simbuild 2012

7/27/2019 Simbuild 2012

1/18

Mezi Research Group

Mezi Research Group

Creating zoning approximationsto building energy models using

the Koopman operatorMichael Georgescu

Bryan EisenhowerIgor Mezi

Department of Mechanical EngineeringUniversity of California, Santa Barbara

SimBuild 2012Madison, WI

August 1st, 2012

7/27/2019 Simbuild 2012

2/18

Mezi Research Group

Mezi Research Group

Introduction

When creating building energy models, approximations are

made to manage model complexity.

One approximation is zoning, i.e., how the volume of a

building is divided into regions where properties areassumed to be uniform.

In practice, creating zoning approximations is performed

cut-and-try.

A systematic approach to zoning is introduced and used to

study how model accuracy is influenced by a coarser

building representation.

1/16

7/27/2019 Simbuild 2012

3/18

Mezi Research Group

Mezi Research Group

Modal Decompositions

Structures

http://www.lusas.com/case/bridge/images/bosphorus_mode_shapes.jpghttp://www.acs.psu.edu/drussell/Demos/2-dof-coupled/2-dof.html

Spring Mass System

In oscillatory systems, motion can be

decomposed into normal modes.

These modes depend on the structure,

materials, and boundary conditions andexpress the motion of a system in terms

of their characteristic behavior.

2/16

Courtesy of Jiazeng Shan
http://www.lusas.com/case/bridge/images/bosphorus_mode_shapes.jpghttp://www.acs.psu.edu/drussell/Demos/2-dof-coupled/2-dof.htmlhttp://www.acs.psu.edu/drussell/Demos/2-dof-coupled/2-dof.htmlhttp://www.lusas.com/case/bridge/images/bosphorus_mode_shapes.jpg

7/27/2019 Simbuild 2012

4/18

Mezi Research Group

Mezi Research Group

Introduction (cont.)

Temperature output from a building simulation can also be

decomposed into modes.

With a modal decomposition, influential features a building

temperature evolution can be identified.

Zoning approximations are created by calculating modes

produced by a building simulation, and determining

approximations based off of similarities in these modes.

3/16

7/27/2019 Simbuild 2012

5/18

Mezi Research Group

Mezi Research Group

The Koopman Operator

)(1 jj xfx =+

)())(()( 1+== jjj xgxfgxUg

Given a finite dimensional nonlinear system

(i.e. 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

[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

4/16

7/27/2019 Simbuild 2012

6/18

Mezi Research Group

Mezi Research Group

Koopman Modes


Because the operator is unitary, the eigenvalue equation

holdskkkU =

Koopman modes describe the dynamics of observables at different frequencies,

and will be the basis for model order reduction of building models

is assumed to be in the span of eigenfunctions)(xg

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 expressed in terms of

eigenfunctions, , and

eigenvalues , , of the operator

k

Mk :

5/16

7/27/2019 Simbuild 2012

7/18

Mezi Research GroupMezi Research Group

Koopman Modes (cont.)


Koopman modes are calculated by taking harmonic averages of observables

over the spatial field

is in the span of eigenfunctions if is on the attractor)(xg 0x

)())((

1

lim

))((1

lim)(

*21

0

)1(22

1

0

2*

xgexfgene

xfgen

xUg

in

j

j

ji

n

i

n

j

j

ji

n

=

+

=

==

=

)5.0,5.0[jie 2

*g is a harmonic average

are eigenvalues

=

=1

0

2* )(1

lim)(n

j

j

ji

nxge

nxg Note that

6/16

7/27/2019 Simbuild 2012

8/18


ApproachTemperature Data

Zoning Approximations

Koopman Modes

7/16

7/27/2019 Simbuild 2012

9/18


Name:Location:

Size:Function:

Floors:HVAC:

Engineering Science Building

Santa Barbara80,500 Square FeetUniversity Administration and Multi-functional Spaces3Combined mechanical and

natural ventilation

Building model created of the Engineering Science Building Model parameters determined from design drawings and

measurement

Zoning Case Study

8/16

7/27/2019 Simbuild 2012

10/18


Detailed Model Floorplan

# ofZones

Area(sqft)

Total Building 191 80,500

Offices 94 15,000

Clean Room 14 10,000

Laboratories 49 21,000

Simulated Surfaces 2247

Simulated Windows 478

Floor 3

Floor 1

Floor 29/16

7/27/2019 Simbuild 2012

11/18


Koopman Operator Spectrum

Koopman modes arecalculated based off

of zone temperature Reduced zoning

determined from

modes which are

largest in magnitude

Period Max

Min

Mean

8760 Hrs 3.459 1.000 2.040

24 Hrs 2.055 0.121 0.626

12 Hrs 0.788 0.022 0.136

8 Hrs 0.371 0.004 0.048

6 Hrs 0.221 0.006 0.029

10/16

7/27/2019 Simbuild 2012

12/18


Comparison of Koopman Modes

Koopman modes

illustrate thermalbehavior of zones

at different time

scales

11/16

7/27/2019 Simbuild 2012

13/18


Zoning approximations are created by merging adjacentzones similar KM magnitude and phase

and are adjacent zones

If zone and zone are within a pre-specified interval

r, the zones are merged as one effective zone Shared walls of previously unmerged zones become

internal masses to the new zone (if desired) By adjusting the value ofr, models with a desired

amount of zone coarseness can be created

Creating Zoning Approximations

rxx jk

ik

7/27/2019 Simbuild 2012

14/18


Detailed Model Floorplan

# ofZones

Area(sqft)

Total Building 191 80,500

Offices 94 15,000

Clean Room 14 10,000

Laboratories 49 21,000

Simulated Surfaces 2247

Simulated Windows 478

Floor 3

Floor 1

Floor 213/16

7/27/2019 Simbuild 2012

15/18


Comparison of Zoning Approximations

N

191 Zones

112 Zones

60 Zones

Floor 3 Floor 2 Floor 1

14/16

7/27/2019 Simbuild 2012

16/18


Results of Simplified Models

Each model was simulated with a continuously operatingHVAC system (EnergyPlus Ideal Loads HVAC)

Error is percentage difference in predicted HVAC energy

HVAC Prediction Error

15/16

7/27/2019 Simbuild 2012

17/18


Summary

Using properties of the Koopman operator, a systematicapproach to creating zoning approximations is shown

A 191 zone model was reduced to 60 zones before a

sharp increase occurs in the rate of prediction error

With this method, neglecting the internal mass of

unmodeled walls causes error to increase linearly

Future work is to understand what causes fast growth of

prediction error with coarser zoning

16/16

M i R h G

M i R h G

7/27/2019 Simbuild 2012

18/18


Thank You

Questions?

[email protected]

This work was partially funded by Army Research OfficeGrant W911NF11-1-0511, with program manager Dr.

Sam StantonCollaborators:Erika EskenaziValerie EacretKazimir Gasljevic

Amorette Getty

simbuild 2012

Documents