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

75 “Al-Farg‘oniy avlodlari” elektron ilmiy jurnali ISSN 2181-4252. Tom: 1 | Son: 3 | 2025-yil "Descendants of Al-Farghani" electronic scientific journal. ISSN 2181-4252. Vol: 1 | Iss: 3 | 2025 year Электронный научный журнал "Потомки АльФаргани" ISSN 2181-4252. Том: 1 | Выпуск: 3 | 2025 год https://al-fargoniy.uz/ OPTIMIZATION OF HVAC ENERGY CONSUMPTION UNDER UNCERTAINTY OF EXTERNAL CONDITIONS: A PROBABILISTIC MODELING APPROACH Varlamova Lyudmila Petrovna, Doctor of Technical Sciences, professor Department of Computational Mathematics and Information Systems Faculty of Applied Mathematics and Intelligent Technologies National University of Uzbekistan named after Mirzo Ulugbek Tashkent, Uzbekistan [email protected] Rahimova Mohira Muzaffar qizi, Master's student of "Information Systems" Department of Computational Mathematics and Information Systems Faculty of Applied Mathematics and Intelligent Technologies National University of Uzbekistan named after Mirzo Ulugbek Tashkent, Uzbekistan [email protected] Abstract: Heating, Ventilation, and Air Conditioning (HVAC) systems constitute a dominant share of building energy consumption, accounting for approximately 40% of total energy use. The inherent uncertainty in external conditions, including weather variability and stochastic occupancy patterns, significantly affects both energy efficiency and indoor environmental quality. Deterministic control strategies are limited in their ability to address these uncertainties, which often results in suboptimal performance and increased operational costs. This study develops a probabilistic optimization framework for HVAC energy management that explicitly incorporates uncertainty in external inputs. The proposed approach models occupancy as a stochastic process and represents weather conditions using probabilistic distributions. Indoor dynamics of temperature and CO₂ concentration are described by stochastic differential equations, which are integrated into a constrained optimization problem. The objective function minimizes the expected value of total energy consumption subject to probabilistic comfort constraints. Keywords: HVAC systems; energy optimization; uncertainty modeling; probabilistic approach; stochastic control; occupancy prediction; weather variability. Introduction Heating, Ventilation, and Air Conditioning (HVAC) systems account for nearly 40% of total building energy use, making them one of the largest contributors to overall energy consumption in the built environment [1]. The dual challenge faced by HVAC systems lies in reducing energy demand while maintaining indoor environmental quality, which directly affects occupant comfort, productivity, and health [2]. Conventional control strategies, such as rulebased and PID control, are primarily designed using deterministic assumptions, where external conditions like weather and occupancy are treated as fixed or perfectly predictable [3]. However, real-world operating environments are inherently uncertain. Weather variability influences heating and cooling loads, while occupancy patterns fluctuate stochastically, directly impacting internal heat gains and CO₂ generation. When these uncertainties are not considered, HVAC systems often operate inefficiently, leading to excessive energy consumption, increased costs, and violations of comfort standards [4-5]. Recent research highlights the potential of advanced approaches such as Model Predictive Control (MPC) for HVAC optimization. MPC provides a systematic framework for optimizing system performance over a prediction horizon while 76 “Al-Farg‘oniy avlodlari” elektron ilmiy jurnali ISSN 2181-4252. Tom: 1 | Son: 3 | 2025-yil "Descendants of Al-Farghani" electronic scientific journal. ISSN 2181-4252. Vol: 1 | Iss: 3 | 2025 year Электронный научный журнал "Потомки АльФаргани" ISSN 2181-4252. Том: 1 | Выпуск: 3 | 2025 год https://al-fargoniy.uz/ considering operational constraints [6]. Nevertheless, most existing MPC strategies remain deterministic in nature, which limits their ability to handle uncertainty effectively. As a result, stochastic and probabilistic modeling techniques are gaining increasing attention. These approaches explicitly represent uncertain factors such as weather conditions and occupancy levels as random processes and employ probabilistic constraints to ensure system robustness [2-8]. The objective of this study is to develop a probabilistic optimization framework for HVAC systems that integrates stochastic weather models, occupancy distributions, and dynamic indoor air quality models. The novelty of the approach lies in combining stochastic differential equations with Monte Carlo sampling and stochastic Model Predictive Control (MPC). The proposed framework minimizes expected energy consumption while ensuring probabilistic compliance with thermal comfort and air quality requirements. Materials and Methodology Research on HVAC control has progressed from simple rule-based systems toward advanced predictive and stochastic frameworks. This section reviews the main approaches, emphasizing their capacity to handle uncertainty in external conditions such as weather variability and occupancy fluctuations. • Classical Control Strategies The earliest HVAC control systems were primarily based on rule-based logic and Proportional– Integral–Derivative (PID) controllers. These methods are simple, cost-effective, and widely adopted in commercial buildings [3-6]. However, they rely on fixed schedules or predefined rules, which makes them poorly suited for dynamic environments. When occupancy or weather deviates from expected patterns, classical controllers are unable to adapt effectively, resulting in energy inefficiency and comfort violations [6-9]. • Deterministic Model Predictive Control (MPC) Model Predictive Control (MPC) represents a significant advancement in HVAC optimization. It utilizes predictive models to optimize control actions over a finite horizon while satisfying operational constraints [6]. Deterministic MPC has demonstrated the ability to reduce energy consumption compared to classical controllers, particularly when reliable forecasts of weather and occupancy are available [2-8]. Nevertheless, the robustness of deterministic MPC is limited, as it assumes exact predictions of external conditions. In real-world scenarios, forecast errors and stochastic variations often degrade performance [8]. • Probabilistic and Stochastic Approaches To overcome the shortcomings of deterministic methods, recent studies have emphasized probabilistic and stochastic optimization. In these frameworks, uncertainties such as occupancy patterns are modeled as random processes (e.g., Poisson or Markov-based models), while weather variables are represented through probabilistic distributions [10]. Stochastic Model Predictive Control (SMPC) incorporates these uncertainties into the optimization process, often using Monte Carlo sampling or chance-constrained formulations to ensure performance reliability [8-11]. Such approaches enable the minimization of expected energy consumption while maintaining comfort with a desired probability level. Moreover, probabilistic modeling supports the integration of indoor air quality constraints, particularly CO₂ concentration dynamics, which directly depend on stochastic occupancy [6-12]. This dual focus on thermal comfort and air quality highlights the growing importance of uncertainty-aware strategies for HVAC systems in modern buildings. In summary, while classical and deterministic methods provide a foundation for HVAC control, they fall short in dynamic and uncertain environments. Probabilistic and stochastic approaches represent a robust alternative, offering improved adaptability, reduced energy costs, and enhanced reliability in maintaining comfort conditions. The proposed methodology is based on a probabilistic framework that integrates stochastic models of weather and occupancy into the optimization of HVAC energy consumption. The framework includes four components: (i) uncertainty modeling, (ii) indoor dynamics representation, (iii) probabilistic 77 “Al-Farg‘oniy avlodlari” elektron ilmiy jurnali ISSN 2181-4252. Tom: 1 | Son: 3 | 2025-yil "Descendants of Al-Farghani" electronic scientific journal. ISSN 2181-4252. Vol: 1 | Iss: 3 | 2025 year Электронный научный журнал "Потомки АльФаргани" ISSN 2181-4252. Том: 1 | Выпуск: 3 | 2025 год https://al-fargoniy.uz/ optimization problem formulation, and (iv) solution using stochastic Model Predictive Control (MPC). • Uncertainty Modeling Two major sources of uncertainty are considered: 1. Weather conditions: Outdoor temperature () out Tt , humidity, and solar radiation are modeled as Gaussian random variables around forecasted values: 2 ( ) ( ( ), ) out T T T t N t  (1) where () Tt  is the forecasted mean temperature and 2 T  represents forecast uncertainty [13]. 2. Occupancy: The number of occupants O(t) is represented as a Poisson process: () () ( ( ) ) , 0,1,2,.... ! kt te P O t k k k   − = = = (2) where λ(t) is the expected occupancy at time t, estimated from historical data [11-14]. • Indoor Dynamics The building’s indoor environment is modeled through coupled thermal and CO₂ balance equations. 1. Thermal balance [13-15]: () ( ) ( ) ( ( )) ( ( ) ( )) in HVAC solar occ in out dT t C Q t Q t Q O t U T t T t dt = + + − − (3) where: • С – thermal capacity of the room, • () HVAC Qt – heating/cooling power supplied by HVAC, • () solar Qt – solar gains, • ( ( )) occ Q O t – internal gains from occupants, • U – overall heat transfer coefficient, • () in Tt – indoor temperature. 2. CO₂ dynamics [15]: ( ) ( ) ( ( )) ( ( ) ) in vent in out dC t Q t G O t C t C dt V V = − − (4) where: • () in Ct – indoor CO₂ concentration, • out C – outdoor CO₂ concentration, • () vent Qt – ventilation airflow rate, • ( ( ))G O t – CO₂ generation rate from occupants, • V – room volume. • Probabilistic Optimization Problem The optimization problem is formulated as: () 0 min [ ( ( ( )) ( ( ), ( ))) ] T in in ut E P u t D T t C t dt  +  (5) subject to probabilistic comfort constraints: min max max { ( ) } 1 , { ( ) } 1 in T in T P T T t T P C t C      −   − (6) where: • ( ) [ ( ), ( )] HVAC vent u t Q t Q t= – control inputs, • ( ( ))P u t – HVAC energy consumption, • ()D – discomfort penalty function, • , TC  – allowable probabilities of violation for thermal comfort and CO₂ concentration. This formulation ensures that expected energy costs are minimized while comfort violations remain within probabilistic limits. • Solution Approach The optimization is solved using Stochastic MPC with scenario-based analysis: 1. At each control step, uncertainty samples for ( ) ( ) out T t and O t are generated via Monte Carlo simulation [3-15]. 2. Indoor dynamics ( ( ), ( )) in in T t C t are simulated under each scenario. 3. Expected cost is calculated as: 10 1( ( ( )) ( ( ), ( ))) T Nii i in in i J P u t D T t C t dt N  = =+  (7) Where: N is the number of scenarios. 4. Optimal control u(t) is chosen by minimizing J 78 “Al-Farg‘oniy avlodlari” elektron ilmiy jurnali ISSN 2181-4252. Tom: 1 | Son: 3 | 2025-yil "Descendants of Al-Farghani" electronic scientific journal. ISSN 2181-4252. Vol: 1 | Iss: 3 | 2025 year Электронный научный журнал "Потомки АльФаргани" ISSN 2181-4252. Том: 1 | Выпуск: 3 | 2025 год https://al-fargoniy.uz/ subject to probabilistic constraints. 5. The first control input is applied, and the process repeats at the next time step. This iterative process enables robust and adaptive operation of HVAC systems under uncertain external conditions. Results The proposed stochastic Model Predictive Control (s-MPC) was evaluated through scenariobased simulations with Monte Carlo sampling (500 runs) for an office building model of 500 m². • Energy Consumption: The s-MPC reduced the expected energy consumption by 12.3% compared to deterministic MPC (d-MPC). The greatest savings were observed during periods of rapid outdoor weather fluctuations, confirming earlier findings on the robustness of stochastic optimization under uncertain conditions [4-14]. • Comfort Violations: The proportion of time with thermal comfort or CO₂ constraint violations was 11.5% for d-MPC and 6.8% for s-MPC. This improvement is consistent with previous research demonstrating the benefits of probabilistic occupancy prediction and adaptive ventilation control [11-13]. • Sensitivity Analysis: Increasing weather forecast uncertainty by ±3 °C amplified the relative energy savings of s-MPC to 15%, which supports earlier observations that stochastic MPC outperforms deterministic approaches under high uncertainty [3-15]. Even under stable weather conditions, the advantage of s-MPC in maintaining comfort reliability was preserved. These results confirm the robustness and efficiency of the probabilistic optimization framework in uncertain operating conditions, highlighting its practical applicability for buildings with highly variable occupancy patterns [5-11]. Discussion The results demonstrate that incorporating uncertainty into HVAC optimization significantly improves both energy efficiency and comfort reliability. Compared to deterministic MPC, the stochastic approach reduced expected energy consumption by 12.3% while halving the probability of comfort violations. This is aligned with recent studies emphasizing the role of probabilistic modeling in resilient building control [4-11]. A key advantage of the proposed s-MPC is its ability to adaptively manage ventilation rates in response to stochastic occupancy variations. Previous work has shown that deterministic methods are highly sensitive to occupancy forecast errors, often leading to CO₂ accumulation and comfort breaches [3-6]. By modeling occupancy as a Poisson process and explicitly considering its randomness, the framework effectively anticipates and mitigates these risks. The sensitivity analysis further highlights the robustness of the probabilistic approach. With increased weather forecast uncertainty (±3 °C), the energy-saving advantage of s-MPC reached 15%, consistent with earlier findings that stochastic formulations provide greater resilience under high uncertainty [8-13]. Even when environmental conditions were relatively stable, s-MPC maintained an edge in reducing comfort violations, supporting its suitability for real-world applications. From a practical perspective, the methodology is especially relevant for educational and office buildings, where occupancy patterns are inherently unpredictable [11-14]. The integration of IoT-based sensing and real-time data processing can further enhance the accuracy of uncertainty modeling, enabling predictive control strategies that bridge the gap between theory and practical deployment. Conclusion This study introduced a probabilistic optimization framework for HVAC energy management under uncertain external conditions. By integrating stochastic weather and occupancy models into a stochastic Model Predictive Control (s-MPC) structure, the framework achieved: • 12–15% reduction in expected energy consumption, 79 “Al-Farg‘oniy avlodlari” elektron ilmiy jurnali ISSN 2181-4252. Tom: 1 | Son: 3 | 2025-yil "Descendants of Al-Farghani" electronic scientific journal. ISSN 2181-4252. Vol: 1 | Iss: 3 | 2025 year Электронный научный журнал "Потомки АльФаргани" ISSN 2181-4252. Том: 1 | Выпуск: 3 | 2025 год https://al-fargoniy.uz/ • Lower frequency of comfort violations (temperature and CO₂), and • Improved robustness to uncertainty in weather and occupancy forecasts. The findings confirm that uncertainty-aware control strategies outperform deterministic approaches, particularly in dynamic environments where forecast errors are inevitable. The probabilistic approach not only enhances energy efficiency but also increases reliability in maintaining indoor comfort and air quality. 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