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Mathematical Modelling of Power Units

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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

fuel

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

Physical approach – a model of a boiler – an example

mass and energy balances

the boiler output and efficiency

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

electricity consumption

boiler blowdown

constraints on temperature, pressure, and flow

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

pressure losses

specific enthalpies

Physical approach

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

– an example

mass and energy balances

Physical approach

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

– an example (cont.)

Steam flow capacity equation

where:

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

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:

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 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

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.

0 – load coefficient at reference conditions;

mC10 – inlet heated water mass flow at reference conditions;

TC10 – inlet heated water temperature at reference conditions.

Coefficients received with a linear regression method:

Covariance

Correlation coefficient

Standard deviation

Random variables

Expected value

X – measured values

Y – simulated values

Correlation coefficient

Sample size

TC1

dw

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

- Modelling based only on an analysis of historical data
- No reason-result relations taken into account
- „Black – box” model based on a statistical analysis

- Linear models
- Neuron nets
- MLP
- Kohonen nets

- Fuzzy neron nets

- 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

y2

y1

x3

x2

x1

- 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

- 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

- 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

- Dynamic optimization (models in control systems)
- Virtual measuring sensors or validation of measuring signals

- 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

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

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

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

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

Linear Programming

- SIMPLEX

Chosen methods of computations

- Linear programming with non-linear criterion function
- MINOS Method (GAMS/MINOS)

Chosen methods of computations

Optimization with non-linear function and non-linear constraints

- Linearization of constraints
- MINOS method

Chosen methods of computations

- Solving a set of non-linear equations
- „open equation method”

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

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

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

variant A

variant B

variant C

variant D

variant E

variant F

variant G