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

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

Simplified scheme

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 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

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: = for impulse turbine = for turbine with a small reaction = 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 Pressure difference (drop) for regulation stage: 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 Condensate outlet to lower exchanger Heated water outlet Heated water inlet CB A Steam cooling zoneCondensate cooling zoneSteam condensing zone Condensate inflow from higher exchanger x Steam inlet Load coefficient (Bośniakowicz):

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 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; mC1 0 – inlet heated water mass flow at reference conditions; TC1 0 – inlet heated water temperature at reference conditions.

An example – an empiric model of a chosen heat exchanger Coefficients received with a linear regression method: X – measured values Y – simulated values Standard deviation Expected value Random variables Correlation coefficient Covariance

Changes of a correlation coefficient Sample size Correlation coefficient

An example of calculations TC 1 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 e k Developed model of ARMAX type Identification – weighted minimal second power

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 x1x1 x2x2 x3x3 y1y1 y2y2

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 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 WM1...4 MW1...4 secondary air OFA air - total re-heated steam live steam CO O 2 NO x fraction combustion chamber temperature energy in steam outlet flue 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 system

Steam turbine – an object for 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

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” Chosen methods of computations f1f1 f2f2 f3f3 x 1 given x 2 given x3x3 x4x4 x5x5 x 6 given

measured: p, t measured: m,p,t measured: 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 – 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

variant A

variant B

variant C

variant D

variant E

variant F

variant G