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OPTIMIZATION OF HVAC ENERGY CONSUMPTION UNDER UNCERTAINTY OF
EXTERNAL CONDITIONS: A PROBABILISTIC MODELING APPROACH
Va lamo a Lyudmila Pe o na,
Doc o o Technical Sciences, p o esso
Depa men o Compu a ional Ma hema ics and In o ma ion Sys ems
Facul y o Applied Ma hema ics and In elligen Technologies
Na ional Uni e si y o Uzbekis an named a e Mi zo Ulugbek
Tashken , Uzbekis an
dimi [email protected]
Rahimo a Mohi a Muza a qizi,
Mas e 's s uden o "In o ma ion Sys ems"
Depa men o Compu a ional Ma hema ics and In o ma ion Sys ems
Facul y o Applied Ma hema ics and In elligen Technologies
Na ional Uni e si y o Uzbekis an named a e Mi zo Ulugbek
Tashken , Uzbekis an
[email p o ec ed]
Abs ac : Hea ing, Ven ila ion, and Ai Condi ioning (HVAC) sys ems cons i u e a dominan sha e
o building ene gy consump ion, accoun ing o app oxima ely 40% o o al ene gy use. The inhe en
unce ain y in ex e nal condi ions, including wea he a iabili y and s ochas ic occupancy pa e ns,
signi ican ly a ec s bo h ene gy e iciency and indoo en i onmen al quali y. De e minis ic con ol
s a egies a e limi ed in hei abili y o add ess hese unce ain ies, which o en esul s in subop imal
pe o mance and inc eased ope a ional cos s.
This s udy de elops a p obabilis ic op imiza ion amewo k o HVAC ene gy managemen ha
explici ly inco po a es unce ain y in ex e nal inpu s. The p oposed app oach models occupancy as a
s ochas ic p ocess and ep esen s wea he condi ions using p obabilis ic dis ibu ions. Indoo
dynamics o empe a u e and CO₂ concen a ion a e desc ibed by s ochas ic di e en ial equa ions,
which a e in eg a ed in o a cons ained op imiza ion p oblem. The objec i e unc ion minimizes he
expec ed alue o o al ene gy consump ion subjec o p obabilis ic com o cons ain s.
Keywo ds: HVAC sys ems; ene gy op imiza ion; unce ain y modeling; p obabilis ic app oach;
s ochas ic con ol; occupancy p edic ion; wea he a iabili y.
In oduc ion
Hea ing, Ven ila ion, and Ai Condi ioning
(HVAC) sys ems accoun o nea ly 40% o o al
building ene gy use, making hem one o he la ges
con ibu o s o o e all ene gy consump ion in he buil
en i onmen [1]. The dual challenge aced by HVAC
sys ems lies in educing ene gy demand while
main aining indoo en i onmen al quali y, which
di ec ly a ec s occupan com o , p oduc i i y, and
heal h [2].
Con en ional con ol s a egies, such as ule-
based and PID con ol, a e p ima ily designed using
de e minis ic assump ions, whe e ex e nal condi ions
like wea he and occupancy a e ea ed as ixed o
pe ec ly p edic able [3]. Howe e , eal-wo ld
ope a ing en i onmen s a e inhe en ly unce ain.
Wea he a iabili y in luences hea ing and cooling
loads, while occupancy pa e ns luc ua e
s ochas ically, di ec ly impac ing in e nal hea gains
and CO₂ gene a ion. When hese unce ain ies a e no
conside ed, HVAC sys ems o en ope a e ine icien ly,
leading o excessi e ene gy consump ion, inc eased
cos s, and iola ions o com o s anda ds [4-5].
Recen esea ch highligh s he po en ial o
ad anced app oaches such as Model P edic i e Con ol
(MPC) o HVAC op imiza ion. MPC p o ides a
sys ema ic amewo k o op imizing sys em
pe o mance o e a p edic ion ho izon while
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conside ing ope a ional cons ain s [6]. Ne e heless,
mos exis ing MPC s a egies emain de e minis ic in
na u e, which limi s hei abili y o handle unce ain y
e ec i ely. As a esul , s ochas ic and p obabilis ic
modeling echniques a e gaining inc easing a en ion.
These app oaches explici ly ep esen unce ain ac o s
such as wea he condi ions and occupancy le els as
andom p ocesses and employ p obabilis ic cons ain s
o ensu e sys em obus ness [2-8].
The objec i e o his s udy is o de elop a
p obabilis ic op imiza ion amewo k o HVAC
sys ems ha in eg a es s ochas ic wea he models,
occupancy dis ibu ions, and dynamic indoo ai
quali y models. The no el y o he app oach lies in
combining s ochas ic di e en ial equa ions wi h Mon e
Ca lo sampling and s ochas ic Model P edic i e
Con ol (MPC). The p oposed amewo k minimizes
expec ed ene gy consump ion while ensu ing
p obabilis ic compliance wi h he mal com o and ai
quali y equi emen s.
Ma e ials and Me hodology
Resea ch on HVAC con ol has p og essed
om simple ule-based sys ems owa d ad anced
p edic i e and s ochas ic amewo ks. This sec ion
e iews he main app oaches, emphasizing hei
capaci y o handle unce ain y in ex e nal condi ions
such as wea he a iabili y and occupancy luc ua ions.
• Classical Con ol S a egies
The ea lies HVAC con ol sys ems we e
p ima ily based on ule-based logic and P opo ional–
In eg al–De i a i e (PID) con olle s. These me hods
a e simple, cos -e ec i e, and widely adop ed in
comme cial buildings [3-6]. Howe e , hey ely on
ixed schedules o p ede ined ules, which makes hem
poo ly sui ed o dynamic en i onmen s. When
occupancy o wea he de ia es om expec ed pa e ns,
classical con olle s a e unable o adap e ec i ely,
esul ing in ene gy ine iciency and com o iola ions
[6-9].
• De e minis ic Model P edic i e Con ol
(MPC)
Model P edic i e Con ol (MPC) ep esen s a
signi ican ad ancemen in HVAC op imiza ion. I
u ilizes p edic i e models o op imize con ol ac ions
o e a ini e ho izon while sa is ying ope a ional
cons ain s [6]. De e minis ic MPC has demons a ed
he abili y o educe ene gy consump ion compa ed o
classical con olle s, pa icula ly when eliable
o ecas s o wea he and occupancy a e a ailable [2-8].
Ne e heless, he obus ness o de e minis ic MPC is
limi ed, as i assumes exac p edic ions o ex e nal
condi ions. In eal-wo ld scena ios, o ecas e o s and
s ochas ic a ia ions o en deg ade pe o mance [8].
• P obabilis ic and S ochas ic App oaches
To o e come he sho comings o de e minis ic
me hods, ecen s udies ha e emphasized p obabilis ic
and s ochas ic op imiza ion. In hese amewo ks,
unce ain ies such as occupancy pa e ns a e modeled
as andom p ocesses (e.g., Poisson o Ma ko -based
models), while wea he a iables a e ep esen ed
h ough p obabilis ic dis ibu ions [10]. S ochas ic
Model P edic i e Con ol (SMPC) inco po a es hese
unce ain ies in o he op imiza ion p ocess, o en using
Mon e Ca lo sampling o chance-cons ained
o mula ions o ensu e pe o mance eliabili y [8-11].
Such app oaches enable he minimiza ion o
expec ed ene gy consump ion while main aining
com o wi h a desi ed p obabili y le el. Mo eo e ,
p obabilis ic modeling suppo s he in eg a ion o
indoo ai quali y cons ain s, pa icula ly CO₂
concen a ion dynamics, which di ec ly depend on
s ochas ic occupancy [6-12]. This dual ocus on
he mal com o and ai quali y highligh s he g owing
impo ance o unce ain y-awa e s a egies o HVAC
sys ems in mode n buildings.
In summa y, while classical and de e minis ic
me hods p o ide a ounda ion o HVAC con ol, hey
all sho in dynamic and unce ain en i onmen s.
P obabilis ic and s ochas ic app oaches ep esen a
obus al e na i e, o e ing imp o ed adap abili y,
educed ene gy cos s, and enhanced eliabili y in
main aining com o condi ions.
The p oposed me hodology is based on a
p obabilis ic amewo k ha in eg a es s ochas ic
models o wea he and occupancy in o he op imiza ion
o HVAC ene gy consump ion. The amewo k
includes ou componen s: (i) unce ain y modeling,
(ii) indoo dynamics ep esen a ion, (iii) p obabilis ic
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op imiza ion p oblem o mula ion, and (i ) solu ion
using s ochas ic Model P edic i e Con ol (MPC).
• Unce ain y Modeling
Two majo sou ces o unce ain y a e
conside ed:
1. Wea he condi ions: Ou doo empe a u e
()
ou
T
, humidi y, and sola adia ion a e
modeled as Gaussian andom a iables a ound
o ecas ed alues:
2
( ) ( ( ), )
ou T T
T N

(1)
whe e
()
T

is he o ecas ed mean empe a u e
and
2
T

ep esen s o ecas unce ain y [13].
2. Occupancy: The numbe o occupan s O( ) is
ep esen ed as a Poisson p ocess:
()
()
( ( ) ) , 0,1,2,....
!
k
e
P O k k
k


−
= = =
(2)
whe e λ( ) is he expec ed occupancy a ime ,
es ima ed om his o ical da a [11-14].
• Indoo Dynamics
The building’s indoo en i onmen is modeled
h ough coupled he mal and CO₂ balance equa ions.
1. The mal balance [13-15]:
() ( ) ( ) ( ( )) ( ( ) ( ))
in HVAC sola occ in ou
dT
C Q Q Q O U T T
d = + + − −
(3)
whe e:
•
С
– he mal capaci y o he oom,
•
()
HVAC
Q
– hea ing/cooling powe supplied by
HVAC,
•
()
sola
Q
– sola gains,
•
( ( ))
occ
Q O
– in e nal gains om occupan s,
•
U
– o e all hea ans e coe icien ,
•
()
in
T
– indoo empe a u e.
2. CO₂ dynamics [15]:
( ) ( )
( ( )) ( ( ) )
in en in ou
dC Q
G O C C
d V V
= − −
(4)
whe e:
•
()
in
C
– indoo CO₂ concen a ion,
•
ou
C
– ou doo CO₂ concen a ion,
•
()
en
Q
– en ila ion ai low a e,
•
( ( ))G O
– CO₂ gene a ion a e om occupan s,
•
V
– oom olume.
• P obabilis ic Op imiza ion P oblem
The op imiza ion p oblem is o mula ed as:
() 0
min [ ( ( ( )) ( ( ), ( ))) ]
T
in in
u E P u D T C d

+

(5)
subjec o p obabilis ic com o cons ain s:
min max
max
{ ( ) } 1 ,
{ ( ) } 1
in T
in T
P T T T
P C C


   −
  −
(6)
whe e:
•
( ) [ ( ), ( )]
HVAC en
u Q Q =
– con ol inpu s,
•
( ( ))P u
– HVAC ene gy consump ion,
•
()D
– discom o penal y unc ion,
•
,
TC

– allowable p obabili ies o iola ion o
he mal com o and CO₂ concen a ion.
This o mula ion ensu es ha expec ed ene gy
cos s a e minimized while com o iola ions emain
wi hin p obabilis ic limi s.
• Solu ion App oach
The op imiza ion is sol ed using S ochas ic
MPC wi h scena io-based analysis:
1. A each con ol s ep, unce ain y samples o
( ) ( )
ou
T and O
a e gene a ed ia Mon e Ca lo
simula ion [3-15].
2. Indoo dynamics
( ( ), ( ))
in in
T C
a e simula ed
unde each scena io.
3. Expec ed cos is calcula ed as:
10
1( ( ( )) ( ( ), ( )))
T
Nii
i in in
i
J P u D T C d
N

=
=+

(7)
Whe e: N is he numbe o scena ios.
4. Op imal con ol u( ) is chosen by minimizing J
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subjec o p obabilis ic cons ain s.
5. The i s con ol inpu is applied, and he p ocess
epea s a he nex ime s ep.
This i e a i e p ocess enables obus and
adap i e ope a ion o HVAC sys ems unde unce ain
ex e nal condi ions.
Resul s
The p oposed s ochas ic Model P edic i e
Con ol (s-MPC) was e alua ed h ough scena io-
based simula ions wi h Mon e Ca lo sampling (500
uns) o an o ice building model o 500 m².
• Ene gy Consump ion: The s-MPC educed
he expec ed ene gy consump ion by 12.3%
compa ed o de e minis ic MPC (d-MPC). The
g ea es sa ings we e obse ed du ing pe iods
o apid ou doo wea he luc ua ions,
con i ming ea lie indings on he obus ness o
s ochas ic op imiza ion unde unce ain
condi ions [4-14].
• Com o Viola ions: The p opo ion o ime
wi h he mal com o o CO₂ cons ain
iola ions was 11.5% o d-MPC and 6.8%
o s-MPC. This imp o emen is consis en
wi h p e ious esea ch demons a ing he
bene i s o p obabilis ic occupancy p edic ion
and adap i e en ila ion con ol [11-13].
• Sensi i i y Analysis: Inc easing wea he
o ecas unce ain y by ±3 °C ampli ied he
ela i e ene gy sa ings o s-MPC o 15%,
which suppo s ea lie obse a ions ha
s ochas ic MPC ou pe o ms de e minis ic
app oaches unde high unce ain y [3-15]. E en
unde s able wea he condi ions, he ad an age
o s-MPC in main aining com o eliabili y
was p ese ed.
These esul s con i m he obus ness and
e iciency o he p obabilis ic op imiza ion amewo k
in unce ain ope a ing condi ions, highligh ing i s
p ac ical applicabili y o buildings wi h highly
a iable occupancy pa e ns [5-11].
Discussion
The esul s demons a e ha inco po a ing
unce ain y in o HVAC op imiza ion signi ican ly
imp o es bo h ene gy e iciency and com o
eliabili y. Compa ed o de e minis ic MPC, he
s ochas ic app oach educed expec ed ene gy
consump ion by 12.3% while hal ing he p obabili y o
com o iola ions. This is aligned wi h ecen s udies
emphasizing he ole o p obabilis ic modeling in
esilien building con ol [4-11].
A key ad an age o he p oposed s-MPC is i s
abili y o adap i ely manage en ila ion a es in
esponse o s ochas ic occupancy a ia ions. P e ious
wo k has shown ha de e minis ic me hods a e highly
sensi i e o occupancy o ecas e o s, o en leading o
CO₂ accumula ion and com o b eaches [3-6]. By
modeling occupancy as a Poisson p ocess and
explici ly conside ing i s andomness, he amewo k
e ec i ely an icipa es and mi iga es hese isks.
The sensi i i y analysis u he highligh s he
obus ness o he p obabilis ic app oach. Wi h
inc eased wea he o ecas unce ain y (±3 °C), he
ene gy-sa ing ad an age o s-MPC eached 15%,
consis en wi h ea lie indings ha s ochas ic
o mula ions p o ide g ea e esilience unde high
unce ain y [8-13]. E en when en i onmen al
condi ions we e ela i ely s able, s-MPC main ained an
edge in educing com o iola ions, suppo ing i s
sui abili y o eal-wo ld applica ions.
F om a p ac ical pe spec i e, he me hodology
is especially ele an o educa ional and o ice
buildings, whe e occupancy pa e ns a e inhe en ly
unp edic able [11-14]. The in eg a ion o IoT-based
sensing and eal- ime da a p ocessing can u he
enhance he accu acy o unce ain y modeling,
enabling p edic i e con ol s a egies ha b idge he
gap be ween heo y and p ac ical deploymen .
Conclusion
This s udy in oduced a p obabilis ic
op imiza ion amewo k o HVAC ene gy
managemen unde unce ain ex e nal condi ions. By
in eg a ing s ochas ic wea he and occupancy models
in o a s ochas ic Model P edic i e Con ol (s-MPC)
s uc u e, he amewo k achie ed:
• 12–15% educ ion in expec ed ene gy
consump ion,
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• Lowe equency o com o iola ions
( empe a u e and CO₂), and
• Imp o ed obus ness o unce ain y in
wea he and occupancy o ecas s.
The indings con i m ha unce ain y-awa e
con ol s a egies ou pe o m de e minis ic app oaches,
pa icula ly in dynamic en i onmen s whe e o ecas
e o s a e ine i able. The p obabilis ic app oach no
only enhances ene gy e iciency bu also inc eases
eliabili y in main aining indoo com o and ai
quali y.
Fu u e esea ch di ec ions include:
1. Real- ime implemen a ion wi h IoT senso
ne wo ks o adap i e model upda es,
2. Mul i-objec i e op imiza ion balancing ene gy,
cos , and ca bon oo p in , and
3. Applica ion o la ge-scale and mixed-use
building complexes.
O e all, he p oposed amewo k con ibu es o
he de elopmen o nex -gene a ion HVAC sys ems
ha combine e iciency, adap abili y, and
sus ainabili y, o e ing a obus pa hway owa d
sma e buildings.
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