Mathematical Modelling of Power Units

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

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

34. 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

35. 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

36. 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

37. 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

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

39. 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

40. 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

42. 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

43. Data from PI system

44. Steam turbine – an object for identification

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

46. 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

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

48. Chosen methods of computations Linear Programming • SIMPLEX

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

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