This paper compares the performances of standard surrogate models in the development of an optimal control framework. The optimal control strategy is implemented on an Active Thermoelectric (ATE) window design. The ATE window design uses thermoelectric units to actively regulate the overall thermodynamic properties of the windows. The optimization of the design is a multiobjective problem, where both the heat transferred through the window and electric power consumption are minimized. The power supplies and the heat transfer are optimized under a reasonable number of randomly sampled environmental conditions. The subsequent optimal designs obtained are represented as functions of the corresponding environmental conditions using surrogate models. To this end, four types of surrogate models are used, namely, (i) Quadratic Response Surface Methodology (QRSM), (ii) Radial Basis Functions (RBF), (iii) Extended Radial Basis Functions (E-RBF), and (iv) Kriging. Their performances are compared using two accuracy measurement metrics: Root Mean Squared Error (RMSE) and Maximum Absolute Error (MAE). We found that any one of the surrogate modeling methods is not superior to the others over the whole domain for the optimal control of the ATE window.

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