )] +}O(flr~~(flrl,)+ll(f~r.)Ko(flri) F¢r s* (-'3 [K' (flr")[II (flr~) (r/'r~)l, ( f i r ) ] + / ! ( f i r . ) [ K , (flrt)+ [r/r,)K,lflr)]j ",i .'o'.~- C, [I, (flr~)K ~ (fir.) - I 1 (flr.)K ~(fir,)] + lo(flr~)K , ([Jr.)+ I, (fir.)Ko(flr ~) where
l]=
Fir, s ) 0,,~%,.
~'a
1+
2(1 -/~)
, C,=
r2J
2ri 2ri fi)4"~D' C2=-fl(r,,-r,)2 7~, C3=fl(r2Lr,.2 )"
(50)
The transfer functions for the principal stresses in the tube wall due to the pressure of the working fluid are given in Table 2. TABLE 2.
TRANSFER FUNCTIONS OF MATERIAL STRESSES DUE TO THE WORKING FLUID PRESSUREPD
F(r.s)=T.
1
i7
the inner surface is represented in Fig. 6. Here one can see the strong influence of the heat transfer coefficient ~ on the thermal stresses. The thermal stresses are increasing with :% and thus the Steam mass flow rate ~/D. Figure 7 shows the relationship between fuel mass flow rate ~/B and the thermal stresses on the inner surface. The relationship between steam mass flow rate and thermal stresses is shown in Fig. 8. It is interesting that the thermal stresses are small for very low and very high frequencies. This means that fast load changes do not necessarily cause high thermal stresses.
1
Fir, s ) = w/-)i%;'"~Zi [1 -(r./r)z] Pl) +°,p
4. C A L C U L A T I O N
OF
THERMAL
STRESSES
FOR
LARGE TEMPERATURE AND LOAD CttANGES
3.2. Evaluation
of trans['er functions cussion of some of the results
and dis-
The following examples are concerned with the outlet header of the final superheater section of the steam generator in a heating power plant (Durmaz, 1970). The thermal stress dynamics are shown in Fig. 4 by Nyquist plots where R = ( r - r i ) / ( r , - r i ) . 5001~0 is the relative distance from the inside wall of the tube. It is obvious that the Nyquist plots are strongly dependent on R. The maximum values of thermal stresses are found at the inside wall (R =00,,). The inner surface is the critical part of the tube wall, therefore the dynamic behaviour of thermal stresses should mainly be investigated there. The thermal stresses also depend strongly on the frequency (o of the input variable, and hence on the rate of change of steam temperature
The thermal stresses are of major interest for large load variations, i.e. when large temperature variations are to be expected. In this case, the nonlinearities of the material properties in the basic equations have to be considered. It is obvious that the solution of this problem calls for numerical methods rather than for an analytical treatment. Therefore, equation (2) is discretized as follows in a general way (Richtmayer and Morton, 5967): O j: k + 1 -- (Jj: k = ~1( 0 )
At
Ar e ×[~(b20)j,k+i+(5--U,)(a20)j,t]
(11)
where (~20b.k)=
5+ i:
0; l,k--2Oj, k
~)1i).
The amplitudes of thermal stresses are presented in Fig. 5 as functions of the frequency /,~ of the steam temperature and relative distance R. As is seen in this figure, there is a definite range of frequency, e.g. 0.006<(o<0.4, where the thermal stress amplitudes are especially large. The dynamic behaviour of thermal stresses at
+ 1-
Oj+l.k.
(12t
The coefficient ~ is bounded as 0 < ~ < 1. For the problem treated here, an optimal value for ~" was found to be ~=0.75 as far as convergence,
l'O:
D y n a m i c b e h a v i o u r o f m a t e r i a l s t a t e s in s t e a m p o w e r p l a n t c o n t r o l
]
bJ ooo/. "o~
/
~
~ /
-0.2
/
~._~'~'>~
~
I
~
~.,r.~
. ~
~ i ' / ~
~,
i
r - 4 - o~
RE
-0.06 " -- -- ~'1~ OJ~3
0.00
-0.1 FIG. 4. Nyquist plots for the thermal stress transfer function
F(R, s), (~o = 3000 kcal/m 2 h °C)
I dzwIR) Oo OJ5
0.1
so /, 0.0~ 2O i
o£)ooi
10
o.ooi
o~)I
o,1
~,o
1o.o
~[{-]
FIG. 5. Magnitude curves for the therma! stress frequency responses ( % = 3000 k c a l / m 2 h °C). IN
O,OS
ag-
-og$
-o.1
o,ol
ot o • oo
FIG. 6. N y q u i s t plots for the t h e r m a l stress t r a n s f e r f u n c t i o n
F(rl, s). (% kcal/m 2 h "C as parameter.)
49
50
A. DURMAZ
?
)-
[,,,,,2,,,1 m31h J
I
""'•0.02
\ l
-1
-2
\ I
/ /
J
! L m31h J
-1 .9
f-'l(;. 7. Nyquist plots for the thermal stress transfer function F.(r. s). (Fuel mass flow rate as input variable, 60% load.)
further increased by the thermal stresses themselves. In state variable notation, equation (2), when discretized by only the spatial variable, reads (Durmaz, 1974)
IM
kp/mm2] t-ST~-J - -
0.002
0.05'
O(t)=AO(t)+g(t)
, 0I'004
o,~ool 0,05
0,1
RE
0'2l
)q l
0,04
[kp/m~] ,-77g--J
/
---0.05
0,013
J
I"lc;. 8. Nyquist plots for the thermal stress transfer function
F(ri,s).
(Steam mass flow rate as input variables, 60 ~i [oad.)
stability and computing time are concerned. With this method, the tube wall is divided into n(j = 1. . . . . n) concentric tubes of thickness Ar. The number of time steps is denoted by k. F r o m the viewpoint of control theory the temperatures in the concentric tubes can be regarded as n additional state variables by which the steam generator dynamics are augmented as indicated in Fig. 1. Note that according to equations (2), (3) and (4) the principal stresses are derived directly from these temperatures, and thus the system in Fig. 1 contains only n additional states. The n u m b e r of states is not
(13)
where 0 is an n-dimensional state vector, A is a n x n three diagonal matrix with the elements depending on the discretization method and g is a n-dimensional vector depending on the boundary conditions. It should be mentioned, however, that equation (13) represents, in general, a nonlinear system, since the matrix A and the vector g depend on 0. The discretization method can also be extended to two and three dimensional heat transfer cases and to the structures other than cylindrical tubes.
5. SIMULATION OF THE DYNAMIC BEHAVIOUR OF THE MATERIAL STATE VARIABLES Up to now, only the development of mathematical models for describing the dynamic behaviour of material state variables (temperature and thermal stresses) has been discussed. The transfer functions represent the mathematical model for these material state variables for small temperature and load changes. The treatment of material temperature and thermal stresses in steam power plant control could only be accomplished by simulation on analog or digital computers.
Dynamic behaviour of material states in steam power plant control 5.1. Simulation of the mathematical models for small temperature and load changes
elements are simulated on the digital computer, can be used (Herbik and Voss, 1969). Along with the simplicity in the programming and error determination, these block oriented programming languages have the advantage of no scaling problems. A disadvantage may be in the demand of too large a computer time which is especially important in optimization problems (Unbehauen and Herbrik, 1970). The simulation of (15) entails the solution of a system of differential equations. This has been done and the step responses of the temperatures O(r~, t), Ore(t) and the thermal stress tTzw(ri, t) are given in Fig. 10.
The transfer functions given in Fig. 3 for describing the dynamic behaviour of material temperatures and thermal stresses for small temperature and load changes are of transcendental type. These transfer functions are not appropriate for simulation purposes. Therefore, they should be approximated by a rational fraction function in the form of
F*(r, s ) =
Z 0 -~-Z 1 s -J- Z 2 $2 -~- ... -~-Zm Sm
No+N1s+N2s2+...
+s n
(14)
with m < n (Unbehauen, 1966; Strobel, 1968). The rational fraction approximation of frequency response plot of the transfer function F(rl, s)o~ . . . . is represented in Fig. 9 (Strobel, 1968). Equation (14) can easily be programmed on the analog computer. Taking the integrator outputs xi in the analog computer block diagram as the new set of state variables (14) can also be converted into the state space representation as
~(t)=Ax(t)+Bu(t).
5.2. Simulation of the discretized nonlinear model
for large temperature and load variations The dynamic behaviour of material temperature and thus the thermal stresses for large temperature and load changes were described in the form of a state space corresponding to (13). Therefore, the simulation involves the solution of a system of nonlinear differential equations. The system matrix A and vector g depend on material temperatures and boundary conditions. Since the system is nonlinear, matrix A and vector g must be determined at every step. The discretized nonlinear model is tested for a step change in steam temperature of A0=25°C. The step response for temperatures O(rl, t), O,,(t) and thermal stress ~r~w(ri,t) are compared with the results of linear model.
(15)
The analog simulation is limited by the availability of computer demands. For very complex multivariable control systems found in steam generators, the block oriented digital programming language, such as A N A L O G 66, MIMIC, where the functions of analog computer
IM kp,om, °C J O,08
0.0,5
Exact
frequency
response O 0
Approximated by using a rational fraction transfer function x x It RE [ k p l m m 2]
L" - ~ J
0
.
0
51
~
~"1 r, SI = ',~O~ZW-.
Zo'Z's *Zzs 2 . . . . . Z6se N0*NlS *N2s:~* *Nss6
-0.05
8.17555.10 -~ NI = 2,705t.6.104
NO =
Z 0 = 1.24814.1(~ s
71 •-2.73282
N3 = 2.16076 103
Z:) = - 3 . 4 8 2 6 0 101
N~ = 1.23140 10j
Z~ • - 5 . 4 8 2 2 2 - 1 0 1
Ns = 1.00220 10~
Zs : °4.1063/.. 101
N G = 1,00000
Z 6 •-1.O9380
FIG. 9. Approximation of the thermal stress frequency response
F(rl, s) by a rational fraction transfer function F*(rz, s).
103
52
A. DURMAZ Om
Otri) I*Cl 25
II
1
f
20 15 10 5
5
Oz.(ri) -1
10
15
~
20
~min_]
t
1
-3 FIG. 10. Responses of temperatures 0(r,,t), O,,,(t) and thermal stress cr:,.(r~,t) for a step change in steam temperature 90 ( - calculated by linear model, , calculated by discretized nonlinear model).
6. CONCLUSIONS
The limitations of the admissible rate of load change in steam power plants are mainly imposed by material states especially due to thermal stresses in the thick wall parts of the steam generator and turbine. These thermal stresses can, however, be evaluated as functions of main controls. Thus an optimal control strategy can be derived with respect to the limitations. A method of simulating and investigating the dynamic behaviours of material temperatures and thermal stresses in the steam power plant control is presented in this study. The mathematical models derived for thermal stresses can also be used for the development and testing of effective measurement methods for thermal stresses. These models can also be used for the development of simulators to be used for the training of power plant operators.
REFERENCES Baehr, H. D. (1955). Die ~ung nichstationaerer Warmeleitungs-probleme mit Hilfe der Laplace Transformation. Forschung lng.-Wesen 21, (2), 33 40. Dettinger, R., E. Welfonder und R. Herbrik (1974). Verbesserung des Regelverhaltens von Dampferzeugern dutch strucktur-optimierte Regelung. Brennst.-Waerme-K rq# 26, {2), 43 54. Durmaz, A. (1970). Messungen mit Hilfe der elektronischen
Datenerfassung-sanlage am Kessel 6 im HKW-pfaffenwald der Universitaet Stuttgart und Auswertung der Messdaten auf dem Digitalrechner. Diplomarbeit am IVD der Universitaet Stuttgart. Durmaz, A. (1973). Regelkreisoptimierung eines im Gleitdruck betreiebenen Bensonkessels. Vergleich der theoretischen und experimentellen Ergebnisse. Energie 25, 213 216. Durmaz, A. (1974). Zur Berechnung der zeitlichen Aenderung der Werkstoff-beanspruchung bei Betriebszustandsaenderungen yon Dampferzeugern. Dissertation, Universitaet Stuttgart. Herbrik, R. and K. Voss (1969), Digitale Simulateion der Regelsyteme von Durchlaufdampferzeugern mit Hilfe blockorientierter Programmiersprachen. Brennst,-Waerme-Krq/'t 21, 13 16. Parkus, H. (1959). lnstationaere Waermespannungen. SpringerVerlag, Wien. Pich, R. (1964). Die Berechnung der elastischen instationaeren Waermespannungen in Platten, Hohlzylindern und Hohlkegeln mit quasisitationaeren Temperaturfeldern. VGB-Mitt. 88, 53 60. Richtmayer, R. D. and K. W. Morton (1967). D!/]erence Method for Initial Value Problems 2nd edition. Interscience, New York. Schittke, H. J. and A. Durmaz (1974). Berechung des dynamischen Verhaltens yon instationaeren Temperaturfeldern in zylindrischen Beuteilen. Waerme- und Stq[li'ibertragung 7, 121 132. Stoll, A. E. Haehle and A. Fischer (19681. Materialgercchter Betrieb grosser Dampferzeuger mit Hilfe eincs Freilastrechners Siemes- Z. 42, (2), 112 116. Strobel, H. (1968). Systemanalyse mit Determmiertetl Testsignalen. Verlag Technik, Berlin. Unbehauen, H. and R. Herbrik (1970). Identification, simulation and optimization of control systems in steam boiler plants. IFAC-Symposium on System Engineering Approach to Computer Control, Kyoto, Paper 9.2. Unbehauen, R. (1966). Ermittlung rationaler Frequenzgaenge aus Messwerten. Regelungstechnik 14, 268 273.
0005-1098,,80/0101-0045$02.00/0
Dynamic Behaviour of Material States in Steam Power Plant Control* A. D U R M A Z t
Mathematical models, developed for the investigation of the dynamic behaviour of material state variables (temperature, strain, stress), indicate that the maximum allowable values of these variables should be approached and maintained in nonstationary operation of steam power plants. Key Words--Control system analysis; computer-aided designs; stress control; material states dynamics; multivariable control system; steam generator; nonstationary operation control; identification; modelling; frequency response; simulation.
Almtract.--The enormous price increase in petroleum and the limited reserves of other fossil fuels have resulted in the increased use of nuclear power plants for the basic load of electricity supply. Consequently, conventional steam power plants have been used to meet the variable and peak loads in electricity generation. The control of fast and large load changes is becoming more and more important for conventional power plant operations. An economical control of the nonstationary operation (start-up, shut-down and large load changes) is possible only by approaching and maintaining the m a x i m u m allowable values of the material state variables (temperature 0, strain ~ and stress a) at the critical points of the power plant components. To achieve this primary control goal, the dynamic behaviour of material states must be well understood. A direct measurement of the changes in material states especially due to thermal stresses, caused by temperature gradients, is not possible. The material states can be calculated, however, by means of mathematical models describing the nonstationary temperature distribution. The working fluid temperature, pressure and mass flow rate are the input variables for these mathematical models. This paper presents a linear mathematical model for the investigation of the dynamic behaviour of material states. Another mathematical model is also developed for calculating the thermal stresses for large temperature and load changes. Thus, the steam generator models so far describing only the states of the working fluid can be extended to the material states. The basic idea is demonstrated for an insulated thick wall tube containing a working fluid. The simulation of the mathematical models is performed. a E F Mn Mo ~/E P PD r ri r,
q,. t cto # v oaD 0 a
heat loss at the outside wall time heat transfer coefficient (steam-tube wall) Poisson's ratio linear thermal coefficient of expansion steam temperature material temperature (tube wall temperature) material strain material stress
Indices a B D i
outside fuel steam inside
m p r v
mean pressure radial resultant
w thermal z axial q~ peripheral a stress
1. I N T R O D U C T I O N
load change in steam power plants is limited by material states, especially due to thermal stresses occurring in the thick wall elements during the transient state of steam generator and turbine. The material temperature distribution 0 and thus the thermal strain ew and thermal stresses a w in the critical parts are dependent mainly on the steam temperature and the heat transfer coefficient, i.e. the steam mass flow rate. The steam generator models that are used in a number of publications, such as the ones for optimization of controller settings (Durmaz, 1973) and driving an optimal control policy for load changes (Dettinger, Wolfonder and Herbrik, 1974) can be extended as shown in Fig. 1 to take care of the limitations imposed by the material state variables such as temperatures 0, strains and stresses a. For achieving this goal, the dynamic behaviour of material states, which depends on the dynamics of the working fluid, must be known. The extension of the dynamics of working fluid variables, such as the steam temperature 0o, steam mass flow rate )f/o and the steam pressure PD, to the dynamics of material states in represented in Fig. 2. The mathematical model used for investigating THE
NOMENCLATURE thermal diffusivity (a=2/pc) modulus of elasticity transfer function fuel mass flow rate steam mass flow rate spray water mass flow rate electric power steam pressure spatial variable in radial direction tube inside radius tube outside radius
*Received 19 October 1978; revised 25 June 1979. The original version of this paper was presented at the 7th IFAC Congress on A Link Between Science and Application of Automatic Control which was held in Helsinki, Finland during June 1978. The published Proceedings of this IFAC Meeting may be ordered from: Pergamon Press Ltd, Headington Hill Hall, Oxford OX3 0BW. U.K. This paper was recommended for publication in revised form by associate editor B. Wollenberg. tMiddle East Technical University, Department of Electrical Engineering, Ankara, Turkey. 45
RATE o f
46
A. DURMAZ turers, plant operators and control system designers is required. The mathematical models derived in this paper can also be utilized for the investigation of the dynamic behaviour of the existing measurement methods for thermal stresses or for the development of the new measurement methods, which are appropriate for on-line or off-line stress calculations (Durmaz, 1974).
Po
I/I w
u=!
Steam
t____ ~ 1 ofDynamics materiat Dynamics states Generator
t Controtter
I-I¢i. 1. Extension
t]
of steam generator control by considering material states.
the material state dynamics is coupled to the working fluid side and the controller side by the operator blocks KG1 and KG2, respectively. K G I gives the heat transfer coefficient c~D and KG2 gives the resultant stress at, Strains and stresses are caused by thermal effects and the other effects due to working fluid pressure and external loads. The material strains and stresses due to the two types of effects are superimposed as c = ~:w+ %, a = c%,+ crp, where the subscripts w and p refer to the thermal effects and internal pressure effects, respectively. In this study, these model extensions are explained by considering a thick wall tube that is completely insulated from the surroundings. This is admissible idealization for the parts which are generally agreed to be the most critical ones, namely the steam headers which operate especially in the high steam temperature range. The material stresses due to the steam pressure are of proportional-behaviour. In the following sections the dynamic behaviour, particularly of material states due to the thermal stresses is investigated by means of the developed mathematical models. These models are useful for the design of control systems and for the economic guidance of nonstationary power plant operations. One of the main problems of stress control system design and its effective utilization is the determination of the critical points and the change of critical sections in the whole power plant. This is particular to every power plant and for this purpose a close cooperation between power plant manufac-
",.hDff)
.
Ctu(tl
2. BASIC EQUATIONS FOR INVESTIGATING THE DYNAMIC BEHAVIOUR OF MATERIAL STATES
2.1. Basic equations for material temperatures The general expression for heat transfer in an isotropic and homogeneous body is givetn by 90
~
c(O)p(O)~t-4(O)dlv(gradO)=O.
(1)
The dimensions of the tube wall geometry as well as the steam mass flow rate and the physical properties of the working fluid are assumed to be constant over the length of the tube element. The axial heat conduction in the wall can be neglected, since the steam temperature is nearly constant along the header. Then, the temperature changes of the tube wall in the radial direction are given by
~O(r,t)
,,~,[t'~ZO(r,t) +l t?O(r,t)]
(2)
with the boundary conditions -2(0)
gO(r, , ~ t) ] =%['%(t)-O(r,t) r=r
i
(2a) r=r
on the inside surface of the tube, and
), 0 ?O(r. t) -~() ~ I =0 ('F
12b)
r-r
on the outside surface (0,.,=0 since outside wall is completely isolated). 2.2. Basic equation ,/'or thermal stresses The thermal stresses in the principal directions
I
e I,.,L
,,
~'(r.t}
qv~ FIG. 2. Block diagram for investigating the dynamic behaviour of material states of a thick wall tube with the coupling elements KG1 and KG2 on the steam side and controller side.
Dynamic behaviour of material states in steam power plant control for a tube wall are given as follows (Parkus, 1959):
47
3. INVESTIGATION OF THE DYNAMIC BEHAVIOUR OF THERMAL STRESSES FOR SMALL TEMPERATURE AND LOAD CHANGES
in the axial direction
3.1. Derivation of the transfer functions for ther-
mal stresses
vE
~Ar, t)=i-s~[O.,(t)-O(r,t)]
Linearization of equations (2)-(7) gives the linear equations which describe the dynamic behaviour of the material temperature and thermal stresses for small variations of steam temperature and mass flow rate. The block diagram in Fig. 3 shows how the dynamics of thermal stresses are coupled with the steam generator dynamics and the main controllers. The transfer functions, which represent various parts of the mathematical model, are derived from these linear equations through Laplace transformation. The transfer functions for investigation of the dynamic behaviour of unsteady state temperature fields in cylindrical structures are treated in the literature (Schittke and Durmaz, 1974). The transfer functions for the significant temperatures O(r, t), Ore(t), O,,~(t) of Fig. 3 are given in Table 1 where I 0, It are Bessel functions and K0, K1 are Neuman functions. The remaining transfer functions of Fig. 3 are of proportional type with the following gains:
(3)
in the peripheral directions
vE
r~
+(1
r~
l
OIr
I4,
and in the radial direction vE
~,Ar, t)=~(i~_#) 1-7~ [oAt)-omAt)] (5) where Ore(t) and Om,(t) represent mean temperatures over the tube cross-section, and are given by
Ore(t ) - 2 2 r~ if" O(r,t)rdr r a -ri 2
O(r, Ordr.
(6) (7)
F(r, s)=F(r, s)=F(r, s)= The changes of material properties 2, p, c, a, E, #, v of boiler materials are functions of temperature (Pich, 1964).
Workin 9
f I.uid
dynamics
F(r,s)=F(r,s)=F(r,s) Orelr~qjw
Thermot
stress
Q{r)
_r7~
1~- o.I
om I
Omr~arw
vE 2(1-#)\
dynamics
_ _ _
Oo
Om~r w
~ .
,.
= ~ -1oo- ~,.I
O'zw I t |
0'~ow(r)
.-
vE
2
FIG. 3. Block diagram showing the coupling of the steam generator dynamics with the thermal stress dynamics for small temperature and load changes.
(1
(8)
r~. r2j (9)
48
A. DURMAZ TABLE l.
TRANSFER I~UNCIIONSOF ]'HE SIGNIFICANF FEMPERATURES FOR CALCULATINGTHE IHERMAL STRESSES
F'r "'= lo(flr )K , (fir~)~ l ~ (~r~)Ko(flr ! :}," }~J C, [ l , (flr, )K , ( f l r . ) - l , (flr~)K , (flr~ )] + lo(flr,)K , {flr.) + I i {flr,i )Ko{flr,) C2 [1, (fir,)K, (fir.) - l ~(fir.)K 1 (flrl)]
fi(:],) = c f)i(/~rlK;il~r;,)Zll(fir.-iK;(l~r )] +}O(flr~~(flrl,)+ll(f~r.)Ko(flri) F¢r s* (-'3 [K' (flr")[II (flr~) (r/'r~)l, ( f i r ) ] + / ! ( f i r . ) [ K , (flrt)+ [r/r,)K,lflr)]j ",i .'o'.~- C, [I, (flr~)K ~ (fir.) - I 1 (flr.)K ~(fir,)] + lo(flr~)K , ([Jr.)+ I, (fir.)Ko(flr ~) where
l]=
Fir, s ) 0,,~%,.
~'a
1+
2(1 -/~)
, C,=
r2J
2ri 2ri fi)4"~D' C2=-fl(r,,-r,)2 7~, C3=fl(r2Lr,.2 )"
(50)
The transfer functions for the principal stresses in the tube wall due to the pressure of the working fluid are given in Table 2. TABLE 2.
TRANSFER FUNCTIONS OF MATERIAL STRESSES DUE TO THE WORKING FLUID PRESSUREPD
F(r.s)=T.
1
i7
the inner surface is represented in Fig. 6. Here one can see the strong influence of the heat transfer coefficient ~ on the thermal stresses. The thermal stresses are increasing with :% and thus the Steam mass flow rate ~/D. Figure 7 shows the relationship between fuel mass flow rate ~/B and the thermal stresses on the inner surface. The relationship between steam mass flow rate and thermal stresses is shown in Fig. 8. It is interesting that the thermal stresses are small for very low and very high frequencies. This means that fast load changes do not necessarily cause high thermal stresses.
1
Fir, s ) = w/-)i%;'"~Zi [1 -(r./r)z] Pl) +°,p
4. C A L C U L A T I O N
OF
THERMAL
STRESSES
FOR
LARGE TEMPERATURE AND LOAD CttANGES
3.2. Evaluation
of trans['er functions cussion of some of the results
and dis-
The following examples are concerned with the outlet header of the final superheater section of the steam generator in a heating power plant (Durmaz, 1970). The thermal stress dynamics are shown in Fig. 4 by Nyquist plots where R = ( r - r i ) / ( r , - r i ) . 5001~0 is the relative distance from the inside wall of the tube. It is obvious that the Nyquist plots are strongly dependent on R. The maximum values of thermal stresses are found at the inside wall (R =00,,). The inner surface is the critical part of the tube wall, therefore the dynamic behaviour of thermal stresses should mainly be investigated there. The thermal stresses also depend strongly on the frequency (o of the input variable, and hence on the rate of change of steam temperature
The thermal stresses are of major interest for large load variations, i.e. when large temperature variations are to be expected. In this case, the nonlinearities of the material properties in the basic equations have to be considered. It is obvious that the solution of this problem calls for numerical methods rather than for an analytical treatment. Therefore, equation (2) is discretized as follows in a general way (Richtmayer and Morton, 5967): O j: k + 1 -- (Jj: k = ~1( 0 )
At
Ar e ×[~(b20)j,k+i+(5--U,)(a20)j,t]
(11)
where (~20b.k)=
5+ i:
0; l,k--2Oj, k
~)1i).
The amplitudes of thermal stresses are presented in Fig. 5 as functions of the frequency /,~ of the steam temperature and relative distance R. As is seen in this figure, there is a definite range of frequency, e.g. 0.006<(o<0.4, where the thermal stress amplitudes are especially large. The dynamic behaviour of thermal stresses at
+ 1-
Oj+l.k.
(12t
The coefficient ~ is bounded as 0 < ~ < 1. For the problem treated here, an optimal value for ~" was found to be ~=0.75 as far as convergence,
l'O:
D y n a m i c b e h a v i o u r o f m a t e r i a l s t a t e s in s t e a m p o w e r p l a n t c o n t r o l
]
bJ ooo/. "o~
/
~
~ /
-0.2
/
~._~'~'>~
~
I
~
~.,r.~
. ~
~ i ' / ~
~,
i
r - 4 - o~
RE
-0.06 " -- -- ~'1~ OJ~3
0.00
-0.1 FIG. 4. Nyquist plots for the thermal stress transfer function
F(R, s), (~o = 3000 kcal/m 2 h °C)
I dzwIR) Oo OJ5
0.1
so /, 0.0~ 2O i
o£)ooi
10
o.ooi
o~)I
o,1
~,o
1o.o
~[{-]
FIG. 5. Magnitude curves for the therma! stress frequency responses ( % = 3000 k c a l / m 2 h °C). IN
O,OS
ag-
-og$
-o.1
o,ol
ot o • oo
FIG. 6. N y q u i s t plots for the t h e r m a l stress t r a n s f e r f u n c t i o n
F(rl, s). (% kcal/m 2 h "C as parameter.)
49
50
A. DURMAZ
?
)-
[,,,,,2,,,1 m31h J
I
""'•0.02
\ l
-1
-2
\ I
/ /
J
! L m31h J
-1 .9
f-'l(;. 7. Nyquist plots for the thermal stress transfer function F.(r. s). (Fuel mass flow rate as input variable, 60% load.)
further increased by the thermal stresses themselves. In state variable notation, equation (2), when discretized by only the spatial variable, reads (Durmaz, 1974)
IM
kp/mm2] t-ST~-J - -
0.002
0.05'
O(t)=AO(t)+g(t)
, 0I'004
o,~ool 0,05
0,1
RE
0'2l
)q l
0,04
[kp/m~] ,-77g--J
/
---0.05
0,013
J
I"lc;. 8. Nyquist plots for the thermal stress transfer function
F(ri,s).
(Steam mass flow rate as input variables, 60 ~i [oad.)
stability and computing time are concerned. With this method, the tube wall is divided into n(j = 1. . . . . n) concentric tubes of thickness Ar. The number of time steps is denoted by k. F r o m the viewpoint of control theory the temperatures in the concentric tubes can be regarded as n additional state variables by which the steam generator dynamics are augmented as indicated in Fig. 1. Note that according to equations (2), (3) and (4) the principal stresses are derived directly from these temperatures, and thus the system in Fig. 1 contains only n additional states. The n u m b e r of states is not
(13)
where 0 is an n-dimensional state vector, A is a n x n three diagonal matrix with the elements depending on the discretization method and g is a n-dimensional vector depending on the boundary conditions. It should be mentioned, however, that equation (13) represents, in general, a nonlinear system, since the matrix A and the vector g depend on 0. The discretization method can also be extended to two and three dimensional heat transfer cases and to the structures other than cylindrical tubes.
5. SIMULATION OF THE DYNAMIC BEHAVIOUR OF THE MATERIAL STATE VARIABLES Up to now, only the development of mathematical models for describing the dynamic behaviour of material state variables (temperature and thermal stresses) has been discussed. The transfer functions represent the mathematical model for these material state variables for small temperature and load changes. The treatment of material temperature and thermal stresses in steam power plant control could only be accomplished by simulation on analog or digital computers.
Dynamic behaviour of material states in steam power plant control 5.1. Simulation of the mathematical models for small temperature and load changes
elements are simulated on the digital computer, can be used (Herbik and Voss, 1969). Along with the simplicity in the programming and error determination, these block oriented programming languages have the advantage of no scaling problems. A disadvantage may be in the demand of too large a computer time which is especially important in optimization problems (Unbehauen and Herbrik, 1970). The simulation of (15) entails the solution of a system of differential equations. This has been done and the step responses of the temperatures O(r~, t), Ore(t) and the thermal stress tTzw(ri, t) are given in Fig. 10.
The transfer functions given in Fig. 3 for describing the dynamic behaviour of material temperatures and thermal stresses for small temperature and load changes are of transcendental type. These transfer functions are not appropriate for simulation purposes. Therefore, they should be approximated by a rational fraction function in the form of
F*(r, s ) =
Z 0 -~-Z 1 s -J- Z 2 $2 -~- ... -~-Zm Sm
No+N1s+N2s2+...
+s n
(14)
with m < n (Unbehauen, 1966; Strobel, 1968). The rational fraction approximation of frequency response plot of the transfer function F(rl, s)o~ . . . . is represented in Fig. 9 (Strobel, 1968). Equation (14) can easily be programmed on the analog computer. Taking the integrator outputs xi in the analog computer block diagram as the new set of state variables (14) can also be converted into the state space representation as
~(t)=Ax(t)+Bu(t).
5.2. Simulation of the discretized nonlinear model
for large temperature and load variations The dynamic behaviour of material temperature and thus the thermal stresses for large temperature and load changes were described in the form of a state space corresponding to (13). Therefore, the simulation involves the solution of a system of nonlinear differential equations. The system matrix A and vector g depend on material temperatures and boundary conditions. Since the system is nonlinear, matrix A and vector g must be determined at every step. The discretized nonlinear model is tested for a step change in steam temperature of A0=25°C. The step response for temperatures O(rl, t), O,,(t) and thermal stress ~r~w(ri,t) are compared with the results of linear model.
(15)
The analog simulation is limited by the availability of computer demands. For very complex multivariable control systems found in steam generators, the block oriented digital programming language, such as A N A L O G 66, MIMIC, where the functions of analog computer
IM kp,om, °C J O,08
0.0,5
Exact
frequency
response O 0
Approximated by using a rational fraction transfer function x x It RE [ k p l m m 2]
L" - ~ J
0
.
0
51
~
~"1 r, SI = ',~O~ZW-.
Zo'Z's *Zzs 2 . . . . . Z6se N0*NlS *N2s:~* *Nss6
-0.05
8.17555.10 -~ NI = 2,705t.6.104
NO =
Z 0 = 1.24814.1(~ s
71 •-2.73282
N3 = 2.16076 103
Z:) = - 3 . 4 8 2 6 0 101
N~ = 1.23140 10j
Z~ • - 5 . 4 8 2 2 2 - 1 0 1
Ns = 1.00220 10~
Zs : °4.1063/.. 101
N G = 1,00000
Z 6 •-1.O9380
FIG. 9. Approximation of the thermal stress frequency response
F(rl, s) by a rational fraction transfer function F*(rz, s).
103
52
A. DURMAZ Om
Otri) I*Cl 25
II
1
f
20 15 10 5
5
Oz.(ri) -1
10
15
~
20
~min_]
t
1
-3 FIG. 10. Responses of temperatures 0(r,,t), O,,,(t) and thermal stress cr:,.(r~,t) for a step change in steam temperature 90 ( - calculated by linear model, , calculated by discretized nonlinear model).
6. CONCLUSIONS
The limitations of the admissible rate of load change in steam power plants are mainly imposed by material states especially due to thermal stresses in the thick wall parts of the steam generator and turbine. These thermal stresses can, however, be evaluated as functions of main controls. Thus an optimal control strategy can be derived with respect to the limitations. A method of simulating and investigating the dynamic behaviours of material temperatures and thermal stresses in the steam power plant control is presented in this study. The mathematical models derived for thermal stresses can also be used for the development and testing of effective measurement methods for thermal stresses. These models can also be used for the development of simulators to be used for the training of power plant operators.
REFERENCES Baehr, H. D. (1955). Die ~ung nichstationaerer Warmeleitungs-probleme mit Hilfe der Laplace Transformation. Forschung lng.-Wesen 21, (2), 33 40. Dettinger, R., E. Welfonder und R. Herbrik (1974). Verbesserung des Regelverhaltens von Dampferzeugern dutch strucktur-optimierte Regelung. Brennst.-Waerme-K rq# 26, {2), 43 54. Durmaz, A. (1970). Messungen mit Hilfe der elektronischen
Datenerfassung-sanlage am Kessel 6 im HKW-pfaffenwald der Universitaet Stuttgart und Auswertung der Messdaten auf dem Digitalrechner. Diplomarbeit am IVD der Universitaet Stuttgart. Durmaz, A. (1973). Regelkreisoptimierung eines im Gleitdruck betreiebenen Bensonkessels. Vergleich der theoretischen und experimentellen Ergebnisse. Energie 25, 213 216. Durmaz, A. (1974). Zur Berechnung der zeitlichen Aenderung der Werkstoff-beanspruchung bei Betriebszustandsaenderungen yon Dampferzeugern. Dissertation, Universitaet Stuttgart. Herbrik, R. and K. Voss (1969), Digitale Simulateion der Regelsyteme von Durchlaufdampferzeugern mit Hilfe blockorientierter Programmiersprachen. Brennst,-Waerme-Krq/'t 21, 13 16. Parkus, H. (1959). lnstationaere Waermespannungen. SpringerVerlag, Wien. Pich, R. (1964). Die Berechnung der elastischen instationaeren Waermespannungen in Platten, Hohlzylindern und Hohlkegeln mit quasisitationaeren Temperaturfeldern. VGB-Mitt. 88, 53 60. Richtmayer, R. D. and K. W. Morton (1967). D!/]erence Method for Initial Value Problems 2nd edition. Interscience, New York. Schittke, H. J. and A. Durmaz (1974). Berechung des dynamischen Verhaltens yon instationaeren Temperaturfeldern in zylindrischen Beuteilen. Waerme- und Stq[li'ibertragung 7, 121 132. Stoll, A. E. Haehle and A. Fischer (19681. Materialgercchter Betrieb grosser Dampferzeuger mit Hilfe eincs Freilastrechners Siemes- Z. 42, (2), 112 116. Strobel, H. (1968). Systemanalyse mit Determmiertetl Testsignalen. Verlag Technik, Berlin. Unbehauen, H. and R. Herbrik (1970). Identification, simulation and optimization of control systems in steam boiler plants. IFAC-Symposium on System Engineering Approach to Computer Control, Kyoto, Paper 9.2. Unbehauen, R. (1966). Ermittlung rationaler Frequenzgaenge aus Messwerten. Regelungstechnik 14, 268 273.