Optimal decision making in ventilation control

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Energy 34 (2009) 1835–1845

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Energy

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Optimal decision making in ventilation control

Andrew Kusiak*, Mingyang LiDepartment of Mechanical and Industrial Engineering, 3131 Seamans Center, The University of Iowa, Iowa City, IA 52242-1527, USA

a r t i c l e i n f o

Article history:Received 24 February 2009Received in revised form23 July 2009Accepted 24 July 2009Available online 15 August 2009

Keywords:VentilationAir qualityMulti-objective optimizationSchedulingEvolutionary strategyEnergy saving

* Corresponding author. Tel.: þ1 31 9 3355934; faxE-mail address: [email protected] (A. Ku

0360-5442/$ – see front matter � 2009 Elsevier Ltd.doi:10.1016/j.energy.2009.07.039

a b s t r a c t

In this paper, a two-mode ventilation control of a single facility is formulated as a scheduling model overmultiple time horizons. Using the CO2 concentration as the major indoor air quality index and expectedroom occupancy schedule, optimal solutions leading to reduced CO2 concentration and energy costs areobtained by solving the multi-objective optimization model formulated in the paper. A modifiedevolutionary strategy algorithm is used to solve the model at different time horizons. The optimizedventilation schedules result in energy savings and maintain an acceptable level of indoor CO2

concentration.� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Maintaining air quality and providing thermal comfort isimportant for facilities supported by heating, ventilating and air-conditioning (HVAC) systems. According to published statistics,HVAC systems account for almost 31% of the electricity consumedby U.S. households [1]. Therefore, appropriate consumption ofenergy while maintaining the desired air quality has an impact onenergy cost and indoor comfort.

The traditional approach to ventilation is to provide a fixedminimum ventilation rate per person based on the maximumoccupancy of a facility. To provide air quality guidelines, ASHRAEStandard 90.1 [2] specifies the minimum ventilation rate of 2.5 l/sper person, while ASHRAE Standard 62-2004 [3] has been revisedto the minimum ventilation rate of 10 l/s per person [4]. Thenumber of occupants in any facility varies over time, and it is rarethat the facility is fully occupied. This provides a good opportu-nity to save energy by ventilating facilities on demand [5]. Thus,the demand-control ventilation (DCV) is a commonly usedstrategy in HVAC systems based on signals from the indoorsensors, e.g., a CO2 sensor. Both simulations and field tests of theCO2-based DCV have demonstrated the potential to save energy[6], especially in facilities with a high occupancy density. A major

: þ1 319 3355669.siak).

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difficulty with this approach is that CO2 can only be used asa surrogate of human generated pollutants, whereas a CO2 sensorcannot respond to pollutants such as emissions from furniture orpainted materials. The location and stability of CO2 sensors arealso problematic. Therefore, different control strategies [7,8] havebeen developed to deal with these issues. Other sensors like theVOC (volatile organic compound) sensor, occupancy sensor,humidity sensor, particle sensor, and so on, are used to modulatethe ventilation rate over time under various conditions. Inaddition, devices such as air-side economizers are also used inventilation systems to reduce energy consumption [9]. Thequantity of fresh air supply is determined on the basis of theoutside air dry-bulb temperature, enthalpy or other thermalproperties. These approaches are usually cost-effective in areaswhere the heating or cooling cost is high.

Various optimization models [10–12] and algorithms [13,14]have been discussed in the HVAC literature. In this paper, the on-offventilation control is formulated as an optimization model. Themodel involves three objectives, namely the fan-on time period, theaverage CO2 above threshold, and the time period corresponding tothe CO2 above a threshold. The model is solved by an evolutionaryalgorithm.

By optimizing fan on-and-off schedules on the basis of thetrade-off among the three objectives, energy savings can be ach-ieved, while proper air quality can be ensured by maintaining theCO2 concentration in an acceptable range without installing anyanalog indoor sensors.

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Entering Exhaust

Nomenclature

C indoor CO2 concentration, ppmG CO2 generation rate of occupants, l/slv air exchange rate, m3/hrCout outside air CO2 concentration, ppmA area of surfaces on which indoor pollutants are

removed by deposition, m2

vd deposition velocity for the pollutantQac air flow rate through an air cleaner, m3/hreac efficiency of an air cleanerQ overall air ventilation rate, m3/hrV volume of the facility, m3

Clast relative indoor CO2 concentration at the start of a timestep, ppm

Cnext relative indoor CO2 concentration at the end of a timestep, ppm

DT sampling time interval, st0, t1,., tN�1, tN time stamps of evenly divided intervals, sDt time interval length, sSi random variable denoting the number of occupants in

time interval, (ti�1,ti)fi(,) probability density function of Si

xi, Dxi fan start time and running time period in time interval,(ti�1, ti)

IQ status of ventilation fanQmech mechanical ventilation rate, m3/hrQnat natural ventilation, m3/hr

Clim CO2 limit, ppmCti relative CO2 concentration at time stamp ti, ppmNp the expected number of occupantsP0 initial populationPexternal external populationPoffspring offspring populationPparent parent populationNcurrent current population size cluster limitNparent parent population sizeMi the number of individuals in Pcurrent that solution i

dominatesFi fitness of ith elite individual in the external population

Pexternal

Fj fitness of jth individual in the current populationPcurrent

A elite set including all elite solutions that dominatea certain solution in the current population Pcurrent

ri ith solution variable before mutationDri Gaussian noise imposed on ith solution variableri’ ithsolution variable after mutations Standard deviation of Gaussian noiseObj objective valuewi ith weightN the number of intervals in the generalized model

(see Fig. 2)n the number of sampling points of the CO2

concentration above the threshold (see Fig. 4).

A. Kusiak, M. Li / Energy 34 (2009) 1835–18451836

2. Problem formulation

Carbon dioxide concentration in indoor air is commonly used asan indicator of the outside air ventilation rate [15]. CO2 is a practicaland widely used metric for measuring air quality. Though it doesnot reflect all air containments, a high level of CO2 concentrationpoints to insufficient ventilation of indoor space. In facilities, suchas classrooms with relatively stable occupancy rates during certaintime periods, a high concentration of CO2 can degrade theproductivity of students [16,17]. In this paper, CO2 is used as theindex to optimize ventilation control.

Air Infiltration

Airexfiltration

CO2 generation

Indoor air CO2 concentration

Outside air CO2concentration

air air

Supplyair

Returnair

Fig. 1. Single facility ventilation system.

2.1. CO2 predictive model

The equilibrium CO2 concentration in a single facility can bederived based on the number of occupants, the CO2 generation rateof the occupants, and the supply quantity of the outside air.A diagram of a single facility ventilation system is shown in Fig. 1.

Define the air exchange rate lv¼Q/V, where Q is the overalloutside air ventilation rate and V is the volume of the facility. Thesteady-state of indoor CO2 concentration is obtained from the massbalance in equation (1) [16].

dCdt¼ G

Vþ lvCout � lvC � vd

AV

C � Qac

VCeac (1)

where: C: the indoor CO2 concentration; G: the CO2 generation rateof occupants; lv: the air exchange rate (defined above); Cout: theoutside air CO2 concentration; A: the surface area on which indoorpollutants are deposited; vd: the deposition velocity of thepollutant; Qac: the air flow rate in an air cleaner; eac: the efficiencyof an air cleaner.

Equation (1) is based on the assumption that the indoor air CO2

is completely mixed and the air flow rate in and out of the facility,including mechanical ventilation, infiltration, exfiltration, and soon, is balanced.

Note that the unit of C and Cout is a fractional concentration (v/v).A conversion factor between the fractional concentration and ppmis 106. All concentration-related variables used in this paper have

A. Kusiak, M. Li / Energy 34 (2009) 1835–1845 1837

been expressed in ppm. Assuming no pollutant deposition and noair cleaning is taking place, equation (2) is derived.

VdCdt¼ Gþ QðCout � CÞ (2)

Replacing G with the product of the number of people Sand theaverage CO2 generation rate R, equation (2) can be transformed intoan iterative form which is convenient for simulation with a fixedtime step [18].

Cnext ¼ ðRS=QÞ�

1� e�QDT=V�þ Claste

�QDT=V (3)

where Cnext is the difference between the CO2 concentration in theindoor and the outside air at tnext, Clast is the difference between theCO2 concentration in the indoor and the outside air at tlast.

Using the CO2 predictive model, the optimal ventilation rate canbe determined as a trade-off between the energy savings and theair quality.

2.2. Optimizing ventilation of a single facility

A facility ventilation control can be represented as a fan on-offscheduling model. Unlike demand-control ventilation relying onindoor sensor measurements, the indoor CO2 concentration can beestimated by applying the CO2 steady-state equation (3) and thestatistically described occupancy pattern drawn from the occupancyinformation, e.g., meeting or course schedules. In this case, venti-lation can be determined without using any sensor feedback signals.Based on simplifying assumptions a scheduling model is formulated.

Denote: t0, tN the time stamps when a facility opens and closesfor a work day, e.g., 8:00 AM to 6:00 PM. In other words,occupants may only appear in the interval between t0

and tN (see Fig. 2).xi, Dxi in the time period (ti�1,ti) represents the fan-on timeperiod.Ct is the relative CO2 concentration at time stamp t.

Assumption 1. Assume the time horizon is divided into N equaltime intervals, t0, t1,., tN�1, tN. The time interval length isDt¼ (tN� t0)/N. During each time period (ti�1, ti), define a randomvariable Si, i¼ 1,.,N. Si¼ 0 represents the number of occupants inthis period with a probability density function of fi(,). Forexample, if the probability of Si¼ 0 is 1, it refers to the unoccupiedfacility during the time period (ti�1,ti). If the probability ofSi¼ constant is high, it implies meetings or courses where thenumber of occupants is relatively stable over a certain timeperiod. With a more complex probability density function, Si maydescribe a more dynamic occupant pattern during a certain timeperiod.

Assumption 2. Assume during time period (ti�1, ti) the ventilationfan is to be turned on at most once. This assumption is practical ifthe time interval Dt is relatively short. Note that a user may defineDt as short as needed. Another reason is that frequently turning the

8>><>>:

Ct ¼ Ct0

Ct ¼ gðQnat; t � ti�1;Cti�1ÞCt ¼ gðQnat þ Qmech; t � xi; gðQnat; xi � ti�1;Cti�1ÞÞCt ¼ gðQnat; t � xi � Dxi; gðQnat þ Qmech;Dxi; gðQnat; xi � ti�1;Cti�1Þ

ventilation fan on and off may adversely impact the lifetime of themechanical motor.

Assumption 3. Assume a fan operates in two modes, on and off.Furthermore, assume that the ventilation speed of the fan isconstant. The proposed methodology applies to multi-modeventilation as well as continuous ventilation fans by discretizingcontinuous values into discrete ones. Thus, the overall ventilationrate Q discussed in [18] is shown in equation (4).

Q ¼ IQ,Qmech þ Qnat (4)

where IQ ¼�

0 t˛½ti�1; x1ÞW½xi þ Dxi; tiÞ1 t˛½x1; xi þ DxiÞ

i ¼ 1;2;.;N

IQ : denotes the status of the ventilation fan;

Qmech: is the mechanical ventilation rate of the fan; and

Qnat: is the natural ventilation:

In a single facility ventilation system, optimal fan controlinvolves determining the start time of the ventilation fan and theduration of its run. Minimizing the run time of the fan decreases thepower consumption of the fan and reduces the air handling unit(AHU) heating/cooling load due to the reduced amount of air usedwhile maintaining air quality at the desired level. Quantification ofenergy savings could be provided once the proposed methodologyis fully implemented. Assume a maximum allowable level of anindoor CO2 concentration of 1600 ppm and 100 occupants using anindoor facility for an hour. Vary the fan start time in five-minuteintervals from the first hour to the second hour with the occupantspresent. The fan run time is fixed at 30 min. The simulation result isshown in Fig. 3.

As shown in Fig. 3, different start times of the fan result invarious CO2 curves, some of which reach but others do not, thepreset threshold of 1600 ppm. Therefore, an appropriate start timefor the fan not only can maintain indoor CO2 concentration underthe specified threshold, but offers a potential for operating the fanfor much less time than the original 30-min-long run time. Thus,formulating an optimization model to determine the optimalschedule is worthy of further research.

2.3. Optimization model

For a specific facility and time period (ti�1, ti), V is constant.Assume Si is the number of occupants and the average generationrate is determined in (ti�1, ti). Equation (3) can be generalizedas (5).

Cnext ¼ gðQ ; tnext � tlast;ClastÞ (5)

where Cnext is the difference between the CO2 concentration in theindoor and the outside air at tnext, and Clast is the differencebetween the CO2 concentration in the indoor and the outside air attlast. Therefore, the CO2 concentration at any time during theinterval (t0, tN) can be represented as follows:

t ¼ t0t˛ðti�1; xi�t˛ðxi; xi þ Dxi�

ÞÞ t˛ðxi þ Dxi; ti�; i ¼ 1;2;.;N

(6)

Fig. 3. The simulated CO2 curve.

0t 1t Nt1Nt −1x 1 1x x+ Δ N Nx x+ ΔNx

tΔ

0tC

1xC

1 1x xC +Δ 1tC

1NtC

− NxCN Nx xC +Δ Nt

C

Fig. 2. On–off schedule of the ventilation fan.


As the total run time of all intervals is included in the objectivefunction, the single facility ventilation control can be formulated asmodel (7).

minxi;Dxi

PNi¼1 Dxi

s:t:hðCtÞ � Climti�1 � xi � titi�1 � xi þ Dxi � tiSiwfið,Þ

(7)

In this model (7), constraint h(Ct)� Clim limits the CO2 concen-tration. For example, when h(Ct)¼ Ct, the constraint indicates thatthe CO2 concentration at any time interval will reach the set limit

iCΔ

1 2 i

2CΔ1CΔ

abovTΔ

CO2 threshold

CO2 curve

Fig. 4. CO2 exceedin

Clim. In a typical application scenario, the upper CO2 threshold Clim

may be replaced with a tolerance or a preference function. Themore it exceeds a certain value, the higher dissatisfaction valueproduced. In Section 3, a tri-objective optimization model ispresented.

3. Computational study

A single room ventilation model is formulated by evenlydividing working time intervals into N intervals with a specificprobability density function of the number of occupants (seeSection 2.2). Two scenarios, a single-time interval and two-timeintervals, are considered. The model is generalized to a ten-intervalmodel discussed towards the end of this section.

1n − n

1nC −ΔnCΔ

e−threshold

g a threshold.

Table 1Parameters’ descriptions.

Variable Value Description Unit

V 2000 Facility volume m3

Np 100 Number of occupants personQnat 240 Natural ventilation rate m3/hrQmech 3000 Mechanical ventilation rate m3/hrCout 400 Outside CO2 concentration ppmCt0 500 Initial indoor CO2 concentration ppmCthreshold 1500 CO2 threshold ppmR 0.01 Average CO2 generation rate l/s


3.1. Optimization model for a single-time interval

Assume the expected number of occupants to appear at thistime interval is known and fixed. This is a reasonable assumption inpractice, e.g., for classrooms and conference rooms, because thenumber of students or conference participants registered can beused as the expected value in the probability density function. Asthe number of occupants to appear in the interval is defined asa random variable S, consider the constraint function h(Ct)¼ E(Ct)and equation (3) shown in (8).

EðCnextÞ ¼ ðREðSÞ=QÞ�

1� e�QDT=V�þ EðClastÞe�QDT=V (8)

Let Np¼ E(S), where Np is the number of occupants.To include different additional situations when indoor CO2

exceeds the threshold, the CO2 constraint in optimization model (7)can be transformed into two objective functions.

Based on Fig. 4, define the average CO2 concentration above thethreshold. Assume n sampling points of the CO2 concentrationabove the threshold; then the second optimization objective isexpressed in (9). Note the first objective represents the totalamount of the fan run time.

Obj2 ¼ Average CO2 concentration above the threshold

¼ 1n

Xn

i¼1

DCi (9)

where DCi denotes the difference between the ith point CO2

concentration and CO2 threshold.The third objective is expressed in (10)

Obj3 ¼ Elapsed time ¼ DTabove-threshold (10)

where DTabove-threshold, shown in Fig. 4, is the total time the indoorCO2 concentration is above the threshold.

The three objectives, Obj1, Obj2, and Obj3, are integrated intothe optimization model for time interval (t0,t1) shown in (11):

minx1;Dx1

fObj1;Obj2;Obj3gs:t:t0 � x1 � x1 þ Dx1 � t1

(11)

where x1, Dx1 are the start time and running time of the fan (bothintegers):

Table 2Description of the weight assignment and the results.

Curve w1 w2 w3

A 0 0 0B 1, w2¼w3¼ 0 0, Obj2 ˛ [0,100] 0, Obj3

0.002, w2¼w3¼ 0.499 0.499, Obj2 ˛ (100,þN) 0.499, OC 0.5 0.4 0.1D 1, w2¼w3¼ 0 0, Obj2 ˛ [0,50] 0, Obj3

0.002, w2¼w3¼ 0.499 0.499, Obj2 ˛ (50,þN) 0.499, OE 0.002 0.499 0.499

Obj1 ¼ Dx1

Obj2 ¼ 1n

Xn

i¼1

DCi

Obj3 ¼ DTabove-threshold

For Obj2¼ 0 and Obj3 ¼ 0, model (11) ensures that theindoor CO2 concentration is below the threshold value at anytime. Transforming the constrained model into a non-con-strained one, a bi-objective optimization problem can bebuilt by constructing objective functions Obj1 ¼Dx1

and Obj2 ¼ maxð0; ð1=nÞPn

i¼1 DCiÞ þmaxð0;DTabove-thresholdÞ.In this paper, a more general tri-objective optimization modelis built to include all these situations. In solving this model,a Pareto-based evolutionary strategy is proposed for findingan elite set which includes non-dominated solutions. To findsolutions for CO2 strictly under the CO2 threshold, simplychoose solutions in this elite set with both Obj2¼ 0 andObj3¼ 0.

3.2. Model solving by the evolutionary strategy algorithm

To solve model (11), some parameters need to be initialized. Theparameters used in this research are listed in Table 1. The unitassociated with the solution variables, the start time and the runtime of the fan (shown in Tables 2,3 and 5), is ten seconds.

The simplest way of solving the multi-objective optimizationmodel is by aggregating several objectives into a single objective[19]. This approach has been used in numerous applications[20,21]. However, whenever the weights associated with theobjectives are poorly selected, the model needs to be solvednumerous times. To overcome this issue, a Pareto-based approachis used in this paper. The concept of Pareto-optimality was firstproposed by Goldberg [22]. It implies a search for a family ofsolutions that cannot dominate each other in the presence ofmultiple objectives. The solution to a tri-objective optimizationmodel converges to the Pareto-optimal front. Different solutionstrade off differently among the three objectives. Optimal solutionschange with the weights assigned to the objectives. It is possiblethat one solution is better than the other for one objective (e.g.,run time) but is worse for another objective (e.g., the average CO2

above the threshold or the elapsed time). Neither can dominatethe other, and they are called non-dominated solutions. Fonsecaet al. [19] reviewed several multi-objective evolutionary algo-rithms and grouped them into three categories: aggregatingapproaches, population-based, non-Pareto approaches, and Par-eto-based approaches. In this paper, one of the Pareto-basedapproaches, the Strength Pareto Evolutionary Algorithm (SPEA),[23,24], is used to search the space of non-dominated solutionsand update them to the elite set at each generation. To solve themodel at hand, the modified evolutionary strategy algorithmpresented next has been used.

x1 Dx1 Description

0 0 Fan is not on˛ [0,20] 202 124 Obj2 admitted in bound [0,100]bj3 ˛ (20,þN) Obj3 admitted in bound [0,20]

208 128 Different weights˛ [0,10] 199 142 Obj2 admitted in bound [0,50]bj3 ˛ (10,þN) Obj3 admitted in bound [0,10]

182 171 Obj2¼ 0, Obj3¼ 0

Table 4Occupancy schedules for ten-time periods.

8:00AM–9:00A M

9:00 AM–10:00 AM

10:00 AM–11:00AM

11:00AM–12:00 AM

12:00 AM–1:00 PM

1:00 PM–2:00PM

2:00PM–3:00PM

3:00PM–4:00PM

4:00PM–5:00PM

5:00PM–6:00 PM

1 50 10 100 0 0 40 0 30 0 102 30 20 35 15 10 43 0 30 0 273 10 50 45 9 0 17 40 30 14 54 13 25 14 9 5 17 16 21 14 35

Table 3The weight assignment and the results.

Curve A B C D E

w1 0 1, w2¼w3¼ 0 0.5 1, w2¼w3¼ 0 0.0020.002, w2¼w3¼ 0.499 0.002, w2¼w3¼ 0.499

w2 0 0, Obj2 ˛ [0,100] 0.4 0, Obj2 ˛ [0,50] 0.4990.499, Obj2 ˛ (100,þN) 0.499, Obj2 ˛ (50,þN)

w3 0 0, Obj3 ˛ [0,100] 0.1 0, Obj3 ˛ [0,50] 0.4990.499, Obj3 ˛ (100,þN) 0.499, Obj3 ˛ (50,þN)

x1 0 198 178 172 160Dx1 0 159 175 183 197x2 0 75 14 65 22Dx2 0 80 112 91 105Total run time 0 239 287 274 302Description Fan is not on Obj2 admitted in range [0,100] Different weights Obj2 admitted in range [0,50] Obj2¼ 0, Obj3¼ 0

Obj3 admitted in range [0,100] Obj3 admitted in range [0,50]


Step 1. Initialize a population P0 as the current population Pcurrent

and create an empty external population Pexternal to storeelite solutions;

Step 2. Find non-dominated solutions in Pcurrent and copy theminto Pexternal;

Step 3. Find non-dominated solutions in Pexternal and update theelite population Pexternal;

Step 4. Cluster solutions, if the size of Pexternal exceeds the limit N.Step 5. Assign fitness values to each individual in Pcurrent and

Pexternal;Step 6. Select Nparent individuals into Pparent from Pcurrentþ Pexternal

by using the binary tournament selection scheme withreplacement;

Step 7. Randomly select two individuals and retain the fitter indi-vidual for inclusion in Poffspring. Poffspring has the samepopulation size as Pcurrent;

Step 8. Apply recombination and mutation operators to Poffspring;replace Pcurrent with Poffspring and go back to Step 2 until thestopping criterion (here the maximum number of genera-tions) is met.

To reduce computational effort, clustering takes place in Step 4.Euclidean distance is used as a distance metric between the datapoints. If the size of the external population is larger than the limitN, clustering will be performed by setting a distance limit to filtersimilar points in each cluster while remaining a representativepoint in each group. For solutions with Obj2¼ 0 and Obj3¼ 0, allpossible solutions are kept until the final generation by choosingthe lowest Obj1.

The fitness functions assigned in Step 5 to the individuals inPcurrent and Pexternal differ. For individuals in the elite populationPexternal, the fitness function is

Table 5Computed ventilation schedules.

x1, Dx1 x2, Dx2 x3, Dx3 x4, Dx4 x5, Dx5

1 131, 41 111, 119 64, 291 0, 0 0, 02 218, 19 27, 53 48, 96 161, 36 228, 743 258, 18 194, 119 118, 129 40, 77 0, 04 32, 9 281, 10 149, 33 17, 49 226, 47

Fi ¼Mi

N þ 1(12)

current

where Fi is the fitness of ith elite individual in the external pop-ulation Pexternal, Mi is the number of individuals in Pcurrent thatsolution i dominates, Ncurrent is the population size of Pcurrent.

For the individuals in the current population Pcurrent , the fitnessfunction is

Fj ¼ 1þXi˛A

Fi (13)

where Fj is the fitness of jth individual in the current populationPcurrent, Fi is the fitness of ith elite which dominates the solution j, Aisan elite set including all elite solutions that dominate the solution j.

Individuals with smaller fitness values have a higher probabilityto reproduce (Step 6). The mutation operation of Step 8 is realizedby adding noise Dri to ri, where ri is the ith solution variable and Dri

is a Gaussian distribution with a mean of zero and standard devi-ation s [25]. Solution riis updated to ri’ by r0i ¼ ½ri þ Nð0; sÞ�, where[$] is the nearest integer. The value of s is selected for each variable,and it remains fixed for all generations. The mutated solutions arechecked for possible constraint violations. When a constraint isviolated, the value of the violating solution is replaced with a cor-responding constraint-bound value to make sure all solutionsremain in the specified search space. In the one-time intervalscenario, solution variables r1,r2 are x1, Dx1 respectively. Set s¼ 2.The maximum number of generations (the stopping criterion) is setas 30, as there is no significant difference in the solution qualitywhen the number of generations exceeds 30. The values of theparameters used in the evolutionary strategy algorithm, such as theratio of the parent and offspring size and the initial population size,

x6, Dx6 x7, Dx7 x8, Dx8 x9, Dx9 x10, Dx10

67, 103 0, 0 93, 114 0, 0 214, 8373, 117 0, 0 78, 121 0, 0 196, 42194, 101 207, 85 80, 83 128, 16 146, 54245, 66 5, 4 56, 6 11, 195 261, 98

140

160

180

200

220

240

30 50 70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390

Obj

1 (1

0 se

cond

s)

Initial population size

#1 #2 #3 #4

Fig. 5. Values of the objective function Obj1 for different initial population size basedon four runs of the algorithm.


are set as follows. The evaluation criterion is based on solutionswith Obj2¼ 0 and Obj3¼ 0 in the final elite set.

For each of the four runs of the SPEA algorithm, the initialpopulation size varies from 30 to 390. Fig. 5 demonstrates Obj1 ofthe solution in the final elite set, with Obj2¼ 0 and Obj3¼ 0, fordifferent values of the initial population size. Fig. 6 shows theaverage values of Obj1 based on four runs of the algorithm.

As illustrated in Fig. 6, the initial population size of 250 hasa relatively low average value and therefore is selected as the initialsize. Fig. 7 demonstrates the change of Obj1 for different parent andoffspring ratios.

The parent offspring ratio of 1/4 is selected, as it has producedthe best quality results (the lowest value Obj1, as shown in Fig. 7).The algorithm is run using the tuned parameters above. Distribu-tions of the offspring and the elite objective values at differentgenerations are shown in Figs. 8 and 9.

As shown in Fig. 8, with the increase of the number of iterations,the points in the offspring set move towards the origin point (0, 0) inthe two dimensional objective space. It indicates that the distribu-tion front of the offspring moves towards the origin point (0, 0, 0) inthe three dimensional space as the number of iteration increases.Fig. 9 outlines the shape of the Pareto front in the space of objectivevalues It also demonstrates the difference in the elite set before andafter the clustering. Each point in the elite set cannot dominateanother. Points with a small Euclidean distance are removed, whileone remains to represent that cluster for computational costconsideration. Here the elite size limit N is set as 10. The Euclideandistance limit is set to 50, which means that if a point’s distance isless than 50, it will be considered for clustering. When consideringthe diversity of solutions with both Obj2¼ 0, Obj3¼ 0, points on theObj1 axis are not clustered. At the last iteration, the smallest Obj1value is kept, while the larger ones are filtered out.

160

165

170

175

180

185

190

195

30 50 70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390

Ave

rage

Obj

1(10

sec

onds

)

Initial population size

Average Obj1 for four runs of the algorithms

Fig. 6. Average Obj1 for four runs of the algorithm for different initial population size.

3.3. Optimal solution selection

The selection of the optimal solution from the elite set dependson the importance of each objective. Assigning weights to eachobjective and transforming them into a single objective isa commonly used approach. The final solution corresponds to theminimum value of Obj in (14).

Obj ¼ w1Obj1�Obj1min

Obj1max�Obj1minObj1þw2

Obj2�Obj2min

Obj2max�Obj2minObj2

þw3Obj3�Obj3min

Obj3max�Obj3minObj3 ð14Þ

where w1,w2,w3 are the user-defined weights indicating theimportance of each objective and Obj1max and Obj1min are themaximum and the minimum values of Obj1 in the final elite set.Similar notation is used for Obj2max, Obj2min, Obj3max, and Obj3min.

Note thatP3

m¼1 wm ¼ 1, with w1,w2,w3 being eitherconstants or functions of other objectives. For example, if theindoor CO2 concentration is required to be below a certain CO2

threshold, w2 and w3 are assigned relatively large valuescompared to w1. If the value of Obj2 is the range [0,a], w2isconstructed as follows:

w2 ¼�

0 Obj2˛½0; a�large compared to w1 Obj2˛ða;þNÞ

Fig. 10 shows five cases of the change in the indoor CO2

concentration. The ventilation schedule varies among the fivecases. In all cases, the occupants arrive at 8:00 AM and leave thefacility at 9:00 AM.

The details of each of the five cases are illustrated in Fig. 10. Theweights, optimal solutions, and the experiment description areprovided in Table 2.

As presented in Table 2, by assigning different weights to thethree objectives, the fan-on time period varies. Maintaining theCO2 concentration strictly below the CO2 threshold, e.g.,1500 ppm, implies higher fan energy consumption, whileallowing the CO2 concentration to remain at a certain thresholdinterval can reduce the run time of the fan, thus resulting inhigher energy savings.

3.4. Statistical analysis

In the one-time-interval optimization model, the expectation ofthe number of occupants is used. Other assumptions based ondifferent probability functions and probability of the number ofoccupants can also be made. For example, assume the occupancydistribution function is the Poisson distribution [26].

155

160

165

170

175

180

185

190

1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10

Obj

1 (1

0 se

cond

s)

Parent/offspring ratio

#1 #2 #3

Fig. 7. Values of the objective function Obj1 for different parent offspring ratios forfour runs of the algorithm.

Fig. 8. Distributions of the offspring in two dimensional space of objective values at different iterations.


f ðS1Þ ¼lS1 e�l

S1!S1 ¼ 0;1;2;. (15)

If P(S1�100)¼ 0.95, l is 85. Consider the optimal solutionchosen in Section 3.3 to keep the CO2 strictly under threshold.

Fig. 9. Distributions of the offspring and the elite objective

Because equation (3) is a non-decreasing function, fewer peoplethan 100 can always satisfy the constraints. The maximumnumber of people is 101. Therefore, statistical confidence of thisoptimal decision making with Obj2¼ 0, Obj3¼ 0 can becomputed as

values before and after clustering at 30th generation.

Fig. 10. Five cases of the indoor CO2 concentration.


PðS1 � 101Þ ¼ 0:96

The probability of the number of people less than 102 is 0.96. Ifthe number of people less than that particular number is true, theindoor air CO2 concentration will be strictly under 1500 ppm, usingthe proposed optimal solution of x1¼1820s, Dx1¼1710s. DefinefCO2�MaxðzÞ as the maximum CO2 concentration for S1¼ z. ThenPðfCO2�MaxðzÞ � 1500Þ ¼ 0:96. If the CO2 threshold is lowered, theprobability will decrease. Assume the acceptance probability is 0.95and introduce the variation of the CO2 threshold.

P�fCO2�MaxðzÞ � 1500� DCO2 threshold

�� 0:95

Fig. 11. Distributions of the offspring and elite o

From that calculation, DCO2 threshold� 19 ppm. In other words,statistical confidence of this optimal decision making can also holdif the variation of the CO2 threshold is less than a limited value. Thedistribution of the occupants can be any other probability densityfunction, e.g., exponential family or probability density functionslearned from the occupancy data.

3.5. Optimization model for two-time intervals

Like the one-time interval, the optimization model of twoconsecutive time intervals can be formulated as follows:

bjective values before and after clustering.

Fig. 12. Change of indoor CO2 concentration in five cases.


minx1;Dx1;x2;Dx2

fObj1;Obj2;Obj3gs:t:

t0 � x1 � x1 þ Dx1 � t1t1 � x2 � x2 þ Dx2 � t2

(16)

where: x1, Dx1 are the start time and run time of the fan during thefirst interval [t0,t1]; x2, Dx2 are the start time and run time of the fanduring the second interval[t1,t2]; Obj1¼Dx1þDx2, Obj2, Obj3 aredefined the same as in the one-time interval.

Searching four solution variables by applying the evolutionaryalgorithm mentioned above, distribution of points in the offspringand the elite and clustered elite set at the last iteration is shown inFig. 11. The maximum iteration here is 40.

Fig. 13. CO2 concentration for di

The optimal solutions selecting criteria is the same by assigningweights to each objective. Fig. 12 demonstrates the change ofindoor CO2 concentrations in five cases. People appear from 8:00AM to 10:00 AM. The number of people is 100 and 30, respectively,in each one-hour interval.

The weight assignment and solutions to model (16) are shownin Table 3.

3.6. The scenario with ten-time intervals

For Nconsecutive intervals, the number of solution variables is2n and Obj1 ¼

PNi¼1 Dxi. For example, assume the working hours

are from 8:00 AM to 6:00 PM. The occupancy schedule is

fferent optimized schedules.


established on an hourly basis (see Table 4). Therefore, there are tenintervals and N¼ 10. Some intervals may not include occupants,e.g., the lunch time period, 12:00 AM to 1:00 PM.

Optimal solutions for the scenario with Obj2¼ 0, Obj3¼ 0 are tobe determined. To reduce the solution search space, the fan isturned off during intervals in which the number of occupants is 0.This is feasible because the CO2 concentration can never exceed thethreshold in these intervals provided that the CO2 concentration atthe start point of such intervals is below the threshold. For a pop-ulation size to 2000, the number of generations at 30, and an elitepopulation size of 100, the optimal solutions for each of the fouroccupancy schedules of Table 4 are shown in Table 5. The units ofvalues below are listed in 10s.

Note that search is performed in parallel so that the entireschedule is considered at the same time. Solutions can be alsodetermined based on the time interval by a time interval basis. Theoptimal solution for the first interval is determined first, and thena solution for the second interval is determined based on theprevious one. In this paper, the parallel approach is followed. TheCO2 charts for the four schedules are shown in Fig. 13.

4. Conclusion

In this paper, a scheduling model was developed for a ventila-tion control problem. Optimal ventilation was determined fordifferent occupancy schedules. The fan start and run times weredetermined to maintain the CO2 level below a pre-selectedthreshold. The optimized run time of the mechanical ventilationsystem leads to savings due to the reduced power consumption ofthe fan and the reduced amount of heating and cooling. Thesesavings were accomplished while the indoor air quality was guar-anteed without installing any analog indoor sensors. Futureresearch will focus on modeling cases where the fan operates inmore than two modes. The CO2 model needs to be improved byconsidering the quasi-steady state and CO2 spatial distribution.

Acknowledgement

This research has been supported by the Iowa Energy Center,Grant No. 08-01.

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