ATE_MAO_2010_Jun
- 1. Comparison of Surrogate Models Used for Adaptive Optimal Control of Active Thermoelectric Windows Junqiang Zhang*, Achille Messac#, Jie Zhang*, and Souma Chowdhury * * Rensselaer Polytechnic Institute, Department of Mechanical, Aerospace, and Nuclear Engineering # Syracuse University, Department of Mechanical and Aerospace Engineering 13th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference Sep 13 – 15, 2010 Fort Worth, Texas
- 2. 2 Outline • Motivation • Design and Modeling of Active Thermoelectric (ATE) Window • Optimization of Energy Efficiency • Surrogate Models of Optimal Operations • Comparison of Surrogate Model Performance • Concluding Remarks
- 3. Simulation and optimization Surrogate models 3 Motivation Optimal control under varying conditions • Select appropriate surrogate models • Increase global and local accuracy Sample Condition space Sampling Condition 1 Optimal Solution 1 Condition n Optimal Solution n Optimal solutions
- 4. 4 Thermoelectric (TE) Units • A TE unit will form hot and cold sides when subjected to electric current. Melcor Center Hole Series Thermoelectric Cooler VTE Hot Side Cold Side Heat Flow TE Unit TE Unit
- 5. 5 ATE Window Design Window Front View Cross Section of Side View 5 Side Channel Clear glass Side Channel Fans Heating air flow in Winter 1 m .5 m TE Units & Fin Dividing Wall 6 mm Air Clear Glass Tinted pane 24 mm Out In 12 mm Inner Pane Middle Pane Outer Pane
- 6. Fan Model 6 Top Left Top Right Bottom Left Bottom Right Left Most Center Right Center Left Right Most
- 7. Fan Model 7 The Pressure Jumps of Fans Group Left Most Center Left Center Right Right Most Top Left (1-3k) Δpavg (1-k) Δpavg (1+k) Δpavg (1+3k) Δpavg Top Right (1+3k) Δpavg (1+k) Δpavg (1-k) Δpavg (1-3k) Δpavg Bottom Left (1-3k) Δpavg (1-k) Δpavg (1+k) Δpavg (1+3k) Δpavg Bottom Right (1+3k) Δpavg (1+k) Δpavg (1-k) Δpavg (1-3k) Δpavg Δpavg : average pressure jump produced by fans k : the slope of the pressure jumps • A linear pressure profile is assumed. • The total electric power consumption of all fans is negligible compared with that of the TE units. It is not minimized.
- 8. Weather Conditions 8 Indoor weather conditions: • Indoor temperature: 297 K (75 ℉) • Indoor heat transfer coefficient: 3.6 W/m2 Outside weather conditions: • Outside temperature: 259 to 309 K (7 to 97 ℉) • Wind speed: 0 to 21.5 m/s • Solar radiation: 0 to 1000 W/m2 Heating condition: Outside T < Inside T (usually in winter) Cooling condition: Outside T > Inside T (usually in summer)
- 9. Sample Weather Conditions 9 • Commonly used sampling methods • Latin hypercube sampling • Hammersley sequence sampling • Sobol’s quasirandom sequence generator algorithm • Number of weather conditions as training data: 10 x 3 (the small set population size) • Number of weather conditions as testing data: 10
- 10. 10 Optimization Problems Heating Cooling
- 11. Optimization Results 11 No. 1 2 3 4 …… 28 29 30 Red: heating conditions (Outside T < Inside T) Blue: cooling conditions (Outside T > Inside T) Weather Condition Tout (K) 284.0 296.5 271.5 277.8 …… 298.8 273.8 280.1 Tout (℉) 51.5 74.0 29.0 40.3 …… 78.2 33.2 44.5 Vwind (m/s) 10.75 5.38 16.13 8.06 …… 0.34 11.09 3.02 Esolar (W/m2) 500 750 250 625 …… 359 859 234 Optimal result VTE (V) 1.14 0.97 3.28 1.34 …… 1.37 1.86 1.84 Δpavg (Pa) 0.96 0.95 1.77 1.00 …… 1.75 1.89 2.00 k 0.07 0.07 0.02 0.08 …… 0.01 0.01 0.02
- 12. 12 Inputs and Outputs of Surrogate Models • Inputs are the three weather conditions (normalized): 1. Outside temperature, Tout 2. Wind speed, vwind 3. Solar radiation , Esolar • Outputs are the optimal values of 1. The absolute value of voltage supplied to the TE units, VTE 2. The average pressure jump over one fan, Δpavg 3. The pressure slope, k (scaled by 100) Outside Temperature Wind Speed Solar Radiation Thermometer Anemometer Light Sensor Thermostat Processor Memory TE Units Power Controller Fan Power Controller TE Units Fans
- 13. 13 Surrogate Models of Optimal Operation • Surrogate modeling methods • Quadratic Response Surface Methodology (QRSM) • Radial Basis Functions (RBF) • Extended Radial Basis Functions (E-RBF) • Kriging
- 14. Quadratic Response Surface Methodology 14 • Polynomial regression models using the method of least squares • Readily to implement • Show global trends Response Polynomial term Coefficient Error
- 15. Radial Basis Functions 15 The RBFs are expressed in terms of the Euclidean distance, One of the most effective forms is the multiquadric function: y (r) = r2 + c2 where c > 0 is a prescribed parameter. r = x - xi The final approximation function is a linear combination of these basis functions across all data points. Unknown parameters
- 16. Extended Radial Basis Functions 16 Extended Radial Basis Functions (E-RBF) are linear combinations of Radial Basis Functions (RBFs) and Non-Radial Basis Functions (N-RBFs). N-RBFs: The non-radial basis functions are functions of each individual coordinate instead of the Euclidean distance. E-RBF:
- 17. Kriging 17 The basic Kriging method estimates the summation of two parts. A linear model A systematic departure from the polynomial A popularly used exponential correlation model is
- 18. Surrogate Models of VTE 18 5 4 3 2 1 Tcr 260 270 280 290 300 T out V TE QRSM RBF E-RBF Kriging Median values: vwind = 10.75 m/s, Esolar = 500 W/m2 The explanation of the heat transfer process: • Tout < Tcr, TE units heat up the window. • Tout > Tcr, TE units cool down the window.
- 19. Surrogate Models of VTE 19 1.6 1.4 1.2 1 0.8 vcr1 vcr2 0 5 10 15 20 v wind V TE QRSM RBF E-RBF Kriging Median values: Tout = 284 K, Esolar = 500 W/m2 A possible explanation of the heat transfer process: • Factors: convection and thermal resistance of fins. • vwind < vcr2, the loss of the volumetric heat in the panes increases as vwind increases • vwind > vcr2, the thermal resistance of heat sink decreases as vwind increases
- 20. Surrogate Models of VTE 20 2 1.8 1.6 1.4 1.2 1 0.8 Ecr 0 200 400 600 800 1000 E solar V TE QRSM RBF E-RBF Kriging Median values: Tout = 284 K, vwind = 10.75 m/s The physics behind the models: • Esolar < Ecr, TE units heat up the window • Esolar > Ecr, TE units cool down the window
- 21. Surrogate Models of Δpavg 21 1.8 1.6 1.4 1.2 1 0.8 260 270 280 290 300 T out avg D p 1.3 1.2 1.1 1 p D 0.9 0.8 0.7 v wind 0 5 10 15 20 avg 2 1.5 1 0.5 0 200 400 600 800 1000 E solar avg D p QRSM RBF E-RBF Kriging QRSM RBF E-RBF Kriging QRSM RBF E-RBF Kriging vwind = 10.75 m/s Esolar = 500 W/m2 Tout = 284 K Esolar = 500 W/m2 Tout = 284 K Esolar = 500 W/m2
- 22. Surrogate Models of k 22 0.1 0.05 0 -0.05 -0.1 260 270 280 290 300 T out k 0.1 0.05 0 -0.05 -0.1 0 5 10 15 20 v wind k 0.08 0.06 0.04 0.02 0 -0.02 0 200 400 600 800 1000 E solar k QRSM RBF E-RBF Kriging QRSM RBF E-RBF Kriging QRSM RBF E-RBF Kriging vwind = 10.75 m/s Esolar = 500 W/m2 Tout = 284 K Esolar = 500 W/m2 Tout = 284 K Esolar = 500 W/m2
- 23. Comparison of Surrogate Model Performance 23 Two standard accuracy measures: • Root Mean Squared Error (RMSE) • Maximum Absolute Error (MAE)
- 24. Comparison of Surrogate Model Performance 24 Ten sets of testing data are used to evaluate the two performance metrics. Output Range Metric QRSM RBF E-RBF Kriging VTE 2.9245 RMSE 0.3056 (10%) 0.2754 (9%) 0.4564 (16%) 0.2855 (10%) MAE 0.5623 (19%) 0.5669 (19%) 1.1492 (39%) 0.537 (18%) Δpavg 16 RMSE 0.6132 (4%) 0.6858 (4%) 0.7198 (4%) 0.6234 (4%) MAE 1.2303 (8%) 1.3257 (8%) 1.5004 (9%) 1.3247 (8%) k 0.8 RMSE 0.0534 (7%) 0.0341 (4%) 0.0337 (4%) 0.0548 (7%) MAE 0.0962 (12%) 0.0578 (7%) 0.0595 (7%) 0.0996 (12%) • The values of RMSE and MAE for each output are at the same order of magnitude. • Surrogate models with higher accuracy for different outputs • VTE: RBF and Kriging • Δpavg: QRSM and Kriging • k: E-RBF and RBF
- 25. Trust Region based Local Accurate Models 25 • Divide the whole domain into appropriate regions. • Sample the space accordingly. • Latin hypercube sampling method x1 x2 • Evaluate surrogate models in the different regions at the sample points using testing data. • Based on the evaluation, select locally accurate models.
- 26. 26 Concluding Remarks • The values of the two error metrics are at the same order of magnitude. None of the surrogate modeling methods has an overall better performance than the others in accuracy. • For the different outputs of the optimal operation, particular surrogate modeling methods are preferable to represent the optimal operation.
- 27. 27 Thank you
- 28. Questions 28
Editor's Notes
- #4: Current energy production methods are unsustainable. Comment on the demand side.
- #5: We chose thermoelectric units for our system because they are very small and are solid state.
- #13: Introduce the heat sinks. Start with heat sinks, then move to single TE units, then to cascades of units
- #14: Introduce the heat sinks. Start with heat sinks, then move to single TE units, then to cascades of units