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An Optimum Start/Stop Control

Algorithm for Heating and Cooling

Systems in Buildings

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Washington, DC 20234

May 1983

I

Sponsored by: — QC

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03-2720

1983

Office of Building and Community Systems

U.S. Department of Energy

U.S. Navy Civil Engineering Laboratory

U.S. Department of Defense

NATIONAL BUREAU. OF STANDARDS

LIBRARY AUG 1 1983,

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NBSIR 83-2720

AN OPTIMUM START/STOP CONTROL

ALGORITHM FOR HEATING AND COOLING

SYSTEMS IN BUILDINGS

Cheol Park

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Washington, DC 20234

May 1983

Sponsored by:

Office of Building and Community Systems

U.S. Department of Energy

U.S. Navy Civil Engineering Laboratory

U.S. Department of Defense

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U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

ABSTRACT

When a building structure is occupied intermittently, energy savings can be

realized from the optimal start-up and shut-down of the heating or cooling

system. This strategy, known as optimum start/stop control, reduces energy

consumption by delaying the start-up of the space conditioning system until

the last moment and then initiating shut-down as early as possible, while

maintaining a preset level of comfort during the period of building occupancy.

Based on the bang-bang control theory, a simple optimum start/stop control

algorithm is developed for computerized control systems in buildings. The

optimum start time is obtained by finding the intersection of cool-down and

heat-up curves that are approximated by exponential fitting of the previous

and current day's data.

Information is presented in this report on the input and output variables,

logic flow, and methodology employed in developing the algorithm. A computer

program listing of the optimum start/stop control algorithm written in FORTRAN

77 and sample input and output data are included in the appendices.

Key words: digital control systems; energy conservation; energy management

and control systems; heating and cooling systems; optimum

start/stop time; preheat time.

iii
TABLE OF CONTENTS

ABSTRACT iii

LIST OF FIGURES v

LIST OF TABLES v

1. INTRODUCTION 1

2. OPTIMUM START/STOP ALGORITHM 4

2.1 OVERVIEW 4

2.2 INPUT AND OUTPUT DATA 6

2.3 ASSUMPTIONS AND LIMITATIONS 8

2.4 METHODOLOGY 9

2.5 APPLICATION NOTES 15

3. CONCLUSIONS 24

ACKNOWLEDGMENTS 25

REFERENCES 26

APPENDIX A. DETERMINATION OF OPTIMUM START TIME FOR HEATING 27

APPENDIX B. FLOW CHARTS OF THE COMPUTER PROGRAMS 34

APPENDIX C. COMPUTER PROGRAM LISTING 41

APPENDIX D. SAMPLE INPUT AND OUTPUT OF THE PROGRAM 53

iv
LIST OF FIGURES

Page

Figure 1. Room air temperature with respect to time of day

under an ideal situation 5

Figure 2. Room air temperature with respect of time of day

under a realistic situation 5

Figure 3. A case with the on-off control operation 12

Figure 4. A typical temperature history during the

initial cycle 12

Figure 5. Relationships of subprograms 14

Figure 6. Flow chart of the main algorithm, OPTSS 16

LIST OF TABLES

Table 1. Cross reference table of notation 17

v

1 . INTRODUCTION

When a building is occupied intermittently, energy savings can be realized by

minimizing the time the heating or cooling system is operated. The interior

building temperature is maintained at a comfortable level only during occupancy. For example, during the heating season, the heating system is turned off

shortly before the end of an occupied period and the space temperature decays

until the heating equipment is reactivated. This reactivation occurs either

because a new occupied period is approaching or because the indoor temperature

has reached a predetermined minimum temperature which must be maintained to

prevent freezing and condensation problems. Building control algorithms that

try to determine the start and stop times for the heating and cooling equipment such that start-up is delayed until the last moment and shut-down takes

place as early as possible before the end of occupancy, are commonly referred

to as "optimum start/stop" algorithms [1,2). One such algorithm is the subject of this report.

Due to thermal inertia of both the building structure and its heating and

cooling equipment, a preheat or precool time is almost always required to

raise or lower the space temperature to the desired level before the beginning

of occupancy. This start-up time depends on the outdoor environment, the

thermal response of the building envelope, and the thermal performance of

space conditioning equipment [3-5]. The starting time of the equipment is the

main value to be determined in optimum start control. Optimum stop control

makes use of the dead time after the heating or cooling equipment is turned

off when thermal inertia prevents the indoor temperature from dropping below

1
or rising above comfort levels. This allows the system to be turned off

before the end of an occupied period.

Practical use of optimum start control was accomplished by Jackson [3,4] on a

heating system using an analog control device. With an intuitively derived

relationship between indoor and outdoor air temperatures, the start time of

preheat period vas determined. Jackson claimed substantial energy savings

from the use of optimum start control. He also indicated that a nighttime

setback technique was not economically justified in control of intermittently

occupied buildings. A field study was made by Henderson [5] comparing optimum

start control with nighttime setback. Bloomfield and Fisk [6] described the

optimization of intermittent heating with a detailed mathematical presentation

and mace simulations using prior knowledge of the thermal response factors of

building structures.

Using recursive least squares estimation, a self-tuning optimum start control

algorithm for a heating plant was developed by Dexter [7]. The preheat start

time is expressed in terms of bulk structure temperature and unknown parameters that are to be determined adaptively using daily data. Both analog and

digital simulations were made. The estimate of bulk temperature vas calculated using a first order difference equation with a bulk time constant

representing the thermal behavior of the structure.

As microprocessors became available, the HVAC industry began to implement

microcomputers to control building equipment. Although software developed by

industry exists on the market, details of the algorithms are not generally

available in the public domain. The purpose of this report is to introduce a

2
simple optimum start/stop algorithm which has been developed for digital

control of heating systems. With minimal changes, the same algorithm can be

used to optimally start and stop building cooling systems.

In the algorithm presented, the starting tine for the heating system is determined by using both the temperature rise characteristics during the preheat

period of the previous day and the temperature decay behavior during the

current cool-down period. The stop time is determined from the delay in the

temperature cool-down cycle. A provision for prevention of freezing and

condensation problems is also incorporated. The nature of the algorithm used

is adaptive, but the approach is much simpler than that used by Dexter [7].

Neither a time series analysis nor recursive least squares estimation is

involved

.

3
2. OPTIMUM START/ STOP ALGORITHM

2.1 OVERVIEW

According to optimum control theory, minimization of energy use can be

achieved if the control is of the "bang-bang" type [8]. This means that, in

the heating mode, the heating system operates at full power during the preheat

period and at zero power during the cool-down period. Similar operating

procedures are applied in the cooling mode, but for simplicity only heating

operation will be described in detail. In addition, it will be assumed that

the minimum temperature reached during the off-period is greater than the

space temperature which would result in freezing or condensation problem.

The bang-bang control scheme is applied only during the unoccupied period. If

the outdoor environmental conditions remain constant during the unoccupied

period, an ideal situation is illustrated as shown in figure 1. Room air

temperature, TRA, responds immediately on the starting or stopping of the

heating equipment. Thermal inertia of the building structure or of the

heating system is not accounted for. This kind of ideal response is

impossible in the real world. Figure 2 represents a more realistic response.

The outdoor environment and thermal response characteristics of the building

structure determine the length of the preheat period.

In the optimum start/stop algorithm presented in this report, the start time

for heating is determined by using the temperature rise characteristics during

the heat-up period of the previous day and the temperature decay characteristics during the current cool-down period. The stop time is determined from

the delay in the temperature decay.

4
tra

OCCUPIED UNOCCUPIED OCCUPIED

TIME OF DAY

Figure 1. Room air temperature with respect to time of day

under an ideal situation

TIME OF DAY

Figure 2. Room air temperature with respect to time of

day under a realistic situation

5
2.2 INPUT AND OUTPUT DATA

It is assumed that the optimum start/stop algorithm is not executed continuously, but at discrete time intervals. The time interval is called the

sampling period. The sampling period used depends upon the building response

characteristics. The sampling period assigned in this report is 15 minutes.

At each sampling instant, the following data are sampled:

t: time of day in military time expressed in hours

T

Ra

: room air temperature

Toa

: outdoor air dry-bulb temperature

Units for temperature data can be either metric or English units. Instead of

room air and outdoor air temperatures, interior and exterior surface temperatures of a building structure may be used. Surface temperature measurements

usually contain less noisy and more stable signals.

As initial input data, the main algorithm (OPTSS) requires the following

information

:

t occ: start time of occupancy (military time)

t unoc

: ending time of occupancy (military time)

t e: initial guess of starting time for use in the learning cycle

s (military time)

y set : set point temperature during the occupied period

In addition, the room air temperature must be maintained above a minimum

temperature to prevent freezing and condensation problems during the heating

season, and it should not be higher than a maximum temperature during the

6
cooling season. Because of this, the following additional input data are also

needed:

HEAT: logical variable

For the heating mode, HEAT is true and for the cooling mode, HEAT

is false.

ymin : minimum temperature allowed during the heating season

ymax : maximum temperature allowed during the cooling season

While the most important output is the start time, t_, the following logical

outputs are also provided for the purpose of system control via supporting

software and hardware:

DAYCNT: logical variable

If DAYCNT is true, the daytime controller is to be called,

otherwise the daytime controller is not called. The daytime

controller is referred as a controller to be used in maintaining

y get during the occupied period assuming that the occupied

period is at daytime.

FSYSON: logical variable

If FSYSON is true, the heating (or cooling) system operates at

full output. Otherwise the heating (or cooling) system operates

at less than full output.

MAINTN : logical variable

If MAINTN is true, a controller is called to maintain a minimum

requirement in heating (or cooling).

7
When a simple on-off controller is used to maintain the minimum unoccupied

temperature during the heating mode (or the maximum permissible temperature

level during the cooling mode). Information on the on-off temperature

differential, Ay, will also be required.

2.3 ASSUMPTIONS AND LIMITATION

The following assumptions were made in the development of the optimum

start/stop algorithm:

(a) Full output operation is carried out only by the heating equipment during

the preheating period and by the cooling equipment during the precooling

period

(b) A daytime controller (e.g., Pi-controller) is available during the

occupied period and both the daytime control and the optimum start/stop

control do not interfere with each other in their operations. If the

occupancy period is at nighttime, the daytime controller should be

considered as a nighttime controller.

(c) Execution of the algorithm is performed once every sampling period of 15

minutes (i.e., four samples per hour).

(d) Maintaining a minimum (maximum) temperature load in the heating (cooling)

mode during the off-period is done by an on-off controller.

The use of the algorithm in an actual implementation is not limited as long

as the computer used in the controls associated with the system to be

controlled has enough capacity to store sampled data and the program.

8
2.4 METHODOLOGY

The optimum start time for heating is determined by using a measured temperature rise curve from the preheat period of the previous day and a measured

temperature decay curve from the current cool-down period, and finding the

intersecting point of the two curves before occupancy. Equations of the two

curves are approximated by simple exponential fits based on actual data. Each

equation contains a single time constant (t), and one predetermined time (x^)

with its corresponding measured temperature (yp. Each time constant is

determined from the exponential fitted equation using two data points. The

intersection of two curves are obtained using a method for finding a root of

the function which is expressed as a difference of two fitted equations.

An x-y coordinate system is employed to express these equations for

convenience. Time and temperature are denoted by x and y, respectively, and

the origin of the x-axis is chosen as a midpoint of the occupied period. When

the occupancy schedule for the next cycle (e.g., next day) is changed (i.e.,

t and t are varied) before the beginning of the occupied period, the

origin of the x-axis is translated by the amount of the change made.

Although it is desirable that the space temperature reaches the target temperature (set point) at the beginning of occupancy, the desired temperature

may not be reached on time due to approximation errors in curve fitting and/or

changes in characteristics of heating (cooling) equipment and building thermal

response from the previous day's state. The temperature will reach the set

point a little earlier or later than the optimum time. The fitted temperature

rise curve is thus corrected by the amount of time offset, x

q

.

9
When the building thermal response is slow and the heating or cooling system

has a transport delay (dead time), then the space temperature does not rise

immediately, but responds after a certain time period is passed. The time

period without the response of temperature rise may be called the "flat

response" period or simply system dead time, d Qn . Under these circumstances,

the start time obtained by finding two curves should be advanced by the amount

of the system dead time. Details on how the start time, x

g

, for heating mode

is determined can be found in Appendix A, where mathematical expressions are

provided

.

At the beginning of the unoccupied period, the heating system is usually

turned off. However, when the decay of indoor temperature is slowed by the

thermal characteristics of the building structure, energy savings can be

realized by advancing the turning-off of the heating system so that the "flat

response" period is included in the occupied period. This advanced stop time

will be referred to as the optimum stop time, even though its determination is

not based on optimum control theory. This approach is simple, compared with

the method developed by Bloomfield and Fisk [6],

When the outdoor air temperature is low enough, the indoor air temperature may

fall below the desired minimum temperature, ym ^ n > during the off-period. The

indoor temperature must be maintained above the minimum temperature to prevent

freezing and condensation problems. A simple on-off control may be utilized

to maintain the minimum level of temperature. The heating system is turned on

when y < ym ^ n , and turned off when y ym i n

+ Ay. In this case, the full

capacity operation of the heating system may not be necessary, and partial

10
load operation may be adequate. This is illustrated in figure 3, where the

dead time is assumed to be zero.

A typical temperature history during the initial cycle (learning cycle) is

illustrated in figure 4. Possible time delays and offset are included.

Assuming the program run starts at near a midpoint time of the occupied period

of a typical weekday in the heating season, the following is a simplified

description of the sequence of operations performed by the program:

(1) Get initial input data and set initial conditions.

(2) Start to take data of t, (=y^) , and Tq^ every 15 minutes.

(3) Turn off the system at the beginning of the unoccupied period.

(4) Determine dead time during the off-period.

(5) Check if on-off control is necessary to maintain a minimum

space temperature (y <_ ym£n any time during the off-period).

(6) Turn on the system at the given initial guess of the start time.

(7) Determine dead time during the on-period.

(8) Turn off the system at the beginning of occupancy and let a daytime

controller take over.

(9) Update constant values, x^, X

2

, x^ , yj , y 2

» and y-j (see Appendix A for

definition)

.

(10) Determine the minimum outdoor temperature.

(11) Update the heat-up curve.

(12) Determine the interval where an intersecting point of the heat-up and

cool-down curves locates.

(13) Begin a new cycle at the midpoint of the following day's occupied period.

11
1

1 OFF- PERIOD

1

1 ON-PERIOD

( *occ ,y — set )

/ * ymin+ ay

\ ( AY

ON-OFF CONTROL

t ymin

X

Figure 3. A case with the on-off control operation

Figure 4. A typical temperature history during the

initial cycle

12
(14) Turn off the system before the beginning of the unoccupied period by the

amount equal to the off-period dead time.

(15) Repeat steps (4) through (5).

(16) Find the intersection of the heat-up and cool-down temperature curves by

using the regula falsi method (false position method) [9].

(17) Predict the new start time incorporating the on-period dead time

determined previously.

(18) Turn on the system at the newly calculated start time.

(19) Go to step (7 )

.

It is possible to generalize the optimum start/stop control algorithm from the

heating mode to the cooling mode by relaxing some of the assumptions made in

Appendix A. The same equations used for heating given in Appendix A can be

used for cooling except that y is taken to be the highest outdoor dry-bulb

temperature during the unoccupied period, the value of ygg is approximated as

0.5 y get and the maximum allowable space temperature, ymax , replaces ym £ n whenever it appears.

Figure 5 shows a block diagram of the relationships between procedures in the

main algorithm. The following are brief descriptions of each routine shown in

this figure:

OPTSS: Primary routine for optimum start/stop control. Each event is

controlled sequentially. Both heating and cooling season

operations are included.

XLTRT: Left-most and right-most values of the time coordinate (x) are

computed to be used as upper and lower limits of the interval in

13
Figure 5. Relationships of subprograms

14
ROOT subroutine.

ROOT:

MAXMIN

XCOORD:

FYDIF:

YONNEW:

YOFF:

This subroutine finds the real root of a function in a given

interval.

A subroutine to obtain the maximum and minimum values of a given

array.

This subroutine converts military time into the modifed time

coordinate, x, and vice versa.

A function that obtains the difference of two

functional values.

A function to update the heat-up profile, yQj,j(x).

(See Appendix A.)

A function to describe the building cool-down profile, Yoff^*

(See Appendix A.)

The flow chart of the main algorithm, the subroutine OPTSS, is shown in figure

6. Further detailed flow charts of the OPTSS and the associate subprograms

are given in Appendix B. Table 1 is a cross reference of the notation used in

the computer program, flow charts, main text, and Appendix A. A complete

listing of the computer program appears in Appendix C. The computer program

was written in FORTRAN 77 for the UNIVAC 1100/82 computer.

2.5 APPLICATION NOTES

The subroutine OPTSS contains four COMMON blocks of shared storage space with

other subroutines and the main program. The DATA statements in the OPTSS

contain the following pre-assigned numerical data:

EPS: small positive real number specifying the accuracy required of

the ROOT routine (=0.005).

15
OPTSS

16

Figure

6.

Flow

chart

of

the

main

algorithm,

OPTSS
Table 1. Cross reference table of notation

Notation Notation

in Programs Descriptions in Flow Charts

A(I) array values A(I)

AMAX maximum value of array values amax

AMIN minimum value of array values ^MIN

DATE date DATE

DAY 0 for non-working day

1 for working day

DAY

DAYCNT true if the daytime controller

operates

DAYCNT

false if the daytime controller

does not operate

DELY temperature difference AY

DX time increment used in XLTRT AX

EPS positive small number for accuracy

used in ROOT

e

EPSOFF positive small number

EPSON positive small number

EPSSET positive small number

EPSX positive small number determined

by sampling frequency

FSYSON true if the system operates at full

power

FSYSON

false if the system is off

FX scalar function values

FXL functional value of FX with the

argument of XL

f

L

FXR functional value of FX with the

argument of XR

f

R

FX2 intermediate value of FX £'

Notation

in the Text

DAYCNT

Ay

c

FSYSON

f (x)

17
Table 1 (cont.)

Notation

in Programs

HEAT

ICYCLE

I END

IFLAG

I START

ITDOWN

ITER

I TIME

ITMAX

ITOFF

ITON

ITSET

ITUNOC

ITUP

IW

MAINTN

Description

true if heating mode

false if cooling mode

number of cycles (e.g., one cycle per day)

index number of ending of search

0 if finding a root is successful

Notation

in Flow Charts

HEAT

ICYCLE

I END

IFLAG

1 if finding a root is not

successful

index number of beginning of search ISTART

index number where temperature ITDOQN

decay begins

number of iteration

index number of modified time of I day

maximum number of iteration to IMAX

obtain a root

index number where X = XOFF ITOFF

index number where X = XSTART ITON

index number where temperature ITSET

reaches at set point

index number where X = XUNOC ITUNOC

index number where temperature ITUP

rise begins

index to control printing out

true if the on-off control is MAINTN

needed

false if the on-off control is

not needed

Notation

in the Text

HEAT

MAINTN

18
Table 1 (cont.)

Notation

in Programs

N

NMAX

NPTHR

NSKIP

PARTON

RSCHDL

SWI

TAUOFF

TAUON

TBEGIN

TDEDOF

TDEDON

TEMP

TIME

TIMEX

TITLE

TOA(I)

Notation Notation

Description in Flow Charts in the Tex t

number of intervals used in XLTRT N

total number of samples a day

number of samples in one hour

number of records to be skipped

true if the system operates with

a partial capacity

false if the system is off

time when the schedule file is

read (military time)

event control switches

(I = 0, . .

. ,7)

system time constant during

off-period

system time constant during

on-period

origin of x-axis (military time)

dead time during off-period

dead time during on-period

indoor or internal surface

temperature

time of day (military time)

modified time of day in scalar

title of input data set in two rows

outdoor temperature

PARTON

SWi

toff

ton S

d 0FF

d

0N

TIME

toa

( i)

t off

t on

d

ON

T

RA

t

T

OA

19
Table 1 (cont.)

Notation Notation Notation

in Programs Description in Flow Charts in the Text

TOCC beginning time of occupancy

(military time)

tocc

TOFF optimum stop time (military time)

TOUT dry-bulb outdoor temperature

TSTART optimum start time (military time) t

s TUNOC beginning time of unoccupancy

(military time)

t unoc

XD X-value where Y-value is equal X X

to YSET

o o

XDATA(I) modified time of day (X-value) x(i) X

XL left-most value of X X

L

X

L

XOCC beginning of occupancy xocc

Xocc XOCC1 past value of XOCC

XOFF optimum off time X

0FF

XR right-most value of X

XROOT root XR00T

XSTART optimum start time x

s x

s XUNOC beginning of unoccupancy ^NOC Xunoc XUNOC1 past value of XUNOC

XX intermediate value of X XX

X(I) specific time of day with its

origin TBEGIN (I = 1,2, 3, 4)

X.

i

x.

1

20
Table 1 (cont.)

Notation

in Programs

X2 '

YDATA(I)

YINF

YMAX

YMIN

YOFF

YONNEW

YSET

YSS

Y(I)

Notation Notation

Descriptions in Flow Charts in the Text

intermediate value used in ROOT X’

temperature (Y-value) Y (I)

y

steady-state temperature as X goes

to infinity during off -period

YCO Yco

maximum temperature Y

MAX ymax

minimum temperature Y

* MIN y . J mm

temperature curve during cool-down

period

yoff yOFF

updated temperature curve during

on-period

y

on y0N C

set point temperature (target

temperature)

Y

SET y set

steady-state temperature as X

goes to infinity during on-period

Y

ss y

ss

temperature (I = 1,2, 3, 4) Y

.

l yi

21
ITMAX: maximum number of iterations permitted in ROOT to obtain a

root within the specified accuracy (=20)

EPSOFF: small positive real number for the temperature tolerance used

in determining the dead time during the off-period (=0.25 °C)

EPSON: small positive real number for the temperature tolerance used

in determining the dead time during the on-period (=0.25 °C)

EPSSET: small positive real number for the temperature tolerance used

in reaching the daytime set point temperature (=0.25 °C)

The above numerical values were assigned to be appropriate for the simulated

system under control. Depending upon the characteristics of the building and

the heating/cooling system, the values in the DATA statements may be changed

accordingly.

For testing purposes, the main program, OPMAIN, can be used. This program,

which is listed at the end of Appendix C, calls the main routine, OPTSS, and

the on-off control routine, ONOFF. In the OPMAIN program, the on-off control

routine is the only routine called during non-working days. The on-off controller may be replaced with another type controller if desired. If occupancy

schedule changes are needed, new values of t occ and t unoc

for the next cycle

must be provided before the ending of the current cycle.

A simulation run of the program has been made using a predetermined temperature history of the outdoor air and the room air to demonstrate input-output

operation. Input data and resulting output data of this run are presented in

Appendix D, and may be used for test purposes.

22
If the sampling frequency is to be changed, the integer number NPTHR in the

PARAMETER statement of the OPTSS and OPMAIN routines must be changed. For

example, if the sampling interval is 10 minutes, the sampling frequency

(NPTHR) will be 6 (samples per hour).

Implementation of the optimum start/stop control algorithm on an actual

building control computer requires that the main program, OPMAIN, is to be

converted into a subroutine which serves as an interface routine between the

main routine, OPTSS, and a supervisory program. The supervisory program may

govern data collection and transmission activity and perform high level

operations; for example, choosing a proper algorithm for a given situation

based on the level of priority.

23
8. CONCLUSIONS

An optimum start/stop control algorithm is described for use in on-line

digital control of HVAC systems. The start time for a heating system is

determined using the temperature rise profile from the previous day's heat-up

period and the temperature decay profile from the current cool-down period,

and finding the point (time before occupancy) at which they intersect. The

profiles are approximated from actual data using a simple exponential fit.

With minor modifications, the algorithm is also capable of finding the optimum

start and stop times for a building cooling system. If this algorithm is

applied to lightweight, intermittently occupied buildings with low thermal

inertia, energy savings should be realized, although the exact magnitude of

these savings has not yet been determined. Future research involves implementation, testing, and verification of this algorithm on an energy management

and control system in an actual building.

24
ACKNOWLEDGEMENTS

The author is indebted to Dr. George E. Kelly and Mr. William B. May for their

helpful suggestions.

25
REFERENCES

1. Shih, J.Y., ' ’Energy Conservation and Building Automation," ASHRAE Trans.,

Vol. 81, Part 1, 1975, pp. 419-435.

2. Liptak, B.G., "Reducing the Operating Costs of Buildings by the Use of

Computers," ASHRAE Trans., Vol. 83, Part 1, 1977, pp. 343-360.

3. Jackson, T.B., '*The Theory and Economics of Optimum Start Programming of

Heating Plant," J.I.H.V.E., Vol. 39, May 1971, pp. A24-A38.

4. Jackson, T.B., "Optimum Start Control - Why Now?," B.S.E., Vol. 44,

October 1976, pp. A24-A44.

5. Henderson, P.E., "Optimum Start Control: Principle and Practice," Energy

World, February 1975, pp. 3-6.

6. Bloomfield, D.P. and Fisk, D.J., '*The Optimization of Intermittent

Heating," (CP 14/77) Practical Studies from the Building Research Establishment, Vol. 4, Energy Heating and Thermal Comfort , the Construction

Press, Lancaster, England, pp. 101-113.

7. Dexter, A.L., "Self-turning Optimum Start Control of Heating Plant,"

Automatica, Vol. 17, No. 3, 1981, pp. 483-492.

8. Sage, A.P. and White, C.C., Optimum Systems Control , 2nd Ed. PrenticeHall, 1977 .

9. Carnahan, B., Luther, H.A., and Wilkes, J.O., Applied Numerical Methods ,

John Wiley, 1969.

26
APPENDIX A. DETERMINATION OF OPTIMUM START TIME FOR HEATING

If measured data are available on the building interior space temperature

with respect to time, approximate equations can be constructed to fit both

the temperature decay and rise curves that result from turning the heating

equipment off and on. Assuming that the structure is very light (fast

thermal response) and that the heating equipment has no dead time (immediate

response), a temperature history similar to the one shown in figure A-l may

be assumed. The off- and on-periods are defined as the period when heating

equipment is off and full on, respectively. For convenience, an x-y

coordinate system described in section 2.4 is used. Time and temperature

are denoted by x and y, respectively. Military time at the origin of the

x-y coordinate system is taken as a midpoint time of the occupied period.

During the on-period, an approximate equation yields:

( 1 )

where

x^ = an arbitrary point of x in the on-period (the chosen value of x^

is one-third of the on-period plus x-value at the intersection)

y^

= measured temperature at x^,

y = steady-state temperature as x -* °°, and

u J = time constant during the on-period.

27
Figure A-l. Temperature history of a lightweight structure

with immediate response of heating equipment

28
Using two data points, (x^, y^) and (x^, y2 )» the time constant can be

obtained from:

where

x^ = an arbitrary point of x in the on-period which is greater than

x^ (the chosen value of x^ is two-thirds of the on-period plus

x-value at the intersection) , and

y^

= measured temperature at x^.

An expression for the off-period is:

y0FF (x > = y°o + (Y, - yoo) e ( 3 )

where

x^ = an arbitrary point of x in the off-period (the chosen value of

x^ is one-fourth of the off-period plus x-value at the beginning of

the unoccupied period. ),

y^

= measured temperature at x^,

yoo = steady-state temperature as x -»• and

x OFF = time constant during off-period.

The off-period time constant is given by

t off ( 4 )

29
If y<» and are predetermined, the decay and rise temperature curves can be

determined using the given data points (x^ y^, (x

2

, y2 >, (x 3> y

3

), and (x^

,

y^). The value of yoo is taken to be the lowest allowable indoor dry-bulb

temperature during the unoccupied period (heating mode) and y is approximated

as 1.5 y , where y is the set point temperature for occupancy. Since the

u C L. O C L time domain of interest is finite, the effect on y (x) and y (x) of errors

UN Or F

in the estimated asymtotic values yoo and y^g is small.

After the functions yn (x) and y (x) are determined, the point of their

intersection can be obtained using a method for finding a root of a function

f(x), where f (x) = yQN

(x) - yQFF(x). The method used was the regula falsi

method (false position method [9]). For this method, if a function is bounded

and has only one root in a given interval, the solution converges within a

given tolerance after a number of iterations. The time corresponding to their

intersection is then considered to be the optimum start time.

Difficulties can arise in using the regula falsi method with 7q^(x ) if the

whole unoccupied period is taken as the interval of interest. As seen in

figure A-l, y^^(x) is not, in general, bounded in that interval. Thus, the

left-most value, xT , of the interval should be selected so that y^„(x) is

bounded in that interval. The approach used in this algorithm for choosing

x is to set x equal to the value of x which satisfies the condition:

Li Li yrt . y T (x) = y , , where y , is the minimum temperature to maintain.

0N •'min •'min

The optimum start time is determined by finding an intersection of the current

temperature decay curve and the temperature rise curve of the previous day.

Any changes in the heating system characteristics and environment in the

30
current day could result in a change in the temperature rise curve during

the current on-period.

It is desirable that the temperature rise curve reaches the target temperature

(set point) at the beginning of occupancy. If the temperature rise curve

changes during the on-period and there are errors due to approximation, the

space temperature will reach the set point a little earlier or later than the

optimum time. In order to correct this offset, the fitted temperature rise

equation should be corrected by the amount of time offset.

Figure A-2 shows both the fitted curve (solid line) based on measurement and

the updated curve (dotted line) during the on-period. The x-coordinate is

shifted by the difference, x - x . x is the beginning of occupancy and

x^ is the point where y is equal to the set point, y

g

. The updated equation

is then

y,on

(5)

where

x"^ = the updated x^, given by

( 6 )

and

x

o x

1 TON

(7)

31
X

Figure A-2. The fitted curve (solid line) based on measurement

and the updated curve (dotted line) during the

on-period

Figure A-3. The dead time during the on-period

32
Consequently, equation 5 should be used instead of equation 1 to determine the

optimum start time in conjunction with equation 3.

When the building structure has slow thermal response and the heating system

has a transport delay time (dead time), the assumption made earlier about dead

time is not correct. Therefore, it is necessary to take account of the duration

of any "flat response" during the on-period. This "flat response" period can

be defined as the time period that | y — y | < e, where y is the temperature

D J at the beginning of the on-period and e is a small positive number. The dead

time is then defined as the duration of the "flat response" period (see

figure A-3). The calculated start time must then be adjusted by the amount

of dead time, d^. For example, if the calculated start time is x^ under the

assumptions of light structure and no dead time, the correct start time for

/

a system with high thermal mass and a heating system with transport delay will

be

X

S = X

S

d

0N‘

Since the magnitude of e strongly affects the dead time, the better the

selection of e, the better the estimate of the optimum start time. A value

for e of 0.25°C was used in the computer program presented in Appendix C.

33
APPENDIX B. FLOW CHARTS OF THE COMPUTER PROGRAMS

Flow charts of the optimum start/stop control algorithm are shown in

figures B-l through B-9. The most important subroutine is the OPTSS

subroutine. For detailed information, this subroutine is divided into

five blocks, which are given in subsequent figures (figures B-l through

B-3). Descriptions of principal variables can be found in table 1.

34
35

Figure

B-l.

Flow

charts

of

the

block 1

and 2

of

the

subroutine,

OPTSS
Is

it

heating

36

Figure

B-2.

Flow

charts

of

the

block 3

and 4

of

the

subroutine,

OPTSS
Is

IFLAG

37

Figure

B-3.

Flow

chart

of

the

block 5

of

the

Figure

B-4.

Flow

chart

of

the

subroutine,

OPTSS

subroutine,

XCOORD
38

Figure

B-5.

Flow

charts

of

the

functions;

Figure

B-6

.

Flow

charts

of

the

YONNEW,

YOFF,

and

FYDIF

subroutine;

MAXMIN
39

Figure

B-7.

Flow

charts

of

the

subroutines,

ROOT

and

XLTRT
ONOFF

40

Figure

B-8.

Flow

chart

of

the

main

Figure

B-9.

Flow

chart

of

the

on-off

program,

OPMAIN

subroutine,

ONOFF
APPENDIX C. COMPUTER PROGRAM LISTING

Q$Q$Q$*OPT( 1 ) .OPTALL(O)

1 C ••***••••

2 C

3 C OPTSS : OPTIMUM START AND STOP CONTROL ALGORITHM

4 C

5 c MAR. 3. 1983 C.P.

6 C

7 C

e c

9 C daycnt TRUE IF THE DAYTIME CONTOLLER OPERATES

10 C FALSE IF THE DAYTIME CONTROLLER DOES NOT OPERATE

1 1 C DELY TEMPERATURE DIFFERENCE

12 C EPS POSITIVE SMALL NUMBER FOR ACCURACY USED IN ROOT

13 C EPSOFF POSITIVE SMALL NUMBER

14 C EPSON POSITIVE SMALL NUMBER

15 C EPSSET POSITIVE SMALL NUMBER

16 C EPSX TOLERANCE DETERMINED BY SAMPLING FREQUENCY

17 C FSYSON TRUE IF THE SYSTEM OPERATES AT FULL POWER

18 C FALSE IF THE SYSTEM IS OFF

19 C HEAT TRUE FOR HEATING MODE

20 C FALSE FOR COOLING MODE

21 C ICYCLE NUMBER OF CYCLES (ONE CYLCE PER DAY)

22 c ITDOWN INDEX NUMBER WHERE TEMPERATURE DECAY BEGINS

23 c IT I ME INDEX NUMBER OF MODIFIED TIME OF DAY

24 c ITMAX MAXIMUM NUMBER OF ITERATION TO OTAIN A ROOT USED IN ROOT

25 c ITOFF INDEX NUMBER WHERE X=XOFF

26 c ITON INDEX NUMBER WHERE X=XSTART

27 c 1TSET INDEX NUMBER WHERE TEMPERATURE REACHES AT THE SET POINT

28 c ITUNOC INDEX NUMBER WHERE X=XUNOC

29 c I TUP INDEX NUMBER WHERE TEMPERATURE' RISE BEGINS

30 c MAINTN TRUE IF THE ON-OFF CONTROL IS NEEDED

31 c FALSE IF THE ON-OFF CONTROL IS NOT NEEDED

32 c NMAX TOTAL NUMBER OF SAMPLES A DAY

33 c NPTHR NUMBER OF SAMPLES IN ONE HOUR

34 c PARTON TRUE IF THE SYSTEM OPERATES WITH A PARTIAL CAPACITY

35 c FALSE IF THE SYSTEM IS OFF

36 c SWI EVENT CONTROL SWITCHES (1=0 7)

37 c TAUOFF SYSTEM TIME CONSTANT DURING THE OFF-PERIOD

38 c TAUON SYSTEM TIME CONSTANT DURING THE ON-PERIOD

39 c TBEGIN ORIGIN OF X-COORDINATE (MILITARY TIME)

40 c TOE DOF DEAD TIME DURING THE OFF-PERIOD

41 c TOE DON DEAD TIME DURING THE ON-PERIOD

42 c TEMP INDOOR OR INTERIOR SURFACE TEMPERATURE

43 c TIME TIME OF DAY (MILITARY TIME)

44 c TIMEX MODIFIED TIME OF DAY IN SCALAR

45 c TOA ( *

)

OUTDOOR TEMPERATURE

46 c TOCC BEGINNING TIME OF OCCUPANCY (MILITARY TIME)

47 c TOFF OPTIMUM STOP TIME (MILITARY TIME)

48 c TOUT DRY- BULB OUTDOOR TEMPERATURE

49 c TST ART OPTIMUM START TIME (MILITARY TIME)

50 c TUNOC BEGINNING TIME OF UNOCCUPANCY (MILITARY TIME)

51 c X( I ) MODIFIED TIME OF DAY WITH ITS ORIGIN AT TBEGIN

52 c XD X-VALUE WHERE Y-VALUE IS EQUAL TO YSET

53 c XDATA( *

)

MODIFIED TIME OF DAY (X-VALUE)

54 c XOCC BEGINNING TIME OF OCCUPANCY IN X-COORDINATE

55 c X0CC1 PAST VALUE OF XOCC

56 c XOFF OPTIMUM STOP TIME IN X-COORDINATE

57 c XSTART OPTIMUM START TIME IN X-COORDINATE

41
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of)n

ooo

ooo

oooonooooooooo

XUN0C BEGINNING TIME OF UNOCCUPANCY IN X-C00RDINATE

XUN0C1 PAST VALUE OF XOCC

V ( I ) TEMPERATURE

YDATA( * ) TEMPERATURE (Y-VALUE)

YINF STEADY-STATE TEMPERATURE AS X GOES TO INFINITY OURING

THE OFF-PERIOD

YMAX MAXIMUM TEMPERATURE

YMIN MINIMUM TEMPERATURE

YSET SET POINT TEMPERATURE (TARGET TEMPERATURE)

YSS STEADY-STATE TEMPERATURE AS X GOES TO INFINITY OURING

THE ON-PERIOD

SUBROUTINE OPTSS( T IME . TEMP . TOUT

)

LOGICAL SW0.SW1 . SW2 . SW3 . SW4 . SW5 . SW6 . SW7 . HEAT . MAINTN . PARTON

.

4 FSYSON , DAYCNT

EXTERNAL FYDIF

PARAMETER ( NPTHR*4 . NMAX«NPTHR*24

)

DIMENSION XDAT A ( 0 : NMAX ) , YDATA ( 0 : NMAX ) , TOA ( 0 : NMAX

)

COMMON /XY/ X ( 4 ) .Y(4) . XOCC . XUNOC . YSET

4 /CNST / TAUON.TAUOFF .YINF, YSS. XD

4 /LIMIT/ YMIN. YMAX. HEAT. MAINTN. FSYSON. DAYCNT . XST ART

4 /TSET/ TSTART.TOFF.TOCC.TUNOC.TBEGIN

NAMELIST /RESULT/X , Y , XOFF , XST ART . TAUON . TAUOFF . TDEDON , TDEDOF

.

4 YSS. YINF. I CYCLE. I TIME. I TOFF. ITUNOC. ITDOWN. I TON. ITUP.ITSET.

4 T BEG IN , TOF F . TUNOC , TSTART , TOCC

DATA EPS , ITMAX/O . 005 , 20/

DATA SWO/. TRUE. /.EPSOFF. EPSON. EPSSET/3*0. 25/

INITIAL CONDITIONS

EPSX«0.5/NPTHR

IF(SWO) THEN

ICYCLE-0

SW1». FALSE.

SW2>. FALSE.

SW3«. FALSE.

SW4«. FALSE.

SW5». FALSE.

SW6>. FALSE.

SW7«. FALSE.

MAINTN=. FALSE.

PART0N= . FALSE

.

FSYSON=. FALSE.

DAYCNT= .TRUE.

CHANGE TIME COORDINATES

CALL XCOORD (XST ART .TSTART ,T BEGIN. 1

)

CALL XCOORD ( XOCC . TOCC , TBEGIN , 1

)

CALL XCOORD(XUNOC. TUNOC. TBEGIN. 1

)

ENDIF

STORE INPUT DATA AS ARRAYS IN TERMS OF INDEX TIME

CALL XCOORD ( TIMEX, TIME. TBEG IN, 1

)

ITIME«TIMEX*NPTHR

42
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»

no

ooo

ooo

oooo

oooo

T0A( 1 TIME ) »T0UT

XDATA( ITIME)-TIMEX

YDATA( IT I ME )«TEMP

X(4)»XDATA( I TIME)

Y(4)“YDATA( I TIME)

TURN OFF THE SYSTEM AT THE BEGINNING OF UNOCCUPANCY OF INITIAL

CYCLE

IF ( SWO. AND.ABS(XDATA( IT IME ) -XUNOC ) . LT . EPSX ) THEN

DAYCNT*. FALSE.

IT0FF-IT1ME

ITUNOC* IT IME SWO». FALSE.

SW2« . TRUE

.

PRINT 100

ENOIF

TURN OFF THE SYSTEM AT THE BEGINNING OF UNOCCUPANCY AFTER INITIAL

CYCLE

I F ( SW 1 . AND . XDATA( I TIME ) . GE . XOFF ) THEN

DAYCNT=. FALSE.

ITOFF=ITIME

SW1«. FALSE.

SW2= .TRUE.

PRINT 200

ENOIF

DETERMINE DEAD TIME DURING THE OFF-PERIOD

IF ( SW2 . AND . ABS( YDATA( IT IME ) - YDATA( ITOFF ) ) .GE . EPSOFF ) THEN

TDEDOF=XDATA( IT IME ) -XDATA( ITOFF

)

XOFF=XUNOC-TDEDOF

ITDOWN= ITIME

SW2*. FALSE.

SW3« .TRUE

.

SW4* . TRUE

PRINT 300

ENDIF

MAINTAIN MINIMUM LEVEL OF OPREATION

IF(SW3) THEN

I F( (HEAT. AND. (Y(4) . LE . YMI N ) ) . OR . ( . NOT . (HEAT ). AND

.

6 Y ( 4 ) . GT . YMAX ) ) THEN

I F ( ICYCLE.GE. 1 ) XST ART*XLEFT-TDEDON

MAINTN= . TRUE

.

SW3= . FALSE.

PRINT 400

ENDIF

ENDIF

FIND THE INTERSECTION OF ON- AND OFF-TEMPERATURE CURVES

I F ( ICYCLE.GE . 1 ) THEN

IF(ABS(XDATA( ITIME) -X(3) ) .LT.EPSX) THEN

Y(3)«YDATA( ITIME)

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228

229

230

231

nnnn

non

ooo

ENOIF

I F ( ( .N0T.MAINTN) .AND. (XDATA( IT IME ) . GT . X ( 3 ) . AND . XDATA ( IT IME

)

& .LT.XSTART) . AND . ( ABS ( Y( 3

) * Y(4 ) ).GT.EPSSET) ) THEN

I F( HEAT ) THEN

I F ( Y{ 4 ) . GE . Y (3 ) ) THEN

YINF=TOAMAX

ELSE

YINF=TOAMIN

END1F

ELSE

IF(Y(4) . LE.Y(3) ) THEN

YINF'TOAMIN

ELSE

YINF«T0AMAX

ENDIF

ENDIF

X L«X LEFT

XR-XRIGHT

CALL ROOT ( X L . XR , EPS . I TMAX , FYDIF , XROOT . I FLAG

)

I F ( I FLAG . EQ . 0 ) THEN

XST ART*XROOT * TDEDON

ELSE

XST ART “X LEFT * TDEDON

ENDIF

ENDIF

CALL XCOORD( XSTART,TSTART,TBEGIN,2)

ENDIF

TRUN ON THE SYSTEM AT OPTIMUM START TIME

I F ( SW4 . AND . XDATA( I TIME ) . GE . XSTART ) THEN

FSYSON= .TRUE.

I TON= I T IME MAINTN= .FALSE.

SW4». FALSE.

SW5« . TRUE

.

PRINT 500

ENDIF

DETERMINE DEAD TIME DURING THE ON-PERIOD

IF ( SW5 . AND. ABS( YDATA( IT IME ) -YDATA( I TON) ) .GE. EPSON) THEN

TDEDON=XDATA( IT IME) -XST ART

ITUP=ITIME

SW5-. FALSE.

SW6« . TRUE

.

PRINT 600

ENDIF

TURN OFF THE SYSTEM NEAR THE BEGINNING OF OCCUPANCY AND CALL

THE DAYTIME CONTROLLER

IF(SW6. AND. ((ABS(YDATA(ITIME)-YSET). LE.EPSSET)

& .OR. (HEAT. AND.YDATA( ITIME) .GE.YSET)

& ,OR.(.NOT.HEAT.AND.YDATA(IT!ME).LE.YSET))) then

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289

nnn

ooo

oooo

FSYSON* . FALSE

.

DAYCNT * .TRUE

.

ITSET=ITIME

5W6* . FAL5E

.

SW7*= . TRUE

.

PRINT 700

ENDIF

COMPUTE X- AND Y-VALUES BASED ON PREVIOUS OBSERVATION

AND UPDATE THEM USING NEW TOCC AND TUNOC

Z F ( SW7 ) THEN

CALL XC00RD( XOFF .T0FF.TBEGIN.2)

PRINT 900

PRINT RESULT

PRINT 900

X0CC1=X0CC

XUN0C1 =XUNOC

I F( TUNOC. GT .TOCC) THEN

TBEGI N=0 . 5* ( TUNOC+TOCC

)

ELSE

TBEGIN=0 . 5* ( TUNOC+24 . +T0CC

)

I F ( T BEGIN . GE . 24 . ) TBEGIN=TBEGIN-24

.

ENDIF

CALL XC00RD(X0CC.T0CC.TBEGIN.1

)

CALL XCOORD( XUNOC, TUNOC , T BEGIN, 1

)

K1*(ITSET-ITUP)/3+ITUP

K2-(ITSET-ITUP)*2/3+ITUP

I F ( ICYCLE . EQ . 0 ) THEN

K3=( ITON- ITUN0C)/4+ITUN0C

ELSE

K3= ( ITON- lTD0WN)/4+ITD0WN

ENDIF

X(1 ) =XDAT A ( K 1 ) + X0CC-X0CC1

X(2)=XDATA(K2 )+XOCC - XOCC

1

X(3)=XDATA(K3 )+ XUNOC- XUNOC

1

Y ( 1 ) =YDAT A ( K 1 ) Y ( 2

) =YDAT A ( K2

)

Y ( 3

) =YDAT A ( K3

)

XSTART = XSTART+XOCC* X0CC1

X0FF=X0FF+XUN0C - XUNOC

1

CALL XCOORD( XSTART.TSTART, TBEGIN , 2

)

CALL XCOORD( XOFF, TOFF. TBEGIN.2)

DETERMINE MAXIMUM AND MINIMUM OUTDOOR TEMPERATURES

CALL MAXM I N ( TOA , TOAMI N . TOAMAX , NMAX

)

COMPUTE CONSTANTS FOR ON- AND OFF-PERIOD TEMPERATURE EQUATIONS

I F ( HEAT ) THEN

YSS= 1 . 5* YSET

YINF=TOAMIN

ELSE

YSS=0.5*YSET

YINF-TOAMAX

ENDIF

45
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347

C DETERMINE XLEFT AND XRIGHT

c CALL XLTRT(XLEFT. XRIGHT)

ICYCLE=ICYCLE+1

SW7«. FALSE.

PRINT 800

PRINT 900

PRINT RESULT

PRINT 900

ENDIF

C

C RESET SWITCH SW1 AT THE BEGINNING OF NEXT CYCLE

C

I F ( ICYCLE.GE. 1 . AND. ABS( TIME -TBEGIN) . LT.EPSX) SW1 * .TRUE

.

C

C FORMAT STATEMENTS

C

100 FORMAT (5X, '---S WO ---SYSTEM OFF AT THE FIRST CYCLE')

200 FORMAT (5X, ' - - -SW1 - --SYSTEM OFF')

300 FORMAT (5X,'---SW2 ---DECAY/RISE RESPONSE OCCURS')

400 FORMAT (5X, '**-SW3--»0N-0FF CONTROLLER TAKES OVER')

500 F0RMAT(5X. '---SW4-- -SYSTEM ON AT FULL POWER')

600 FORMAT ( 5X , ' SW5 - - -UPR I SE/DECAY RESPONSE OCCURS')

700 FORMAT (5X. '---SW6--- DAYTIME CONTROLLER TAKES OVER')

BOO F0RMAT(5X. -SW7- --CALCULATIONS FOR NEXT CYCLE ARE DONE')

900 FORMAT ( 80 ( ' *

' ) /

)

C

c RETURN

END

C

c

c

c

c

cc

c

c

cc

c

c

c

c

cc

c

c

cc

FYDIF : TEMPERATURE DIFFERENCE

YONNEW UPDATED TEMPERATURE DURING THE HEAT-UP PERIOD

YOFF TEMPERATURE DURING THE COOL-DOWN PERIOD

************•**•»**»•******»***•••***•****+*»*•»**••****#••*»***#•**•

FUNCTION FYDIF ( XX

)

TEMPERATURE DIFFERENCE BETWEEN YONNEW AND YOFF

TO BE USED IN DETERMINATION OF INTERSECTION. XROOT

FYDI F« YONNEW ( XX )-YOFF(XX)

RETURN

END *********************************************************************

YONNEW, YOFF : THE ON- AND OFF-PERIOD TEMPERATURES

FUNCTION YONNEW(XX)

COMMON /XY/ X(4) ,Y(4) , XOCC , XUNOC , YSET

A /CNST/ TAUON , TAUOFF . YINF , YSS , XD

46
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349

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351

352

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400

401

402

403

404

405

C

C THE ON-PERIOD TEMPERATURE WITH ADJUSTMENT

C

TAUON« ( X ( 2

) -X ( 1 ))/ALOG( ( YSS- Y ( 1 ) ) /( YSS- Y( 2 ) )

)

XD*X( 1 )-TAU0N*AL0G((YSS-YSET)/(YSS-Y(1 ) )

)

X1P=X(1 )+XOCC- XD

YONNEW=YSS- (YSS-YI 1 )

) *EXP (

( XI P-XX )/TAUON

)

RETURN

C

ENTRY YOFF(XX)

C

C THE OFF-PERIOD TEMPERATURE

C

TAU0FF=(X(4)-X(3) )/ALOG( (Y(3)-YINF)/(Y(4)-YINF)

)

Y0FF*(Y(3)-YINF)*EXP( ( X ( 3

) - XX )/TAUOFF )+YINF

RETURN

END

£ 4*a*»********a**************r***********************«********»«*«**«*

c

C ROOT : FIND A ROOT OF A FUNCTION F(X)«0 IN A GIVEN INTERVAL

C BY THE REGULA-FALSI METHOD

C

C REFERENCE : C CARNAHAN. LUTHER. AND WILKES

C * APPLIED NUMERICAL METHODS ", JOHN WILEY. 1969. P.193

C

C THE ORIGINAL PROGRAM WAS MODIFIDED IN F0RTRAN77

C

c

•• c

c EPS A POSITIVE SMALL NUMBER FOR ACCURACY

c FX SCALAR FUNCTION VALUE

c I FLAG 0 IF FINDING A ROOT IS SUCCESSFUL

c

c ITMAX

1 IF FINDING A ROOT IS NOT SUCCESSFUL

MAXIMUM NUMBER OF ITERATIONS

c XL LEFT-MOST X-VALUE

c XR RIGHT-MOST X-VALUE

c XROOT A REAL ROOT

c

£ a*********************************************************************

c SUBROUT I N E ROOT ( X L . X R . EPS . I TMAX . FX . XROOT . I F LAG

)

C

C* PRINT 100. XL.XR.EPS, I TMAX

100 FORMAT ( 3F1 0 . 5 , 15)

C

C SET LEFTMOST AND RIGHTMOST FUNCTION VALUES

C

FXL-FX(XL)

FXR*FX( XR

)

C

C CHECK FOR PRESENCE OF A ROOT

C

IF(FXL*FXR.LT.O. ) THEN

C

C BEGIN REGULA FALSI ITERATION

C

DO 10 XTER=1 , ITMAX

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X2«(XL*FXR-XR*FXL)/(FXR-FXL)

FX2«FX(X2)

C

C CHECK FOR CONVERGENCE

c IF(ABS(FX2).LE.EPS) GOTO 20

C

C KEEP RIGHT OR LEFT SUBINTERVAL

C

IF(FX2*FXL. LT.O. ) THEN

XR = X2

FXR= FX2

ELSE

XL*X2

FXL-FX2

ENDIF

10 CONTINUE

PRINT 200.ITMAX

200 F0RMAT(5X. 'NO CONVERGENCE AFTER' . 15 . 2X ITERATIONS '

)

ELSEIF ( FXL*FXR.EO.O. ) THEN

ITER=1

I F( FXL . EQ . 0 . ) THEN

X2 = XL

FX2=0

.

ELSE

X2 = XR

FX2=0

.

ENDIF

ELSE

IFLAG=1

PRINT 300

300 FORMAT (5X. 'POSSIBLY NO ROOT ')

RETURN

ENDIF

20 XROOT «X2

IFLAG-0

C* PRINT 400. ITER. XROOT. FX2

400 F0RMAT(5X. ' ITER « '. I5/5X .' XROOT » ' , FI 0 . 6/5X . ' FX2 «'.F10.6/)

C

RETURN

END

C

C XLTRT : LEFTMOST AND RIGHTMOST X-VALUES FOR THE SUBROUTINE. ROOT

C

C

C DX TIME INCREMENT

C N NUMBER OF INTERVALS

C XL LEFT-MOST VALUE OF X C XR RIGHT-MOST VALUE OF X c

c SUBROUTINE XLTRT ( XL , XR

)

LOGICAL HEAT

COMMON /XY/ X ( 4 ) , Y ( 4 ) , XOCC , XUNOC , YSET

A /LIMIT/ YM1 N . YMAX , HEAT . MAINTN , FSYSON . DAYCNT , XST ART

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C

C TIME BASIS IS HOUR. AND TIME INCREMENT IS 5 MIN.

C

DX*5./60.

N= ( XOCC-XUNOC ) /DX

XX*X( 2

}

DO 10 1=1 .N

I F ( HEAT ) THEN

I F ( YONNEW ( XX ) . LE . YMIN ) GOTO 20

ELSEIF( YONNEW(XX) .GE. YMAX) THEN

GOTO 20

ENDIF

XX*XX-DX

10 CONTINUE

20 XL*XX

XR=X0CC

PRINT 100.XL.XR

100 FORMAT ( 5X , ' XLEFT* '. FI 0 . 4 . 5X . ' XRIGHT- ' . F10 . 4/

)

C

RETURN

END

C *****************************************************

cc

cc

c

c

c

cc

c

c

c

cc

c

MAXMIN : EVALUATION OF MAXIMUM AND MINIMUM VALUES

A ARRAY VALUES

AMAX MAXIMUM VALUE OF ARRAY VALUES

AMIN MIMINUM VALUE OF ARRAY VALUES

IEND INDEX NUMBER OF ENDING OF SEARCH

ISTART INDEX NUMBER OF START OF SEARCH

NMAX NUMBER OF SAMPLES A DAY

NPTHR SAMPLEING FREQUENCY ( SAMPLES PER HOUR )

• •*#*«,-*•**»**********.,*•*•*•*****•«**•*»***•»»***•#•**•**•**••**•

SUBROUT I NE MAXMIN ( A . AMI N . AMAX . NMAX

)

DIMENSION A ( 0 : NMAX

)

COMMON /XY/X ( 4 ) . Y ( 4 ) , XOCC . XUNOC . YSET

NPTHR=NMAX/24

I ST ART = XUNOC* NPTHR

IEND=XOCC*NPTHR

AMIN=A( ISTART- 1

)

AMAX* A ( ISTART- 1

)

DO 10 I=ISTART , IEND

I F ( A ( I ) . LT . AMIN ) AMIN*A(I)

IF(A(I) .GT.AMAX) AMAX*A(I)

10 CONTINUE

C

RETURN

END

C * + * + **********##*•«******.)>«•**•**• + *•*»*•*••***•***» + *•«*»*••*••*•••

c

C XCOORD : X-COORDINATE TRANSFORM

C

C I FLAG *1 FOR TRANSFORM OF TIME OF DAY INTO X-VALUE

C -2 FOR TRANSFORM OF X-VALUE INTO TIME OF DAY

49
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579

oonoonoonooooooo

o

o

oooooo

n

nnn SUBROUTINE XC00RD( X . TIME . TBEGIN . I FLAG)

IF( IFLAG.E0. 1 ) THEN

I F(TIME.GE. TBEGIN. AND. TIME. LE. 24. ) THEN

X«TIME-TBEGIN

ELSE

X-TIME+(24. -TBEGIN)

ENDIF

ELSE

TIME-X+TBEGIN

I F ( TIME . GE . 24 . 0 ) TIME-TIME-24.0

ENDIF

RETURN

END

ONOFF : THE ON-OFF CONTROL ALGORITHM

SUBROUTINE ONOFF(TEMP)

LOGICAL HEAT.MAINTN, FSYSON . DAYCNT , PARTON

COMMON /LIMIT/ YMI N , YMAX . HEAT MAINTN . FSYSON , DAYCNT , XSTART

A /ONOFFC/ PARTON. DELY

I F ( HEAT ) THEN

I F ( TEMP . LE . YM1N ) PARTON- . TRUE

I F( TEMP .GE . YMIN+DELY ) PARTON- . FALSE.

ELSEIF(TEMP.GE.YMAX) THEN

PARTON- .TRUE.

ELSEI F (TEMP . LE . YMAX- DELY ) THEN

PARTON- .FALSE.

ENDIF

RETURN

END

***+*#*******•********» •»** •***************#***•***»****** •-#**•*

OPMAIN : MAIN PROGRAM OF OPTIMUM START/STOP CONTROL

FOR BOTH HEATING AND COOLING SEASONS

DATE DATE OF YEAR

DAY -0 FOR NON-WORKING DAY

-1 FOR WORKING DAY

NSKIP NUMBER OF RECORDS TO BE SKIPPED IN PRINTING OUT

RSCHDL TIMES WHEN THE SCHEDULE FILE IS READ (MILITARY TIME)

TITLE TITLE OF INPUT DATA SET IN TWO ROWS

»**•*************•***•********•***«# +** *#**********•*•***#*»*****

PROGRAM OPMAIN

LOGICAL HEAT . MAINTN . FSYSON . DAYCNT . PARTON

INTEGER DATE. DAY

50
580

581

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586

587

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C

C

C

C

cc

c

c

c

c10

cc

c

c

CHARACTER TITLE(2)*80

PARAMETER ( NPTHR=4

)

DIMENSION RSCHDK2)

COMMON /XY/ X ( 4 ) , Y ( 4 ) , XOCC , XUNOC , YSET

& /CNST/ TAUON , T AUOFF .YINF.YSS.XD

& /LIMIT/ YMI N . YMAX , HEAT , MA I NTN , FSYSON „ DAYCNT , XST ART

& /ONOFFC/ PARTON . DELY

& /TSET/ T START, TOFF, TOCC . TUNOC . T BEG IN NAMELIST /INIT/ DATE . DAY . TOCC . TUNOC , TSTART

.

« YSET. YMIN, YMAX. DELY, HEAT. NSKIP

& /DAYOCC/ DATE, DAY. TOCC. TUNOC

& /OUTPUT/ TIME, MAINTN. PARTON. FSYSON. DAYCNT,

& TSTART. TOFF. TOCC. TUNOC. TBEGIN

DATA RSCHDL/6. 0.18.0/

OPEN FILE 7 FOR INITIAL AND SCHEDULE INFORMATION

OPEN FILE 8 FOR SIMULATED DATA OF MEASURED VALUES

OPEN ( 7 . FILE 3 ' OPT IN')

OPEN (8,FILE='0PTDATA')

REWIND 7 REWIND 8

READ INITIAL INPUT DATA

READ( 7 , 1 000 ) TITLE

READ(7. INIT)

PRINT 1000. TITLE

PRINT INIT

IW = 0

1*0

READ SIMULATED MEASURED DATA AND SCHEDULE INFORMATION

READ(8.*,EN 0=999) TIME . TEMP , TOUT

XF(Z.EO.O) TBEGIN=TIME

I F ( ( ABS( T IME - RSCHDL( 1 ) ) . LT . 0. 5/NPTHR ) . OR

.

& ( ABS ( T IME* RSCHDL ( 2 ) ) . LT . 0 . 5/NPTHR ) ) THEN

READ( 7. DAYOCC)

PRINT DAYOCC

ENDIF

MAINTAIN MINIMUM REQUIREMENT DURING NOT-WORKING DAYS

I F ( DAY . EQ • 0 ) THEN

MAINTN* .TRUE

.

ELSE

CALL OPTSS(TIME.TEMP.TOUT)

ENDIF

I F ( MAINTN ) CALL ONOFF (TEMP

)

I F ( IW.EQ.NSKIP+1 ) THEN

PRINT *. TIME. TEMP. TOUT

PRINT OUTPUT

IW»0

ENDIF

IW-IW-M

1-1

51
638 GOTO 10

639 C

640 C FORMAT STATEMENTS

641 C

642 1000 FORMAT( A80/A80

)

643 c 644 999 STOP

645 ENO
INPUT

DATA

APPENDIX D SAMPLE INPUT AND OUTPUT OF THE PROGRAM

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64
NE.S- > 4A (REV. 2-ao>

3. Publication Date June 1983

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See Instructions)

1. PUBLICATION OR

REPORT NO.

83-2720

2. Performing Organ. Report No.

4. TITLE AND SUBTITLE

An Optimum Start/Stop Control Algorithm for Heating and Cooling

Systems in Buildings

5. AUTHOR(S)

Cheol Park

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

national bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

Office of Buildings and Community Systems U.S. Navy Civil Engineering Laboratory

U.S. Department of Energy U.S. Department of Defense

1000 Independence Avenue, SW Washington, DC 20310

Washington, DC 20585

10.

SUPPLEMENTARY NOTES

2] Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

When a building structure is occupied intermittently, energy savings can be

realized from the optimal start-up and shut-down of the heating or cooling system.

This strategy, known as optimum start/stop control, reduces energy consumption by

delaying the start-up of the space conditioning system until the last moment and then

initiating shut-down as early as possible, while maintaining a preset level of comfort

during the period of building occupancy.

Based on the bang-bang control theory, a simple optimum start/stop control algorithm

is developed for computerized control systems in buildings. The optimum start time is

obtained by finding the intersection of cool-down and heat-up curves that are approximated by exponential fitting of the previous and current day's data.

Information is presented in this report on the input and output variables, logic

flow, and methodology employed in developing the algorithm. A computer program

listing of the optimum start/stop control algorithm written in FORTRAN 77 and sample

input and output data are included in the appendices.

12.

KEY WORDS (S/x to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) digital control systems; energy conservation; energy management and control systems;

heating and cooling systems; optimum start/stop time; preheat time.

13. AVAILABILITY

pXl Uni imited

| | For Official Distribution. Do Not Release to NTIS

[" Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

- 20402.

QX| Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

69

15. Price

$ 10.00

USCOMM-DC 8043-P80