Mathematical modelling of power units
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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

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Mathematical modelling of power units

MathematicalModelling of Power Units


Mathematical modelling of power units1

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 units2

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 units3

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 units4

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


Mathematical modelling of power units

fuel

SURROUNDINGS

electricity

SYSTEM

steam


Mathematical modelling of power units5

Mathematical Modelling of Power Units

choice of the modelling approach - determination ofthe 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 units6

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


Simplified scheme

Simplified scheme


Mathematical modelling of power units7

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 units8

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 units9

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 units10

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 units11

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 units12

Mathematical Modelling of Power Units

Empiric approach - basic relations:

  • empiric process characteristics

  • parameters constraints

  • other relations - economical, ecological, technological


Mathematical modelling of power units

Physical approach – a model of a boiler – an example

mass and energy balances

the boiler output and efficiency


Mathematical modelling of power units

Physical approach – a model of a boiler – an example (cont.)

electricity consumption

boiler blowdown

constraints on temperature, pressure, and flow


Mathematical modelling of power units

Physical approach – a model of a boiler – an example (cont.)

pressure losses

specific enthalpies


Mathematical modelling of power units

Physical approach

– a model of a group of stages of a steam turbine boiler

– an example

mass and energy balances


Mathematical modelling of power units

Physical approach

– a model of a group of stages of a steam turbine boiler

– an example (cont.)

Steam flow capacity equation

where:


Mathematical modelling of power units

Physical approach

– a model of a group of stages of a steam turbine boiler

– an example (cont.)

internal efficiency characteristic

where:

= 0.000286for impulse turbine

= 0.000333for turbine with a small reaction 0.15 - 0.3

= 0.000869for turbine with reaction about 0.5


Mathematical modelling of power units

Physical approach

– a model of a group of stages of a steam turbine boiler

– an example (cont.)

enthalpy behind the stage group

Pressure difference (drop) for regulation stage:


Empiric description of a 3 zone heat exchanger

empiric description of a 3-zone heat exchanger

Heating steam inlet

U – pipes of a steam cooler

U – pipes of the main exchanger

Steam-water chamber

Condensate level

Condensate inflow from a higher exchanger

Heated water outlet

Heated water inlet

U – pipes of condensate cooler

Condensate outlet to lower exchanger

Water chamber


Scheme of a 3 zone heat exchanger

Steam cooling zone

Steam condensing zone

Condensate cooling zone

Steam inlet

4

3

Heated water outlet

Heated water inlet

3

2

1

2

1

2

1

A

B

C

4

4

3

Condensate outlet to lower exchanger

x

Condensate inflow from higher exchanger

Scheme of a 3-zone heat exchanger

Load coefficient (Bośniakowicz):


The heat exchanger operation parameters

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

Load coefficient for 3-zone heat exchanger with a condensate cooler

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

Empiric relation for load coefficient in changing operation conditions (according to Beckman):

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 a n empiric model of a chosen heat exchanger

An example – an empiric model of a chosen heat exchanger

Coefficients received with a linear regression method:

Covariance

Correlation coefficient

Standard deviation

Random variables

Expected value

X – measured values

Y – simulated values


Changes of a correlation coefficient

Changes of a correlation coefficient

Correlation coefficient

Sample size


An example of calculations

TC1

dw

An example of calculations

Load coefficient changes in relation to inlet water temperature and reduced value of the pipes diameter.


Empiric modelling of processes

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

Most popular empiric models

  • Linear models

  • Neuron nets

    • MLP

    • Kohonen nets

  • Fuzzy neron nets


Linear models

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


Neuron nets mlp

y2

y1

x3

x2

x1

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


Neuron nets fnn

Neuron Nets - FNN

  • 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


Empiric models where to use

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

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

Empiric models – an example of application Combustion in pulverized-fuel boilerDynamic 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


Mathematical modelling of power units

Accessible measurements used only

live steam

combustion

chamber

temperature

energy

in

steam

re-heated

steam

PW1...4

CO

O2

NOx

fraction

OFA

WM1...4

MW1...4

outlet

flue gases

temperature

secondary

air

air - total


Mathematical modelling of power units13

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 system

Data from PI system


Steam turbine an object for identification

Steam 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 units14

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 units15

Mathematical Modelling of Power Units

  • the system structure and the parameters writing in – numerical support


Chosen methods of computations

Chosen methods of computations

Linear Programming

  • SIMPLEX


Chosen methods of computations1

Chosen methods of computations

Chosen methods of computations

  • Linear programming with non-linear criterion function

  • MINOS Method (GAMS/MINOS)


Mathematical modelling of power units

Chosen methods of computations

Optimization with non-linear function and non-linear constraints

  • Linearization of constraints

  • MINOS method


Mathematical modelling of power units

Chosen methods of computations

  • Solving a set of non-linear equations

    • „open equation method”


Mathematical modelling of power units

Chosen methods of computations

  • Solving a set of non-linear equations

    • „path of solution method”

f2

f3

f1

1

4

5

3

2

x1

given

x2 given

x3

x4

x5

x6 given


Mathematical modelling of power units

Example of use of a mathematical model of a power system – determining of unmeasured parameters

measured: p, t

possible calculation: m

measured: m,p,t

measured: p,t


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

Electricity output – not optimized

Electricity output – optimized

Optimal electricity output – computed

Thermal output


Mathematical modelling of power units

Example of use of a mathematical model of a power system – a chose of structure of CHP unit

present situation


Mathematical modelling of power units

variant A


Mathematical modelling of power units

variant B


Mathematical modelling of power units

variant C


Mathematical modelling of power units

variant D


Mathematical modelling of power units

variant E


Mathematical modelling of power units

variant F


Mathematical modelling of power units

variant G


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