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# Mathematical Modelling of Power Units - PowerPoint PPT Presentation

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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MathematicalModelling 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

What for (cont.):

• Optimization of services and maintenance scope

• Optimization of a being constructed or modernized system - structure fixing and devices selecting

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

The system finding out:

• coincidence

• invariability

• completeness of a division into subsystems

• separable subsystems

• done with respect to functional aspects

SURROUNDINGS

electricity

SYSTEM

steam

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

choice of the modelling approach

- determination of the system structure

• Simplification and aggregation – a choice between the model correctness and calculation possibilities and effectiveness

choice of the modelling approach

- way of description of the elements

• basing on a physical relations

• basing on an empirical description

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.

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

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

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

Empiric approach - basic relations:

• empiric process characteristics

• parameters constraints

• other relations - economical, ecological, technological

mass and energy balances

the boiler output and efficiency

electricity consumption

boiler blowdown

constraints on temperature, pressure, and flow

pressure losses

specific enthalpies

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

– an example

mass and energy balances

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

– an example (cont.)

Steam flow capacity equation

where:

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

– an example (cont.)

internal efficiency characteristic

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

– 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:

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

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

mass flows

inlet and outlet temperatures

heat exchanged

heat transfer coefficient

load coefficient

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):

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 conditions (according to Beckman):n 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 conditions (according to Beckman):

Correlation coefficient

Sample size

conditions (according to Beckman):

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 conditions (according to Beckman):

• 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 conditions (according to Beckman):

• Linear models

• Neuron nets

• MLP

• Kohonen nets

• Fuzzy neron nets

Linear Models conditions (according to Beckman):

• 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

y conditions (according to Beckman):2

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 conditions (according to Beckman):

• 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 conditions (according to Beckman):

• 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 conditions (according to Beckman):

• Dynamic optimization (models in control systems)

• Virtual measuring sensors or validation of measuring signals

Empiric models – an example of application conditions (according to Beckman):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

Accessible measurements used only conditions (according to Beckman):

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 Units conditions (according to Beckman):

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 conditions (according to Beckman):

Steam turbine – an object for identification conditions (according to Beckman):

Mathematical Modelling of Power Units identification

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 identification

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

Chosen methods of computations identification

Linear Programming

• SIMPLEX

Chosen methods of computations identification

Chosen methods of computations

• Linear programming with non-linear criterion function

• MINOS Method (GAMS/MINOS)

Chosen methods of computations identification

Optimization with non-linear function and non-linear constraints

• Linearization of constraints

• MINOS method

Chosen methods of computations identification

• Solving a set of non-linear equations

• „open equation method”

Chosen methods of computations identification

• 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

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

Electricity output – not optimized

Electricity output – optimized

Optimal electricity output – computed

Thermal output

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

present situation