Thermodynamic-Based Perceived Predictive Power Control for Renewable Energy Penetrated Resident Microgrids

Abstract

Heating, ventilation, and air conditioning (HVAC) systems and microgrids have garnered significant attention in recent research, with temperature control and renewable energy integration emerging as key focus areas in urban distribution power systems. This paper proposes a robust predictive temperature control (RPTC) method and a microgrid control strategy incorporating asymmetrical challenges, including uneven power load distribution and uncertainties in renewable outputs. The proposed method leverages a thermodynamics-based R-C model to achieve precise indoor temperature regulation under external disturbances, while a multisource disturbance compensation mechanism enhances system robustness. Additionally, an HVAC load control model is developed to enable real-time dynamic regulation of airflow, facilitating second-level load response and improved renewable energy accommodation. A symmetrical power tracking and voltage support secondary controller is also designed to accurately capture and manage the fluctuating power demands of HVAC systems for supporting operations of distribution power systems. The effectiveness of the proposed method is validated through power electronics simulations in the Matlab/Simulink/SimPowerSystems environment, demonstrating its practical applicability and superior performance.

1. Introduction

1.1. Motivation

The penetration rate of renewable energy sources, such as photovoltaic (PV) generation, has increased dramatically in urban distribution power systems (DPSs), which have fundamentally reshaped the dynamics and changed the symmetrical operating conditions of DPSs [1,2]. It is reported by the International Energy Agency (IEA) that the installed capacity of solar PVs has expanded from 2.5 GW to 213 GW globally, representing an increase of 85 times between 2006 and 2018 [2]. Solar PV installations will account for up to 40% of total electricity output from PV generation by 2050 [3]. Nevertheless, PV generation on building rooftops is heavily influenced by sunlight availability, which fluctuates in real time because of time-dependent meteorological conditions such as cloud cover, precipitation, and atmospheric clarity [4]. The intermittent nature of solar generation may result in sudden changes and sharp variations in power output, potentially causing voltage fluctuations and frequency deviations in DPSs [1,4]. In this context, inherent asymmetrical characters, such as the intermittency and volatility of PV outputs, can pose significant threats to the stability and reliability of power supply to consumers. More and more solar renewable energy generations have to be curtailed to maintain the required level of operational reliability for DPSs, thereby highlighting concerns over the effective utilization of high-penetration solar PVs [5]. Battery energy storage systems (BESSs) that use chemicals to store and discharge energy on demand have emerged as a viable option to mitigate fluctuations from renewable energy, like solar and wind [6]. However, the aging process resulting from frequent charging and discharging would lower the energy storage capacity and service life of the battery, and thus, the operational cost of BESSs would be significantly increased [7]. Therefore, a cost-effective symmetrical technological solution is becoming a pressing need to improve the utilization of solar energy.

Residential microgrids have become the common practice to address various operation challenges resulting from the increasing penetration level of renewable energy resources in DPSs [8]. This is mainly because the thermal capacitance of buildings in residential microgrids is a powerful tool for storing excess energy generated from renewable sources [8,9]. Specifically, thermal capacitance is the ability of building materials and air to absorb and store heat. Thus, excess renewable electricity can power heating or cooling systems, converting electrical energy into stored thermal energy that reduces peak energy demand. Meanwhile, recent advancements in smart meters; building energy management systems; heating, ventilation, and air conditioning (HVAC) systems; and other smart appliances offer substantial operation and control flexibility for residential microgrids [10,11,12,13,14,15,16,17,18]. Various power electronic devices, including variable frequency drives, are installed in most residential microgrids to regulate the operating state of HVAC and other electric appliances. For example, building energy management systems can continuously and frequently adjust the speed of the HVAC supply fan every few seconds to modulate fan power by tracking the control signal [18]. However, one of the crucial challenges for residential power control arises from the inherent intermittence of renewable generation (e.g., PV generation), which introduces considerable forecast uncertainty on the supply side [13,18]. On the other hand, the energy demand profile of small-scale residential microgrids is generally influenced by user behavior preferences and environmental conditions, such as temperature. The power control of residential microgrids should also take into account the impact of variations in external temperature. In this context, uncertain renewable generation and changing external environmental factors can dramatically reduce economic benefits, compromise customer comfort, and pose technical challenges for the power management of residential microgrids. Consequently, this paper aims to investigate a perceived predictive power control strategy for resident microgrids to reduce the curtailment of renewable generations considering the varying environmental temperatures in buildings.

1.2. Literature Review

Numerous researchers have been engaged in investigating residential microgrid operation and planning with a focus on renewable energy integration. Several studies have explored the application of different energy scheduling algorithms and capacity planning methods for residential microgrids, such as optimization-based methods [13,19] and machine learning techniques [20], to address the complexities of renewable energy sources, load variability, and distributed energy generation. The scheduling objectives in these studies include cost minimization, system stability, and emission reduction. To name a few. a strategic planning and optimization framework was introduced in [13] to facilitate the integration of energy storage systems and PV generation within the residential microgrid. In [20], a multiagent deep reinforcement learning-based method was introduced for determining the optimal operational strategy of residential microgrid clusters. This method aims to reduce energy costs for each residence while maintaining resident comfort and preventing transformer overload. A multiobjective optimization method was adopted in [21] to determine the optimal capacity of renewable generation and battery energy storage. Nonetheless, the existing literature predominantly addresses deterministic scenarios, neglecting the dynamic nature of renewable energy sources. To overcome this drawback, recently, advanced stochastic programming [22] and robust optimization [23] have been employed in the literature to tackle the challenges of the uncertainties related to renewable energy sources. For instance, in [22], a two-stage stochastic mixed-integer nonlinear programming model was presented for the optimal planning and operation of community-scale microgrids, and uncertainties in solar irradiation and wind speed were represented by a group of stochastic scenarios. A bilayer coordinated operation scheme was developed in [23] to obtain the economic operation of the multienergy building microgrid, where an uncertainty set was provided to address the uncertainty. However, all these mentioned works mainly focused on microgrid energy management on an hourly or minimum basis. Real-time control mechanisms of residential microgrids to integrate temporal adaptability and robustness against renewable fluctuations were neglected.

Driven by the rapid development of smart grid technologies and the integration of the Internet of Things, the energy management and control of residential electrical appliances have evolved significantly [24]. In existing works [20,21,22,23,25], electrical appliances, including washing machines, tumble dryers, and dishwashers have been modeled to run automatically during user-defined times. Modern home energy management systems (HEMSs) facilitate the management and control of different devices such as refrigerators, washing machines, and lighting systems. Based on dynamic electricity pricing and user preferences, the power consumption of residential electrical appliances can be optimized by employing demand response strategies [26,27,28,29]. For example, a bilevel optimization strategy for park-level integrated energy systems was developed in [26] to increase user satisfaction and enhance economic performance considering uncertainties of price-based load demand response. In addition, HVAC systems represent a critical component of residential energy consumption. In [28], a multiobjective demand response model was established for a smart residential house to reduce electrical load demand and minimize consumer costs. HVAC loads were controlled in [27] via a smart thermostats integrated into a building microgrid, and the room inertia was considered to schedule heating/cooling and manage power consumption. The demand response potential of HVAC systems in medium-sized commercial office buildings located in four cities representing diverse humid climate zones has been studied in [29]. To guarantee energy efficiency and execute demand response programs, building energy modeling has proven invaluable in facilitating the forecasting and regulation of energy consumption. A resistance–capacitance (R-C) thermal dynamics model was presented in [17] to evaluate the potential flexibility in commercial buildings for providing a flexible power reserve to the main grid. Moreover, model-free and model-based control methods used in HVAC have been widely developed [30]. For instance, a deep reinforcement learning-based HVAC system control method was developed in [31] to fully utilize uncertain weather forecast information. In [32], a strategic control learning framework was proposed for HVAC systems to reduce energy use. However, variations in climate zones influence the energy consumption behavior of occupants, and the model-free works failed to consider the integrated power and temperature control strategies considering the thermodynamic effects for renewable accommodation within residential microgrids. Model-free control methods typically rely on large volumes of high-quality historical and real-time data. This data dependency poses a significant challenge to their deployment and limits their robustness under real-world conditions.

Microgrid control has gained significant attention, and various control strategies have been developed to enhance grid resilience, support renewable energy integration, and improve energy efficiency [13,22,33]. A centralized control strategy has been studied in [34], which relies on a single control entity to make management decisions for renewable generations and loads within the entire residential microgrid. Nevertheless, the centralized control strategy is dependent on an effective and steady communication network, and the computational burden on the central controller increases with system complexity. To address the limitations of centralized control, decentralized and distributed control strategies have been developed. A decentralized control strategy divides the residential microgrid into smaller, self-contained subsystems. Then, each subsystem is governed and managed by an independent local controller with minimal intercommunication. Distributed control employs a network of controllers that communicate and collaborate to achieve system-wide objectives. For instance, in [35], a decentralized control strategy was proposed for islanded microgrids to reduce the unbalance in load bus voltages. However, the mentioned methods struggle with handling the dynamic and uncertain nature of microgrids, particularly with fluctuating renewable generation and load demand. The recently emerging model predictive control (MPC) strategy provides an excellent alternative to optimize control actions of residential microgrids over a finite horizon while incorporating constraints and uncertainties. To sum up, MPC stands out as a highly versatile control strategy due to its unique advantages [17,19,34]. First, MPC can seamlessly incorporate different types of constraints, such as control-related and physical constraints. Second, MPC delivers exceptional dynamic performance under varying operating conditions. Third, control signals can be generated by MPC directly, which offers a straightforward and efficient implementation process. Finally, the open architecture of MPC allows to integration of diverse optimization algorithms. Motivated by the outstanding features of the MPC, this strategy is used to address challenging control tasks in residential microgrids. Specifically, there are many significant results for MPC-based research targeting building microgrids [36,37,38]. For example, the authors presented in [37] introduced optimization-based and optimization-free model predictive control algorithms, offering valuable insights into enhancing energy efficiency through multienergy microgrid deployment.

1.3. Contribution

Traditional research on residential microgrid control methods often focuses only on local optimization of temperature regulation while ignoring its actual coupling with the energy system [17,32]. The existing work has not unified the consideration of HVAC systems with microgrid systems, leading to uncertainties in many scenarios regarding how HVAC systems impact the voltage output of the microgrid. To the best of our knowledge, no similar studies have been explored in the existing literature. This paper proposes a thermodynamics-based symmetrical power predictive control method for residential microgrids to accommodate the high penetration of variable renewables in DPSs. The work presented in this paper expands the theoretical boundaries of coordinated control between HVAC systems and residential microgrids for harvesting fluctuating renewable energy sources. The main contributions are threefold:

(1) An R-C thermodynamics-based robust predictive control method is proposed for the real-time power regulation in residential microgrids so as to maintain indoor temperatures of microgrids stably around preset desired values under external disturbances such as sudden weather changes or drastic load fluctuations. The proposed method incorporates a multisource disturbance compensation mechanism that accounts for user comfort levels, external weather conditions, and heat gains from both internal and external heat sources. Then, a thermal R-C model is developed to accurately characterize the heat transfer and thermal storage dynamics of building envelopes within the residential microgrids; the system demonstrates exceptional robust performance and control accuracy.

(2) An HVAC load control model is established for residential microgrids to enhance local accommodation of intermittent renewable energy. By combining the complementarity and flexibility of power loads with the thermal inertia of building structures, the model enables real-time dynamic regulation of air mass flow into buildings to achieve second-level HVAC load response to support the operations of DPSs while maintaining indoor temperature stability and occupant thermal comfort. Through coordinated utilization of user energy consumption characteristics and precise airflow modulation of HVAC loads, the proposed control model successfully achieves a complete renewable energy supply for residential microgrids.

(3) A power tracking and voltage support secondary controller is designed for residential microgrids with time-varying constant power loads. Using a simplified power model, the dynamic power demand under regional temperature robust predictive control is derived and further modeled as a time-varying constant power load in the microgrid so as to accurately capture real-time fluctuations in HVAC system power demands. The developed controller provides an important theoretical foundation for subsequent microgrid controller designs.

2. Thermodynamics-Based Symmetrical Power Predictive Control

2.1. Framework Structure of Residential Microgrids with HVAC Systems

In this paper, the microgrid system operates in islanded mode to supply power to the HVAC system within a building. The system architecture is illustrated in Figure 1. As shown in the figure, the system consists of renewable energy generation units, power converters, buses, transmission lines, and loads. Among these components, the smart building equipped with an HVAC system represents a critical load in the microgrid. The microgrid is designed to achieve two primary objectives: (1) providing precise power supply according to the building’s demand and (2) maintaining the voltage of each bus within the allowable rated range.

The renewable energy generation units, which may include photovoltaic (PV) panels, wind turbines, and energy storage systems, serve as the primary power sources for the microgrid. These units are connected to the buses through power converters, which ensure efficient energy conversion and regulation. The buses act as central nodes for power distribution, linking the generation units, loads, and transmission lines. It should be noted that addressing the output fluctuation of new energy sources is a fundamental issue in microgrid control research. Following common practices in the field (as in reference [10,11,12]), it is assumed in this paper that energy storage systems have smoothed out the variability of renewable energy output. In practice, ESSs will be paired with an active control strategy to work effectively, as described in [39]. The HVAC system, as a key load, exhibits nonlinear and time-varying power consumption characteristics, posing unique challenges for the microgrid’s control and stability. HVAC systems often experience variations in power demand based on factors such as temperature setpoints, time of day, and external environmental conditions. It should be noted that we model the HVAC system as a time-varying constant power load because it simplifies the complex dynamics of HVAC operation while still capturing the essential features of its power consumption. By treating the HVAC system as a time-varying constant load, variations in power demand can be considered without the need to account for complex dynamics like voltage or current fluctuations.

2.2. Mathematical R-C Network Model for Room Thermodynamic Analysis

Thermal models have been widely utilized for simulating transient temperature dynamics and estimating thermal demands, which is a critical factor influencing the energy consumption of HVAC systems in residential microgrids. Thus, the building’s transient thermal behavior is characterized and modeled as an R-C thermal network in [13,17], where thermal resistance is adopted for mapping the heat transfer paths, while thermal capacitance quantifies heat storage capacitances. Figure 2 illustrates the schematic of the R-C network model used for building thermal analysis.

Unlike conventional building energy models in the literature prioritizing temperature control, the HVAC airflow into building zones is formalized as an additional control variable to realize HVAC load adjustments at small timescales, such as second-level. It should be noted that the developed model systematically integrates occupancy patterns, ambient climate variations, heat exchange, and envelope thermal inertia. Every wall or zone of buildings in residential microgrids is depicted by a node characterized by a thermal potential (i.e., temperature), and the nodes are interconnected through thermal capacitors to the ground and via thermal resistors to their neighboring nodes. In addition, each internal wall node connects to a zone node through two resistances $R_{W_i}$/2 and $R_{\mathrm{i}\mathrm{n}i}$. A peripheral wall node links to an outside air node via two resistances $R_{W_i}$/2 and $R_{\mathrm{o}\mathrm{u}\mathrm{t}i}$. The window is modeled as a pure resistance, denoted as $R_{\mathrm{w}\mathrm{i}\mathrm{n}i}$, which is placed in parallel with the thermal resistance of its adjacent wall. This simplification arises from ignoring the thermal capacitance of the window, which is justifiable due to the insignificant mass of the window mass in comparison to that of the wall. In the building and construction industry, the R-value refers to the relative thermal resistance. The correlation between these thermal resistances and their respective R-values are outlined in Equations (1) and (2). Assuming the R-values remain consistent throughout the same building, the number of independent model parameters that require identification is significantly decreased in our developed R-C network model.

$$R_{Wi}=R_{\mathrm{value}}^{\mathrm{wall}}/A_W^i,R_{\mathrm{in}i}=R_{\mathrm{value}}^{\mathrm{in}}/A_W^i,R_{\mathrm{out}i}=R_{\mathrm{value}}^{\mathrm{out}}/A_W^i$$

(1)

$$R_{\mathrm{win}i}=\left(R_{\mathrm{value}}^{\mathrm{in}}+R_{\mathrm{value}}^{\mathrm{glass}}+R_{\mathrm{value}}^{\mathrm{out}}\right)/A_{\mathrm{win}}^i$$

(2)

where $R_{\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}}^{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}}$, $R_{\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}}^{\mathrm{i}\mathrm{n}}$, $R_{\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}}^{\mathrm{o}\mathrm{u}\mathrm{t}}$, and $R_{\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}}^{\mathrm{g}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s}}$ are the R-value of the wall, internal surface film, external surface film, and glass window, respectively; $A_W^i$ and $A_{\mathrm{w}\mathrm{i}\mathrm{n}}^i$ denote the area of wall i and window area in building room i.

The equation governing the energy balance within the building room is given in (3). The governing equation for the temperature of wall i is shown in (4).

$$C_{Zi}{\displaystyle \frac{d{T}_{Zi}}{dt}}=c_a{\dot{m}}_{Zi}(T_{Si}-T_{Zi})+{\displaystyle \sum _{j\in {\mathbb{N}}_{Zi}}\frac{T_j-T_{Zi}}{R_{i,j}}}+{\dot{Q}}_{\mathrm{int}}^i+n_{\mathrm{win}}^i{\epsilon }_{\mathrm{win}}^i{A}_{\mathrm{win}}^i{Q}_{\mathrm{rad}}^i$$

(3)

$$C_{Wi}{\displaystyle \frac{d{T}_{Wi}}{dt}}={\displaystyle \sum _{j\in {\mathbb{N}}_{Wi}}\frac{T_j-T_{Wi}}{R_{i,j}}}+{\phi }_{\mathrm{W}}^i{\alpha }_{\mathrm{W}}^i{A}_{\mathrm{W}}^i{Q}_{\mathrm{rad}}^i$$

(4)

where $T_{Zi}$, $T_{Wi}$, and $T_j$ represent the temperature of room i, wall i, and node i, respectively. $R_{i,j}$ denotes thermal resistance between node i and node j; $Q_{\mathrm{i}\mathrm{n}\mathrm{t}}^i$ is the internal heat gain from appliances, apparatuses, etc.; $Q_{\mathrm{r}\mathrm{a}\mathrm{d}}^i$ is the density of solar radiative heat flux that is radiated into node i; $n_{win}^i$ represents a binary variable indicating the presence or absence of window i; ${\epsilon }_{win}^i$ is the glass transmissivity of the window; ${\phi }_{\mathrm{W}}^i$ is a binary variable, which equals 1 if wall i is an exterior wall and equals 0 when wall i is an interior wall; ${\alpha }_{\mathrm{W}}^i$ is the absorption coefficient of solar radiation on wall surfaces; $C_{Zi}$ and $C_{Wi}$ are thermal capacitance values of room i and wall i; $c_a$ is the heat capacity of air; ${\dot{m}}_{Zi}$ is the air mass; $T_{Si}$ is the temperature of the HVAC air supply entering room i; and ${\mathbb{N}}_{Zi}$ and ${\mathbb{N}}_{Wi}$ denote the set of neighboring nodes to building room node i and wall node i.

Considering an indoor area of the building depicted in Figure 2, the above thermal energy balance Equations (3) and (4) can be reformulated as the following form:

$$\begin{array}{l}{C}_{Z1}{\displaystyle \frac{d{T}_{Z1}}{dt}}=c_a{\dot{m}}_{Z1}(T_{S1}-T_{Z1})+{\displaystyle \frac{T_{W1}-T_{Z1}}{R_{W1}/2+R_{\mathrm{in}1}}}+{\displaystyle \frac{T_{W2}-T_{Z1}}{R_{W2}/2+R_{\mathrm{in}2}}}\\ +{\displaystyle \frac{T_{W3}-T_{Z1}}{R_{W3}/2+R_{\mathrm{in}3}}}+{\displaystyle \frac{T_{W4}-T_{Z1}}{R_{W4}/2+R_{\mathrm{in}4}}}+{\displaystyle \frac{T_{\mathrm{out}}-T_{Z1}}{R_{\mathrm{win}1}}}+{\epsilon }_{\mathrm{win}}^1{A}_{\mathrm{win}}^1{Q}_{\mathrm{rad}}^1+{\dot{Q}}_{\mathrm{int}}^1\end{array}$$

(5)

$$C_{Wi}{\displaystyle \frac{d{T}_{W1}}{dt}}={\displaystyle \frac{T_{Z_1}-T_{W_1}}{R_{W1}/2+R_{\mathrm{in}1}}}+{\displaystyle \frac{T_{\mathrm{out}}-T_{W_1}}{R_{W1}/2+R_{\mathrm{out}1}}}+{\alpha }_W^1{A}_W^1{Q}_{\mathrm{rad}}^1$$

(6)

To obtain the whole residential microgrid model and decrease the computational efforts, a linearization method in [40] is adopted to linearize heat transfer equations (5) and (6) for each building room and wall. Hence, differential Equations (5) and (6) can be written and transformed into a state space format, and the dynamics of the residential microgrid are linearized around the nearest equilibrium point to the specified operating point. This state space form enables the efficient implementation of real-time predictive control algorithms while preserving thermodynamic consistency across the whole residential microgrid. Subsequently, the linearized state space model is converted into a discrete form for predictive power control, as outlined below:

$$x_{k+1}=A{x}_k+B{u}_k+E{d}_k$$

(7)

$$y_k=C{x}_k$$

(8)

where $x_k$ is represents the state vector indicating the nodal temperatures in the thermal network at time slot k; $y_k$ denotes the output vector encompassing all room temperatures; $u_k$ is the controllable HVAC input vector, and $d_k$ captures the system disturbance and uncontrollable inputs. It should be pointed out that only airflow is treated as the controllable input, while other inputs are classified as disturbance variables. The impact of the above linearization on the model accuracy or stability of the R-C thermal model is detailed in [17] and will not be repeated here.

2.3. Design of Robust and Symmetrical Predictive Temperature Control for HVAC

Next, the design and analysis of robust predictive temperature control (RPTC) for HVAC systems will be presented in detail. First, for (7), the following incremental system can be derived:

$$\delta x_{t+1}=A\delta x_t+B\delta u_t+\delta w_t$$

(9)

where $\delta$ is the incremental operator, i.e., $\delta x_{t+1}=x_t-x_{t+1}$. Note that all subsequent $\delta$ values indicate this meaning. In order to design robust predictive control, the temperature tracking error is given here as

$${\epsilon }_t=y_t-\hat{r}$$

(10)

where $\hat{r}$ is a virtual reference value related to the actual tracking value that needs to be subsequently designed. Further, according to

$$\delta {\epsilon }_t=\delta y_t=C\delta x_t$$

(11)

it can be obtained that

$$\delta {\epsilon }_{t+1}=CA\delta x_t+CB\delta u_t+C\delta w_t$$

(12)

Combining (9) and (12), one gets the dynamic system in the following matrix form:

$$\left[\begin{array}{c}\delta x_{t+1}\\ {\epsilon }_{t+1}\end{array}\right]=\left[\begin{array}{cc}A& 0\\ CA& I_m\end{array}\right]\left[\begin{array}{c}\delta x_t\\ {\epsilon }_t\end{array}\right]+\left[\begin{array}{c}B\\ CB\end{array}\right]\delta u_t+\left[\begin{array}{c}{I}_n\\ C\end{array}\right]\delta w_t$$

(13)

To present the design process in the compact form, define the new state variable as

$${\xi }_t=\left[\begin{array}{c}\delta x_t\\ {\epsilon }_t\end{array}\right]$$

(14)

Furthermore, one has

$${\xi }_{t+1}=\mathcal{A}{\xi }_t+\mathcal{B}\delta u_t+d_t$$

(15)

where

$$\mathcal{A}=\left[\begin{array}{cc}A& 0\\ CA& I_m\end{array}\right],\mathcal{B}=\left[\begin{array}{c}B\\ CB\end{array}\right],{\mathcal{B}}_w=\left[\begin{array}{c}{I}_n\\ C\end{array}\right],d_t={\mathcal{B}}_w\delta w_t$$

(16)

It is well known that for HVAC systems, the nominal system is

$$\begin{array}{l}{\hat{x}}_{t+1}=A{\hat{x}}_t+B{\hat{u}}_t\\ {\hat{y}}_t=C{\hat{x}}_t\end{array}$$

(17)

A nominal system refers to the dynamical model that remains after removing the uncertainty from the original system model. Similarly, for the augmented system (13) of the HVAC system, the nominal system is written as

$${\hat{\xi }}_{t+1}=\mathcal{A}{\hat{\xi }}_t+\mathcal{B}\delta {\hat{u}}_t$$

(18)

We design a RPTC scheme with the following structure:

$$\delta u_t=\delta {\hat{u}}_t+\mathcal{K}({\xi }_t-{\hat{\xi }}_t)$$

(19)

where $\mathcal{K}$ is the control matrix to be designed.

Substituting the above controller into the augmented system, the closed-loop error system for HVAC temperature control can be obtained as

$$({\xi }_{t+1}-{\hat{\xi }}_{t+1})=\mathcal{F}({\xi }_t-{\hat{\xi }}_t)+d_t$$

(20)

where

$$\mathcal{F}=\mathcal{A}+\mathcal{B}$$

In order to design RPTC for HVAC systems, there are three more steps that need to be completed as follows:

(I) Firstly, the system state and the control quantity $(x_t,u_t),({\hat{x}}_t,{\hat{u}}_t)$ need to be expressed in the following form

$$\left[\begin{array}{c}{x}_t\\ u_{t-1}\end{array}\right]=C^{\ast }\left[\begin{array}{c}{\xi }_t\\ {\hat{r}}_t\end{array}\right]+C_w{w}_{t-1},\left[\begin{array}{c}{\hat{x}}_t\\ {\hat{u}}_{t-1}\end{array}\right]=C^{\ast }\left[\begin{array}{c}{\hat{\xi }}_t\\ {\hat{r}}_t\end{array}\right]$$

(21)

where $C^{*}$ and $C_w$ are matrices to be determined. Recalling the state variable (14), one has

$$\left[\begin{array}{ccc}{I}_n& 0& 0\\ 0& I_m& I_m\end{array}\right]\left[\begin{array}{c}{\xi }_t\\ {\hat{r}}_t\end{array}\right]=\left[\begin{array}{c}\delta x_t\\ {\epsilon }_t+{\hat{r}}_t\end{array}\right]=\left[\begin{array}{c}{x}_t-x_{t-1}\\ y_t\end{array}\right]$$

(22)

Due to

$$\left[\begin{array}{c}{x}_t\\ u_{t-1}\end{array}\right]=\left[\begin{array}{cc}A& B\\ 0& I_m\end{array}\right]\left[\begin{array}{c}{x}_{t-1}\\ u_{t-1}\end{array}\right]+\left[\begin{array}{c}{I}_n\\ 0\end{array}\right]w_{t-1}$$

(23)

it can be obtained that

$$\left[\begin{array}{c}{x}_t-x_{t-1}\\ y_t\end{array}\right]=\left[\begin{array}{cc}A-I_n& B\\ CA& CB\end{array}\right]\left[\begin{array}{c}{x}_{t-1}\\ u_{t-1}\end{array}\right]+\left[\begin{array}{c}{I}_n\\ C\end{array}\right]w_{t-1}$$

(24)

which indicates that

$$\left[\begin{array}{c}{x}_{t-1}\\ u_{t-1}\end{array}\right]={\left[\begin{array}{cc}A-I_n& B\\ CA& CB\end{array}\right]}^{-1}\left(\left[\begin{array}{c}{x}_t-x_{t-1}\\ y_t\end{array}\right]-\left[\begin{array}{c}{I}_n\\ C\end{array}\right]w_{t-1}\right)$$

(25)

Substituting the above equation into (23), one can obtain

$$\left[\begin{array}{c}{x}_t\\ u_{t-1}\end{array}\right]=\left[\begin{array}{cc}A& B\\ 0& I_m\end{array}\right]{\left[\begin{array}{cc}A-I_n& B\\ CA& CB\end{array}\right]}^{-1}\left(\left[\begin{array}{c}{x}_t-x_{t-1}\\ y_t\end{array}\right]-\left[\begin{array}{c}{I}_n\\ C\end{array}\right]w_{t-1}\right)+\left[\begin{array}{c}{I}_n\\ 0\end{array}\right]w_{t-1}$$

(26)

Denote

$$\mathsf{\Sigma }=\left[\begin{array}{cc}A-I_n& B\\ CA& CB\end{array}\right]$$

Comparing the forms of (21) and (26), the matrices $C^{*}$ and $C_w$ can be determined to be

$$\begin{array}{l}{C}^{\ast }=\left[\begin{array}{cc}A& B\\ 0& I_m\end{array}\right]{\mathsf{\Sigma }}^{-1}\left[\begin{array}{ccc}{I}_n& 0& 0\\ 0& I_m& I_m\end{array}\right],\\ C_w=\left[\begin{array}{c}{I}_n\\ 0\end{array}\right]-\left[\begin{array}{cc}A& B\\ 0& I_m\end{array}\right]{\mathsf{\Sigma }}^{-1}\left[\begin{array}{c}{I}_n\\ C\end{array}\right]\end{array}$$

(II) Secondly, it needs to be shown that there exists a robust positive definite invariant set, denoted as $\tilde{\mathbb{Z}}$, to which $({\xi }_t-{\hat{\xi }}_t,w_{t-1})$ is guaranteed to converge. According to (21), splitting of $C^{*}$ results in

$$C^{\ast }=\left[\begin{array}{cc}{C}_{\xi }& C_y\end{array}\right]$$

Thus, the following closed-loop dynamic system can be obtained as

$$\left[\begin{array}{c}{x}_t\\ u_{t-1}\end{array}\right]=C^{\ast }\left[\begin{array}{c}{\xi }_t\\ {\hat{r}}_t\end{array}\right]+C_w{w}_{t-1}$$

(27)

$$\begin{array}{c}\hfill \left[\begin{array}{c}{x}_t-{\hat{x}}_t\\ u_{t-1}-{\hat{u}}_{t-1}\end{array}\right]=C^{\ast }\left[\begin{array}{c}{\xi }_t-{\hat{\xi }}_t\\ 0\end{array}\right]+C_w{w}_{t-1}\\ \hfill =C_{\xi }({\xi }_t-{\hat{\xi }}_t)+C_w{w}_{t-1}\\ \hfill =\left[\begin{array}{cc}{C}_{\xi }& C_w\end{array}\right]\left[\begin{array}{c}{\xi }_t-{\hat{\xi }}_t\\ w_{t-1}\end{array}\right]\end{array}$$

(28)

According to (20), it can be obtained as

$$({\xi }_{t+1}-{\hat{\xi }}_{t+1})=\mathcal{F}({\xi }_t-{\hat{\xi }}_t)+{\mathcal{B}}_w{w}_{t+1}-{\mathcal{B}}_w{w}_t$$

(29)

Further, writing $w_t$ and (29) in vector form, one can deduce that

$$\left[\begin{array}{c}{\xi }_{t+1}-{\hat{\xi }}_{t+1}\\ w_{t+1}\end{array}\right]=\left[\begin{array}{cc}\mathcal{F}& -{\mathcal{B}}_w\\ 0& 0\end{array}\right]\left[\begin{array}{c}{\xi }_t-{\hat{\xi }}_t\\ w_t\end{array}\right]+\left[\begin{array}{c}{\mathcal{B}}_w\\ I_n\end{array}\right]w_{t+1}$$

For the sake of presentation, let

$$\tilde{\mathcal{F}}=\left[\begin{array}{cc}\mathcal{F}& -{\mathcal{B}}_w\\ 0& 0\end{array}\right],{\tilde{\mathcal{B}}}_w=\left[\begin{array}{c}{\mathcal{B}}_w\\ I_n\end{array}\right]$$

In this way, (29) can be further written as

$$\left[\begin{array}{c}{\xi }_{t+1}-{\hat{\xi }}_{t+1}\\ w_{t+1}\end{array}\right]=\tilde{\mathcal{F}}\left[\begin{array}{c}{\xi }_t-{\hat{\xi }}_t\\ w_t\end{array}\right]+{\tilde{\mathcal{B}}}_w{w}_{t+1}$$

(30)

Since $\mathcal{F}$ is Schur, so is $\tilde{\mathcal{F}}$. Thus, it can be said that $w_{t+1}\in \mathbb{W}$. Find the minimal invariant set $\tilde{\mathbb{Z}}$ for the above system. In particular, the expression for this minimal invariant set can be given as

$$\tilde{\mathbb{Z}}=\underset{k=0}{\overset{\infty }{\oplus }}{\tilde{\mathcal{F}}}^k{\tilde{\mathcal{B}}}_w\mathbb{W}$$

(31)

Alternatively, it can be calculated by the equation

$$d_t={\mathcal{B}}_w\delta w_t\in {\mathcal{B}}_w(\mathbb{W}\oplus (-\mathbb{W}))$$

(32)

for the system

$$({\xi }_{t+1}-{\hat{\xi }}_{t+1})=\mathcal{F}({\xi }_t-{\hat{\xi }}_t)+d_t,d_t={\mathcal{B}}_w\delta w_t$$

which can be used to compute the invariant set $({\xi }_t-{\hat{\xi }}_t)\in \mathbb{Z}$.

Remark 1. 

The significance of obtaining an invariant set is that when $\left[\begin{array}{c}{\xi }_t-{\hat{\xi }}_t\\ w_t\end{array}\right]\in \tilde{\mathbb{Z}}$, since $w_{t+j}\in \mathbb{W}$ for all $j\ge 0$, there is $\left[\begin{array}{c}{\xi }_{t+j}-{\hat{\xi }}_{t+j}\\ w_{t+j}\end{array}\right]\in \tilde{\mathbb{Z}}$, which indicates that $\left[\begin{array}{c}{\xi }_{t+j}\\ w_{t+j}\end{array}\right]\in \left[\begin{array}{c}{\hat{\xi }}_{t+j}\\ 0\end{array}\right]\oplus \tilde{\mathbb{Z}}$. Eventually, ${\hat{\xi }}_{t+j}$ is going to converge to 0, so the range of ${\xi }_{t+j}$ can be obtained.

(III) Finally, the tightened set ${\hat{\mathbb{X}}}_{\mathbb{U}}$ in which the predicted states and control quantities will be lying needs to be computed from $\tilde{\mathbb{Z}}$ to prove that the constraints on $({\hat{x}}_t,{\hat{u}}_{t-1})$ can be satisfied. Within the invariant set, i.e., when $\left[\begin{array}{c}{\xi }_t-{\hat{\xi }}_t\\ w_t\end{array}\right]\in \tilde{\mathbb{Z}}$, there is

$$\left[\begin{array}{c}{x}_t-{\hat{x}}_t\\ u_{t-1}-{\hat{u}}_{t-1}\end{array}\right]=\left[\begin{array}{cc}{C}_{\xi }& C_w\end{array}\right]\left[\begin{array}{c}{\xi }_t-{\hat{\xi }}_t\\ w_{t-1}\end{array}\right]\in \left[\begin{array}{cc}{C}_{\xi }& C_w\end{array}\right]\tilde{\mathbb{Z}}$$

(33)

which indicates that

$$\left[\begin{array}{c}{x}_t\\ u_{t-1}\end{array}\right]\in \left[\begin{array}{c}{\hat{x}}_t\\ {\hat{u}}_{t-1}\end{array}\right]\oplus \left[\begin{array}{cc}{C}_{\xi }& C_w\end{array}\right]\tilde{\mathbb{Z}}$$

Therefore, it can be ensured that

$$\left[\begin{array}{c}{x}_t\\ u_{t-1}\end{array}\right]\mathbb{X}\times \mathbb{U}$$

There must be

$$\left[\begin{array}{c}{\hat{x}}_t\\ {\hat{u}}_{t-1}\end{array}\right]\in {\hat{\mathbb{X}}}_{\mathbb{U}}\subseteq (\mathbb{X}\times \mathbb{U})-\left[\begin{array}{cc}{C}_{\xi }& C_w\end{array}\right]\tilde{\mathbb{Z}}$$

From $\left[\begin{array}{c}{\hat{x}}_t\\ {\hat{u}}_{t-1}\end{array}\right]=C^{\ast }\left[\begin{array}{c}{\hat{\xi }}_t\\ {\hat{r}}_t\end{array}\right]$, it can be obtained that

$$C^{\ast }\left[\begin{array}{c}{\hat{\xi }}_t\\ {\hat{r}}_t\end{array}\right]\in {\hat{\mathbb{X}}}_{\mathbb{U}}\subseteq (\mathbb{X}\times \mathbb{U})-\left[\begin{array}{cc}{C}_{\xi }& C_w\end{array}\right]\tilde{\mathbb{Z}}$$

It is therefore necessary to calculate a Maximum Output Allowable Set (MOAS), denoted as $\mathbb{O}$, for the following auxiliary system as

$$\left[\begin{array}{c}{\hat{\xi }}_{t+1}\\ {\hat{r}}_{t+1}\end{array}\right]=\left[\begin{array}{cc}\mathcal{F}& 0\\ 0& I_m\end{array}\right]\left[\begin{array}{c}{\hat{\xi }}_t\\ {\hat{r}}_t\end{array}\right]$$

(34)

Suppose that the for MOAS there exists a fitted ${\mathbb{O}}_{\epsilon }$. The definition for ${\mathbb{O}}_{\epsilon }$ is as follows. When $(\hat{\xi },\hat{r})\in {\mathbb{O}}_{\epsilon }\in {\mathbb{R}}^{n+2m}$, for all $t\ge 0$ and $C_y{\hat{r}}_t\in {\hat{\mathbb{X}}}_{\mathbb{U}}(\epsilon )$, there is $C_{\xi }{\mathcal{F}}^t\hat{\xi }+C_y\hat{r}\in {\hat{\mathbb{X}}}_{\mathbb{U}}$, where ${\hat{\mathbb{X}}}_{\mathbb{U}}(\epsilon )$ is a closed tight set that satisfies ${\hat{\mathbb{X}}}_{\mathbb{U}}(\epsilon )\oplus B_{\epsilon }^{n+m}(0)\subseteq {\hat{\mathbb{X}}}_{\mathbb{U}}$. Furthermore, the fitted MOAS and the true MOAS satisfy ${\mathbb{O}}_{\epsilon }\subset \mathbb{O}$. From

$$\left[\begin{array}{c}{\hat{x}}_t\\ {\hat{u}}_{t-1}\end{array}\right]=\left[\begin{array}{cc}{C}_{\xi }& C_y\end{array}\right]\left[\begin{array}{c}{\hat{\xi }}_t\\ {\hat{r}}_t\end{array}\right]$$

(35)

one can deduce that the feasible ${\hat{r}}_t$ is $C_y{\hat{r}}_t\in {\hat{\mathbb{X}}}_{\mathbb{U}}(\epsilon )$. That is, one can find a set for the reference value ${\hat{r}}_t$, and the nominal states and inputs that satisfy the constraint wherein ${\hat{\mathbb{X}}}_{\mathbb{U}}$ exists only when ${\hat{r}}_t$ is within this set. The preparation for RPTC is completed above, and the specific design of the controller is carried out below. Notice that the decision variables in robust temperature predictive control are ${\hat{\xi }}_t,\delta {\hat{u}}_{[t:t+N-1]}$, where ${\hat{\xi }}_t=\left[\begin{array}{c}\delta {\hat{x}}_t\\ {\hat{y}}_t-{\hat{r}}_t\end{array}\right]$ denotes $\delta {\hat{x}}_t$, and ${\hat{y}}_t$ is also a decision variable. Meanwhile, ${\hat{r}}_t$ is also considered as a variable in order to ensure feasibility.

Remark 2. 

The significance of not using a true reference but creating an ${\hat{r}}_t$ is that since the feasibility of the robust temperature predictive control has requirements for r as well, there is a good chance that the true r will not be satisfied, and especially if the tracking error is large in the initial phase, there may be no solution at all.

To sum up, the robust and symmetrical temperature predictive control problem can be attributed to an optimization problem:

$$V_N^{*}(r^o,\delta x_t,y_t)=\underset{\delta {\hat{x}}_t,{\hat{y}}_t,{\hat{r}}_t,\delta {\hat{u}}_{[t:t+N-1]}}{\min }{V}_N(\delta {\hat{x}}_t,{\hat{y}}_t,{\hat{r}}_t,\delta {\hat{u}}_{[t:t+N-1]};r^o,\delta x_t,y_t)$$

(36)

and

$$V_N={‖{\hat{r}}_t-r^o‖}_T^2+{\displaystyle \sum _{k=t}^{t+N-1}({‖{\hat{\xi }}_k‖}_Q^2+{‖\delta {\hat{u}}_k‖}_R^2)}+{‖{\hat{\xi }}_{t+N}‖}_P^2$$

where $T,Q,R$, and $P$ are given in the next subsection, which is subject to the following:

(i)

The dynamics constraint of the HVAC system, i.e., the nominal system of the augmented system:

$${\hat{\xi }}_{t+1}=\mathcal{A}{\hat{\xi }}_t+\mathcal{B}\delta {\hat{u}}_t$$

(ii)

Initial value constraints on the invariant set $\left[\begin{array}{c}{\xi }_t-{\hat{\xi }}_t\\ w_t\end{array}\right]\in \tilde{\mathbb{Z}}$ defined as

$$\left[\begin{array}{c}{I}_{n+m}\\ 0\end{array}\right]({\xi }_t-{\hat{\xi }}_t)\in \left[\begin{array}{c}0\\ -w_{t-1}\end{array}\right]\oplus \tilde{\mathbb{Z}}$$

(iii)

Essentially, it is the state constraints of x and u. In order for x and u of the original system to satisfy the sets X and U, the state constraints of the nominal system of the original system need to satisfy an indentation set, and hence equivalently, the conditions that the state constraints of the nominal system of the augmented system need to satisfy the following:

$$C^{\ast }\left[\begin{array}{c}{\xi }_{t+k}\\ {\hat{r}}_t\end{array}\right]\in {\hat{\mathbb{X}}}_{\mathbb{U}}$$

for each time instant $k=1,\cdots ,N-1$.

(iv)

Terminal constraints that satisfy the MOAS

$$\left[\begin{array}{c}{\hat{\xi }}_{t+N}\\ {\hat{r}}_t\end{array}\right]\in {\mathbb{O}}_{\epsilon }$$

Up to this point, $\delta {\hat{x}}_{t|t},{\hat{y}}_{t|t},{\hat{r}}_{t|t},\delta {\hat{u}}_{[t:t+N-1]|t}$ can be obtained by solving the above optimization problem and obtaining the robust temperature predictive control $\delta {\hat{u}}_t=\delta {\hat{u}}_{t|t}$ as

$${\hat{\xi }}_t=\left[\begin{array}{c}\delta {\hat{x}}_{t|t}\\ {\hat{y}}_{t|t}-{\hat{r}}_{t|t}\end{array}\right]$$

2.4. Parameter Selection Guidelines for Robust and Symmetrical Predictive Temperature Control

The careful selection of tuning parameters plays a pivotal role in establishing the convergence properties of the proposed control algorithm. Specifically, the weighting matrices Q and R are designed to be symmetric and positive definite, ensuring that the optimization problem remains well posed and that the control objectives are properly prioritized. The matrix P, which is central to the stability analysis, is selected as the positive definite solution to the Lyapunov equation

$$\tilde{\mathcal{F}}P\mathcal{F}-P=-(Q+{\mathcal{K}}^T\mathcal{F}\mathcal{K})$$

(37)

This choice guarantees that the control algorithm not only achieves the desired performance but also ensures asymptotic stability of the closed-loop system. The Lyapunov equation provides a rigorous mathematical framework for analyzing the system’s stability, and the positive definiteness of P ensures that the energy function associated with the system decays over time, thereby confirming the convergence of the proposed control strategy. This systematic approach to parameter selection underscores the robustness and reliability of the algorithm in practical applications. For the matrix T in the cost function, decompose $P\in R^{(n+m)\times (n+m)}$ as

$$P=\left[\begin{array}{cc}{P}_{xx}& P_{xy}\\ P_{xy}^T& P_{yy}\end{array}\right]$$

where $P_{xx}\in R^{n\times n}$, and T is chosen to satisfy the inequality

$$T-P_{yy}>0$$

The selection of $T$ ensures optimal closed-loop performance, as presented in [41].

2.5. Consumption Interface Between HVAC System and Microgrid System

In this paper, all of the energy described above for realizing the regulation of the HVAC system is provided by a distributed renewable energy-based microgrid. Utilizing the HVAC system to consume new energy is equivalent to matching the load power provided by the microgrid system to the power consumed by the HVAC system. According to the paper [13], the simplified power model of the HVAC system can be given as follows:

$$\begin{array}{l}{P}_t^f=c_1{\dot{m}}_{f,t}^3+c_2{\dot{m}}_{f,t}^2+c_3{\dot{m}}_{f,t}+c_4\\ P_t^h={\displaystyle \sum _{i=1}^n{c}_a}{\dot{m}}_{Z_i,t}{\displaystyle \frac{T_{S_i,t}-T_{out,t}}{CO{P}_h}}\\ P_t^c={\displaystyle \sum _{i=1}^n{c}_a}{\dot{m}}_{Z_i,t}{\displaystyle \frac{T_{out,t}-T_{S_i,t}}{CO{P}_c}}\end{array}$$

(38)

where $P_r^f$ is the fan power of the HVAC system; $P_t^h$ is the heating coil power of the HVAC system; and $P_t^c$ is the cooling coil power of the HVAC system during the robust and symmetrical predictive temperature regulation process. This yields the total power consumption of the HVAC system as

$$P_{cpl}^{hvac}=P_t^f+P_t^h+P_t^c$$

(39)

In this paper, the above loads are modeled as constant power loads of the microgrid system, as shown in Figure 1. It is worth noting that the microgrid supplies resistive loads in addition to the HVAC system. According to the secondary control of the microgrid, power tracking and voltage balancing of microgrids containing HVAC systems can be realized. The theoretical method proposed in this paper is similar to a class of classical model predictive methods, and the stability of this method has been established in several articles [14,15,16]. Reference [15] is a theoretical research study that provides a profound analysis of the MPC but does not address a practical issue. Both [17,32] solely focus on partial aspects of power scheduling in HVAC systems without investigating the regulation control of microgrid output with the involvement of HVAC. In summary, the research presented in this paper differs from the issues tackled in these references.

3. Results

This section presents the simulation results of the thermodynamics-based robust and symmetrical predictive power control for renewable energy residential microgrids. The parameters of the simulation system are as follows: Rw = 1.659; Rgl = 0.824; Rin = 0.62; Rout = 2.149; Cr1 = 10.673; Cr2 = Cr1; Cr3 = Cr1; Cw1 = 27.07; Cw5 = Cw1; Cw9 = Cw1; Cw2 = 27.30; Cw6 = Cw2; Cw10 = Cw2; Cw3 = 18.95; Cw7 = Cw3; Cw4 = 38.98; and Cw8 = Cw4. The filter parameters for the microgrid were 1 mH for inductance, 1000 uF for the capacitance of the bulk converter, and 100 ohms for both resistive loads. The droop coefficients were [0.4, 0.4, 0.2, 0.2]. The specific simulation environment details were as follows: the model was built using Simulink version 2023a (integrated with Simscape Electrical 23.1.2 toolbox), employing a fixed-step discrete solver to match the 1 ms execution period of the actual controller. The simulation step size was set to 1μs to accurately capture the transient characteristics of SiC MOSFET switches (switching frequency of 100 kHz). Particularly configured in the Model Configuration Parameters were a relative tolerance of 1 × 10⁻⁵, an absolute tolerance of 1 × 10⁻⁷, and the activation of the algebraic loop auto-decomposition feature to ensure the stability of numerical computations. The logical structure of the validation model is shown in Figure 3. As can be seen from the signal flow shown in the figure, the HVAC is modeled as a constant power load, which is the key to the joint simulation of the microgrid subsystem and the HVAC subsystem.

Based on the theoretical analysis and design discussed earlier, the research in this paper is divided into two core components: (1) RPTC for residential building environments and (2) power tracking and voltage balance control for microgrid systems with time-varying constant power loads. Therefore, the simulation results will focus on these two aspects, providing a comprehensive validation of the effectiveness and superiority of the proposed method through quantitative analysis.

3.1. Results of Robust and Symmetrical Predictive Temperature Control for the HVAC System

The results of the RPTC strategy for the HVAC system are illustrated in Figure 4, Figure 5 and Figure 6. Figure 6 demonstrates the effectiveness of temperature regulation in different zones of the HVAC system. As shown in Figure 6, the temperatures in all three zones of the HVAC system quickly reach their desired setpoints. To visually depict the regulation process, Figure 6 presents the variations in the control inputs during the temperature adjustment. Additionally, Figure 6 shows the power consumption curve calculated from (39) during the HVAC regulation process. The temperature reference peaks at minutes 2 and 6 min in Figure 6 reflect abrupt changes in user comfort preferences, likely due to behavior or external events. Given that the HVAC system is modeled as a constant power load, these preference changes lead to step variations in power demand, as shown by the corresponding peaks in Figure 6. These results in Figure 4, Figure 5 and Figure 6 collectively highlight the robustness and efficiency of the proposed control strategy in achieving precise temperature regulation while managing power consumption effectively.

To show the superiority of the proposed robust and symmetrical control method, the temperature regulation results of the conventional PID control are shown in Figure 7 and Figure 8, while those of the proposed method are presented in Figure 4 and Figure 5. In practical scenarios, temperature control systems are inevitably subject to disturbances. As shown in Figure 4 and Figure 5, without a robust and symmetrical control method, such disturbances can lead to the oscillatory behavior observed in Figure 7 and Figure 8. The results in Figure 4 and Figure 5 clearly demonstrate that, compared to results obtained from traditional PID control in Figure 7 and Figure 8, the proposed approach exhibits significant advantages in terms of dynamic response speed, temperature control accuracy, and energy utilization efficiency. Additionally, the simulation analyzes the impact of control parameters on system performance, further verifying the robustness of the proposed method. In addition to the waveform comparisons mentioned above, the quantitative comparisons of the proposed method with the existing PI scheme are also given in

Table 1. Quantification of performance improvements over other methods.

	PI Control	The Proposed RPTC
RMSE	3.12 ± 0.22	2.21 ± 0.14
MAE	2.45 ± 0.15	1.78 ± 0.09
Overshoot	22.4%	9.7%
Settling time	4.2 s	2.8 s

, where MAE and RMSE mean absolute error and root mean squared error, respectively. From the results in the table, it can be seen that the proposed method exhibits certain advantages in terms of regulation accuracy, overshoot, and settling time.

3.2. Results of Power Tracking and Voltage Balance for the Microgrid System

The aforementioned results validate the effectiveness of the proposed RPTC strategy. Next, this section will further verify the feasibility of residential microgrids with high penetrations of renewable energy resources in supplying nonlinear slow time-varying loads such as HVAC systems. To comprehensively demonstrate the simulation results, this paper adopts an islanded microgrid model with four generation units to provide power supply for the HVAC system. Considering the dynamic characteristics of the HVAC system (e.g., nonlinearity, slow time-varying behavior, and fluctuating power demand), a local PI-based secondary controller is designed to achieve precise regulation of the microgrid’s voltage and power output.

In the simulation, the microgrid system is configured to operate under time-varying constant power load conditions, as illustrated in Figure 1. Figure 9 and Figure 10 present the voltage and power output results of the microgrid, respectively. The simulation results demonstrate that, despite the significant time-varying characteristics of the HVAC system’s power demand, the residential microgrid can maintain stable operation and provide efficient power supply to the HVAC system. Specifically, the voltage fluctuations of the microgrid are controlled within acceptable limits, and the power output of each generation unit can rapidly respond to load variations, achieving accurate power distribution and stable voltage support. These results fully prove the effectiveness of the proposed scheme and also indicate the potential of microgrids to independently supply HVAC systems, enabling efficient energy utilization without relying on external grids. Particularly in handling nonlinear loads such as HVAC systems, the proposed method can better adapt to dynamic load changes, ensuring system stability and reliability.

In order to verify the feasibility of the proposed method under multiple microgrid synergies, we have done two simulation studies under residential microgrids. The simulation results are shown in Figure 11 and Figure 12. Based on the voltage and power results, it can be seen that the proposed control method synergizes the energy flow between different microgrids well and ensures that the voltage is within the permissible range.

In summary, the proposed robust and symmetrical control method not only effectively addresses the energy accommodation problem between residential microgrids and HVAC systems but also provides an interesting and feasible technical solution for subsequent related research. Through simulation validation, this paper offers theoretical support and practical references for the coordinated optimization of renewable energy microgrids and building energy systems, demonstrating significant academic value and engineering significance. In the future, further exploration of coordinated control strategies for multiple types of loads (e.g., electric vehicles, energy storage systems, etc.) and microgrids could be conducted to promote the widespread application of smart microgrid technologies. The above simulation results and discussion fully validate the effectiveness and scalability of the proposed method.

4. Conclusions

This paper proposes an RPTC method and a coordinated microgrid control strategy for addressing the critical challenges of temperature control and renewable energy integration in residential microgrids with high renewable energy penetration. By integrating a thermodynamics-based R-C model with a multisource disturbance compensation mechanism, the approach ensures stable indoor temperature regulation under varying external conditions. Real-time airflow modulation enhances the flexibility of HVAC systems, allowing for better accommodation of intermittent renewable energy. The designed power tracking and voltage support secondary controller accurately capture the dynamic power demands of HVAC systems, providing a theoretical foundation for future microgrid controller designs. Unlike existing studies, this work uniquely integrates HVAC systems with microgrid power tracking, offering a novel approach to coordinated control. The proposed method’s effectiveness has been rigorously validated through power electronics simulations with the Simulink/SimPowerSystems toolbox, highlighting its practical relevance and superior performance. This research not only advances the theoretical understanding of HVAC-microgrid coordination but also provides a robust framework for real-world applications in renewable energy integration and energy-efficient building management. Future work will focus on extending the method to applications such as peak load shaving and conducting experimental validation using small-scale physical prototypes.

It would be interesting to further investigate the scalability of the proposed method. In the future, we will explore the possibility of building a small hardware testbed or a hardware-in-the-loop testbed based on RT-LAB to evaluate its efficiency in real-world applications. Focusing on HVAC systems and including smart buildings also includes electric vehicles (EVs); battery energy storage systems (BESSs); smart lighting and other loads; and exploring how different smart loads interact with the proposed microgrid control system is of great significance albeit challenging. We will focus on these topics in the future. Modeling HVAC systems as constant power loads simplifies the dynamics of HVAC systems to a certain extent. However, this simplification is reasonable for controller design in terms of achievability. Following the trajectory of this paper, the accuracy loss resulting from the simplification in modeling HVAC systems poses an intriguing and challenging issue for the future. Moving forward, we will continue to focus on this matter.