Abstract — In this paper based on the fundamental laws of physics, thermodynamics principles and energy semi-empirical laws for heat transfer, the mathematical models are developed and applied to electrical power generating plants in order to characterize the essential dynamic behavior of the boiler subsystems. These models are developed for a sub-critical once- through Benson type boiler based on the experimental data obtained from a complete set of field experiments. An optimization approach based on genetic algorithm (GA) is executed to estimate the model parameters and fit the models response on the real system dynamic. Comparison between the responses of corresponding models with the response of plants validates the accuracy and performance of modeling approach.
I. INTRODUCTION
ncreasing demand for electricity and growing need for more and safer power generating has motivated investigation into dynamic analysis of power plants to design more reliable control systems. Adequate information about the system dynamics is a key to design the suitable controllers. Better system performance means increase in power generation efficiency and decrease in the maintenance costs.
The dynamics of most power plants are highly nonlinear with numerous uncertainties. However, no mathematical model can exactly describe such a complicated physical process, and there will always be modeling errors due to un- modeled dynamics and parametric uncertainties [1].
In order to characterize the essential dynamic behavior of power plants with representation of plant components, detailed models are needed [2]. Besides, detailed modeling of plants dynamics is often not efficient for control synthesis.
There are complicated models based on finite element approximations to partial differential equations. These models are in the form of large simulation codes for plant design, simulators and commissioning. However, they are not normally used in control design approach because of their complexity [3].
Manuscript received September 14, 2006.
A. Ghaffari is with theMechanical Engineering Department, K.N. Toosi University of Technology, Tehran, Iran (corresponding author to provide phone: 98-21-77-334-024; fax:98-21-77-334-338).
(e-mail: [email protected]).
A. Chaibakhsh is Ph.D. student at the Mechanical Engineering Department, K.N. Toosi University of Technology, Tehran, Iran, (e-mail:
H. Parsa is Ph.D. student at the Design and Manufacturing Institute (DMI), Stevens Institute of Technology, Hoboken, NJ, USA,
(e-mail: [email protected]).
The plant model should describe the plant dynamics with sufficient accuracy and not describe the microscopic details occurring within individual plant components [1].
The analytical plant model can be formulated based on the fundamental laws of physics such as mass conservation, momentum, and energy semi-empirical laws for heat transfer and thermodynamics state conversion [3]. To build such analytical models, it is necessary to define their parameters with respect to boundaries, inputs, and outputs. Generally, the developed models need to be tuned by performing tests to validate for steady state and transient responses [4] [5].
When the identified model is nonlinear in the parameters, using conventional methods like standard least squares technique will not provide superior results. In these cases, evolutionary algorithm based methodologies are investigated as potential solutions to obtain good estimation of the model parameters [6]. Genetic algorithms have an advantage that it does not require a complete system model and can be employed to globally search for the optimal solution [7].
In this paper, the mathematical models with unknown parameters for subsystems of a once-through Benson type boiler are first developed based on the thermodynamics principles and energy balance. Then the related parameters are either determined from constructional data such as fuel and water steam specification, or by applying genetic algorithm (GA) techniques on the experimental data.
Finally, the responses of the corresponding models are compared with the responses of the real plant in order to validate the accuracy and performance of the models.
II. SYSTEM DESCRIPTION
A 440 MW unit is considered for identification and modeling approach. The steam generator of a sub-critical once through Benson boiler with steam rate 1408 ton per hour is shown in Fig. (1). A single furnace with 14 bottom burners serves to heat all sections of the boiler.
The hot water is converted to steam in evaporator, and gets superheated by passing through superheaters.
The evaporator outlet temperature is about 365oC. The superheater consists of four sections built in the boiler second pass. The heating surface of superheater #1 is located in the boiler walls and jointed to superheater #2. The outlet steam temperature from these sections is about 407oC. The steam leaves superheater #2 toward superheater #3. For rapid steam temperature control, four spray attemperators is considered between the two sections. The outlet temperature should be kept constant at 470oC. The steam for final superheating stage goes to superheater #4. The temperature at outlet header of this part should be constant at 535oC. At full load condition, the outlet superheated steam pressure at boiler outlet header is 18.1MPa.
An Optimization Approach Based on Genetic Algorithm for Modeling Benson Type Boiler
A. Ghaffari, A. Chaibakhsh and H. Parsa
I
The superheated steam from main steam header is fed toward the high pressure (HP) turbine, and from high pressure turbine is discharged into the cold reheat header.
The steam temperature in cold reheat line is 351oC. The outlet reheated steam temperature after RH-A should be constant at 452oC and for RH-B should be constant at 535oC. Also, at the full load condition, the outlet reheated steam pressure is 4.35MPa.
III. MODEL DEVELOPMENT
The boiler system can be decomposed into smaller components that can be analyzed and modeled separately.
The behavior of the subsystems can be captured in terms of balance equations and constructive equations [2] [3].
The steam-generating unit consists of economizer, evaporator, superheaters and reheater sections. In the boiler sections, the heat released by fuel combustion is transferred to the working fluid in the boiler.
According to global mass and energy balances, we have.
[
sVs wVw]
mf msdt
d ρ. +ρ . = & − & ( 1)
[ ]
s s f f a p a w w w s s s
h m h m Q
T C m V u V dt u
d
&
& −
+
= +
+ . . . .
.
. ρ
ρ ( 2)
The right hand side of Eq. (2) represents the energy flow to the system from fuel and feed water and the energy flow from the system via the steam. Since the internal energy isu=h−p/ρ, the global energy balance can be written as;
[ ]
s s f f
a p a w
w w s s s
h m h m Q
T C m pV V h V dt h
d
&
& −
+
= +
−
+ . . . .
.
. ρ
ρ (3)
Multiplying Eq. (1) byhw and subtracting the result from Eq. (3) we have,
[ ]
) (
) ( .
.
. .
) (
w s s
f w f a
p a
w w s w s s s s w s
h h m
h h m dt Q
Vdp T C dt m
d
dt V dh dt
V dh dt
V d h h
−
−
− +
=
− +
+ +
−
&
&
ρ ρ ρ
) 4 (
By considering,
P
p T
C h
∂
= ∂ ,
v
v T
C u
∂
= ∂ (5)
when, dt dT T h dt dh
∂
= ∂ (6)
By the additional simplification, for the superheater parts we have;
[ ]
out out p in in p
out p s s m
a p a s
s s
T m C T m C Q
dt C dT V T C dt m d dt V d h
in
&
&
4 4 4
4 8
4 4 4
4 7
6 &
− +
= + +
≈
. . . . .
) ( f
ρ ρ (7)
It is shown that for drum boiler, the changes in energy content of the water and metal masses are the physical phenomena that dominate the dynamics of the boiler [3].
The simulation results of models with detailed representation of the temperature distribution show that by removing these terms, there is an off-set between the model response and experimental data at the steady state condition.
To take these terms into account, they are expressed as a function of mass flow in order to make the model response close to real system.
[
. .]
( ). s s a p a in
s m C T f m
dt d dt V d
u ρ + = & (8)
This function is chosen as a linear function. This approximation is good enough to fit the model response on the real system response.
in
in k m
m
f(& )= 1.& (9)
Therefore, the equation (7) is captured as follow;
1 ) 0
. ( 1 . .
1
1 2
C k T k T V m C Q V dt
dT
out p in in k
s s k
p s s out
+ +
− +
=
&
3 2 1
43
42 1
ρ ρ
) 10 (
This equation is rewritten as follow;
) ) (
( 1 2
2 C Q m T T B B
dt K dT
out in in
out = p + & − + + (11)
where
s sV
K .
1
2=ρ , Cp
B1= k1 , B2 =k0ρs.Vs (12) The heat flow can be captured by using calorific value / lower heating value (Hv) of the fuel.
fuel vm H
Q= .& (13)
By consideringK1=Hv.Cp, the superheater model is derived;
) ) (
( 1 1 2
2 Km m T T B B
dt K dT
out in in
out = &fuel+ & − + + (14)
The dynamic behavior of the heated surfaces of the risers in the drum type boiler is shown in reference [3]. In the same way, these relations can be used to obtain the dynamics of steam generator.
Fig. 1. Boiler Cross Section
The specific internal energy for the steam and water mixture is,
w
s h
h
h=α. +(1−α). (15)
whereαis the quality of the mixture. The mass and energy balances for the heated surfaces are
=0
∂ +∂
∂
∂ z q A ρt ,
V Q z
h q A t
h =
∂ + ∂
∂
∂ρ 1 (16)
In the steady state condition, it is shown that the quality of mixture can be obtained,
V h q
z A Q
c
α= (17)
The simulation results show that the steam quality can be expressed as a linear function of the fuel flow rate as follow,
fuel
fuel K m
m
f(& )= 3.&
α= (19)
With respect to these equations, the proposed temperature model for the evaporator and superheater sections are presented in Fig. (2) and Fig. (3). It should be noted that only the steam phase is presented in the superheater sections.
It should be noted that for modeling the boiler subsystems, the delay time is needed to be considered. This
delay time is defined as the time required for burning the fuel in the furnace, transferring the released heat and absorbing transferred heat by fluid in boiler subsystems. The delay time is an important factor that affects the dynamics of boiler. The delay times of boiler subsystems are presented in Table (1) [Appendix].
Next, the outlet pressure of the boiler can be calculated based on the fluid dynamics principles. Naturally, the steam pressure will drop by passing through the boiler sections such that in the HP section of this boiler, the pressure loss is about 6MPa. The main pressure drop is observed where vaporization takes place. In the steam side, the pressure loss due to change in flow velocity is prevailing. So, by neglecting the other effects, the pressure loss model can be captured based on the steam areas effect. The mass of accumulative volume is,
out
in m
dt m
dM = & − & (20)
The total accumulative action of any control volume in boiler consists of accumulative action of steam, hot liquid and metal parts. Therefore, differential equation for the total accumulative action can be written as follows.
P M P M P M P
M PS PW PM
∂ +∂
∂ +∂
∂
=∂
∂
∂ (21)
Noting that the effects of second and third terms on the right hand side of Eq. (21), comparing with the first term, are negligible. We have,
P M P
M PS
∂
= ∂
∂
∂ (22)
In this sliding pressure Benson type boiler, the pressure differences are the driving forces for mass flow through the components. The change of steam mass due to the pressure change is given by,
=
) ( . 1
P v dP V d dP
dM (23)
By considering the value of polytrophic exponent is nearly 1.0 (p.v=p0.v0), If
0
0 >
> for dP dP
dM (24)
From Eqs. (20) to (24), the outlet pressure is obtained;
) . (
0 in out
v
m m m
P dt
dP= & − &
τ (25)
This equation can be used as a general relation for pressure drop between turbine stages.
A model for the mass flow responding to steam pressure changes is proposed by Borsi [7]. The swing of main steam flow strictly relies on the change of steam pressure.
dt dP P P
m dt
m d
out in
out out
) (
2 0− 0
= &
& (26)
The superheater and reheater temperatures must be kept constant at specific temperature. The spray attemperators is implemented between these sections to control outlet temperature. Furthermore, de-superheating spray is used to achieve mixing between the superheated steam at the outlet of the preceding component (e.g., the primary superheater).
The water spray is modulated by suitable valves. Because Fig. 3. Temperature Model for the Superheater Sections
Fig. 2. Temperature Model for the Evaporator Section
the attemperator has a relatively small volume, the mass storage inside that is negligible.
Therefore, the steady-state mass and energy balances yield
out spray
in m m
m& + & = & (27)
out out spary spray in
inh m h m h
m& + & = & (28)
During a normal operation, steam flow m&out in the secondary superheater is imposed (over a wide band) by the load controller, the specific enthalpy hinis determined by upstream superheater and hspary is nearly constant. The inlet temperature of the second superheater,Toutis governed by the following equation[2].
spray out p
spray in in out
in out out p
out in out p out
m m C
h T h
m m m m C
h h h
T C
&
&
&
&
) (
) (
1
− − +
= −
∆
=
) 29 (
This completes the dynamics modeling of different boiler subsystems.
IV. PARAMETER ADJUSTMENT BY GENETIC ALGORITHM
In this section, the parameters of the developed models are evaluated and adjusted by using genetic algorithm (GA) training approach.
Genetic algorithm is specifically useful for those systems working under different operating conditions with different behavior. The GA determines the multiple next searching points using the evaluation values of multiple current searching points. The next searching points are determined by using the fitness values of the current searching points which are widely spread throughout the searching space. In conventional optimization methods, like gradient methods, in some cases the gradient information causes searching being trapped at a local minima point. The GA has the mutation operator to escape from these possible trapping points. These advantages cause that GA is preferred to other methods.
For GA training, the error E is given by the mean value of squared difference between the target output y* and actual output y as follows;
∑
=−
= P
p
p
p y
P y E
1
2 ) (
* )
1 ( (30)
wherePis the number of entries used for training process.
The initial parameters of models are determined based on the steady state conditions of the plant during the period of operation. For example, at the steady state condition from Eq. (14) we have,
0 )
2 (
1
1mfuel +Bmin+B +min Tin−Tout =
K & & & (31)
where m&in,m&fuel,TinandTout are the values of inlet steam mass rate, inlet fuel mass rate, inlet and outlet steam temperature at the steady state, respectively.
The parametersK1,B1andB2are to be calculated. For the range of operation between 440 to 195 MW and for three middle set points at 90%, 75% and 60% of load, the response of the actual plant, based on the experimental data, is shown in Fig. (5). Therefore, Eq. (31) is solved for these unknown parameters at three steady state conditions. By solving these three set of linear equations, we obtain the initial values of parameters used in the model training process.
The next step for developing models, is gathering appropriate data for different subsystems of the boiler. In order to have persistent excitation data, the control systems are switched to manual control. The experimental data are recorded for 24 hours. These data include the transient and steady state conditions. Although the parameters are adjusted based on a limited volume of the collected data, but the selected data are well spread and the developed models are exploited in a wide range of load variations. The models are trained with the data when the load is ramped down from 100 to 45 percent, and they are checked with different set of data when the load is ramped up from 45 to 100 percent.
The parameters adjustment is executed through a set of 2000 points of boiler data. The models training process is performed by using Matlab genetic algorithm toolbox and is simulated by Matlab Simulink. Some adjusted parameters of the superheater models are presented in Table (2). The optimization parameters for the proposed models in this paper are presented in Table (3).
Fig. 5. The Actual Power Output at Four Diffrent Setpoints Between 195 to 440 MW.
Fig. 4. Mass Flow- Pressure Model
V. SIMULATION RESULTS
In this section, the trained models are simulated by using Matlab Simulink. The responses of the proposed models and the real responses of different subsystems for steady state and transient conditions are shown in Figures (6) to (10).
The response of the economizer, as shown in Fig. (10), indicates some differences between the temperature of the model and the real system. Noting that the economizer is located at the end part of the boiler, to have more reliable model for economizer, the temperature of the exhaust smoke
should be replaced by the fuel flow signal. However, it should be noted that the maximum difference between the response of the actual system and the models in Fig. (10) is much less than 0.2%.
In a sliding pressure boiler, the phenomenon like shrink and swell effect which occurs immediately when the boiler turns off or when the boiler is in the start up phase (under 35% load). In these cases, the dynamics of once-through boiler is the same as a drum boiler. The start-up control valves are activated to cope with these conditions. So, it is more suitable to develop a different model for these conditions.
Figures (6) to (10) show that the responses of presented models in this paper are significantly closer to the actual system responses.
VI. CONCULSIONS
In this paper, based on the physical and thermo- dynamical laws, different parametric models are developed for boiler subsystem and for the turbine in a steam power generating plant. The genetic algorithm is executed as an optimization method to adjust the model parameters based on the experimental data. The comparison between the responses of the corresponding models with the responses of the plant subsystems validates their accuracy in the steady state and transient conditions. These models can be used for
Fig. 10. Response of the Economizer Fig. 9. Response of the Superheater # 4
Fig. 8. Response of the Superheater # 3 Fig. 7. Response of the Superheater # 1 and 2 Fig. 6. Response of the Evaporator
design a proper temperature control system for superheaters sections.
Simulation results show the effectiveness and feasibility of the developed model in term of more accurate and less deviation in the corresponding subsystems.
These models can be improved for the abnormal conditions such as start-up and shutdown modes. It should be considered in the model by taking into account the heat exchange and combustion process in the furnace. For this conditions, it is more suitable to develop a different model for these conditions.
The further model improvements will make them proper for using in simulators and emergency control system.
APPENDIX
REFERENCES
[1] C.K. Weng, A. Ray and X. Dai, “Modeling of Power Plant Dynamics and Uncertainties for Robust Control Synthesis,” Application of Mathematical Modeling, vol. 20, pp. 501-512, 1996.
[2] C. Maffezzoni, “Boiler-Turbine Dynamics in Power Plant Control,”
Control Engineering Practice. vol. 5, pp. 301-312, 1997.
[3] K.J. Astrom and R.D. Bell, “Drum-Boiler Dynamics,” Journal of Automatica. vol. 36 (3), pp.363-378, 2000.
[4] S. Lu and B.W. Hogg, “Dynamic Nonlinear Modelling of Power Plant by Physical Principles and Neural Networks,” Journal of Electrical Power and Energy Systems, vol. 22,pp. 67-78, 2000.
[5] F.P. De Mello, “Boiler Models for System Dynamic Performance Studies,” IEEE Transaction on Power Systems, vol. 6 (1),pp. 66-74, 1991.
[6] R. Horst, P. Pardalos and N. Thoai, “Introduction to Global Optimization, Second Edition,” Kluwer Academic Publishers, Dordrech, pp. 34-36, 2000.
[7] L. Borsi, “Extended Linear Mathematical Model of a Power Station Unit with a Once-Through Boiler”, Siemens Forschungs und Entwicklingsberichte, vol. 3 (5), pp. 274-280, 1974.
Nomenclature
C Specific Heat (kJ/kgK) h Specific Enthalpy (kJ/kg) Hv Calorific Value (LHV) (kJ/kg)
m& Mass Flow (kg/s)
m Mass (kg)
m
Molecular WeightM
Accumulative MassP Pressure (Mpa) Q Heat Transferred (MJ)
T
Temperature (oC) u Specific Internal Energy (kJ/kg) v Specific Volume (m3/kg) V Volume of Subsystem (m3)ρ
Specific Density (kg/ m3) t Time (s)τ
Time Constant (s)ξ
Coefficient of Heat Absorbing Subscriptsf Feedwater
s Steam
w Water
in
Input out Outa Metal
0 Standard Variable
spray
Water SprayTABLEI
DELAY TIMES FOR BOILER SUBSYSTEMS
Delay (Minute) Outlet Pressure 0.6
Re-heater 1.2
Fuel Firing 0.2
Economizer 2.5 Pre-heater 4.0 HP-Heater 3.0 Evaporator 0.8
SH # 12 0.9
SH # 4 0.9
SH # 3 1.0
TABLEII MODEL PARAMETER
B1 B2 K1 K2
sh12_1 -0.8667 0.9375 0.8654 3.333e-4 sh12_2 -0.8741 0.8914 0.8591 3.333e-4 sh12_3 -0.8522 0.9135 0.8604 3.333e-4 sh12_4 -0.8608 0.9042 0.8638 3.333e-4 sh31 -0.325 0.503 0.551 3.448e-4 sh32 -0.332 0.521 0.542 3.448e-4 sh33 -0.345 0.515 0.548 3.448e-4 sh34 -0.329 0.511 0.553 3.448e-4 sh41 -0.165 0.312 0.307 3.584e-4 sh42 -0.113 0.320 0.305 3.584e-4 sh43 -0.114 0.317 0.308 3.584e-4 sh44 -0.102 0.331 0.308 3.584e-4
TABLEIII
OPTIMIZATION PARAMETERS FOR GA
Population Size 50
Crossover Rate 0.7
Mutation Rate 0.1
Generations 100 Migration Forward, Fraction:0.2 , Int.: 20
Selecting Stochastic Uniform
Reproduction Elite Count :2
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