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Mathematical Modelling of Power Units

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

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  1. MathematicalModelling of Power Units

  2. 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

  3. 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

  4. 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

  5. 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

  6. fuel SURROUNDINGS electricity SYSTEM steam

  7. 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

  8. 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

  9. Simplified scheme

  10. 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

  11. 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.

  12. 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

  13. 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

  14. 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

  15. Mathematical Modelling of Power Units Empiric approach - basic relations: • empiric process characteristics • parameters constraints • other relations - economical, ecological, technological

  16. Physical approach – a model of a boiler – an example mass and energy balances the boiler output and efficiency

  17. Physical approach – a model of a boiler – an example (cont.) electricity consumption boiler blowdown constraints on temperature, pressure, and flow

  18. Physical approach – a model of a boiler – an example (cont.) pressure losses specific enthalpies

  19. Physical approach – a model of a group of stages of a steam turbine boiler – an example mass and energy balances

  20. Physical approach – a model of a group of stages of a steam turbine boiler – an example (cont.) Steam flow capacity equation where:

  21. 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

  22. 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:

  23. 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

  24. 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):

  25. The heat exchanger operation parameters mass flows inlet and outlet temperatures heat exchanged heat transfer coefficient load coefficient

  26. 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.

  27. 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.

  28. 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

  29. Changes of a correlation coefficient Correlation coefficient Sample size

  30. TC1 dw An example of calculations Load coefficient changes in relation to inlet water temperature and reduced value of the pipes diameter.

  31. 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

  32. Most popular empiric models • Linear models • Neuron nets • MLP • Kohonen nets • Fuzzy neron nets

  33. 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

  34. 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

  35. 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

  36. 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

  37. Empiric models – examples of application • Dynamic optimization (models in control systems) • Virtual measuring sensors or validation of measuring signals

  38. 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

  39. 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

  40. 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

  41. Data from PI system

  42. Steam turbine – an object for identification

  43. A characteristic of a group of stages – results of identification

  44. 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

  45. Mathematical Modelling of Power Units • the system structure and the parameters writing in – numerical support

  46. Chosen methods of computations Linear Programming • SIMPLEX

  47. Chosen methods of computations Chosen methods of computations • Linear programming with non-linear criterion function • MINOS Method (GAMS/MINOS)

  48. Chosen methods of computations Optimization with non-linear function and non-linear constraints • Linearization of constraints • MINOS method

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