mathematical modelling of power units. what for: determination of unknown parameters optimization of...

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