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- Niklas Andersson, Dept. of Chemical Engineering, Lund University

- Johan Åkesson, Modelon AB

- KilianLink, Siemens AG

- Stephanie Gallardo Yances, Siemens AG

- Karin Dietl, Siemens AG

- Bernt Nilsson, Dept. of Chemical Engineering, Lund University

A combined cycle power plant is modeled and considered for calibration. The dynamic model, aimed for start-up optimization, contains 64 candidate parameters for calibration. The number of parameter sets that can be created are huge and an algorithm called subset selection algorithm is used to reduce the number of parameter sets.

The algorithm investigates the numerical properties of a calibration from a parameter Jacobean estimated from a simulation of the model with reasonably chosen parameter values. The calibrations were performed with a Levenberg-Marquardt algorithm considering the least squares of eight output signals.

The parameter value with the best objective function value resulted in simulations in good compliance to the process dynamics. The subset selection algorithm effectively shows which parameters that are important and which parameters that can be left out.

Full text at: https://www.modelica.org/events/modelica2014/proceedings/html/submissions/ECP14096809_AnderssonAkessonLinkGallardoyancesDietlNilsson.pdf

http://www.modelon.com/news/news-display/artikel/modelica-conference/

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- 1. Parameter selection in a combined cycle power plant Niklas Andersson*, Johan Åkesson**, Kilian Link***, Stephanie Gallardo Yances***, Karin Dietl***, Bernt Nilsson* * Dept. of Chemical Engineering, Lund University **Modelon AB ***Siemens AG
- 2. Presentation outline • Background - Combined cycle power plant - Process overview • Modelling • Parameter estimation • Parameter selection • Results • Summary
- 3. Scope • The start-up of a combined cycle power plant has been analysed. • The goal has been to calibrate a model, with the purpose to optimize the start-up while maintaining long lifetime of critically stressed components. • The model contains many candidate parameters. An algorithm has been used to assist in the selection of the best parameter sets.
- 4. cooling start-up Why? • The electricity demand varies during a day • Sun and wind variations affect the available amount of electricity • Market determines when the process is profitable to run. How? • Manipulate gas turbine load and by-pass valve to steam turbine • Header and drum are sensitive to rapid temperature changes Why calibration? • Optimization of CCPPs requires a model well tuned to the real process Background
- 5. Process overview
- 6. PHASE 1: • Gas turbine accelerated to full speed, no load • Gas turbine synchronized to grid PHASE 2: • Load of the gas turbine increased • Boiler starts producing steam • Generated steam bypassed to condenser PHASE 3: • Bypass valve closes • Steam drives steam turbine Included in calibration Start-up phases
- 7. Modelling approach • Models of HRSG developed in JModelica.org. • Hot gas side, statically modelled • Water side, dynamically modelled • 14 blocks modelled – Gas turbine – 3 reheaters (RH) – 3 high pressure super heaters (HPSH) – Evaporator – Drum – Header – 4 water injections • 764 eqs. (39 cont. time states) • Simulated as an FMU
- 8. Inputs to model Outputs from model
- 9. - The parameter estimation is done with a Levenberg– Marquardt algorithm. Δp = JT J + 𝜆JT J −1 JT R - The Jacobean matrix 𝐽 is estimated with finite differences (central difference). - The objective function to be minimized is formulated using weighted least squares 𝑄 𝒑 = 𝑖=1 𝑛 𝑡 𝒚𝒊 − 𝑦 𝑡𝑖, 𝒑 𝑇 𝑊( 𝒚𝒊 − 𝑦 𝑡𝑖, 𝒑 ) Calibration procedure
- 10. Candidate model parameters 64 parameters divided in 8 categories - Heat transfer constants 𝑘, 𝑘𝑖𝑛, 𝑘 𝑜𝑢𝑡 - Mass and volume 𝑚 𝐻2 𝑂, mFe, V - Sensor heat capacity 𝑐𝑎𝑝 - Valve dynamics parameter
- 11. Candidate model parameters Merged parameters – to reduce number of parameters parent children 𝑝9 = 𝑣 ⟹ 𝑝28 = 𝑝29 = 𝑝30 = 𝑣 A parent parameter can’t be calibrated together with its children
- 12. Parameter selection Why not choose all 64 parameters? - Large parameter confidence intervals - The sensitivity matrix gets singular (dependent parameters) Which parameters to choose? - There are 64 𝑛 𝑝 unique parameter sets with 𝑛 𝑝 number of parameters. Totally ~2 ⋅ 1018 parameter sets. A parameter selection algorithm is used to rank the parameter sets
- 13. How to choose parameters? Subset selection algorithm (SSA) - Subset Selection Algorithm ranks the parameters based on 𝛼 and 𝜅. (Cintrón et al. 2009) - Sensitivity matrix 𝜒 𝑝 = 𝜕𝑦 𝜕𝑝 calculated from nominal parameter values - Covariance matrix Σ 𝑝 = 𝜎0 2 𝜒 𝑝 𝑇 𝜒 𝑝 −1 - Parameter 𝛼 is the normalized parameter uncertainty, defined as Σ 𝑝 𝑖𝑖 𝑝 𝑖 - Parameter 𝜅 is the condition number of the sensitivity matrix. - An SSA score is introduced 𝜃 = lg 𝛼 + lg 𝜅
- 14. 𝛼 – Decreased accuracy of calibration 𝜅 - Solving difficulty. - Each point is a parameter set. - Low values of 𝛼 and 𝜅 is desirable. - When adding parameters the dot clouds get worse. SSA – ranking parameter sets
- 15. Parameter selection loops 2 loops are iterated for parameter sets for 𝑛 𝑝 = [1 … 7] Population of parameter sets: ℙ0 - all individual parameters ℙ 𝑐𝑜𝑚𝑏1, ℙ 𝑐𝑜𝑚𝑏2 - combination ℙ 𝑆𝑆𝐴, ℙ 𝑄 - filtered ℙ 𝑐𝑎𝑙1, ℙ 𝑐𝑎𝑙2 - To be calibrated SSA loop - Ranks all parameter sets from their SSA score. Best sets are calibrated. Calibration loop - Parameter sets with best Q continue to next iteration and are combined and calibrated
- 16. Combination Combination ℙ0 = {𝑝1, 𝑝2, 𝑝3, 𝑝4}ℙ𝑖𝑛 = {𝑝1,2, 𝑝2,3} All parameters (here 4 parameters) ℙ 𝑜𝑢𝑡 = 𝑝1,2,3, 𝑝1,2,4, 𝑝1,2,3, 𝑝2,3,4 ℙ 𝑜𝑢𝑡 = {𝑝1,2,3, 𝑝1,2,4, 𝑝2,3,4} Input parameter sets population
- 17. SSA Evaluation SSA Evalutation ℙ𝑖𝑛 = {𝑝1,2, 𝑝2,3, … } 𝜃 Input parameter sets population
- 18. Calibration Calibration ℙ𝑖𝑛 = {𝑝1,2, 𝑝2,3, … } 𝑄 Input parameter sets population Two populations to calibrate - ℙ 𝑐𝑎𝑙1 (from SSA loop) - ℙ 𝑐𝑎𝑙2 (from Calibration loop)
- 19. Filter Filter ℙ𝑖𝑛 = {𝑝1,2, 𝑝2,3, … } Input parameter sets population 𝑛 𝑐𝑢𝑡𝑜𝑓𝑓 ℙ 𝑜𝑢𝑡
- 20. ℙ 𝑐𝑎𝑙1 ℙ 𝑐𝑎𝑙2 Calibration results 𝒏 𝒑 = 𝟏
- 21. ℙ 𝑐𝑎𝑙1 ℙ 𝑐𝑎𝑙2 Calibration results calib Loop 𝒏 𝒑 = 𝟏 𝒏 𝒑 =2
- 22. ℙ 𝑐𝑎𝑙1 ℙ 𝑐𝑎𝑙2 Calibration results calib Loop calib Loop calib Loop calib Loop calib Loop calib Loop 𝒏 𝒑 = 𝟏 𝒏 𝒑 =2 𝒏 𝒑 = 𝟑 𝒏 𝒑 = 4 𝒏 𝒑 = 5 𝒏 𝒑 = 𝟔 𝒏 𝒑 =7
- 23. Best parameter set 24 6 6 6 13 13 13 13 13 16 16 16 1616 16 16 17 16 • The objective value is decreasing with increased number of parameters. • When 𝑛 𝑝 > 7, poor calibration convergence. (8 output signals) • Best parameter set covers the whole model. • 3 out of 6 parameters are merged. • Narrow confidence intervals for all parameters except 𝑝24
- 24. Best parameter set • The model responses follow the measurement data well. • All output signals improved • 59 calibrations were performed to reach the result Meas. data Calibrated Uncalibrated
- 25. Summary and Future Work Summary • SSA is a good method for reducing the number of parameters • All output signals were improved • Calibration loop performed better than SSA loop for this case Future Work • Perform optimizations of start-ups with the estimated parameters • Apply optimization result on real plant
- 26. Thank you!

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