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Mezi Research Group
Mezi Research Group
Building Energy Efficiency usingKoopman Operator Methods
Michael Georgescu
Igor Mezi
Department of Mechanical Engineering
University of California, Santa Barbara
SIAM Conference on Dynamical Systems
Snowbird, UT
May 20th, 2013
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Mezi Research Group
U.S. Energy Use
Buildings
(~40% of Energy)
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Mezi Research Group
U.S. Energy Use
Buildings
(~40% of Energy)
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HVAC
(Almost 50%)
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Mezi Research Group
Monday, May 20, 2013 SIAM Conference on Dynamical Systems, May 2013
Introduction
A typical building / virtual model can contain 1000
different outputs (measured or simulated)
Data is collected at 5 30 minute intervals
Outputs include: temperature, airflow, energy
consumption, or comfort
Due to the dimensions (temporal and spatial) of data,
analysis is difficult
Koopman modes allow aggregate data at physically
significant time-scales to be studied
Analysis facilitates: fault detection, zoning, or model
calibration
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Mezi Research Group
The Koopman Operator
)(1 jj xfx
)())(()( 1 jjj xgxfgxUg
Given a finite dimensional nonlinear system
(e.g. a building simulation)
with output
The Koopman operator, U, is defined as:
Spectral properties of the Koopman operator are used to study the evolution of
observables produced by building simulations or building data
[Mezi 2005, Nonlinear Dynamics]
MMf :where
Mg:
The infinite dimensional, linear operator captures nonlinear, finite-
dimensional dynamics
Because the operator is unitary on the attractor, it can be studied through aspectral decomposition
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Mezi Research Group
Koopman Modes
[Mezi 2005, Nonlinear Dynamics]Monday, May 20, 2013 SIAM Conference on Dynamical Systems, May 2013 4/11
From this expression for , are a set of vectors called Koopman
modes, and are coefficients of the projections of observables onto the
eigenfunctions of the Koopman operator.
1}{ kkv)(xg
1
)()(k
kkk vxxg
Observables can be projected onto eigenfunctions, ,
with associated eigenvalues k
Mk:
For the operator, eigenfunctions can be calculated
satisfying the eigenvalue equationkkkU
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is in the span of eigenfunctions if is within the attracting set)(xg 0x
Koopman Modes
[Mezi 2005, Nonlinear Dynamics]Monday, May 20, 2013 SIAM Conference on Dynamical Systems, May 2013 5/11
)())((1
lim
))((
1
lim)(
*21
0
)1(22
1
0
2*
xgexfgen
e
xfgenxUg
in
j
j
ji
n
i
n
j
j
ji
n
)5.0,5.0[2ie
*g is a Fourier average
are eigenvalues
1
0
2* )(1
lim)(n
j
j
ji
nxge
nxg Note that
Koopman modes are calculated by taking Fourier averages of observables over
the spatial field.
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Name:
Location:
Size:
Function:
Levels:
HVAC:
Cost:
scape:
Student Resource Building
Santa Barbara
68,413 Square Feet
University Administration and Multi-functional Spaces
3
Combined mechanical and
natural ventilation
Sasaki Associates, INC.
ARUP
Two years of data collected from building automation system
Building model created in EnergyPlus with parameters
determined from design drawings and measurement
Case Study: Student Resource Building
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Concept
Building Output (Temperature)
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Amplitude of Dominant Koopman Modes
24Hr
12Hr
168Hr
(Wk)
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Spectrum of KO
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Other Building Management Outputs
Comfort (24Hr)Power Consumption (24Hr)
Amplitude
Phase
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Example: Model Zoning using KMs
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24Hr
During modeling,approximations are made
Example: Zoning, i.e.,
the division of buildingvolume into uniform
regions
Using KMs, rooms whichare coherent may be
zoned together
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Results of Zoning Approximations
Error is percentage difference in predicted HVAC energyAs zones are decreased by more than a factor of 2 from
original count, oversimplification begins to occur
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Comparison of Zoning Approximations
Floor1
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Summary
KMs facilitate spatial analysisof building data
In zoning example: zones
reduced by a factor of 2 beforeerror sharply increases
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Model Zoning
Fault DetectionOccupancy Prediction
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Thank You
Questions?
This work was partially funded by Army Research Office Grant
W911NF11-1-0511, with program manager Dr. Sam Stanton
Undergraduate Collaborators:
Erika Eskenazi (B.S. UCSB 12)
Valerie Eacret (B.S. TUFTS 12)
Ruben Diaz (B.S. UCSB 12)
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Animation: 24 HR Koopman Mode
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