Accepted Manuscript

Title: A stochastic approach to modeling the dynamics ofnatural ventilation systems

Author: Anthony Fontanini Umesh Vaidya BaskarGanapathysubramanian

PII: S0378-7788(13)00222-3DOI: http://dx.doi.org/doi:10.1016/j.enbuild.2013.03.053Reference: ENB 4242

To appear in: ENB

Received date: 3-10-2012Revised date: 20-3-2013Accepted date: 30-3-2013

Please cite this article as: Anthony Fontanini, Umesh Vaidya, BaskarGanapathysubramanian, A stochastic approach to modeling thedynamics of natural ventilation systems, Energy & Buildings (2013),http://dx.doi.org/10.1016/j.enbuild.2013.03.053

This is a PDF file of an unedited manuscript that has been accepted for publication.As a service to our customers we are providing this early version of the manuscript.The manuscript will undergo copyediting, typesetting, and review of the resulting proofbefore it is published in its final form. Please note that during the production processerrors may be discovered which could affect the content, and all legal disclaimers thatapply to the journal pertain.

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A stochastic approach to modeling the

dynamics of natural ventilation systems

Anthony Fontanini, Umesh Vaidya,and Baskar Ganapathysubramanian1

Department of Electrical and Computer Engineering, 2215 Coover, Iowa StateUniversity, Ames, IA 50010,USA

Department of Mechanical Engineering, 2100 Black Engineering, Iowa StateUniversity, Ames, IA 50010,USA

Abstract

Heating ventilation and air conditioning (HVAC) systems in residential and com-mercial buildings make up 16% of the United States energy consumption. Utilizingnatural ventilation strategies is a low energy solution to reduce the energy usedby building environmental control systems. Design of effective natural ventilationstrategies is challenging because of inherent stochasticity in interior (machine loads,number of people) and exterior conditions (wind load, outside temperature). How-ever, by exploiting the natural dynamics of building systems, efficient design andcontrol seems possible. We explore this idea by introducing a stochastic approachto analyze the natural dynamics of building systems (under natural ventilation) byexplicitly incorporating the effects of stochastic wind speeds and stochastic internalloads. We show that complex dynamics in the form of bi-stable behavior emergeswhen considering a single zone building with stochastic inputs. We show that ne-glecting these complex stochastic dynamics leads to inaccurate predictions in thethermal response, especially for natural ventilation. We compute the sensitivity ofthe system with respect to various system parameters which provide insight intodeveloping robust design guidelines. The techniques presented aid in the design pro-cess, and are a step towards adaptive, efficient, robust control of natural ventilationsystems.

Key words: natural ventilation; stochastic building analysis; thermal modeling;uncertainty quantification; stochastic modeling; bifurcation analysis

1 Corresponding author: Ganapathysubramanian, Fax: 515 294-3261, Email:baskarg@iastate.edu, URL: http://www3.me.iastate.edu/bglab/

Preprint submitted to Elsevier 20 March 2013

*Manuscript

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1 Introduction

Current residential and commercial Heating Ventilation and Air Conditioning(HVAC) systems account for roughly 16% of the United States total energyconsumption [1,2]. Natural ventilation systems are an excellent low energysolution to help reduce the amount of energy used by the HVAC system. Al-though natural ventilation systems are a natural choice to help save energy,designing and controlling natural ventilation systems is difficult [3,4]. This dif-ficulty arises due to natural fluctuations of outside temperatures, wind speeds,internal heating loads, and occupant activities. This is further amplified bythe lack of capabilities to adjust for large fluctuations in exterior and inte-rior conditions as well as the propensity of the system to get stuck in a sta-bly stratified pattern [4]. Understanding how these fluctuations affect naturalventilation systems and leveraging these natural dynamics [5] for control canpotentially help make natural ventilation systems more attractive. This is theprimary motivation for the present study.

Building designs are traditionally evaluated by energy modeling techniques.The industry standard for energy modeling techniques is to incorporate theflow of energy, thermodynamics, psychrometrics, and air leakage by the useof an energy balance [6,7,8,9,10,11,12]. This technique is best known as theHeat Balance Method (HBM). The HBM method is used (and extensively val-idated) by energy modeling software (AtticSIM, EnergyPlus, TRNSYS, andmany others) [13,14,15]. Using the energy balance approach, or lumped pa-rameter analysis, results in a system of coupled time dependent ordinary differ-ential equations (ODEs). The ODE formulation has many benefits, includingthe fact that a system of ODEs are relatively easy to solve compared to partialdifferential equations used in Computational Fluid Dynamics (CFD) simula-tions, and a system of ODEs can easily be manipulated to form a state spacerepresentation (dynamical system). Advanced techniques from dynamical sys-tems theory can then be leveraged to analyze and control these systems.

Such lumped parameter analysis has been shown to work very well in thecontext of forced ventilation systems, where the effects of fluctuations are in-variably inconsequential [6,16]. However, in natural ventilation systems, fluc-tuations greatly effect the building physics, and a deterministic approach maynot give a complete picture of the dynamics. For instance, fluctuations in ex-terior temperature, internal heat loads, and wind speed can result in a largedivergence from the desired zone temperatures. These fluctuations can becaused by variations in the number of people within the space, the activitiesbeing performed by the people in the space, equipment/appliances/lightingturning on and off, or by the stochastic nature of exterior wind conditions.In this context, incorporating weather data into thermal analysis of buildingsis a standard practice [17]. EnergyPlus allows the inclusion of meteorological

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data for a number of different analysis [15]. While this accounts for the ex-ternal weather conditions, the techniques are deterministic i.e. a set of inputsgive a unique temperature response and does not account for randomness in-herent in building systems. Thus, there exists a need for a numerical processto incorporate both internal and external fluctuations, as well as treating thebuilding system as a stochastic system for a more robust statistical descrip-tion of the building. Such an approach will provide valuable insight into theeffect of random fluctuations on natural ventilation dynamics. In this context,there has been some recent work on transition dynamics for natural ventilationsystems [18]. Minimum perturbation magnitude and perturbation time havebeen established for the deterministic dynamical system to transition betweendifferent flow regimes. Uniform fluctuations have been analyzed in natural ven-tilation systems to determine how small amounts of noise affect the stabilityof the fixed points of the dynamical system [18]. Analyzing this bifurcationbehavior gives design insight on limiting fluctuations to stop the bifurcationsor, alternatively, to determine when the bifurcation results in a more com-fortable space. This analysis has to be performed to ensure that ASHRAE’scomfort requirements are satisfied [19,20,21], while minimizing the amount ofenergy used.

Our contributions in this paper are the following: 1) We formulate and an-alyze the stochastic temperature dynamics of natural ventilation systems, 2)We develop methods for representation of the stochastic inputs necessary foranalysis of natural ventilation systems using a data-driven approach, 3) Weshow how uncertainty in the stochastic inputs propagates though the system –both under steady state behavior and during transient dynamics, 4) We showthat complex dynamics in the form of bi-stable behavior emerges with stochas-tic input parameters, 5) We investigate the sensitivity of the system to thesestochastic input parameters, and 6) We illustrate visualization techniques tobetter visualize the stochastic solution. These new techniques and the data-driven numerical framework allows us to set bounds on the variability of inputparameters. Statistical descriptions of variables allows for seamless merging ofsix-sigma design analysis with numerical analysis. We envision that designerswill be able to specify variability limits on the designs of natural ventilationsystems. Limits on the designs reduce the zonal response to the fluctuationsand result in more controllable zone dynamics. Stochastic parameter sensitiv-ity analysis allows for simulation based optimization of building systems.

The organization of the paper is as follows: the next section details the dynam-ical system (lumped parameter model) formulation of the natural ventilationsystem. We outline our numerical framework that includes (a) development ofthe various stochastic inputs, (b) uncertainty propagation in the steady statesystem, (c) modeling the stochastic transient dynamics. The results sectionshowcases application of this framework to the analysis of a single-zone natu-ral ventilation system. We conclude by discussing implications of the developed

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framework and outlining future avenues of analysis.

2 Building Model

The building used for this investigation closely resembles a small office spacedescribed in U.S. Department of Energy commercial building reference [22].We assume that the office is located in central Iowa. The office occupies thelower level of a multistory office building. The small office is broken up intofour perimeter zones and one core zone. Of the five zones in the office space,a corner zone having two exterior walls facing south and west is analyzed.The office space is intended for a workspace for individuals. For descriptionof the variables the reader is directed to the nomenclature, Table 1. Moreinformation about the office space can be seen in Table 2.

2.1 Dynamical System: Natural Ventilation

To set up the deterministic dynamical system equations, using the Heat Bal-ance Method (HBM), we start by performing an energy balance on each zoneof the building. For a natural ventilated building, the time rate of change of thetemperature of a single-zone building, represented in Fig. 1, can be describedas energy entering or leaving a building by air moving through the building,conduction into or out of the building, and energy being generated within thebuilding (Eq. 1). The energy balance can be mathematically written as:

MacpdT

dt= ρacpq (Te − T ) + (UA) (Te − T ) + E (1)

where q =(

CdA)

√

∣

∣

∣2ghβ (T − Te)− 2pwρa

∣

∣

∣, with pw = 12ρa (Cpu − Cpl) v

2ref

and 1

(CdA)=

√

1(CduAu)

2 +1

(CdlAl)2 .

This mathematical model is discussed at length by Yuan and Glicksman [18].The energy balance can be further simplified by expressing it in a dimensionlessform. Such a formulation is useful because the equation is no longer dependenton the exterior temperature Te, but instead on the temperature differencefrom outside to inside T − Te. Using dimensionless quantities, γ = (UA)

Macp, α =

EMacp

, φ =(

CdA)

ρacp√2gh, p∗ = pw

ρagh, and T ∗ = β (T − Te) yields a

dynamical system of the form, T ∗ = f (T ∗), given by:

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dT ∗

dt= α− T ∗ ×

(

φ√

|T ∗ − p∗|+ γ)

; (2)

The mathematical model is a one dimensional non-linear system with the stateof the system being the non-dimensional temperature, T ∗. It is importantto note that this formulation is derived under certain assumptions: (1) Theair in the zone is at one single temperature T at a given time t (uniformmixing). Although this is seldom true [23,4], the HBM is an excellent validatedreduced order model for determining the thermal response of a building [15],(2) the model includes only sensible heat [18], (3) the air flow rate throughthe windows is sufficiently larger than air leakage/infiltration through the restof the building’s envelope, and (4) the air properties are considered constant.

2.2 Evaluation of Stochastic Inputs

The parameters in Eq. 1 can be classified into four distinct groups: air prop-erties, environmental variables, building design parameters, and occupancybased variables. The air properties and building parameters are fixed for aspecific system. The dependence of the outdoor temperature, Te, is removedvia non-dimensionalization (Eq. 2). The other environment variable – windvelocity, vref , is stochastic and can change rapidly throughout the day. Fur-thermore, wind can have weekly, monthly or seasonal effects. The occupantbased variable consists of the energy generation term, E. This energy gener-ation term is stochastic, since energy generated within the room depends onthe number of people within the room, occupant activities, miscellaneous of-fice equipment, and other base loads. The energy generation term also variesthroughout a given day, and may be different during different days of theweek. As a first step, we consider these two variables (wind load, energy gen-eration) to be stochastic and construct a data-driven stochastic representationfor them.

2.2.1 Stochastic Energy Generation

The energy generated within a space is dependent on a number of differ-ent internal loads, Fig 2. The energy generation term can be broken upinto two categories; energy generation from a base load due to office equip-ment/appliances/lighting, and energy generation due to the occupants cur-rently in the space:

E = Eocc + Ebase (3)

Both terms in Eq. 3 are stochastic. The base load within the space is decom-posed into random loads (cycling of miscellaneous office equipment) and a

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deterministic lighting load that follows a given schedule throughout the day(Fig. 2). The occupancy of the room depends on the arrival and departurerate of people. Each person has different metabolic rates based on their hered-ity, height, weight, and the activity being performed. All of these loads arezero or strictly positive in the zone. Each of these energy generation variablesfluctuates differently throughout the day. We discuss each individually.

Base loads: Baseline loads are loads that are not affected by occupants. Thebase load is determined by miscellaneous internal loads (office appliances andlighting loads). Lighting loads, Elight, for small offices have been shown to beroughly 11.8 (W/m2) during working hours and approximately 15 percent ofthe peak load during unoccupied times of the day [24]. Loads from miscella-neous office equipment, Emisc, (computers, monitors, printers, copy machines,water coolers, refrigerators, coffee makers, etc.) have been shown to vary be-tween 2.15 − 8.61 W/m2 during working hours and 0 - 4.31 W/m2 duringnon-working hours [24]. The lighting load is considered deterministic, sincelights typically output a constant amount of heat. The load from the officeequipment is taken to be a random variable, since a number of different ap-pliances are lumped together and cycle times are different. The load from theoffice equipment is a normal random variable, with mean µmisc (6.46 [W/m2]during the working hours and 2.12 [W/m2] during non-working hours), stan-dard deviation σmisc (1.08 [W/m2] during working hours and 0.72 [W/m2]during non-working hours), and standard normal random variable Z. Treat-ing the variable as a normal variable allows the load to fluctuate around theaverage expected load and between the data dependent range. Each base loadis then multiplied by the floor area, Af .

Ebase = Af × (σmiscZ + µmisc + Elight) (4)

Energy generation per occupant: The energy generated within the roomis impacted by the number of people and their activities. The number of peoplewithin a space changes throughout the day. Some of the activities include walk-ing, typing, or any leisure activity. For an office space the metabolic rate foran occupant can fluctuate from 60 to 150 (W/m2)[19]. We take the metabolicrate to be a normal random variable that fluctuates around a mean µmet (115[W/m2]), with a standard deviation σocc (15 [W/m2]). This approach allowsthe metabolic rate per person, Eocc/person, to fluctuate throughout the datadependent range. The DuBois body surface area, As, (usually taken to be 1.8m2 [19]) is the area that the metabolic flux is applied, Eq. 5.

Eocc/person = As × (σoccZ + µmet) (5)

Number of occupants: After the internal loads per person have been esti-

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mated, the number of people, Np, within the space must be determined. Thenumber of people within a space can be mathematically represented as a se-ries of events, with an event being defined as one person entering or leavingthe space. The number of people within the room is then a stochastic processbased on two parameters:

Np = f (〈λrate〉, 〈Np〉) (6)

The first parameter is the expected rate at which individuals enter or leave aspace, 〈λrate〉. The second parameter is the expected number of people withinthe space, 〈Np〉. Note that the expected rate of arrival and the expected num-ber of people can be time dependent. These variables can be fully defined asa combination of two processes. The first process (1) enumerates the times atwhich people arrive/depart the space. The second process (2) identifies whatevent occurred (an arrival or departure).

(1) Arrival/Departure Times: The times at which people enter or leave theroom can be mathematically represented as a Poisson process [25]. The Pois-son process is a counting process that counts the number of events (arrivals ordepartures) in a given time period. The time between events are independentand are exponentially distributed [26]. The parameter defining the Poissonprocess is the expected rate at which arrivals/departures occur. The times be-tween successive events can then be calculated by sampling from an exponen-tial distribution with mean equal to the expected rate of arrivals/departures,〈λrate〉.

The expected rate can be determined from an occupancy schedule. An occu-pancy schedule displays the number of the people in the space (normalized bythe maximum expected number of people, 〈Nmax〉) at different times of theday. Studies done by the National Renewable Energy Laboratory (NREL) andthe US Department of Energy (DOE) have estimated occupancy schedules fora number of commercial buildings. Figure 3 shows the occupancy density persquare foot of a commercial small office space [22]. We fit this data with apiecewise function, Eq. 7 to enable subsequent analysis. A piecewise fit hasseveral advantages, including (a) the fit can easily approximate the rate atwhich people enter and leave the space during the morning commute, thelunch period, and the evening commute hours; and (b) specifying a functionfor the occupancy schedule analytically defines the rate of arrivals/departures.The expected rate is calculated by differentiating the occupancy schedule withrespect to time, as in Eq. 8. This is plotted in Fig. 3.

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〈Np〉〈Nmax〉

=

0.95e−([t−8.5]/1.5)2 , 0 ≤ t < 8.5

0.95− 0.45e−([t−12]/0.5)2, 8.5 ≤ t < 15.5

0.05 + 0.90e−([t−15.5]/1.75)2 , 15.5 ≤ t < 24

(7)

〈λrate〉〈Nmax〉

=

∣

∣

∣

∣

∣

d

dt

[

〈Np〉〈Nmax〉

]∣

∣

∣

∣

∣

(8)

(2) Arrival/Departure Process: The Poisson process is used to modelthe time interval between events. An estimate of the change in the numberof people in the room (transition) when an event occurs is required to fullymodel the occupancy schedule. Recently, Page et al. [27] has shown that thistransition is only dependent on the current number of occupants – i.e. itcan be represented as a Markov process [25]. Markov processes have recentlybeen successfully used to model space occupancy [26,27]. Given that i peopleoccupy the space at an instant, the Markov process assigns a probability Pij tothe number of people becoming j after the next event. These probabilities arecalled the transition probabilities and are collected into a transition probabilitymatrix [25].

We construct such a transition matrix for a system satisfying these properties:(1) we define an event as an individual entering or leaving the room, so thetransitional probabilities follow a birth or death process [25] (i.e. Pii = 0),(2) when there are zero people in the room, the only event that can occur iswhen an individual enters the room, (3) we assume that an event consists ofone individual entering (or leaving) the room. This assumption is made formathematical simplicity, but can be relaxed in a very straightforward way.The transition matrix is given by;

P =

0 1 0 0 0 · · ·P1 0 1− P1 0 0 · · ·0 P2 0 1− P2 0 · · ·...

. . .. . .

. . .. . .

. . .

(9)

Here, the off-diagonal entries follow a Poisson distribution Pk =Nk

p e−Np

k!, with

Np being the number of expected people in the space. The expected numberof people within the office space throughout the day is deduced directly fromthe study done by NREL and the DOE discussed above, Fig. 3. A realizationof the number of people based on a Markov process is shown in Fig. 4.

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2.2.2 Stochastic Wind Speed

In order to approximate the wind speed fluctuations in our office space, weutilize wind data collected over a five year period (2007 − 2011). We focuson a stochastic model that encodes wind speed variability during the summermonths (June − August). The data was collected by a WS-2308U La CrosseTechnology weather station positioned in Ames Iowa at the global coordinatesN 42o 3’ 37” and W 93o 38’ 31”. The weather station used an anemometer tomeasure the wind speed accurately to 0.05 m/s for wind speeds greater than0.5 m/s. The weather station reported the data at 5 − 10 minute averages. Thedata-set considered has a total of 105,857 measurements over the five summers,2007 - 2011. We assumed this to be realizations of a random variable andconstructed the probability density function (PDF) and the cumulative densityfunction (CDF) of this data (Fig. 5). The data was fitted to an exponentialdistribution using MATLAB’s statistical toolbox. 2 The fit resulted in anexponential distribution with rate of decay parameter λwind = 1.52 (m/s). 3

Buildings are usually partially obstructed, and may be in the wake or partialwake of other buildings. The obstructions can change the effective wind loadacting on the building. Thus, the effect of these obstructions have to be takeninto account into the wind load model. A natural way for accounting for thisis via the ’coefficient of interference’. The mean wind speed on a building,λbuild, can be represented as a coefficient of interference, CI , multiplied by theuninterrupted mean wind speed: λbuild = CIλwind.

2 The choice of an exponential PDF was made because the distribution passed theKolmogorov-Smirnov test to a significance level of 5%. Note that the distributioncould possibly also be a distorted Weibull distribution with the removal of windspeeds less than 0.5 m/sec. However, we choose an exponential PDF to account forthe fact that the wind speed as ’seen’ by the windows in the building are essentiallyobstructed. Obstructions existing in the area around the building’s exterior are anormal occurrence, whether they be landscaping or other buildings. To account forsuch obstructions, we assume that there is a large probability that the wind speedis near zero at the windows, and hence the choice of an exponential distribution.Note however, that since the proposed approach is data driven, any continuous ordiscrete PDF for the wind speed can be used that fits the data.3 The fit produces an L2 error norm (calculated between the actual wind speed CDFand the fitted exponential CDF) of 0.7%. The L2 error represents the differentialarea between two data sets as a percentage of the area of the fitted data set. Most ofthe error is at low wind speeds at which the measurement device does not accuratelymeasure the wind speed.

vL2 =

√

∑ns

i=1 (vex,i − vfit,i)2

∑ns

i=1 v2fit,i

(10)

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2.3 Uncertainty Propagation: Steady State Analysis

Buildings are almost always in some dynamic state, but a lot of insight duringearly stages of design can be gained from analyzing the steady state system.This essentially entails solving for the fixed points of the system dynamics.The fixed points can also be used to predict if bifurcation (mode-shifts) arepossible [28]. By incorporating stochastic effects into this analysis, more real-istic analysis can be performed. This is particularly useful to construct boundsover which the temperature fluctuates by propagating the uncertainty of thestochastic inputs through the system. This will subsequently allow for the de-sign to embrace and utilize the fluctuations to help in temperature regulation.

We utilize a stochastic collocation approach [29] to construct the stochasticsteady state response of the system. The collocation approach is based onsampling a finite set of realizations of the stochastic input variables (windspeed and energy generation in this work) and constructing the steady stateresponse for these realizations. The probability of each steady state responseis simply the (joint) probability of each of the stochastic inputs [29]. Thecollocation approach essentially converts a stochastic ODE into a finite set ofdeterministic ODE’s where each ODE (and its solution) is associated with aweight. This weight is the probability of its inputs – energy generation, PE ,and wind speed, Pv –occurring. A detailed outline of this approach is given inthe Appendix.

2.4 Stochastic Transient Formulation

With the introduction of the stochastic energy generation term, ξE , and stochas-tic wind speed, ξv, the time response of the system (given by Eq. 1) is no longerdeterministic. The time response is now a function of the temperature differ-ence as well as the stochastic energy generation, and the stochastic wind speed,Eq. 11. The stochastic dynamical system formulation results in a stochasticfirst order ODE which can be integrated (marching in time) to model thetransient behavior of the temperature within the zone. Time integration isdone numerically by using a 1st order explicit Euler time discretization, Eq.12.

dT ∗

dt= f (T ∗, ξE, ξv) (11)

T ∗

t+∆t = T ∗

t + f (T ∗

t , ξE, ξv)∆t (12)

We start the simulations by initializing the random number generator. Weutilize a random number generator that has a very large time period of 2144

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[30]. The framework proceeds to step through time, where at every time stepa realization of the next event, the stochastic energy generation term, and thestochastic wind generation term is computed.

2.5 Stochastic sensitivity

We perform a sensitivity analysis to investigate the effect of the model param-eters on the temperature response of the building. We extend the sensitivityanalysis to include both the building parameters and the stochastic terms. Thesensitivity analysis is performed by running a series of transient simulations todetermine the statistical description (PDF) of the temperature in each zone.Each parameter of interest is perturbed independently (with varying magni-tudes) resulting in a series of PDFs. Once all the PDFs have been created, theL1 distance (Eq. 13) is then calculated between the reference parameter val-ues and the perturbed parameter values. The L1 distance represents the totalchange of the PDF with respect to the base PDF. Parameters with a higherL1 distance are stochastically more sensitive than parameters with lower L1

distance.

dL1 =npdf∑

i=1

∣

∣

∣PDF(base)i − PDF

(purt)i

∣

∣

∣ (13)

3 Results and Discussion

The result section is broken up into two sections to cover the steady stateanalysis and the transient analysis. First, the steady state system is solved toshow the different regions in which the steady state solution can exist overthe range of the inputs. The uncertainty in the stochastic inputs is then prop-agated through the steady state system to show where the non-linear systemwill naturally gravitate. Once the steady state system has been explored andunderstood, the transient system is investigated. The transient analysis showsthe dynamic changes and the thermal response of the building throughoutthe day with respect to the inputs. The differences between the deterministicmodel and the stochastic model is outlined. Finally, changes in the parame-ters of the stochastic inputs and the building are investigated to determinethe sensitivity of the stochastic variables on the building model.

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3.1 Solution Space for Steady State Analysis

The solution space representation under two parameter variation – energy gen-eration and wind speed is shown in Fig. 6. The solution space was collocatedusing a grid of 501×501 points. At each of the collocated input points a set ofsolutions are possible. The surfaces represent the temperature difference be-tween the room and the exterior temperature as a function of the two randomvariables. The purple surface represents the values at steady state in whichthe only possible steady state value is the buoyancy driven steady state. 4 Inthis state, the wind load is not large enough to overcome the buoyancy forces.In contrast, for the three surfaces [blue, green, and red], three possible steadystate points exist. Of these three states, two are stable [red and blue], while oneis unstable [green]. The red surface denotes the buoyancy driven steady statewhere the wind speed is not great enough to overcome the buoyancy forces.However, the system could still make a transition to the wind driven steadystate under a large enough perturbation. The blue surface represents the winddriven steady state. In this state, the forces caused by the wind blowing onthe building are stronger than the buoyancy forces. The system can transitionfrom the wind dominated state to the buoyancy driven state under a largeenough perturbation. The green surface is unstable and any perturbation willresult in the system moving to either the wind or buoyancy steady state. Forthe building parameters in this system, the unstable surface is extremely closeto the [red] buoyancy surface. Under small perturbations the system couldmove to the wind dominated state.

Propagating the uncertainty through the solution space shows how likely thesystem is to be in the different steady state regions seen in Fig. 6. The single-zone system can be thought as a naturally fluctuating system around a setof fixed points. During the non-working hours, the fluctuations are small andsporadic. On the other hand, during the working hours the system under-goes a long (8 hour) sequence of perturbations in input parameters to givethe thermal response during the work day. From the analysis point of view,we seperate the two behaviors and analyze the system behavior due to twodifferent sets of inputs (unoccupied hours, and occupied hours).

By sampling from the occupancy schedule (and using the Markov processframework), the probability density (PDF) for the energy generation for boththe unoccupied and occupied times of the day was constructed, Fig. 7. Theenergy generation PDF during the mostly unoccupied hours has a large spike

4 The definitions of buoyancy driven flow and wind driven flow are explained inthe J. Yuan [18]. Briefly, the wind driven flow is defined as air entering the spacethough the upper opening and exiting the space through the lower opening in Fig. 1.Buoyancy driven flow is defined as air entering the space through the lower openingand exiting through the upper opening in Fig. 1.

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around 0.4 (kW) and a smaller spike around 1.7 (kW). The larger spike is dueto the fluctuating office equipment. The smaller spike is due to the arrival ofan individual(s) in the space. The energy generation PDF during the occu-pied time of the day has a similar behavior with a bimodal distribution. Thespike around 0.4 (kW) is due to office equipment. The second section of thedistribution is due to people moving in and out of the space throughout theday. The wind speed PDF is independent of the time of day and is analyticallydefined by the experimental data, Fig. 5. The random variable is exponentiallydistributed with a mean wind speed 1.52 (m/s).

The (joint) PDF of the system was calculated based on the PDFs for theenergy generation term and the wind speed. The joint PDF was calculated at501 × 501 points to match directly with the collocated points of the solutionspace. The joint distribution values at the collocated points are applied to thesolution space to give the contoured surfaces of the fixed points. The contoursrepresent the probability of the particular fixed point occurring, Fig. 8. Duringthe mostly unoccupied times of the day the system is most likely to be eitherin the buoyancy driven state and the wind driven state, if there are zero peoplein the room, Fig. 8a. If there is a person in the space the system tends to bein the buoyancy driven state, Fig. 8a. The temperature difference between thewind driven state and the buoyancy driven state in this region is small, 2.0 -2.5 (C), so a large gust of wind or a individual entering the room could causethe system to switch states.

During the mostly occupied time of the day the same spike occurs becausethere are some instances in which there are zero people in the space, Fig. 8b.Otherwise, the system tends to frequent either the buoyancy driven state orthe region with three steady state solutions. When the system is in the regionin which three steady states solutions are possible, a bifurcation is likely tooccur from the buoyancy state to the wind driven state. The likely bifurcationis a result of the unstable fixed point being extremely close to the buoyancydriven fixed point. A large gust of wind could potentially cause the system toswitch states (even when there are people in the space). During the switchingof states the system would increase in temperature first before decreasing.The temperature swing during the switching of states would be roughly 5-10(C). It is worth emphasizing that such a mode-switch could cause extremediscomfort to the occupants.

3.2 Transient Thermal Building Analysis

The transient analysis is broken up into two sections. The thermal response ofthe office space is simulated using the deterministic approach and the stochas-tic approach. From the simulations of the building, benefits and shortcomings

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of the different approaches are illustrated. Then stochastic input parametersand building design parameters of the stochastic system are studied to deter-mine the effect of each variable on the building dynamical system (sensitivityanalysis).

3.2.1 Deterministic Approach vs. Stochastic Approach

The office building (see Table 2 for building information) was simulated foran entire work week. The time step, ∆t, chosen for both the deterministic ap-proach and the stochastic approach is 1 second. The chosen time step was seento be sufficient for temporal convergence for the deterministic approach as wellas for convergence of the probability density functions for the stochastic ap-proach. For the deterministic case, the wind speed remained constant through-out all the simulations and the energy generation rate was time-varying, butstatistically deterministic (i.e. same variations each day). For the stochasticcase, both the wind speed and the energy generation are modeled as stochas-tic (section 2.2). The initial condition used to start the simulation is the winddominated steady state value. For multiple day simulations of the stochasticsystem the initial condition was set to the final temperature of the previousday. A realization of the temporal response of the building for the stochasticsystem and an example of the deterministic system with their respective in-puts can be seen in Fig. 9. The deterministic system seems to get stuck at thesteady state temperature (T ∗) around 4.4 (C). The stochastic system does notget stuck at the buoyancy driven state around 4.4 (C), but does bifurcate tothe buoyancy state during the evening hours (around hour 18 - 26, where thestochastic model results in an individual in the space).

The differences in the two different approaches is easily seen when varyingthe parameters slightly. By varying the wind speed, multiple behaviors canbe seen for the deterministic system, Fig. 10a, while the stochastic systemdisplays more consistent and physically more meaningful behavior, Fig 10b.The deterministic system displays three different behaviors: The deterministicsystem can experience a perturbation (due to occupant energy generationfor cases λwind = 1.52(m/s)) during the day that is not strong enough toforce the system to the buoyancy state. The system then returns to the samestate when the perturbation is removed (evening hours). For the same energygeneration rate, but at a different wind speed value (λwind = 0.76 (m/s)), thedeterministic system experiences a perturbation that is large enough to forcethe system to the buoyancy steady state. Furthermore, when the perturbation(evening hours) is removed the system stays in at the buoyancy state. Finally(for certain wind speeds, λwind = 1.14 (m/s), 0.38 (m/s) ) the deterministicsystem can experience a perturbation that forces the system to the buoyancysteady state and is forced back to the wind driven state. In contrast, thestochastic system displays a more predictable and smooth result, Fig. 10. The

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temperature response curve rise and fall due to an increase and decrease inenergy generation. Note that the temperature response curves increases witha decrease in wind speed (since increased air speed enhances heat transfer).This monotonic behavior is not displayed by the deterministic system. In thestochastic simulations, the stochastic fluctuations destabilize the buoyancysteady state when the unstable fixed point is near the buoyancy steady state.The destabilization of the buoyancy steady state due to the introduction ofnoise causes the expected monotonic behavior.

The introduction of noise through the stochastic processes produces the physi-cally meaningful and expected result. The stochastic system displays bi-stablebehavior due to the perturbation introduced by occupants entering the spaceduring the day and leaving during the evening hours. Bi-stable behavior is thejumping between the buoyancy and wind dominated states over the course ofthe simulation. Note that the deterministic system cannot capture the behav-ior that the stochastic system displays. For instance, after the energy gener-ation during the day hours was removed, the system stayed at the buoyancysteady state. This is not physically realistic, and such deterministic analysis ofthe underlying dynamics could lead drawing incorrect conclusions from mod-eling results – especially with natural ventilation systems. These misleadingresults can further be obfuscated when the results are plotted in dimensionalform by adding the external temperature data to thermal response curves.

3.2.2 Sensitivity Analysis

Sensitivity of the stochastic variables was evaluated with respect to a basecase with parameters given in Table 3. The base case is run with all variableson the low side of the expected range. A total of 1,000 simulations were per-formed for each set of parameters to construct the PDFs necessary to calculatethe sensitivity. All the following sensitivities are compared to the base PDF,Fig. 11. The base PDF has three peaks. The first peak (near 2.25 Celsius) isthe temperature at which the system will likely fluctuate around during theevening and early morning hours. The second peak (near 7.75 Celsius) is dueto an individual entering the building or staying late during the early morningor evening hours. The third peak (near 9.75 Celsius) is where the tempera-ture is most likely to fluctuate during the work day, and the office is mostlyoccupied with people.

The parameters that are attributed to the building design are the effectivedischarge coefficient/window opening area, floor area, building conductance,and window differential height. The building parameters that are most sensi-tive to perturbations are the window opening area as well as the floor area.The effective discharge coefficient and window opening area is directly relatedto the wind speed of the system. The wind speed has a large effect on the

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system, therefore management of how far the windows are open is importantin controlling the temperature within the space. By increasing the floor area,the amount of mass in the system increases. Adding more mass dampens theeffect of the fluctuations on the system.

The parameters that are attributed to stochastic behavior are the mean windspeed, the maximum expected number of people, the mean metabolic rate of anindividual, and the standard deviation of the metabolic rate of an individual.The most sensitive parameter is the mean wind speed. The maximum expectednumber of people effects the system more than increasing the activity level ofthe occupants. The least sensitive parameter is the standard deviation of themetabolic rate of an individual. The fluctuations in standard deviation of anindividual’s metabolic rate does not effect the system significantly.

Based on the results of the sensitivity analysis, Fig. 12, the design of a build-ing should first consider the building parameters. The building parametersare much more sensitive to the temperature response of the building thanthe stochastic variables with the exception of the mean wind speed. The meanwind speed can be controlled by adding obstructions to the windows or manag-ing the degree at which the windows are open. The number of people occupy-ing a space should be considered first before the activities that the individualsare expected to perform. Variation in metabolic rates of individuals can bebasically be neglected when considering other stochastic variables.

4 Conclusions

We have formulated a stochastic approach to account for the inherent fluctu-ations in the natural ventilation systems. We developed a data-driven frame-work necessary to analyze the stochastic processes and natural dynamics ofnatural ventilation systems. The stochastic variables in the building model,internal energy load and wind speed, are fit based on data. A methodologyfor introducing these stochastic variables into the steady state analysis andtransient analysis was detailed.

We see that the introduction of stochastic fluctuations gives insight into thecomplex dynamics of natural ventilation systems. The regions in which thenatural ventilation system fluctuates is easily visualized though uncertaintypropagation through the steady state system. The complex bi-stable dynamicsof natural ventilation system emerge in the transient analysis with stochasticinputs. The introduction of noise destabilizes some of the effects of the sta-ble fixed points in the natural ventilation system. The destabilization of thefixed points allows for the stochastic system to produce more physical andmeaningful results. On the other hand, the deterministic system leads to inac-

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curate predictions of the thermal response. The complex dynamics due to thepresence of noise is instrumental in accurate predictions of natural ventilationsystems.

The techniques developed in this paper provides insight for designers to buildmore statistically robust natural ventilation systems. The framework allows forsix-sigma design analysis to be incorporated into natural ventilation systems.The statistical descriptions of the variables allow for bounds of the tempera-ture response to be controlled. The developed numerical processes gives theability to precisely predict the thermal response of natural ventilation systems.Understanding the stochastic processes that lead to the natural dynamics ofnatural ventilation systems is a significant step towards adaptive, efficient,and robust control of natural ventilation systems. The correct utilization ofnatural ventilation system will lead to the reduction of energy use of the entirebuilding.

5 Appendix

Propagation of the uncertainty in the random inputs gives the designer thetemperatures in which the system is most likely to fluctuate. For the single-zone office space, the uncertainty in the energy generation and the uncertaintyin the wind speed is both propagated through the dynamical system to givethe uncertainty in the zonal temperature. The uncertainty of the randomvariables is described by their probability distribution function (PDF). For asingle random variable the PDF values can be directly propagated throughthe dynamical system, but for multiple random variables the joint PDF needsto be obtained. In our system, the stochastic energy generation and stochasticwind speed are independent random variables. For two independent variablesthe joint PDF is the product of the two PDFs, Eq. 14.

PE,v = PEPv (14)

The uncertainty of all the variables is then propagated through the steadystate system using a three step process, Fig. 13. First, the support of PE isdiscretized into a finite set, (Ei, PE,i) (for i = 1 : m), similarly the support ofPv is discretized into a set, (vref,j, Pv,j) (for j = 1 : n). Second, them×n valuesPE,i and Pv,j are used to calculate the joint PDF, PE,v(i, j) = PE,iPv,j . Third,the m × n pairs of Ei and vref,j are used to calculate the roots that satisfythe steady state problem f (T ∗, Ei, vref,j) = 0. Finally, the values of the jointprobability function, PE,v(i, j) correspond to the values of f (T ∗, Ei, vref,j) =0 ∀ i = 1, 2, ..., m & j = 1, 2, ..., n. The result is a surface of the steady state

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temperatures of the natural ventilation system that is contoured by the jointprobability distribution of the stochastic variables.

References

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[12] M. Zaheer-Uddin, G. Zheng, ”A dynamic model of a multizone VAV system forcontrol analysis”, Transactions: American Society of Heating Refrigerating andAir Conditioning Engineers, 100 (1994), 219.

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[14] P. Bhaskoro, S. Gilani, M. Aris, ”Simulation of intermittent transient coolingload characteristic in an academic building with centralized HVAC system”,International Conference on Environment Science and Engineering, Singapore,2011.

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[17] S. Privara, J. Siroky, L. Ferkl, J. Cigler, ”Model predictive control of abuilding heating system: The first experience”, Energy and Buildings, 43 (2011),564−572.

[18] J. Yuan, ”Transition Dynamics between the Multiple Steady States in NaturalVentilation Systems: From Theories to Applications in Optimal Controls”,Massachusetts Institute of Technology, (2007).

[19] ANSI/ASHRAE Standard 55-2004, Encyclopedia of Energy Engineering andTechnology, Taylor & Francis Group, Boca Raton, FL, (2004).

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[22] M. Deru, K. Field, D. Studer, K. Benne, B. Griffith, P. Torcellini, B. Liu, M.Halverson, D. Winiarski, M. Rosenberg, M. Yazdanian, J. Huang, D. Crawley,”U.S. Department of Energy Commercial Reference Building Models of theNational Building Stock”, 2011.

[23] A. Fontaini, M. Olsen, B. Ganapathysubramanian, ”Thermal comparisonbetween ceiling diffusers and fabric ductwork diffusers for green buildings”,Energy and Buildings, 43 (2011), 2973−2987.

[24] W. Wang, Y. Huang, MD Lane, B. Liu, ”Technical Support Document:50% Energy Savings for Small Office Buildings”, Pacific Northwest NationalLaboratory, (2010).

[25] S. Ross, ”Introduction to Probability Models”, Elsevier (Singapore) Pte Ltd,2007.

[26] D. Wang, C. Federspiel, F. Rubinstein, ”Modeling occupancy in single personoffices”, Energy and Buildings, 37 (2005), 121−126.

[27] J. Page, D. Robinson, N. Morel, J.-L. Scartezzini, ”A generalised stochasticmodel for the simulation of occupant presence”, Energy and Buildings, 40(2008), 83−98.

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[28] H. Khalil, ”Nonlinear Systems”, Prentice-Hall Inc., Upper Saddle River, NJ,2002.

[29] B. Ganapathysubramanian, N. Zabaras, ”A seamless approach towardsstochastic modeling: Sparse grid collocation and data driven input models”,Finite Elements in Analysis and Design, 44 (2008), 298−320.

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Fig. 1. Single-zone building.

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Fig. 2. Flow chart for calculation of the stochastic energy generation term.

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Fig. 3. Above: Occupancy schedule presented by the DOE and NREL along withthe piecewise fit used for determining the number of people in the room during theday. Below: The rate of change of the piecewise fit occupancy schedule to determinethe rate at which people enter the room during the day.

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Fig. 4. A realization of the number of people within the office space during a dayperiod, according the the expected number of people curve.

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Fig. 5. Probability density function and cumulative density function of the windspeed data that was collected.

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Fig. 6. Multiple view of the solution space for two parameter variation: E, energygeneration, and vref , wind speed.

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Fig. 7. Energy generation PDFs for the non-working hours of the day (8 pm - 5 am)and the working hours of the day (5 am - 8pm).

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(a)

(b)

Fig. 8. Multiple views of the joint uncertainty in the stochastic inputs propagatedthrough the steady state system for (a) the system during mostly unoccupied time(8 pm and 5 am), and (b) the system during mostly occupied time (5 am to 8 pm).

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Fig. 9. (a) The temperature response of the office space using the deterministicapproach for an entire work week, and (b) a realization of the temperature responseof the stochastic approach for an entire week.

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(a) (b)

Fig. 10. (a) The temperature response of the office space for different mean windspeeds for the deterministic system. (b) The average temperature response for dif-ferent wind speeds of the stochastic system for 10,000 simulation days.

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Fig. 11. Temperature probability density function of the base design case for thesensitivity analysis.

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(a) (b)

Fig. 12. The stochastic sensitivity of (a) the building design parameters and (b) thestochastic design parameters.

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Fig. 13. The three step process for propagating the uncertainty of the input variablesthrough the steady state system.

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Table 1Nomenclature

Af Floor area of the zone

Al Open area of the lower window

As DuBois body surface area of an individual

Au Open area of the upper window

CdA Effective discharge coefficient and window opening area

Cdl Discharge coefficient for the lower window

Cdu Discharge coefficient for the upper window

CI Coefficient of interference

cp Specific heat capacity of air

Cpl Wind pressure coefficient for the lower opening

Cpu Wind pressure coefficient for the upper opening

dL1 L1 distance between two PDFs

E Overall energy generated within the zone

Ebase Energy generation within the zone due to base loads

Elight Energy generation within the zone due to lighting

Eocc Energy generation within the zone due to all the occupants

Eocc/person Energy generation within the zone due to one individual

g Acceleration due to gravity

h Height differential between the windows

Ma Mass of the air within the zone

Nexpect The expected number of people in the space at the current time

Nmax Maximum expected number of people within the zone

Np Number of people within the space

ns Number of experimental samples

npdf Number bins in discrete PDF

P Transitional probability matrix

Pi Transitional probability of row i

PE Energy generation PDF

PE,v Joint PDF of the energy generation and wind speed

Pv Wind speed PDF

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PDF (base) Base case PDF

PDF (purt) PDF which is being compared to the base case PDF

p∗ Dimensionless pressure difference

pw Wind pressure differential between inlet and outlet

q Volumetric airflow rate

T Overall temperature of zone

T ∗ Dimensionless temperature of the zone

Te Exterior temperature

t Time

UA Mean conductance of the zone

V Volume of zone

vex Experimental wind speed

vL2 L2 error of the theroretical wind speed probability distribution function

vref Reference wind speed

vfit The fitted wind speed

Z Standard unit normal random variable

α Dimensionless energy generation

β Thermal expansion coefficient of air

∆t Time step

γ Dimensionless conductance of the building

λbuild Mean wind speed seen by the building

λrate Expected rate at which individuals enter or leave the space

λwind Mean wind speed

φ Buoyancy momentum coefficient

µmet Individual’s mean metabolic rate

µmisc Mean generation within the zone due to miscellaneous office equipment

ρa Density of air

σocc Standard deviation of an individual’s generation rate

σmisc Standard deviation of the energy generation due to the office equipment

ξE Stochastic energy generation

ξv Stochastic wind speed

〈X〉 Expected value of variable X

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Table 2Single-zone building simulation variables.

cp 1005 [J/kg −K] UA 50 [W/K]

ρa 1.205[

kg/m3]

h 2 [m]

β 3.43E-03 [1/K] V 948.55[

m3]

g 9.81[

m/s2]

CdA 0.35 [−]

Te 0 [C] Cpu = Cpl 1 [−]

〈Nmax〉 6 [−] Af 311[

m2]

Table 3The parameters used for the base case of the stochastic sensitivity analysis.

〈λwind〉 0.5 [m/s]

µmet 75[

W/m2]

σmet 15[

W/m2]

Af 100[

m2]

CdA 0.2 [−]

〈Np〉 4 [−]

h 0.5 [m]

UA 50 [W/K]

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Figure captions

Figure 1: Single-zone building.

Figure 2: Flow chart for calculation of the stochastic energy generation term.

Figure 3: Above: Occupancy schedule presented by the DOE and NREL alongwith the piecewise fit used for determining the number of people in the roomduring the day. Below: The rate of change of the piecewise fit occupancyschedule to determine the rate at which people enter the room during the day.

Figure 4: A realization of the number of people within the office space duringa day period, according the the expected number of people curve.

Figure 5: Probability density function and cumulative density function of thewind speed data that was collected.

Figure 6: Multiple view of the solution space for two parameter variation: E,energy generation, and vref , wind speed.

Figure 7: Energy generation PDFs for the non-working hours of the day (8pm - 5 am) and the working hours of the day (5 am - 8pm).

Figure 8: Multiple views of the joint uncertainty in the stochastic inputs prop-agated through the steady state system for (a) the system during mostly un-occupied time (8 pm and 5 am), and (b) the system during mostly occupiedtime (5 am to 8 pm).

Figure 9: (a) The temperature response of the office space using the deter-ministic approach, and (b) a realization of the temperature response of thestochastic approach.

Figure 10: (a) The temperature response of the office space for different meanwind speeds for the deterministic system. (b) The average temperature re-sponse for different wind speeds of the stochastic system for 10,000 simulationdays.

Figure 11: Temperature probability density function of the base design casefor the sensitivity analysis.

Figure 12: The stochastic sensitivity of (a) the building design parameters and(b) the stochastic design parameters.

Figure 13: The three step process for propagating the uncertainty of the inputvariables through the steady state system.

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Explore and formulate a stochastic approach to analyze the dynamics of natural ventilation

systems

Illustrate the emergence of bi-stable behavior under stochastic conditions, which is not seen in

the deterministic case

Construct the sensitivity of the systems to stochastic inputs.

*Highlights (for review)