Application of energy combined thermal comfort in intelligent building management in complex environments

Abstract

The efficient operation of heating ventilation and air conditioning systems relies on advanced control strategies. However, current control methods are often limited by issues such as uncertain system parameter information and spatial coupling constraints related to the supply rate of the air supply fan. To this end, an energy joint thermal comfort management method for complex environments in multiple regions is proposed. The long-term total cost minimization of the system is established, and then the Lyapunov optimization technology is used to design the distributed control algorithm. Simulation validation shows that the proposed method reduces the energy cost by an average of 11.24% compared to other methods with a thermal discomfort cost coefficient of 0. The average temperature deviation in the area is improved by 0.15 °C and 0.68 °C, respectively. The method saves more than 10% of the total energy cost under different thermal perturbations with an average total temperature deviation of 0.04 °C. The results indicate that the proposed energy joint thermal comfort management method can flexibly balance energy costs and user thermal comfort without knowing any prior information of system parameters, which can also greatly protect user privacy information. This method has application value in the control of heating ventilation and air conditioning systems in complex environments such as commercial buildings.

Introduction

With the development and application of intelligent buildings, building energy consumption has is a main component of national energy consumption, which mainly includes the entire process energy consumption from building material manufacturing and construction to building usage, which is closely related to the usage of traditional energy and the daily consumption of building owners (Ding et al. 2022a; Oh 2023; Fnais et al. 2022). However, the limitations of traditional energy sources have led to a national energy crisis and increased economic burden on building owners, while reducing the input power of Heating Ventilation and Air Conditioning System (HVACS) will in turn reduce user thermal comfort (Kajjoba et al. 2022). How to effectively balance the contradiction between the two has become a new issue in current intelligent building management. Scholars at home and abroad adopts methods such as model predictive control, robustness optimization, and dynamic programming to solve the balance between energy and thermal comfort. However, these methods all rely on prior information of system uncertainty parameters (Vanos et al. 2022). The thermal dynamic models of buildings in real life are limited by various factors, making it very difficult to obtain characterization information of various uncertain parameters. In most of the current related studies, the system parameters of future uncertainty need to be predicted or explicit models are needed to characterize the uncertainty in terms of probability distributions, maximum and minimum values. However, the prediction error is proportional to the prediction length and the existing methods are overly dependent on the prediction accuracy of the parameters. In addition, it is more difficult to obtain joint probability distributions of multiple uncertainty system parameters for practical management. Therefore, in order to achieve the optimization and reduction of the long-term total cost of HVACS in complex environments, such as multi-area commercial buildings, the study uses its establishment of an optimization problem for the design of joint management methods. For the intelligent building management system in complex environments with uncertain parameters, temperature-related time coupling constraints in the region, and spatial coupling constraints related to the total air supply rate of the supply fan, the study proposes a Liapunov optimization-based distributed real-time control algorithm (Liapunov Optimization-Distributed Real- Time Control (LO-DRTC), in order to achieve user privacy protection and system scaling without knowing any a priori information about the system parameters.

The overall structure consists of four parts. The first part summarizes the research achievements and shortcomings of intelligent building energy combined thermal comfort management and HVAC system control algorithms both domestically and internationally. The second part designs an energy joint thermal comfort management method in complex environments. The third part conducts experiments and analysis on the proposed method. The fourth part summarizes the experimental results and points out future research directions.

Related works

With the growth of world population and the increasing purchasing power of emerging economies, the demand for building energy is constantly increasing. This has led to a series of environmental and energy issues, as well as an increased economic burden on building owners (Zhang et al. 2022). The development of information technologies has brought new opportunities for the optimal compromise between user thermal comfort and energy consumption. Therefore, domestic and foreign scholars have conducted various explorations on the joint management of building energy and thermal comfort. Watari et al. built an online collaborative scheduling framework for thermal comfort perception to comprehensively integrate prediction information such as renewable energy generation and thermal comfort. The optimized energy scheduling and photovoltaic power generation and thermal comfort prediction model were combined with the model predictive control method, thereby achieving a reduction in time complexity and considering the balance between electricity bills and thermal comfort (Watari et al. 2023). To improve consumer comfort in building environments, Kumar et al. designed an efficient home energy management system in microgrids. The usage frequency, preferred operating interval, and other user electricity habits were collected. A user centered smart home energy management solution was established, thereby achieving separate waiting factor planning and controllable load for each household appliance (Kumar et al. 2023). Ding et al. proposed a multi-zone residential HVACS thermal comfort control method ground on deep reinforcement learning to address the complex dynamics of HVACS and the balance between energy conservation and thermal comfort. The mixed deep neural network was used for predicting thermal comfort values, thereby minimizing energy consumption (Ding et al. 2022b). Sun et al. proposed an off grid zero energy building with hydrogen storage system to reduce the energy consumption rate of buildings. Steam compression, coolers, and humidifiers provided comfort for residents. Building simulations were conducted using the Fanger model. As a result, a significant reduction in domestic hot water consumption was achieved in transient analysis, while ensuring the thermal comfort (Sun et al. 2022).

The efficient operation of HVACS relies on advanced control strategies. In addition to controlling the daily operation of HVACS, control strategies also include fault-tolerant control, comfort and energy-saving control, and other aspects. Therefore, designing effective control algorithms can regulate the internal environment of HVACS, improve comfort, and enhance the cooling/thermal efficiency of air conditioning. To achieve multi-input multi-output HVACS, Abrazeh et al. developed a new adaptive control method based on digital twins. A nonlinear integral back-stepping controller was designed using deep reinforcement learning algorithms, implemented using physical and virtual controllers. This ensured that the system output of the virtual controller was consistent with that of the physical controller (Abrazeh et al. 2022). Dawood et al. proposed a deterministic strategy based on reinforcement learning technology to minimize the energy consumption of HVACS. Nonlinear autoregressive exogenous neural networks were mixed with deterministic strategies to achieve synchronous control of indoor temperature and carbon dioxide concentration (Dawood et al. 2022). Mohseni et al. proposed a novel reliable digital twin system and a non-singular terminal sliding membrane control method based on proximal strategy optimization to achieve economically efficient energy management in a single region HVACS. The unmodeled system dynamics and disturbances were processed, thereby utilizing hardware in the loop to improve system performance such as unknown uncertainty and fast tracking (Mohseni et al. 2023). Chen et al. proposed a novel control strategy to improve the predictive control performance of HVACS. A neural network planning framework was designed by using supervised learning algorithms to learn the thermal dynamic model of the region, effectively avoiding compound errors and potential divergence during the learning process, and improving learning stability (Chen et al. 2022).

Based on the above, scholars at home and abroad have conducted various studies on the combined thermal comfort management of building energy. However, in the multi-regional commercial buildings, it is mostly necessary to predict future uncertain system parameters or establish models to ensure uncertainty. However, obtaining statistical features such as uncertain system parameters in practical operations requires significant costs and higher operational difficulty. Therefore, a combined energy thermal comfort management method is proposed for the complex environment of commercial buildings in multiple regions. At the same time, to minimize the long-term total cost of HVACS, an innovative distributed control algorithm for HVACS based on Lyapunov optimization is designed, aiming to protect user privacy without predicting any uncertain parameters. In order to demonstrate the feasibility of the proposed method, this study compares it with the joint building energy thermal comfort management method reviewed in the previous section. The results of the comparison are shown in Table 1.

Table 1 Comparison of the proposed methodology of the study with the existing literature

Design of energy combined thermal comfort management method in complex environments

To achieve energy joint thermal comfort management in complex environments such as multi-regional commercial buildings, the study first establishes a long-term total cost minimization problem model for HVACS. Secondly, the LO-DRTC algorithm is designed using Lyapunov optimization. A virtual queue related to temperature in all regions is constructed to decouple the minimization problem, thereby protecting user privacy and improving system scalability without prior knowledge of system parameters.

Modeling of energy joint thermal comfort management in complex environments

In complex environments such as multi-zone commercial buildings, HVACS account for one-half of the building's energy consumption, which often results in high energy costs for the building owner (Gbaarabe and Sodiki 2023). In order to reduce the energy cost of a building, traditional methods have been used mainly by reducing the power input of the HVAC system at the expense of the thermal comfort of the users (Park et al. 2024). Therefore, it is particularly necessary to reduce the energy cost of HVACS in complex environments while guaranteeing the thermal comfort of the users. To intelligently manage the energy combined thermal comfort in complex environments, the long-term total cost minimization problem of HVACS is first modeled. Among them, the distribution of HVACS in complex environments of commercial buildings with multiple regions considered in the study is shown in Fig. 1.

Fig. 1

Schematic of HVACS distribution in a complex environment

In Fig. 1, the complex environment has N regions. The temperature inside a single area is controlled by the variable air volume HVACS. The system includes the Air Handling Unit (AHU) for the entire building and Variable Air Volume (VAV) control boxes associated with each area. Among them, AHU includes Variable Frequency Drive (VFD) air supply fan, air valve, and condensing/heating coil. The AHU air valve is mainly responsible for mixing fresh air outside the area with the air returned from each area to meet the ventilation needs of all areas. The condensing/heating coil is responsible for cooling or heating the mixed air. Then, a VFD blower is used to transfer the cooled or heated mixed air to the VAV control box in each area. The VAV control box is equipped with an air valve and a reheating coil. The air valve is mainly responsible for regulating the air supply rate. The reheating coil will reheat the supplied air depending on the situation. Therefore, the expression function of the building dynamics model for the complex environmental thermal region considered in the study is shown in Eq. (1).

$$A_{i} \frac{{dT_{i,l} }}{dl} = \frac{{T_{o,l} - T_{i,l} }}{{B_{i} }} + A_{a} m_{i,l} (T_{s} - T_{i,l} ) + q_{i,l} ,\forall i,t$$

(1)

In Eq. (1), $B_{i}$ and $A_{a}$ respectively represent the region related temperature parameters thermal resistance and air specific heat value. $d( \cdot )$ represents differentiation. $l$ represents time. $T_{i,l}$ shows the temperature in region $i$ at time $l$. $T_{o,l}$ represents the outdoor temperature at time $l$. $T_{s}$ indicates the supply air temperature of the supply fan. $A_{i}$ represents the heat capacity within the region. $q_{i,l}$ represents the thermal disturbance in region $i$ at time $l$. $m_{i,l}$ represents the air supply rate in region $i$ at time $l$. $t$ denotes the time slot. The study sets the control range of the temperature in the region according to the user's thermal comfort preference, and the supply air rate in the region is regulated by the VAV control box in the region. Therefore, the upper and lower limits of the control range of the temperature in the zone, the supply air rate and the total supply air rate constraint equations are shown in Eq. (2).

$$\left\{ \begin{gathered} T_{i}^{\min } \le T_{i,t} \le T_{i}^{\max } \hfill \\ m_{i}^{\min } \le m_{i,t} \le m_{i}^{\max } \hfill \\ \sum\nolimits_{i} {m_{i,t} \le \overline{m}} \hfill \\ \end{gathered} \right.$$

(2)

In Eq. (2), $T_{i}^{\max }$ indicates the upper limit of comfortable temperature in region $i$. $T_{i}^{\min }$ denotes the lower limit of comfortable temperatures in region $i$. $m_{i}^{\min }$ represents the minimum rate of air supply in region $i$.$m_{i}^{\max }$ represents the maximum rate of wind supply in region $i$. $T_{i,t}$ denotes the temperature in region $i$ under time slot $t$. $m_{i,t}$ denotes the supply air rate in region $i$ under time slot $t$. $\overline{m}$ denotes the total upper supply air rate limit, which is less than the sum of the upper supply air rate limits for all zones. For the total cost of HVACS in complex environments, the study set it to consist of three main components: thermal discomfort cost, energy cost related to the supply fan, and energy cost related to the condensing coil. Where the expression function of the three costs is shown in Eq. (3).

$$\left\{ \begin{gathered} {\mathbb{R}}_{1,t} = \sum\limits_{i} {\varphi_{i} } (T_{i,t + 1} - T_{i,t + 1}^{ref} )^{2} \hfill \\ {\mathbb{R}}_{2,t} = u(\sum\limits_{i} {m_{i,t} } )^{3} S_{t} \tau \hfill \\ {\mathbb{R}}_{3,t} = p_{t} S_{t} \tau \hfill \\ \end{gathered} \right.$$

(3)

In Eq. (3), ${\mathbb{R}}_{1,t}$ denotes the user thermal discomfort cost under time slot $t$. ${\mathbb{R}}_{2,t}$ represents the energy cost bound up with the supply fan under time slot $t$. ${\mathbb{R}}_{3,t}$ denotes the energy cost bound up with the condensing coil under time slot $t$. $\varphi_{i}$ represents the cost coefficient. $T_{i,t + 1}$ represents the temperature within the time slot $t + 1$ in region $i$. $T_{i,t + 1}^{ref}$ represents the user thermal comfort preference temperature within time slot $t + 1$ in region $i$. $u$ represents the power consumption correlation coefficient of the supply fan. $u$ represents the electricity price during the time slot $t$. $\tau$ represents the unit time slot. $p_{t}$ represents the power consumption of the condensing coil, which is expressed as shown in Eq. (4).

$$p_{t} = \sum\nolimits_{i} {m_{i,t} } \frac{{A_{a} }}{\eta COP}\left( {\delta T_{i,t} + \left( {1 - \delta } \right)\left( {T_{o,t} - T_{s} } \right)} \right)$$

(4)

In Eq. (4), $\eta$ denotes the efficiency factor of the condensing coil. $COP$ denotes the performance factor of the condensing coil, and $\delta$ denotes the AHU damper position. Based on the above, the mathematical expression of the long-term total cost minimization problem model for HVACS in complex environments is shown in Eq. (5).

$$P1:\mathop {\min }\limits_{{m_{i,j} }} \mathop {\lim \sup }\limits_{M \to \infty } \frac{1}{M - 1}\sum\limits_{t = 1}^{M - 1} {{\rm E}\left\{ {\sum\limits_{t = 1}^{3} {{\mathbb{R}}_{l,t} } } \right\}}$$

(5)

In Eq. (4), $P1$ represents problem 1 that needs to be solved for the management method proposed in the study. $\lim \sup$ represents the upper limit. ${\mathbb{R}}_{l,t}$ represents the total cost of user thermal discomfort, energy costs related to the supply fan, and energy costs related to the condensing coil. $M$ signifies the total time slots. ${\rm E}\left\{ \cdot \right\}$ signifies expectation, which can affect the randomness of indicators such as electricity price, outdoor temperature, and user thermal comfort preference temperature. In addition, the decision variable for this minimization problem model is $m_{i,t} \left( {\forall i,t} \right)$.

Energy joint thermal comfort management method in complex environments

To overcome the uncertainty of system parameters and achieve effective management of energy combined thermal comfort, Lyapunov optimization technology is introduced to design the LO-DRTC algorithm. The Lyapunov optimization technique utilizes Lyapunov functions for optimal control of dynamical systems, which has been widely applied in control theory to ensure the stability of different forms of systems (Wu et al. 2022; Battiloro et al. 2022). The problem-solving approach is shown in Fig. 2.

Fig. 2

Solution ideas of Liapunov optimization technique

As can be seen from Fig. 2, the Liapunov optimization technique first transforms the original problem into multiple stability problems by constructing virtual queues associated with time-averaged constraints. Finally, it is then transformed into a single time-slot online optimization decision problem through the stability queues. Therefore, the calculation steps of the LO-DRTC proposed in the study are shown in Fig. 3.

Fig. 3

Solution ideas for the LO-DRTC algorithm

In Fig. 3, the LO-DRTC algorithm first constructs virtual queues associated with temperatures in all regions. Second, the solution problem is converted to a single time slot online optimization problem according to the minimum offset-penalty term algorithm of the Liapunov optimization technique, and the upper bound calculation of the right-hand side of the minimization offset-penalty term is carried out to update the virtual queue by taking the supply air rate as the solution of the solution problem. In this case, the unit time slots of the building thermal dynamics model for the thermal region are converted to a discrete form by the finite difference method. The discrete approximation is first substituted for the continuous inverse by noting the temperature within the next time slot $t + 1$ region $i$ as $T_{i,t + 1}$, which leads to $\frac{{dT_{i,l} }}{dl}$ which can be approximated as $\frac{{dT_{i,t} }}{dl} \approx \frac{{T_{i,t + 1} - T_{i,t} }}{\uptau }$. The result of this approximation is carried over to the building thermodynamic model, which is expressed as shown in Eq. (6).

$$A_{i} \frac{{dT_{i,t + 1} - T_{i,t} }}{\tau } = \frac{{T_{o,l} - T_{i,t} }}{{B_{i} }} + A_{a} m_{i,t} (T_{s} - T_{i,t} ) + q_{i,t}$$

(6)

Based on Eq. (6), $T_{i,t + 1}$ is further rearranged to obtain the final discrete form as shown in Eq. (7).

$$T_{i,t + 1} = [1 - \tau /(B_{i} A_{i} )]_{i} T_{i,t} + (\tau A_{a} /A_{i} )m_{i,t} (T_{s} - T_{i,t} ) + [\tau /(B_{i} A_{i} )]T_{o,t} + \frac{\tau }{{A_{i} }}q_{i,t}$$

(7)

In Eq. (7), to ensure the controllability of the system, the study first sets three constraint conditions related to regional temperature, control parameters, and LO-DRTC. The specific mathematical expressions are shown in Eq. (8).

$$\left\{ \begin{gathered} gT_{i}^{\min } + hm_{i}^{\min } (T_{s} - T_{i}^{\min } ) + kT_{o}^{\min } + \frac{\tau }{{A_{i} }}q_{i}^{\min } \ge T_{i}^{\min } \hfill \\ (T_{i}^{\max } - T_{i}^{\min } ) + k(T_{o}^{\max } - T_{o}^{\min } ) + \frac{\tau }{{A_{i} }}(q_{i}^{\min } - q_{i}^{\max } ) \hfill \\ + h[m_{i}^{\max } (T_{s} - T_{i}^{\min } ) + m_{i}^{\min } (T_{s} - T_{i}^{\min } )) > 0 \hfill \\ \frac{{k(T_{i}^{\max } - T_{o}^{\max } ) - \frac{\tau }{{A_{i} }}q_{i}^{\max } }}{{h(T_{s} - T_{i}^{\min } )}} \le m_{i}^{\max } \hfill \\ \end{gathered} \right.$$

(8)

In Eq. (8), $g$ represents $1 - \tau /(B_{i} A_{i} )$. $h$ represents $\tau A_{a} /A_{i}$. $k$ represents $\tau /(B_{i} A_{i} )$. $gT_{i}^{\min } + hm_{i}^{\min } (T_{s} - T_{i}^{\min } ) + kT_{o}^{\min } + \frac{\tau }{{A_{i} }}q_{i}^{\min } \ge T_{i}^{\min }$ indicates that the temperature drop in region $i$ can be terminated by the minimum supply air rate $m_{i}^{\min }$ with minimum in-region temperature $T_{i}^{\min }$, minimum outdoor temperature $T_{o}^{\min }$, and minimum thermal perturbation $q_{i}^{\min } = \min_{t} q_{i,t}$. And $(T_{i}^{\max } - T_{i}^{\min } ) + k(T_{o}^{\max } - T_{o}^{\min } ) + \frac{\tau }{{A_{i} }}(q_{i}^{\min } - q_{i}^{\max } ) + h[m_{i}^{\max } (T_{s} - T_{i}^{\min } ) + m_{i}^{\min } (T_{s} - T_{i}^{\min } )) > 0$ is to ensure that the control parameters of the system are positive. $\frac{{k(T_{i}^{\max } - T_{o}^{\max } ) - \frac{\tau }{{A_{i} }}q_{i}^{\max } }}{{h(T_{s} - T_{i}^{\min } )}} \le m_{i}^{\max }$ then ensures that the algorithm is able to ensure that the temperature in region $i$ does not exceed the maximum outdoor temperature $T_{i}^{\max }$ using the maximum supply air rate $m_{i}^{\max }$ given the upper comfort temperature $T_{i}^{\max }$, the maximum outdoor temperature $T_{o}^{\max }$, and the maximum thermal perturbation $q_{i}^{\max }$. Based on the set conditions, the study performed the construction of a virtual cohort in order to ensure the feasibility of the upper and lower bound constraints on the comfort temperature. Among them, the virtual queue related to regional temperature is shown in Eq. (9).

$$Q_{i,t} = T_{i,t} + \beta_{i}$$

(9)

In Eq. (9), $Q_{i,t}$ represents a virtual queue. $\beta_{i}$ represents a constant. According to Eqs. (7) and (8), the virtual queue expression formula for the next time slot can be further obtained, as shown in Eq. (10).

$$Q_{i,t + 1} = (1 - k)Q_{i,t} + hm_{i,t} (T_{s} - T_{i,t} ) + k(\beta_{i} + T_{o,t} ) + \frac{\tau }{{A_{i} }}q_{i,t}$$

(10)

Secondly, the problem is transformed using Lyapunov optimization techniques. In order to stabilize the proposed virtual queue shown in Eq. (10), the study defines the Liapunov function formula as shown in Eq. (11).

$$H_{t} = \frac{1}{2}\sum\limits_{i = 1}^{N} {Q_{i,t}^{2} }$$

(11)

In Eq. (11), $H_{t}$ represents the Lyapunov function with a time slot of $t$. $N$ represents the number of regions. Therefore, in all regions, a single time slot Lyapunov offset expression function is shown in Eq. (12).

$$\left\{ \begin{gathered} \Delta_{t} = {\rm E}\left\{ {H_{t + 1} - H_{t} \left| {{\mathbf{Q}}_{t} } \right.} \right\} \hfill \\ {\mathbf{Q}}_{t} = \left\{ {Q_{1,t} ,Q_{2,t} ,...,Q_{N,t} } \right\} \hfill \\ \end{gathered} \right.$$

(12)

In Eq. (12), ${\mathbf{Q}}_{t}$ represents the vector matrix of the virtual queue. $\Delta_{t}$ represents the Lyapunov shift function. The ability to act on the price of electricity, the outdoor temperature, the user's thermal comfort preference temperature, the stochastic nature of thermal perturbations, and the control decisions taken, based on the Liapunov shift function for a single time slot. The Lyapunov offset-penalty term expression function is shown in Eq. (13).

$$\begin{gathered} \Delta Y_{t} = \Delta_{t} + L{\rm E}\left\{ {\sum\limits_{l = 1}^{3} {{\mathbb{R}}_{l,t} } \left| {{\mathbf{Q}}_{t} } \right.} \right\} \hfill \\ \le \frac{1}{2}\sum\limits_{i = 1}^{N} {hm_{i}^{\max } \left( {T_{s} - T_{i}^{\max } } \right) + k(\left| {\beta_{i} } \right| + T_{o}^{\max } ) + (\frac{\tau }{{A_{i} }}q_{i}^{\max } )^{2} } \hfill \\ + 2(1 - k)(\left| {\beta_{i} } \right| + T_{o}^{\max } )[k(\left| {\beta_{i} } \right| + T_{o}^{\max } ) + \frac{\tau }{{A_{i} }}q_{i}^{\max } ] + \hfill \\ {\rm E}\left\{ {\sum\limits_{l = 1}^{3} {(1 - k)Q_{i,t} h\left( {T_{s} - T_{i,t} } \right)m_{i,t} } \left| {{\mathbf{Q}}_{t} } \right.} \right\} + L{\rm E}\left\{ {\sum\limits_{l = 1}^{3} {{\mathbb{R}}_{l,t} } \left| {{\mathbf{Q}}_{t} } \right.} \right\} \hfill \\ \end{gathered}$$

(13)

In Eq. (13), $L$ represents a positive number. Finally, ground on the key idea of Lyapunov optimization technology, the upper limit of the right formula for the offset-penalty term is calculated (Biswas et al. 2023; Jia et al. 2022). The specific calculation steps are the detailed calculation steps of the LO-DRTC algorithm, as shown in Fig. 4.

Fig. 4

Computational flowchart of the LO-DRTC algorithm

In Fig. 4, the time slot index and total number of time slots are first set, followed by the calculation of virtual queues and initial zone comfort temperature. Then the number of time slots is used as the solution to Eq. (7). Finally, according to Eq. (10), the virtual queue related to regional temperature is updated. In addition, for the problem of spatially coupled constraints on the supply air rate $m_{i,t}$ in all zones, the study utilizes Cauchy's inequality $\left( {\left( {\sum\nolimits_{i} {m_{i,t} } } \right)^{3} \le \overline{m}\left( {\sum\nolimits_{i} {m_{i,t} } } \right)^{2} \le N\overline{m}\sum\nolimits_{i} {m_{i,t} }^{2} } \right)$ to approximate the energy cost associated with the supply fan into an upper bound $LuS_{t} \tau N\overline{m}m_{i,t}^{2}$ and transforms the problem as shown in Eq. (14). Where $N$ denotes the total number of regions.

$$\begin{gathered} P2:\mathop {\min }\limits_{{m_{i,j} }} \sum\nolimits_{i} {[(1 - k)Q_{i,t} } h\left( {T_{s} - T_{i,t} } \right)m_{i,t} \hfill \\ + L\varphi_{i} (T_{i,t + 1} - T_{i,t + 1}^{ref} )^{2} + Lf_{i,t} m_{i,t} + LuS_{t} \tau N\overline{m}m_{i,t}^{2} ] \hfill \\ \end{gathered}$$

(14)

In Eq. (14), $P2$ represents solving problem 2. $P2$ represents $S_{t} \tau (A_{a} /\eta COP)[\chi T_{i,t} + (1 - \chi )T_{o,t} - T]$. $\chi$ indicates the position of the AHU damper. The centralized solution Eq. (14) using an energy management system may lead to user privacy leakage. Therefore, a distributed solution is proposed in the study. Set user thermal comfort preference temperatures $T_{i,t + 1}^{ref}$ and thermal perturbations $q_{i,t}$ are obtained from actual local measurements and used to solve for the supply air rate $m_{i,t}$. On this basis, the calculated $m_{i,t}$ is returned to the energy management system for testing and verification. The specific solution method is shown in Fig. 5.

Fig. 5

Distributed implementation of solving problem 2

In Fig. 5, each iteration completes three steps. Firstly, the energy management system broadcasts the initial non-negative dual variable values to all control agents in the region. Secondly, each intelligent agent calculates the air supply rate and sends it to the energy management system. Finally, the energy management system checks whether the iteration termination condition is met. Therefore, in order to solve Eq. (14), the first-order partial derivatives of the objective function of the problem with respect to the supply air rate are first taken to zero, as shown in Eq. (15).

$$\left\{ \begin{gathered} m_{i,t}^{\pi } = \frac{{h\left( {T_{s} - T_{i,t} } \right)v - Lf_{i,t} }}{{2LuS_{t} \tau N\overline{m} + 2L\varphi_{i} h^{2} \left( {T_{s} - T_{i,t} } \right)^{2} }} \hfill \\ v = 2L\varphi_{i} [T_{i,t + 1}^{ref} - gT_{i,t} - kT_{o,t} - (\tau /A_{i} )q_{i,t} ] - (1 - k)Q_{i,t} \hfill \\ \end{gathered} \right.$$

(15)

In Eq. (15), when the total air supply rate constraint equation is ignored, then the optimal value of $m_{i,t}$ is $\max [m_{i}^{\min } ,\min (m_{i}^{\max } ,m_{i,t}^{\pi } )]$. The optimal solution of problem 2 is obtained when $\sum\nolimits_{i = 1}^{N} {m_{i,t}^{\pi } } \le \overline{m}$, where $m_{i,t}^{\pi }$ denotes the optimal value of the air supply rate $m_{i,t}$. Conversely, when $\sum\nolimits_{i = 1}^{N} {m_{i,t}^{\pi } } > \overline{m}$, the solution of Problem 2 is transformed according to the optimization solution method Karush–Kuhn–Tucker (KKT) condition.The KKT condition, as a dyadic optimization method, is utilized to enable the constrained optimization problem to be solved exactly (Su and Luu 2022). Therefore, the exact transformation is shown in Eq. (16).

$$\left\{ \begin{gathered} m_{i,t}^{ + + } = \max [m_{i}^{\min } ,\min (m_{i}^{\max } ,m_{i,t}^{ + } )] \hfill \\ m_{i,t}^{ + } = \frac{{h\left( {T_{s} - T_{i,t} } \right)v - Lf_{i,t} - \alpha }}{{2LuS_{t} \tau N\overline{m} + 2L\varphi_{i} h^{2} \left( {T_{s} - T_{i,t} } \right)^{2} }} \hfill \\ \end{gathered} \right.$$

(16)

In Eq. (16), $m_{i,t}^{ + + }$ represents the optimal solution. $\alpha$ represents the non-negative dual variable of the total air supply rate constraint formula for all regions. Ground on the complementary relaxation condition of KKT, the optimal $\alpha$ can satisfy $\sum\nolimits_{i} {m_{i,t}^{ + + } } = \overline{m}$. Therefore, the study carries out the search for the optimal value $\alpha$ by dichotomization with an associated complexity of $\hbar \left( {N_{epoch} N} \right)$. Where, $N_{epoch}$ denotes the total number of iterations. Combining the above, the proposed energy joint thermal comfort management method in the study achieves the joint management of energy and thermal comfort in multiple smart building zones and enhances the system scalability by protecting the user's privacy without the need to know any a priori information about the uncertain system parameters.

Verification and analysis of energy combined thermal comfort management methods in complex environments

To verify the effectiveness of the energy joint thermal comfort management method ground on the LO-DRTC algorithm proposed in the study, the hourly commercial electricity price in Beijing, China in July 2022 is used as the cost basis. Simulation experiments are conducted with the outdoor temperature trajectory in Beijing in July 2022. Firstly, the feasibility of the method in complex environments is verified, and secondly, the performance analysis and robustness analysis of the method under different maximum outdoor temperatures as well as thermal discomfort costs are carried out.

Feasibility analysis of energy combined thermal comfort management method in complex environments

The total number of time slots during the verification process is 8928, with a unit time slot of 5 min. The environmental area is 4, the AHU air valve is spaced 0.95 m apart, the lower limit of the comfortable temperature in the area is 18 °C, the supply air temperature of the supply fan is 12.8 °C, the condensing coil coefficient is 5.9153, the efficiency factor of the condensing coil is 0.8879, the value of the regional air supply rate is [0, 450]g/s, the upper limit of the total air supply rate is 1400 g/s, and the blower power consumption correlation coefficient of 2 × 10⁻⁶W/(g/s)³. In addition, the study sets that the user's thermal comfort preference temperature in all regions follows a uniform distribution of 21.0, 23.0 °C per hour. The thermal disturbance parameters in each unit time slot follow a uniform distribution of [0.1, 0.2] W. The length of the time horizon considered for the research simulation experiments is one month (31 days) due to the fact that in real environments, outdoor temperatures and thermal perturbations can vary within minutes, and electricity prices and user thermal comfort preference temperatures can vary within hours. At the same time, the hourly commercial electricity price in July 2022 in Beijing, China was used as the tariff information, and the outdoor temperature trajectory in Edmonton, Canada was used to increase the hourly temperature data in July 2022 by 8.5 °C, which is close to the temperature range in Beijing. The changes in commercial electricity prices and outdoor temperature trajectories in Beijing in July 2022 are shown in Fig. 6.

Fig. 6

Changes in commercial electricity prices and outdoor temperature trajectories in Beijing, July 2022

Figure 6a shows the commercial electricity price in Beijing in July 2022. Figure 6b shows the outdoor temperature changes in Beijing in July 2022. Meanwhile, the study utilizes Matlab2021a to perform simulations to verify the feasibility of the LO-DRTC algorithm with the control results of region 1, which are shown in Figs. 7 and 8.

Fig. 7

Validation of virtual queues and air supply rates

Fig. 8

Verification of total air supply rate and temperature in the zone

Figure 7a shows the fluctuation variation of the virtual queue within region 1 within the set range. Figure 7b shows the variation of the air supply rate within region 1. Combined with Fig. 7, it can be seen that the virtual queue variation is always within the range of the maximum virtual queue $Q_{1}^{b}$ = 5, and the air supply rate $m_{i,t}$ within region 1 is always less than the set maximum air supply rate $m_{i}^{\max }$ within the region.This indicates that there exists an optimal decision of the air supply rate $m_{i,t}^{ + + }$ that is equal to the minimum air supply rate $m_{i}^{\min }$ within the region when $Q_{1,t} < Q_{1}^{b}$, and an optimal decision of the air supply rate $m_{i,t}^{ + + }$ that is not more than the maximum air supply rate $m_{i}^{\max }$ within the region when $Q_{1,t} > Q_{1}^{b}$. At the same time, this further illustrates that the rationality of the minimum cost model established by the study.

Figure 8a shows the variation of the total supply air rate in region 1 and Fig. 8b shows the variation of the temperature in region 1 over a 9000 time slot. It can be seen that the total air supply rate variations obtained from the simulation experiments of the management method proposed in the study are all lower than the maximum air supply rate. And with the increase of the time slot, the temperature inside region 1 shows different increases and decreases, but the average is around 22.23 °C, and the temperature inside the region is between the lower and upper comfort temperature limits. This indicates that the energy joint thermal comfort management method based on the LO-DRTC algorithm proposed in the study is able to satisfy all the constraints of the original problem. Meanwhile, the proposed energy combined thermal comfort management method is feasible in the thermal comfort management of complex environments with multiple zones. It is worth mentioning that the simulated operating parameters and conditions assumed by the study may affect the scope of LO-DRTC in real-world applications, but the above validation results show that LO-DRTC is feasible in real environments for the combined energy thermal comfort management approach for multiple zones. The above parameters are set to take into account the random perturbations as well as the environmental changes in the actual operation, and are replicable in real scenarios.

Performance analysis of energy combined thermal comfort management methods in complex environments

To further confirm the superior performance of the proposed method, the Comfort-Aware Scheme (CAS) and Modified LO-DRTC (MLO-DRTC) with temperature related shift parameters are introduced to compare the performance. CAS and MLO-DRTC differ from LO-DRTC in that there is a difference in the management parameters between the user's thermal comfort preference temperature and the zone temperature, and the performance of LO-DRTC can be further verified by comparison. Firstly, the thermal discomfort cost coefficient is set to 0. The upper limit of the comfortable temperature is within the [5.0, 28.0 °C. MLO-DRTC and LO-DRTC are the same and the total cost is equal to the energy cost. The energy cost generated by the system, the Average Temperature Deviation (ATD) profile, and the average temperature change in the area under the three methods were first compared to validate the performance of the algorithms in managing the user's thermal comfort preference temperature. The validation results under the fixed cost coefficient are shown in Fig. 9.

Fig. 9

Comparison of 3 methods when the thermal discomfort cost factor is 0

Figure 9a shows the energy cost comparison of three methods. As the upper limit of comfortable temperature increases, the overall change of CAS remained stable, with an average energy cost of 773.54 RMB. The energy cost of MLO-DRTC and LO-DRTC was inversely proportional to the upper bound of comfort temperature. When the upper bound of comfortable temperature was 28 °C, the energy cost of LO-DRTC was reduced by 19.62% and 2.85% compared with CAS and MLO-DRTC, respectively. Comparison of the changes in the ATD curves of the three methods in Fig. 9b shows that LO-DRTC has the highest ATD value, which indicates that the proposed algorithm of the study reduces the energy cost by sacrificing the ATD. While the ATD of CAS is not significantly affected by the change in the upper limit of comfort temperature. In addition, the ATD curves of MLO-DRTC and LO-DRTC did not monotonically increase with the increase of comfort temperature limit. This may be because the average temperature of all areas controlled by both methods first approaches the user's thermal comfort preference temperature. Figure 9c shows the average temperature changes obtained by controlling all regions using three methods. A the upper limit of comfortable temperature increased, the average temperature of MLO-DRTC and LO-DRTC was directly proportional to it, while the overall CAS remained stable.

As can be seen from the comparison of the temperature changes in the region under the control of HVACS by both CAS and LO-DRTC methods in Fig. 10, with the increase in the number of time slots, the temperature in the region under the control of LO-DRTC exhibits a variety of changes such as an increase or a decrease, whereas the temperature in the region under the control of CAS is generally stable. Due to user preferences for thermal comfort and energy costs, the energy consumption of HVACS is affected. Therefore, various temperature changes in the area are beneficial for reducing energy costs and energy consumption. This further confirms the effectiveness of the energy combined thermal comfort method ground on LO-DRTC.

Fig. 10

Comparison of 3 methods based on fixed cost coefficients for temperature in the region

In energy combined thermal comfort management, the thermal discomfort cost coefficient reflects the importance of thermal discomfort costs relative to energy costs. Therefore, the study further compares the performance in various thermal discomfort cost coefficients. Figure 11a shows the total cost comparison of three methods under different thermal discomfort cost coefficients. When users did not care about thermal comfort, the optimal control strategy for HVACS was LO-DRTC. From the energy costs of the three methods in Fig. 11b and the thermal discomfort costs in Fig. 11c, the energy costs of MLO-DRTC and LO-DRTC were inversely proportional to the thermal discomfort cost coefficient, while the thermal discomfort cost was directly proportional to the thermal discomfort cost coefficient. Combining the ATD changes of the three methods shown in Fig. 11d, it can be seen that LO-DRTC and MLO-DRTC are more able to flexibly compromise the relationship between energy cost and ATD than CAS. Compared with MLO-DRTC, LO-DRTC had a superior ATD value. The regulation and control of energy cost and user thermal comfort were more balanced.

Fig. 11

Comparison of 3 methods at different thermal discomfort cost factors

In order to illustrate the applicability of the proposed algorithm of the study to building dynamics modeling in complex environments, further method robustness validation was performed. A building dynamics model without building dynamics for joint thermal comfort management of energy in multiple zones was constructed using Matlab 2021a software and a random perturbation was added to the model, setting the maximum thermal perturbation to 1.8, 3.6 and 5.4 for robustness validation and obtaining the information on outdoor temperature and electricity price from the Pecan Street database, respectively. Figure 12a shows the average total energy cost of three methods under different disturbances. Compared with CAS, LO-DRTC and MLO-DRTC had lower total energy costs. This indicates that the method proposed in the study has better performance. From the average total temperature deviation of the three methods in Fig. 12b, the LO-DRTC saved more than 10% of total energy cost compared with the other two methods, while maintaining a small deviation from the increase in average total temperature. Based on this, the study further compares the energy cost as well as the temperature change in the region for the three scenarios with different maximum thermal perturbations, and the results are shown in Table 2.

Fig. 12

Robustness verification

Table 2 Comparison of the regulation results of the three methods under different thermal perturbations

From Table 2, it can be seen that the indoor temperature obtained from LO-DRTC regulation is more suitable for the user's thermal comfort temperature under different thermal perturbations. And as the thermal perturbation increases, LO-DRTC requires less energy cost and more stable cost gain. This indicates that the algorithm proposed in the study provides a more flexible compromise strategy between reducing energy costs and thermal comfort, without knowing the prior information of all uncertain system parameters.

Discussion and conclusion

In complex environments, there are issues such as multi-source uncertainty in HVACS and time coupling constraints related to temperature within the region. A combined energy thermal comfort management method based on LO-DRTC was proposed. Firstly, The long-term total cost minimization problem of HVACS was established. Secondly, a distributed control algorithm was designed using Lyapunov optimization technology. From the simulation experiments, the designed method could meet the constraints of the upper and lower limits for temperature control range, air supply rate, and total air supply rate within the set area. This was feasible in HAVCS control across multiple regions. Compared with CAS and MLO-DRTC, the proposed method reduced energy costs by 19.62% and 2.85%, respectively. Under different thermal disturbances, LO-DRTC achieved a total energy cost savings of over 10% compared with the other two methods, while maintaining a small increase in average total temperature deviation. The results indicated that the proposed energy combined thermal comfort management method could reduce energy costs while ensuring user thermal comfort. This method can demonstrate superior scalability and user privacy security without knowing any prior information about system parameters.

However, the study assumes that the building thermal dynamics model is known during the methodological design process. However, in practical joint building energy thermal comfort management, the AHU damper position needs to be dynamically adjusted to meet the dual goals of indoor air quality needs and energy savings. This leads to an increase in the cost overhead between managing energy and thermal comfort regulation in smart building environments, while the proposed LO-DRTC has high requirements for the fixing of AHU damper positions, the distribution of infrastructure, and the degree of sophistication are all factors that influence the performance of LO-DRTC in regulating between energy costs and thermal comfort. Therefore, in the subsequent work, the study will investigate the HVACS control method in the case of unknown building thermal dynamics model and consider the integration of deep neural networks and other technologies to explore and overcome the negative effect of the degree of infrastructure sophistication on the HVACS control, with a view to realizing a more efficient design of the HVACS control strategy.