Mathematical Modelling of Power Units

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Mathematical Modelling of Power
Units
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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

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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

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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

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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

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fuel
SURROUNDINGS
electricity
SYSTEM
steam
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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

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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

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Simplified scheme
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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

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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.

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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

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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

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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

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Mathematical Modelling of Power Units

• Empiric approach - basic relations
• empiric process characteristics
• parameters constraints
• other relations - economical, ecological,
  technological

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Physical approach a model of a boiler an
example
mass and energy balances
the boiler output and efficiency
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Physical approach a model of a boiler an
example (cont.)
electricity consumption
boiler blowdown
constraints on temperature, pressure, and flow
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Physical approach a model of a boiler an
example (cont.)
pressure losses
specific enthalpies
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Physical approach a model of a group of stages
of a steam turbine boiler an example
mass and energy balances
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Physical approach a model of a group of stages
of a steam turbine boiler an example (cont.)
Steam flow capacity equation
where
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Physical approach 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
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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
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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
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Scheme of a 3-zone heat exchanger
Load coefficient (Bosniakowicz)
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The heat exchanger operation parameters

• mass flows
• inlet and outlet temperatures
• heat exchanged
• heat transfer coefficient
• load coefficient

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

• TC4 outlet condensate temperatureTx
  inlet condensate temperatureTC1 inlet heated
  water temperaturemA3 inlet steam mass
  flowmx inlet condensate mass flowmC1
  inlet heated water mass flow.

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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.

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An example an empiric model of a chosen heat
exchanger
Coefficients received with a linear regression
method
Covariance
Correlation coefficient
Standard deviation
Expected value
Random variables
X measured values Y simulated values
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Changes of a correlation coefficient
Correlation coefficient
Sample size
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An example of calculations
Load coefficient changes in relation to inlet
water temperature and reduced value of the pipes
diameter.
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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

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Most popular empiric models

• Linear models
• Neuron nets
• MLP
• Kohonen nets
• Fuzzy neron nets

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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

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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

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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

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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

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Empiric models examples of application

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

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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

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Accessible measurements used only
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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

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Data from PI system
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Steam turbine an object for identification
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A characteristic of a group of stages results
of identification
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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

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Mathematical Modelling of Power Units

• the system structure and the parameters writing
  in numerical support

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Chosen methods of computations

• Linear Programming
• SIMPLEX

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Chosen methods of computations
Chosen methods of computations

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

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Chosen methods of computations

• Optimization with non-linear function and
  non-linear constraints
• Linearization of constraints
• MINOS method

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Chosen methods of computations

• Solving a set of non-linear equations
• open equation method

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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
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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
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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
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Example of use of a mathematical model of a power
system a chose of structure of CHP unit
present situation
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variant A
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variant B
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variant C
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variant D
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variant E
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variant F
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variant G
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