mathematical modelling of power units. what for: determination of unknown parameters optimization of...
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Mathematical Modelling
of Power Units
Mathematical Modelling of Power Units
What for:• Determination of unknown parameters
• Optimization of operational decision:– a current structure choosing - putting into
operation or turn devices off– parameters changing - correction of flows,
temperatures, pressures, etc.; load division in collector-kind systems
Mathematical Modelling of Power Units
What for (cont.):
• Optimization of services and maintenance scope
• Optimization of a being constructed or modernized system - structure fixing and devices selecting
Mathematical Modelling of Power Units
How – main steps in a modelling process:
• the system finding out
• choice of the modelling approach; determination of:– the system structure for modelling; simplifications and aggregation– way of description of the elements– values of characteristic parameters – the model identification
• the system structure and the parameters writing in
• setting of relations creating the model
• (criterion function)
• use of the created mathematical model of the system for simulation or optimization calculations
Mathematical Modelling of Power Units
The system finding out:
• coincidence
• invariability
• completeness of a division into subsystems
• separable subsystems
• done with respect to functional aspects
K I
1.4 Mpa
9.6
/3.2
9.6
/0.2
5
3 .2/1.4
9.6
/1.4
3.2 Mpa 0.25 Mpa
TG1 TG2 TG3 TG4
9.6 Mpa
1.4/0 .25
K II
En. e lektryczna
fuel
electricity
steam
SYSTEM
SURROUNDINGS
Mathematical Modelling of Power Units
choice of the modelling approach - determination of the system structure
A role of a system structure in a model creation:
• what system elements are considered – objects of „independent” modelling
• mutual relations between the system elements – relations which are to be taken into account and included into the model of the system
• additional information required: parameters describing particular elements of the system
Mathematical Modelling of Power Units
choice of the modelling approach
- determination of the system structure
• Simplification and aggregation – a choice between the model correctness and calculation possibilities and effectiveness
TG1KI KII
TG1KI KII
Simplified scheme Simplified scheme
1,70 MPa0,65 MPa0,20 MPa
3,40 MPa
TG2 TG4TG1 TG5TG3
K 2 K 4K 1 K 6K 3 K 7K 5
Mathematical Modelling of Power Units
choice of the modelling approach
- way of description of the elements
• basing on a physical relations
• basing on an empirical description
Mathematical Modelling of Power Units
Basic parameters of a model:
• mass accumulated and mass (or compound or elementary substance) flow
• energy, enthalpy, egzergy, entropy and their flows
• specific enthalpy, specific entropy, etc.• temperature, pressure (total, static, dynamic, partial),
specific volume, density, • temperature drop, pressure drop, etc.• viscosity, thermal conductivity, specific
heat, etc.
Mathematical Modelling of Power Units
Basic parameters of a model (cont.):
• efficiencies of devices or processes• devices output• maximum (minimum) values of some technical
parameters• technological features of devices and a system
elements - construction aspects• geometrical size - diameter, length, area, etc.
• empiric characteristics coefficients
• a system structure; e.g. mutual connections, number of parallelly operating devices
Mathematical Modelling of Power Units
Physical approach - basic relations:
• equations describing general physical (or chemical) rules, e.g.:– mass (compound, elementary substance) balance– energy balance – movement, pressure balance– thermodynamic relations– others
Mathematical Modelling of Power Units
Physical approach - basic relations (cont.):
• relations describing features of individual processes– empiric characteristics of processes, efficiency
characteristics– parameters constraints
• some parameters definitions
• other relations – technological, economical, ecological
Mathematical Modelling of Power Units
Empiric approach - basic relations:
• empiric process characteristics
• parameters constraints
• other relations - economical, ecological, technological
Physical approach – a model of a boiler – an example
0 fcba mmmm0 kffccbbaa Qhmhmhmhm
ckudk WmQ
0 ksk Qf
sksksks cQbQaQf 2)(
minmax KKK QQQ
mass and energy balances
the boiler output and efficiency
0 kee QfP
eekk baQQf
Physical approach – a model of a boiler – an example (cont.)
electricity consumption
0)( caoo mmabboiler blowdown
maxmin bbb ttt constraints on temperature, pressure, and flow
maxmin aaa ppp
znbbznb mmm 21
Physical approach – a model of a boiler – an example (cont.)
pressure losses
specific enthalpies
02
2
oznbznb
bba pp
m
mpp
),( aawa pthh
),( bbpb pthh
ffwf pthh ,
cc phh '
0 ba mm
h m h ma a b b 0
Physical approach – a model of a group of stages of a steam turbine boiler
– an example
mass and energy balances
Steam flow capacity equation
0, ),(
2
2),(
2
),(),(
yxzn
yx
xxp
yxx
yyxx
m
m
phv
p
pp
where:
2
),(),(
,
yxzn
zn
yxzn
znznp
x
y
p
pp
phv
Physical approach – a model of a group of stages of a steam turbine boiler
– an example (cont.)
internal efficiency characteristic
i x y
i zn
y
x
y zn
x zn
y
y
y zn
x zn
x y
p
pp
p
p
p
p
p
,
_
_
_
_
,
1 5 4 0
4 1
where:
= 0.000286 for impulse turbine= 0.000333 for turbine with a small reaction 0.15 - 0.3= 0.000869 for turbine with reaction about 0.5
Physical approach – a model of a group of stages of a steam turbine boiler
– an example (cont.)
enthalpy behind the stage group
0,,1 ,, yxixxpypyxixy phsphhh
Pressure difference (drop) for regulation stage:
p p pax a a
Physical approach – a model of a group of stages of a steam turbine boiler
– an example (cont.)
empiric
description of
a 3-zone heat
exchanger
Heating steam inlet
U – pipes of a steam cooler
Steam-water chamber
Condensate inflow from a higher exchanger
Condensate level
Water chamber
Heated water outlet
U – pipes of the main exchanger
U – pipes of condensate cooler
Heated water inlet
Condensate outlet to lower exchanger
Scheme of a 3-zone heat exchanger
3
4
12
Condensate outlet to lower exchanger
Heated water outlet
Heated water inlet
CBA 4
3
21 2
4
3
1
Steam cooling zone Condensate cooling zoneSteam condensing zone
Condensate inflow from higher exchanger
x
Steam inlet
Load coefficient (Bośniakowicz):TB1TB4
TB1TA2Φ
The heat exchanger operation parameters
• mass flows
• inlet and outlet temperatures
• heat exchanged
• heat transfer coefficient
• load coefficient
Load coefficient for 3-zone heat exchanger with a condensate cooler
TC1TB4mC1mA3
1 Tx mC1mx
TC4mC1
mxmA3TB4TA2
Φ
TC4 – outlet condensate temperature;Tx – inlet condensate temperature;TC1 – inlet heated water temperature;mA3 – inlet steam mass flow;mx – inlet condensate mass flow;mC1 – inlet heated water mass flow.
Empiric relation for load coefficient in changing operation conditions (according to Beckman):
ν
0
μ
00 TC1
TC1
mC1
mC1ΦΦ
0 – load coefficient at reference conditions;
mC10 – inlet heated water mass flow at reference conditions;
TC10 – inlet heated water temperature at reference conditions.
An example – an empiric model of a chosen heat exchanger
0,31-0,20
323
TC1
380
mC10,82Φ
Coefficients received with a linear regression method:
2Y
2X
XYσσ
YX,covρ
n
1iii yyxx
n
1YX,cov
X – measured values
Y – simulated values
Standard Standard deviation deviation
Expected value Expected value
n
1iix
n
1x
n
1iiy
n
1y
Random variablesRandom variables
Correlation coefficientCorrelation coefficient
2n
1ii
2X xx
n
1σ
2n
1ii
2Y yy
n
1σ
Covariance Covariance
Changes of a correlation coefficient
0,94
0,95
0,96
0,97
0,98
0,99
1
0 500 1000 1500 Nr próbki
Sample size
Correlation coefficient
An example of calculations
TC1
dw
Load coefficient changes in relation to inlet water temperature and reduced value of the pipes diameter.
Empiric modelling of processes
• Modelling based only on an analysis of historical data
• No reason-result relations taken into account
• „Black – box” model based on a statistical analysis
Most popular empiric models
• Linear models
• Neuron nets– MLP– Kohonen nets
• Fuzzy neron nets
Linear Models• ARX model (AutoRegressive with eXogenous
input) – it is assumed that outlet values at a k moment is a finite linear combination of previous values of inlets and outlets, and a value ek
• Developed model of ARMAX type
• Identification – weighted minimal second power
kmdkmdkdknknkk eubububyayay ........... 11011
)()()( ieA
Ciu
A
Bziy k
Neuron Nets - MLP
• Approximation of continuous functions; interpolation
• Learning (weighers tuning) – reverse propagation method
• Possible interpolation, impossible correct extrapolation
• Data from a wide scope of operational conditions are required
x 1 x 2 x 3
y 1 y 2
• Takagi – Sugeno structure – a linear combination of input data with non-linear coefficients
• Partially linear models– Switching between ranges
with fuzzy rules– Neuron net used for
determination of input coefficients
• Stability and simplicity of a linear model
• Fully non-linear structure
x1
x2
y
(F) (G) (H) (I) (J) (K) (L)
1
1
1wa
ws
1
wa
ws
1
wa
K O N K L U Z J E
(A) (B) (C) (D) (E)
P R Z E S Ł A N K I
1
1 wgwc
1
1
-1
1
1
1 wgwc
1
1
-1
1
1
Neuron Nets - FNN
Empiric models – where to use
• If a physical description is difficult or gives poor results
• If results are to be obtained quickly
• If the model must be adopted on-line during changes of features of the modelled object
Empiric models – examples of application
• Dynamic optimization (models in control systems)
• Virtual measuring sensors or validation of measuring signals
Empiric models – an example of application Combustion in pulverized-fuel boiler
Dynamic Optimization
• Control of the combustion process to increase thermal efficiency of the boiler and minimize pollution
• NOx emission from the boiler is not described in physical models with acceptable correctness
• Control is required in a real-time; time constants are in minutes
PW1...4
WM
1...
4
MW1...4secondaryair
OFA
air - total
re-heatedsteam
live steam
COO2
NOx
fraction
combustion chamber temperature
energyinsteam
outletflue gases temperature
Accessible measurements used only
Mathematical Modelling of Power Units
Choice of the modelling approach
Model identification• Values of parameters in relations used for the object
description– technical, design data– active experiment– passive experiments
• (e.g. in the case of empiric, neuron models)
• data collecting on DCS, in PI
Data from PI systemData from PI system
Steam turbine – an object for identificationSteam turbine – an object for identification
A characteristic of a group of stages – results of identification
A characteristic of a group of stages – results of identification
Mathematical Modelling of Power Units
Model kind, model category:
• based on physical relations or empiric
• for simulation or optimization
• linear or non-linear• algebraic, differential, integral, logical, …• discrete or continuous
• static or dynamic• deterministic or probabilistic (statistic)• multivariant
Mathematical Modelling of Power Units
• the system structure and the parameters writing in – numerical support
Chosen methods of computations
Linear Programming
• SIMPLEX
xcxf
IIjgx
IIjdx
bx
T
c
BGjj
BDjj
)(
,
,
A
• Linear programming with non-linear criterion function
• MINOS Method (GAMS/MINOS)
)()(
,
,
xfxcxf
IIjgx
IIjdx
bx
nl
T
c
BGjj
BDjj
A
Chosen methods of computationsChosen methods of computations
Optimization with non-linear function and non-linear constraints
• Linearization of constraints
• MINOS method
)()(
,
,
)(
xfxcxf
IIjgx
IIjdx
bxF
nl
T
c
BGjj
BDjj
nl
*,
0
bx
f
jxj
inl
Chosen methods of computations
• Solving a set of non-linear equations– „open equation method”
Chosen methods of computations
min))((
)(
02,
xjojnl
nl
bxf
bxF
• Solving a set of non-linear equations– „path of solution method”
Chosen methods of computations
f1f2 f3
x1
given
x2 given
x3 x4 x5 x6 given
12 3
45
TG
measured: p, t
measured: m,p,tmeasured: p,t
possible calculation: m
Example of use of a mathematical model of a power system – determining of unmeasured parameters
Example of use of a mathematical model of a power system – determining of unmeasured parameters
Example of use of a mathematical model of a power system – operation optimization of a CHP unit
Example of use of a mathematical model of a power system – operation optimization of a CHP unit
time
pow
er -
out
put
Mocelektryczna -bezoptymalizacji
Mocelektryczna - zoptymalizacją
Obliczonawartość mocyoptymalnej
Mocciepłownicza
Electricity output – not optimized
Electricity output – optimized
Optimal electricity output – computed
Thermal output
K I
1.4 Mpa
9.6
/3.2
9.6
/0.2
5
3 .2/1.4
9.6
/1.4
3.2 Mpa 0.25 Mpa
TG1 TG2 TG3 TG4
9.6 Mpa
1.4/0 .25
K II
En. e lektryczna
Example of use of a mathematical model of a power system – a chose of structure of CHP unit
Example of use of a mathematical model of a power system – a chose of structure of CHP unit
present situation
p1
p2
TG4
En. elektryczna
K I
1.4 Mpa
9.6/
3.2
9.6/
0.25
3.2/1.4
9.6/
1.4
3.2 Mpa 0.25 Mpa
TG1 TG2 TG3
9.6 Mpa
3.2/1.4
K II
TG N
EKO
SP
KSTG
variant A
p1
p2
TG4
En. elektryczna
K I
1.4 Mpa
9.6/
3.2
9.6/
0.25
3.2/1.4
9.6/
1.4
3.2 Mpa 0.25 Mpa
TG1 TG2 TG3
9.6 Mpa
3.2/1.4
K II
EKO
SP
KSTG
variant B
p2
TG4
En. elektryczna
K I
1.4 Mpa
9.6/
3.2
9.6/
0.25
3.2/1.4
9.6/
1.4
3.2 Mpa 0.25 Mpa
TG1 TG2 TG3
9.6 Mpa
1.4/0.25
K II
TG N
EKO
SP
KSTG
variant C
11
TG4
En. elektryczna
K I
1.4 Mpa
9.6/
3.2
9.6/
0.25
3.2/1.4
9.6/
1.4
3.2 Mpa 0.25 Mpa
TG1 TG2 TG3
9.6 Mpa
1.4/0.25
K II
TG N
EKO
SP
KSTG
etap I
etap II
p =14 bar2
variant D
11
TG4
En. elektryczna
K I
1.4 Mpa
9.6/
3.2
9.6/
0.25
3.2/1.4
9.6/
1.4
3.2 Mpa 0.25 Mpa
TG1 TG2 TG3
9.6 Mpa
1.4/0.25
K II
TG N
EKO
SP
KSTG
etap I
etap II
p =14 bar2
variant E
TGN TG4
En. elektryczna
K I
1.4 Mpa
9.6/
3.2
9.6/
0.25
3.2/1.4
9.6/
1.4
3.2 Mpa 0.25 Mpa
TG1 TG2 TG3
9.6 Mpa
1.4/0.25
K II
variant F
TGN 1TG4
En. elektryczna
K I
1.4 Mpa
9.6/
3.2
9.6/
0.25
3.2/1.4
9.6/
1.4
3.2 Mpa 0.25 Mpa
TG1 TG2 TG3
9.6 Mpa
1.4/0.25
K II
TGN 2
variant G
TG1
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