OPTIMIZATION OF HVAC ENERGY CONSUMPTION UNDER UNCERTAINTY OF EXTERNAL CONDITIONS: A PROBABILISTIC MODELING APPROACH

75 “Al-Fa g‘oniy a lodla i” elek on ilmiy ju nali ISSN 2181-4252. Tom: 1 | Son: 3 | 2025-yil "Descendan s o Al-Fa ghani" elec onic scien i ic jou nal. ISSN 2181-4252. Vol: 1 | Iss: 3 | 2025 yea Электронный научный журнал "Потомки Аль- Фаргани" ISSN 2181-4252. Том: 1 | Выпуск: 3 | 2025 год h ps://al- a goniy.uz/ 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 76 “Al-Fa g‘oniy a lodla i” elek on ilmiy ju nali ISSN 2181-4252. Tom: 1 | Son: 3 | 2025-yil "Descendan s o Al-Fa ghani" elec onic scien i ic jou nal. ISSN 2181-4252. Vol: 1 | Iss: 3 | 2025 yea Электронный научный журнал "Потомки Аль- Фаргани" ISSN 2181-4252. Том: 1 | Выпуск: 3 | 2025 год h ps://al- a goniy.uz/ 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 77 “Al-Fa g‘oniy a lodla i” elek on ilmiy ju nali ISSN 2181-4252. Tom: 1 | Son: 3 | 2025-yil "Descendan s o Al-Fa ghani" elec onic scien i ic jou nal. ISSN 2181-4252. Vol: 1 | Iss: 3 | 2025 yea Электронный научный журнал "Потомки Аль- Фаргани" ISSN 2181-4252. Том: 1 | Выпуск: 3 | 2025 год h ps://al- a goniy.uz/ 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 78 “Al-Fa g‘oniy a lodla i” elek on ilmiy ju nali ISSN 2181-4252. Tom: 1 | Son: 3 | 2025-yil "Descendan s o Al-Fa ghani" elec onic scien i ic jou nal. ISSN 2181-4252. Vol: 1 | Iss: 3 | 2025 yea Электронный научный журнал "Потомки Аль- Фаргани" ISSN 2181-4252. Том: 1 | Выпуск: 3 | 2025 год h ps://al- a goniy.uz/ 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, 79 “Al-Fa g‘oniy a lodla i” elek on ilmiy ju nali ISSN 2181-4252. Tom: 1 | Son: 3 | 2025-yil "Descendan s o Al-Fa ghani" elec onic scien i ic jou nal. ISSN 2181-4252. Vol: 1 | Iss: 3 | 2025 yea Электронный научный журнал "Потомки Аль- Фаргани" ISSN 2181-4252. Том: 1 | Выпуск: 3 | 2025 год h ps://al- a goniy.uz/ • 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. Re e ences 1. Варламова Л.П & Рахимова М.М (2025) Математическое моделирование влияния микроклимата на продуктивность учащихся// De elopmen o science Volume 3 –pp. 174-179 2. 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