Snowbird DS13

7/27/2019 Snowbird DS13

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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

Mezi Research Group

U.S. Energy Use

Buildings

(~40% of Energy)

Monday, May 20, 2013 SIAM Conference on Dynamical Systems, May 2013 1/11


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Mezi Research Group

Mezi Research Group

U.S. Energy Use

Buildings

(~40% of Energy)


HVAC

(Almost 50%)


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Mezi Research Group

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

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

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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Mezi Research GroupMezi Research Group

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)


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


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


Model Zoning

Fault DetectionOccupancy Prediction


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Thank You

Questions?

[email protected]

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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