Analog Duty Cycle Peak-Shaving Control for Inverter Air Conditioners Considering User Comfort Under Prolonged High Temperatures

Abstract

Current research on the participation of inverter-based air conditioners in demand response often prioritizes system performance during regulation periods yet frequently overlooks the prolonged high indoor temperatures that follow. Furthermore, oversimplified user comfort constraints limit the accurate evaluation of peak-shaving potential. To address these limitations, this paper proposes a novel control framework. First, a differential user comfort evaluation model is established to quantify the adjustable temperature range under varying scenarios. Second, an analog duty cycle grouped rotation control model is developed. By leveraging the variable-frequency characteristics of inverter ACs, this method optimized peak-shaving potential while preventing indoor temperatures from remaining at their upper limits for extended durations. Third, to ensure fairness, a user selection model incorporating a User Impact Factor is introduced as a dynamic ranking criterion for participation priority. Finally, to address the inevitable parameter mismatch in practical engineering, the control strategy is upgraded to a feedforward–feedback closed-loop framework. Simulation results demonstrate the superiority of the proposed ADC strategy over existing methods. Specifically, compared to existing methods, it achieved a 45–50% reduction in the high-temperature influence factor and a 67% decrease in the standard deviation of user impact, indicating significantly improved thermal comfort and fairness. Furthermore, the framework exhibits strong robustness; even under 20% parameter uncertainty, it restricted the duration of temperature exceedance to within 0.8%, strictly outperforming traditional open-loop approaches in preventing user discomfort. These improvements ensure a more uniform distribution of comfort impacts among users, thereby enhancing both the precision and sustainability of demand-side peak shaving.

1. Introduction

The rapid proliferation of distributed renewable energy, particularly wind and solar, has introduced significant volatility to modern power grids [1]. With generation-side regulation reaching its limits, demand-side resources offer a crucial buffer for system stability [2]. In this context, Thermostatically Controlled Loads (TCLs) and specifically inverter-based air conditioners have been widely recognized as a critical flexibility resource. Foundational studies by Hao et al. [3] have mathematically quantified the aggregate flexibility of TCLs, establishing a theoretical basis for their participation in demand response. Complementing this, Vrettos and Andersson [4] demonstrated the efficacy of aggregating commercial building loads for providing secondary frequency control reserves, validating the potential of TCLs in maintaining grid stability. Wu et al. explored the coordination between air conditioning loads and electric vehicles, demonstrating that optimizing these resources under price guidance can effectively mitigate peak grid demands [5]. Focusing specifically on building air conditioning systems, various control architectures have been proposed. For instance, Yuan et al. developed a control strategy for AC clusters based on a cyber-physical system (CPS), which enhances the reliability and response speed of the aggregation [6]. Among these resources, air conditioners (ACs) stand out due to their substantial adjustable potential [7,8,9]. Their ability to respond swiftly to control signals makes them a superior adjustable resource [10]. However, characterizing these loads remains challenging due to their wide geographical distribution, vast quantity, and small individual capacity [11]. Consequently, a reasonable evaluation of the aggregate peak-shaving capacity of air conditioner clusters is a fundamental prerequisite for effective demand response [12].

As a critical factor, user comfort has a fundamental impact on the accurate evaluation of the peak-shaving capacity of air conditioning loads [13]. Hou et al. quantified the temperature impact on thermal comfort via the linearized PMV, which improves control accuracy and mitigates the effects of power rebound. However, the main shortcomings lie in the oversimplification of user’s comfort. The control strategy focused on the symptom management of the power drop rather than the optimization of comfort, precision and rebound trade-offs [14]. Bao et al. proposed a probabilistic temperature-setpoint control strategy to achieve smooth regulation of aggregated ACs. But their work is limited by the assumption of homogeneous parameters and a uniform adjustable temperature range, which does not adequately account for the heterogeneity of ACLs [15]. Li et al. improved load smoothness and reduced user electricity cost by analyzing the diversity of electricity consumption behavior. Unfortunately, it fails to establish a connection with the physical thermal comfort metric that directly governs human perception. The absence of explicit thermal comfort constraints in its optimization model undermines its practical viability [16]. Most existing control strategies rely on static temperature constraints, often overlooking the potential benefits of flexible operation. However, strict adherence to static temperature setpoints limits the flexibility of air conditioning clusters. In recent research, as highlighted in the international review by O’Brien et al. [17], there is a growing consensus towards “occupant-centric” control, which emphasizes that building energy codes should account for the adaptive behaviors and diverse thermal needs of users rather than relying solely on rigid standards.

The International Energy Agency (IEA) has underscored that deploying energy-efficient air conditioning is no longer optional but a prerequisite for sustainable urban growth and grid resilience [18]. In this context, inverter-based air conditioners (IACs) have systematically displaced conventional fixed-frequency units [19], owing to their inherent energy-saving advantages and superior thermal regulation [20,21]. Consequently, research on the DR of inverter-based air conditioner load (IACL) clusters is crucial for enhancing grid flexibility and regulation [22,23,24]. Early research on IACs often modeled them as virtual batteries [9]. While this approach achieved compatibility between IACs and conventional battery storage, it largely neglected user comfort requirements and the inherent flexibility and adjustability of IACs themselves [25]. Yu et al. proposed a control strategy for peak shaving by directly reducing the operating frequency of IACs and discussed the feasibility of large-scale implementation. However, the effect of IACs will be detrimental to optimization on longer future timescales [26]. Zhou et al. proposed an analog on/off state-switching (AOSS) control method that elucidates the mechanism of on–off cycling to utilize building thermal inertia and validated its effectiveness through physical experiments. But the proposed strategy is overly conservative and fails to fully exploit the dispatch potential of IACs [27]. In summary, current approaches to assessing the peak-shaving potential of IAC clusters are problematic: strategies based on a single control event severely affect user comfort due to prolonged load support without sufficient rebound, thereby limiting effective long-term response. Moreover, the recent grouped rotation strategies are too conservative to utilize the full adjustable potential. To unlock this potential, a shift in perspective is required. Challenging the traditional view that comfort requires a strict isothermal state, recent physiological research offers a new perspective for demand response. Parkinson et al. [28] introduced the concept of alliesthesia in built environments, demonstrating that corrective thermal transients can actually generate thermal pleasure and are perceived as more comfortable than static conditions. Leveraging this physiological insight, our method operates within the extended admissive range framework established by Hoyt et al. [29]. Rather than clamping the temperature to a single setpoint, we allow indoor temperatures to oscillate within these pre-defined bounds, thereby utilizing the building’s thermal inertia as a virtual storage buffer. This aligns with the transactive control frameworks proposed by Hao et al. [30], where commercial loads dynamically adjust power profiles across timescales to accommodate grid needs.

By these physiological findings and to address the limitations discussed above, this paper proposes an analog duty cycle (ADC) control method for IACs that incorporates differentiated user comfort for efficient peak shaving. First, a differential user comfort evaluation model is established to calculate individualized optimal temperatures based on metabolic rates. Unlike uniform constraints, this model quantifies permissible adjustment ranges for each user, allowing for a granular and personalized assessment of regulation potential. Second, a novel ADC grouped rotation strategy is developed to refine the traditional AOSS method. By enabling the analog duty cycle operation of IACs, this strategy effectively resolves power rebound during restoration and overcomes the conservatism of existing methods, thereby unlocking the full latent regulation capacity of the cluster. Third, to ensure enhanced long-duration response capability, the framework integrates a “User Impact Factor” to prioritize participation. This mechanism ensures a fair distribution of comfort impacts, enabling sustainable, long-duration demand response (DR) without compromising user satisfaction. Finally, unlike existing studies that assumed perfect knowledge of building thermal parameters, a feedforward–feedback control mechanism is designed to compensate for parameter errors, ensuring that the indoor temperature strictly adheres to user comfort constraints during the demand response period. Simulation results demonstrate that the proposed method can effectively exploit the adjustable potential of air conditioning loads while guaranteeing user comfort, achieving satisfactory DR performance.

2. Difference User Comfort Model

The most prevalent model for assessing user comfort is based on the Predicted Mean Vote (PMV) for thermal comfort calculation [31]. A PMV index of zero indicates that the user feels optimal thermal comfort. The larger the absolute value of the PMV index, the greater the thermal discomfort perceived by the user. The calculation formula for the PMV index I_PMV is formulated as follows:

$$\begin{array}{l}{I}_{\mathrm{PMV}}=[0.303\exp (-0.036M)+0.028]\{M-W-3.05\ast {10}^{-3}[5733-6.99(M-W)-p_{\mathrm{a}}]\\ -0.42\ast (M-W-58.15)-1.7\ast {10}^{-5}M(5867-p_{\mathrm{a}})-0.0014M(34-T_{\mathrm{in}})\\ -3.96\ast {10}^{-8}{f}_{\mathrm{cl}}[{(T_{\mathrm{cl}}+273)}^4-{({\overline{T}}_{\mathrm{r}}+273)}^4]-f_{\mathrm{cl}}{h}_{\mathrm{c}}(T_{\mathrm{cl}}-T_{\mathrm{in}})\}\end{array}$$

(1)

where $M$ represents the metabolic rate. $W$ represents the effective mechanical power. $p_{\mathrm{a}}$ represents the water vapor pressure. $T_{\mathrm{in}}$ represents the indoor air temperature. $f_{\mathrm{cl}}$ represents the clothing surface area factor. $T_{\mathrm{cl}}$ represents the clothing surface temperature. $\overline{T_{\mathrm{r}}}$ represents the mean radiant temperature. $h_{\mathrm{c}}$ represents the convective heat transfer coefficient. The calculations formulated for $p_{\mathrm{a}}$, $f_{\mathrm{cl}}$, $T_{\mathrm{cl}}$, $h_{\mathrm{c}}$ refer to reference [31]. The primary parameters affecting residential thermal comfort are metabolic rate, indoor air temperature, relative humidity and air velocity. For indoor air conditioning users, the influence of metabolic rate and indoor temperature on comfort is significantly greater than that of humidity and air velocity [31]. This paper focused on studying the impact of indoor/outdoor temperature variations and differences in human metabolic rate on air conditioning regulation. The changes in indoor humidity and air velocity are temporarily disregarded. It is assumed that after the air conditioner is activated, indoor humidity and air velocity remain constant. Furthermore, data such as outdoor temperature, humidity and wind speed can be monitored and acquired via smart devices.

The main factors influencing the adjustable temperature range of IACs under different scenarios include differences in the initial temperature setpoint, variations in user thermal tolerance and occupant density. These factors collectively lead to differentiated adjustable temperature ranges across IACs. Without loss of generality, the subsequent analysis in this paper will focus on upward temperature adjustments of IACs for peak shaving in summer. The corresponding case for downward temperature adjustments in winter can be derived analogously and will not be discussed further. The difference between initial temperature setpoints arises because of varying environmental conditions, work activities, and occupant densities which lead to differences in metabolic rates. These lead to different temperature setpoints of IACs. It is assumed that before DR each user’s initial temperature setpoint (ITS) represents their most comfortable temperature. In this situation, the user’s PMV index is equal to zero. By substituting the metabolic rate $M$ and setting $I_{\mathrm{PMV}}=0$ into Equation (1) and then solving for the temperature variable, (ITS) $T_{\mathrm{s}}$ can be derived. The upper comfort boundary is defined to account for differing user sensitivities to temperature variations, as $I_{\mathrm{PMVup}}\in \left[0.4,0.6\right]$. By substituting the PMV comfort constraint and the metabolic rate into Equation (1) and solving for the temperature variable, the upper limit of the IAC temperature setpoint $T_{\mathrm{c}}$ that satisfies user comfort can be obtained. From this, the corresponding upward adjustable temperature (UAT) $\Delta T_c$ for different IACs is derived. Furthermore, the metabolic rate $M$ typically falls within the range of [58, 93] W/m² based on common indoor activities in summer such as sitting, standing, office work, and resting [29]. The absolute value of the adjustable temperature range is discretized with a step size of 0.1 °C by taking the lower bound. The actual adjustable temperature range $\Delta T_c$ that satisfies the PMV constraint can then be derived, as listed in

Table 1. Metabolic rate, ITS and UAT range of different IACs.

Metabolic Rate (W/m2)	ITS (°C)	UAT (°C)
[89, 93]	[22, 22.5]	[1.9, 2.8]
[85, 89]	[22.5, 23]	[1.8, 2.7]
[81, 85]	[23, 23.5]	[1.7, 2.5]
[77, 81]	[23.5, 24]	[1.6, 2.4]
[72, 77]	[24, 24.5]	[1.5, 2.2]
[68, 72]	[24.5, 25]	[1.4, 2.1]
[63, 68]	[25, 25.5]	[1.3, 1.9]
[58, 63]	[25.5, 26]	[1.2, 1.8]

.

The value for other input parameters are formulated as follows: the clothing insulation $I_{\mathrm{cl}}$ ranges from 0.5 to 1.0 clo; the relative humidity $RH$ is 40%; the air velocity $v_{\mathrm{ar}}$ is 0.1 m/s; and the mean radiant temperature $\overline{T_{\mathrm{r}}}$ is defined as the air temperature indoor $T_{\mathrm{in}}$ plus 1.8 °C. It is noted that humidity and air velocity are treated as constants in this study. While this simplifies the model, it is a common practice in residential DR optimization due to the difficulty of real-time measurement for these parameters [32]. As shown in Table 1, the metabolic rate increases stepwise, and the user’s ITS rises from 22 °C to 26 °C, while the mean value of the UAT decreases correspondingly. Specifically, a higher metabolic rate leads to a higher mean UAT range and vice versa. The adjustable temperature range gradually narrows as the initial setpoint increases. In general, the modeling approach to setting a uniformly adjustable upper temperature has limitations. For users with a higher UAT, their regulation potential remains underutilized, leading to an underestimation of the aggregate adjustable capacity. For users with a lower UAT, their thermal comfort may be violated during regulation. Hence, it is necessary to establish a differentiated user comfort model to accurately assess the adjustable temperature potential.

3. IACL Grouped Rotation Control Model and Aggregation Model

3.1. IACL Model

The Equivalent Thermal Parameter (ETP) model, founded on a circuit analogy, is the most widely used framework for single ACLs [7]. The ETP model is employed to simulate the heat transfer process in a room, as shown in following equation:

$$C{\displaystyle \frac{\mathrm{d}{T}_{\mathrm{a}}(t)}{\mathrm{d}t}}={\displaystyle \frac{T_{\mathrm{o}}(t)-T_{\mathrm{a}}(t)}{R}}-Q_{\mathrm{a}}(t)$$

(2)

where $C$ represents the equivalent thermal capacitance of the building in kJ/°C. $T_{\mathrm{a}}(t)$ represents the indoor air temperature at time $t$. $T_{\mathrm{o}}(t)$ represents the outdoor air temperature at time $t$. $Q_{\mathrm{a}}(t)$ represents the cooling capacity of IAC at time $t$. $R$ represents the equivalent thermal resistance of the building in °C/kW. It is demonstrated in references [26,33,34] that the electrical power, cooling capacity, and working frequency of an IAC are approximately linearly correlated, as shown in the following equation:

$$\left\{\begin{array}{l}{P}_{\mathrm{a}}(t)=k_{1}f(t)+l_1\\ Q_{\mathrm{a}}(t)=k_{2}f(t)+l_2\end{array}\right.$$

(3)

where $P_{\mathrm{a}}(t)$ represents the electrical power of IAC at time $t$. $f(t)$ represents the working frequency of IAC at time $t$. $k_1$, $l_1$, $k_2$ and $l_2$ are electrical power coefficients and cooling capacity coefficients, as shown in the following equation:

$$\left\{\begin{array}{l}{k}_1={\displaystyle \frac{P_{\mathrm{a}}^{\mathrm{rate}}-P_{\mathrm{a}}^{\min }}{f^{\mathrm{rate}}-f^{\min }}}, l_1=P_{\mathrm{a}}^{\mathrm{rate}}-k_1{f}^{\mathrm{rate}}\\ k_2={\displaystyle \frac{Q_{\mathrm{a}}^{\mathrm{rate}}-Q_{\mathrm{a}}^{\min }}{f^{\mathrm{rate}}-f^{\min }}}, l_2=Q_{\mathrm{a}}^{\mathrm{rate}}-k_2{f}^{\mathrm{rate}}\end{array}\right.$$

(4)

where $f_{\mathrm{set}}(t)$, $f_{\mathrm{set}}(t)$, $f_{\mathrm{set}}(t)$, $f_{\mathrm{set}}(t)$, $f_{\mathrm{set}}(t)$, and $f_{\mathrm{set}}(t)$ represent the rated parameters of electrical power, cooling capacity and working frequencies for the IAC. The control principle of IAC temperature mode is to adjust the operating frequency of the compressor through the difference between the temperature setpoint and the indoor temperature [26], as shown in following equation:

$$f(t)=\left\{\begin{array}{l}{f}^{\mathrm{rate}}{T}_{\mathrm{a}}-T_{\mathrm{set}}\ge n_{+}\\ f^{\min }+U\ast \Delta f n_{-}<T_{\mathrm{a}}-T_{\mathrm{set}}<n_{+}\\ f^{\min }{T}_{\mathrm{a}}-T_{\mathrm{set}}\ge n_{-}\end{array}\right.$$

(5)

$$U=K(T_{\mathrm{a}}(t)-T_{\mathrm{set}}(t)-n_{-})$$

(6)

where $\Delta f$ represents the difference between the rated and minimum operating frequencies. $n_{+}$ and $n_{-}$ are the upper and lower bounds set in the control strategy, respectively. $U$ and $K$ represent the frequency scaling factor and control coefficient of the IAC, respectively. $T_{\mathrm{set}}(t)$ represents the temperature setpoint at time $t$. This strategy is applicable only under normal grid conditions. In contrast, the working frequency of the IAC is determined exclusively by the external DR control strategy during DR.

3.2. ADC Control for Single IACL

To prevent users from remaining at the upper comfort limit for prolonged periods, it is necessary for IACs to perform periodic cooling storage rebound to regularly lower the indoor temperature. The ADC method defines two distinct operating states for IACs. First, ‘Power Support’ state (PSS) refers to the operational state where the air conditioners actively curtail their power consumption to assist the grid in peak shaving. Second, the ‘Thermal Charging’ state (TCS) describes the phase where air conditioners increase power to lower the indoor temperature, utilizing the building’s thermal inertia to store cold energy as a buffer for the regulation period. The IAC operates at a lower power within the user’s comfort range in the PSS, allowing the indoor temperature to rise from its initial value to the upper comfort limit. The IAC operates at a higher power in the TCS to reduce the indoor temperature from the upper comfort limit back to the initial value. The alternating operation of two states for a single IAC is illustrated in Figure 1.

In these two states, $T_1$, $f_1$, and $P_1$ represent the initial indoor temperature which is equal to ITS, the initial compressor working frequency and electrical power of the IAC, respectively. $T_{\mathrm{c}}$ represents the upper limit of the IAC temperature setpoint. $f_{\mathrm{c}}$ and $P_{\mathrm{c}}$ represent the compressor working frequency and electrical power of the IAC when indoor temperature is maintained at $T_{\mathrm{c}}$, respectively. $T_2$ represents a simulated target steady-state temperature setpoint during the TCS. $f_2$ and $P_2$ represent the compressor working frequency and electrical power of the IAC when indoor temperature is maintained at $T_2$, respectively. In contrast to the initial operating point $f_1$ and $P_1$ of the IAC, the IAC operates at a working frequency of $f_{\mathrm{c}}$ and a corresponding electrical power of $P_{\mathrm{c}}$ during PSS. The indoor temperature gradually increases from $T_1$ to $T_{\mathrm{c}}$ during time $t_1$ to $t_2$, simulating the ‘low’ period within the duty cycle. Then, the IAC operates at another working frequency of $f_2$ and a corresponding electrical power of $P_2$ during the TCS. The indoor temperature gradually decreases from $T_{\mathrm{c}}$ to $T_1$ during time $t_2$ to $t_3$, simulating the ‘high’ period within the duty cycle. The duration from time $t_1$ to $t_2$ and time $t_2$ to $t_3$ is ${\tau }_1$ and ${\tau }_2$, respectively. The interval from $t_1$ to $t_3$ constitutes one complete operating cycle, with the next cycle starting from $t_3$ and the process repeating thereafter. The method involves alternating the IAC’s electrical power between low and high states, a characteristic analogous to the duty cycle operation in fixed-frequency air conditioners. Hence, it is named the ADC method. As described by the concept of “alliesthesia” [28], corrective thermal transients such as the slight cooling during the recovery phase after a temperature rise can generate thermal pleasure, which is absent in static environments. In our proposed ADC method, the indoor temperature naturally oscillates within the permitted comfort bounds as shown in Figure 1. Instead of perceiving this fluctuation as a control deviation, it should be viewed as a beneficial dynamic rhythm. This allows the IACs to temporarily operate outside the strict “optimal” point and utilize the building’s thermal inertia as a buffer, thereby achieving a balance between grid peak-shaving needs and the occupants’ dynamic thermal well-being.

The appropriate electrical power level is achieved by varying the compressor’s working frequency to implement the ‘high-low’ electrical power switching of the IAC [26,27]. It is necessary to calculate the required cooling capacity $Q_{\mathrm{a}}(t)$ when the indoor–outdoor heat exchange reaches equilibrium to ensure the user comfort during DR. Assuming the indoor temperature is $T_{\mathrm{c}}$, the relevant IAC parameters are formulated as follows:

$$\left\{\begin{array}{c}{Q}_a\left(t\right)={\displaystyle \frac{T_o\left(t\right)-T_{\mathrm{c}}}{R}}\\ f\left(t\right)={\displaystyle \frac{Q_a\left(t\right)-l_2}{k_2}}\\ P_a\left(t\right)=k_1\ast f\left(t\right)+l_1\end{array}\right.$$

(7)

where $T_{\mathrm{o}}(t)$ represents the outdoor temperature at time $t$. Once $Q_{\mathrm{a}}(t)$ is obtained, it can be substituted into Equation (5) to back-calculate the corresponding IAC working frequency $f(t)$ and electrical power $P_{\mathrm{a}}(t)$. Thus, the operating electrical power of an IAC can be determined based on the outdoor temperature, room thermal resistance parameters, and target steady-state temperature setpoint. In contrast to the AOSS method where ${\tau }_2$ is equal to ${\tau }_1$ and $T_2$ is equal to $T_1$, ${\tau }_2$ is typically less than ${\tau }_1$ and $T_2$ is more than $T_1$ in the ADC method. During TCS, the IAC electrical power exhibits a rebound, quantified as $P_2-P_1$. Because of the same value for $T_1$ and $T_2$, the rebound has been overlooked in the AOSS method. This simplification inherently underestimates the adjustable potential of IACs. To quantitatively describe this rebound power, a derivation is performed as follows. When the system is in the steady state at time $t_2$, the indoor temperature is formulated as follows:

$$T_{\mathrm{c}}=T_{\mathrm{o}}(t)-Q_{\mathrm{c}}R$$

(8)

where $Q_{\mathrm{c}}$ represents the IAC cooling capacity when the electrical power is $P_{\mathrm{c}}$. Then, the dynamic equation for the indoor temperature transition from time $t_1$ to $t_2$ is obtained by substituting this relation into Equation (3). The dynamic equation for the indoor temperature transition from $T_1$ to $T_{\mathrm{c}}$ is formulated as follows:

$$\left\{\begin{array}{c}{T}_{\mathrm{in}}(t)=T_{\mathrm{c}}-\Delta T_c{\mathrm{e}}^{-t/RC}\\ \Delta T_c=T_{\mathrm{c}}-T_1\end{array}\right.$$

(9)

where $T_{\mathrm{in}}(t)$ represents the dynamic indoor temperature. $\Delta T_c$ represents upward adjustable temperature. From Equation (10), it can be concluded that the indoor temperature will not exceed the user comfort limit during time $t_1$ to $t_2$. During the TCS, it is assumed the target temperature $T_2$ is equal to $T_1$. The dynamic equation describing the temperature change from $T_{\mathrm{c}}$ to $T_1$ is formulated as follows:

$$\left\{\begin{array}{c}{T}_{\mathrm{in}1}(t)=T_{ 1}-\Delta T_{\mathrm{c}1}{\mathrm{e}}^{-t/RC}\\ \Delta T_{\mathrm{c}1}=T_1-T_{\mathrm{c}}\end{array}\right.$$

(10)

where $T_{\mathrm{in}1}(t)$ represents the dynamic indoor temperature. $\Delta T_{c1}$ represents the difference between $T_1$ and $T_{\mathrm{c}}$. During the TCS, assuming the target temperature $T_2$ is less than $T_1$. The dynamic equation describing the temperature change from $T_{\mathrm{c}}$ to $T_1$ is formulated as follows:

$$\left\{\begin{array}{c}{T}_{\mathrm{in}2}(t)=T_{ 2}-\Delta T_{\mathrm{c}2}{\mathrm{e}}^{-t/RC}\\ \Delta T_{\mathrm{c}2}=T_2-T_{\mathrm{c}}\end{array}\right.$$

(11)

where $T_{\mathrm{in}2}(t)$ represents the dynamic indoor temperature. $\Delta T_{c2}$ represents the difference between $T_2$ and $T_{\mathrm{c}}$. The difference between $T_1$ and $T_2$ can be obtained from Equations (11) and (12) when both $T_{\mathrm{in}1}(t)$ and $T_{\mathrm{in}2}(t)$ reach $T_1$. This difference is formulated as follows:

$$\left\{\begin{array}{c}\Delta T_{12}={\displaystyle \frac{{\mathrm{e}}^{-{\tau }_2/RC}-{\mathrm{e}}^{-{\tau }_1/RC}}{1-{\mathrm{e}}^{-{\tau }_2/RC}}}\Delta T_{\mathrm{c}1}\\ \Delta T_{12}=T_2-T_1\end{array}\right.$$

(12)

where $\Delta T_{12}$ represents the difference between $T_2$ and $T_1$. Then, with known values ${\tau }_1$ and ${\tau }_2$, the difference between $T_1$ and $T_2$ can be determined. Subsequently, based on this difference and by applying Equation (8), the rebound power relative to $P_1$ and $P_{\mathrm{c}}$ is formulated as follows:

$$\left\{\begin{array}{c}\Delta P_{12}={\displaystyle \frac{{\mathrm{e}}^{-{\tau }_2/RC}-{\mathrm{e}}^{-{\tau }_1/RC}}{1-{\mathrm{e}}^{-{\tau }_2/RC}}}\Delta P_{c1}\\ \begin{array}{l}\Delta P_{12}=P_2-P_1\\ \Delta P_{c1}=P_1-P_c\end{array}\end{array}\right.$$

(13)

where $\Delta P_{12}$ represents the difference between $P_2$ and $P_1$. $\Delta P_{c1}$ represents the difference between $P_1$ and $P_{\mathrm{c}}$. It can derive the electric power during the PSS and the rebound power during the TCS after determining the values ${\tau }_1$, ${\tau }_2$, $R$ and $C$. Additionally, assuming the target temperature $T_2$ is more than $T_1$ would complicate the model and lead to a more conservative estimate of the total adjustable potential in peak-shaving evaluation. This scenario will not be elaborated further here.

3.3. Group Rotation Strategy for IACLs

Following the development of the single-IAC operational model, the focus shifts to enabling continuous power output from an IAC cluster. The ADC’s grouped rotation control is explained using the example of a 2:1 analog duty cycle. Assuming that ${\tau }_1$ is equal to $2{\tau }_2$ and ${\tau }_2$ is equal to $2RC$, the schematic diagram of the electrical power for IACs is illustrated in Figure 2.

Figure 2 shows a group of three parameter-similar IACLs (IACa, IACb, IACc). The plot depicts their electrical power profiles over a specific period. Through alternating operation, these IACLs collectively provide continuous power support $P_{\mathrm{t}}$ from $t_1$ to $t_7$, which is formulated as follows:

$$P_{\mathrm{t}}=2\ast \Delta P_{c1}-\Delta P_{12}$$

(14)

where $P_{\mathrm{t}}$ represents total power reduction provided by the group of three IACs (IACa, IACb and IACc). However, as noted, the cluster size is not fixed at three but depends on the regulation needs. To generalize this for a cluster of $N$ IACs, the aggregate power reduction depends on the duty cycle ratio ${\tau }_1$:${\tau }_2$. This ratio determines the proportion of units in the Power Support State versus the Thermal Charging State at any given moment. Consequently, the generalized total power reduction $P_{\mathrm{total}}$ for $N$ similar IACs operating under a duty cycle of ${\tau }_1$:${\tau }_2$ is formulated as follows:

$$P_{\mathrm{total}}\approx ({\displaystyle \frac{{\tau }_1\ast \Delta P_{c1}-{\tau }_2\ast \Delta P_{12}}{{\tau }_1+{\tau }_2}})\ast N$$

(15)

where $P_{\mathrm{total}}$ represents the total power reduction provided by the IACL cluster. A subset of IACs is put into the thermal charging state in advance to avert the initial power drop at the beginning of DR. This pre-cooling causes the indoor temperature to fall below $T_1$. According to Equation (11), the indoor temperature will stabilize at $T_2$. Ensure that it does not violate the lower limit of $T_2$ during TCS. Furthermore, based on Equation (12), the model’s cycle is set to 30 min for peak-shaving evaluation and the PSS cycle is less than 24 min. The resultant temperature deviation $\Delta T_{12}$ is significantly smaller than $\Delta T_{c1}$. Therefore, this strategy does not excessively compromise user comfort. Additionally, the target temperature is restored to $T_1$ for all IACs once the DR event ends to mitigate post-DR power rebound. IACLs with significantly different parameters, variations in indoor thermal properties lead to distinct dynamic responses. When subjected to a uniform control period, the temperature trajectories within the IAC cluster diverge into the three curves illustrated in Figure 3.

Figure 3 shows the resulting three representative curves for IAC1, IAC2, and IAC3, whose specific parameters are $R_1$, $C_1$, $R_2$, $C_2$, $R_3$, and $C_3$. The indoor temperature of IAC1 reaches a steady state precisely when the PSS or TCS end, indicating that its thermal parameters are perfectly aligned with the control time. The indoor temperature of IAC2 reaches a steady state before the PSS or TCS end, indicating that its thermal parameters are less than the control time, as $R_2{C}_2<R_1{C}_1$. Based on Equation (12), IAC2 is capable of sustaining power support while ensuring the indoor temperature does not surpass its prescribed limit during the PSS and TCS. The indoor temperature of IAC3 cannot reach a steady state when the PSS or TCS end, indicating that its thermal parameters are more than the control time, as $R_3{C}_3>R_1{C}_1$. IAC3 will adjust without reaching the user’s comfort limit throughout its operation. This allows its regulation to stay comfortably within the permissible thermal range. Consequently, the analog duty cycle period is fixed relative to the dispatch timescale within the ADC method. This strategy accommodates IACs with different thermal parameters to work on the same cycle and ensures all IACs remain within their permissible operating bounds during DR.

4. IACL Potential Evaluation and Optimal Scheduling

4.1. System Architecture for IAC Load Regulation

With the bidirectional communication technology and advanced metering infrastructure of the smart grid, a load aggregator can conveniently acquire IAC user parameters and operational status, enabling real-time remote frequency control. On this basis, this paper constructs an air conditioning load regulation framework, as illustrated in Figure 4.

As illustrated in Figure 4, the process comprises three stages. In the potential evaluation stage, the load aggregator partitions the IAC load cluster into several aggregated groups based on IAC and indoor parameters, and it builds an aggregated power model. Combined with the constraints and methods for evaluation from Section 2 and Section 3, the load aggregator assesses the load curtailment and upward regulation potential and reports these to the control center. In the optimal scheduling stage, the control center dispatches a regulation command, mandating the load aggregator to provide a specified amount of regulation capacity through DR. Based on this requirement, the load aggregator formulates a detailed dispatch plan for each aggregated group through an optimization model. In the user response stage, the load aggregator ranks users according to their impact factor. He selects participating AC users in batches over time and implements remote frequency control to accomplish the load curtailment task following the dispatch plan from the previous stage.

4.2. IAC Potential Evaluation Model Based on ADC Method

The primary determinants of IAC potential include UAT, outdoor temperature, parameters with IAC, building, and the specific regulation strategy. Parameters within an IAC cluster such as building thermal properties and user setpoints are typically heterogeneous. However, the ADC method can be applied to model IACLs’ aggregate power dynamics for clusters with plenty of units, consistent parameters and fixed setpoints. A hierarchical multi-step clustering approach is adopted to effectively group IACLs with similar parameters. The procedure is illustrated in Figure 5.

In Figure 5, parameters for each air conditioner and its associated building are identified through a parameter identification method. These parameters include cooling capacity, electrical power, equivalent thermal capacity, equivalent thermal resistance, and UAT. The initial preliminary clustering is performed based on the rated electrical power and cooling capacity of each IAC to ensure homogeneous characteristics of electrical power. Another clustering is based on the adjustable capacity of the IACs, resulting in final aggregated groups where all IACs share a similar temperature adjustment capacity. Because the operating cycle of an IAC is determined by the rate of indoor temperature change, it is characterized by the temperature variation parameter $RC$ and characteristic temperature difference $QR$. The final clustering stage is performed using the parameters $RC$ and $QR$ to ensure that the resulting IAC clusters exhibit highly similar dynamic response characteristics. Then, the aggregate power of each IAC group is formulated as follows:

$$P_{i,\mathrm{agg}}(t)={\displaystyle \sum _{n=1}^{\mathrm{N}}{P}_{i,n,\mathrm{set}}(t)\approx }{P}_{i,\mathrm{set}}(t)\ast N$$

(16)

where $N$ represents the quantity of IAC in each group. $P_{i,agg}(t)$ represents the aggregate power of the IAC cluster in group $i$. $P_{i,n,\mathrm{set}}(t)$ represents the $n$-th IAC electric power in group $i$. $P_{i,\mathrm{set}}(t)$ represents the initial power of a representative single IAC in group $i$, calculated based on its cluster centroid. Then, the aggregate power of all IAC is formulated as follows:

$$P_{\mathrm{agg}}(t)={\displaystyle \sum _{i=1}^{\mathrm{I}}{P}_{i,\mathrm{agg}}(t)}$$

(17)

where $\mathrm{I}$ represents the quantity of IAC group. $P_{\mathrm{agg}}(t)$ represents the aggregate power of the IAC cluster. The clustering method adopted for the aforementioned grouping is based on the K-means algorithm.

4.3. Optimal Dispatch of IAC Groups Based on ADC Method

The decision variables are selected based on the clustering results from Section 4.2 to validate the feasibility of the proposed method in practical dispatch. For the $i$-th group, these variables are the number of participating air conditioners $n_{\mathrm{c},i}$, the temperature upward quantity $\Delta T_{\mathrm{set},i}$, the duty cycle $D_{\mathrm{c},i}$ and the number of regulation cycles $l$. The minimum duration of each control event is limited to one minute for each aggregation group to ensure user comfort while alleviating the computational and communication burden on the load aggregator. All decision variables are formulated as $\mathrm{L}$ dimensional vectors, which are formulated as follows:

$$\left\{\begin{array}{c}\Delta T_{\mathrm{set},i}=\left[\Delta T_{\mathrm{set},i,1},\Delta T_{\mathrm{set},i,2},\dots ,\Delta T_{\mathrm{set},i,l},\dots ,\Delta T_{\mathrm{set},i,\mathrm{L}}\right]\\ n_{\mathrm{c},i}=\left[n_{\mathrm{c},i,1},n_{\mathrm{c},i,2},\dots ,n_{\mathrm{c},i,l},\dots ,n_{\mathrm{c},i,\mathrm{L}}\right]\\ D_{\mathrm{c},i}=\left[D_{\mathrm{c},i,1},D_{\mathrm{c},i,2},\dots ,D_{\mathrm{c},i,l},\dots ,D_{\mathrm{c},i,\mathrm{L}}\right]\\ P_{\mathrm{set},i}(t)=\left[P_{\mathrm{set},i,1}(t),P_{\mathrm{set},i,2}(t),\dots ,P_{\mathrm{set},i,l}(t),\dots ,P_{\mathrm{set},i,\mathrm{L}}(t)\right]\end{array}\right.$$

(18)

where $\Delta T_{\mathrm{set},i,l}$ represents temperature upward quantity for IAC group $i$ during regulation cycle $l$. $n_{\mathrm{c},i,l}$ represents the number of IAC participants in regulation for IAC group $i$ during regulation cycle $l$. $P_{\mathrm{set},i,l}(t)$ represents the electric power for IAC group $i$ during regulation cycle $l$. $D_{\mathrm{c},i,l}$ represents the duty cycle for IAC group $i$ during regulation cycle $l$, which is formulated as follows:

$$D_{\mathrm{c},i,l}={\displaystyle \frac{{\tau }_1}{{\tau }_1+{\tau }_2}}$$

(19)

$P_{\mathrm{set},i}(t)$ is obtained by optimizing the temperature upward quantity $\Delta T_{\mathrm{set},i}$, the number of IACs participants in regulation $n_{\mathrm{c},i}$ and the duty cycle $D_{\mathrm{c},i,l}$ for each aggregated group. Additionally, the optimization model is designed to minimize the deviation between the actual load curtailment $P_{\mathrm{cut}}(t)$ and the required demand reduction $P_{\mathrm{need}}$. This objective function is formulated as follows:

$$\left\{\begin{array}{c}\min F_1={({\displaystyle \sum _{t=t_{\mathrm{start}}}^{t_{\mathrm{end}}}(P_{\mathrm{cut}}(t)-P_{\mathrm{need}})})}^2\\ P_{\mathrm{cut}}(t)={\displaystyle \sum _{i=1}^I(P_{\mathrm{set},i}(t)-P_{\mathrm{initial},i}(t))}\end{array}\right.$$

(20)

where $\min F_1$ represents the deviation between $P_{\mathrm{cut}}(t)$ and $P_{\mathrm{need}}$. $P_{\mathrm{initial},i}(t)$ represents the initial power for IAC group $i$.The restraint condition is formulated as follows:

$$\left\{\begin{array}{l}\Delta T_{\mathrm{set},i,l}\le \Delta T_{\mathrm{c},i}\\ n_{\mathrm{c},i,l}(t)=\left\{\begin{array}{l}{n}_{\mathrm{c},i,l},t_{\mathrm{s},i,l}\le t\le t_{\mathrm{d},i,l}\\ 0\hspace{1em}\hspace{1em},else\end{array}\right.\\ n_{\mathrm{c},i,l}(t)<N_i\\ \hspace{1em}0.5<D_{\mathrm{c},i,l}<0.8\\ \hspace{1em}{t}_{\mathrm{start}}\le t_{\mathrm{s},i,l}\le t_{\mathrm{d},i,l}\le t_{\mathrm{end}}\\ \hspace{1em} \forall t\in \left[t_{\mathrm{start}},t_{\mathrm{end}}\right],l\in \left[1,\mathrm{L}\right],i\in \left[1,\mathrm{I}\right]\end{array}\right.$$

(21)

where $n_{\mathrm{c},i,l}(t)$ represents the number of IACs participants in regulation for IAC group $i$ during regulation cycle $l$, at time $t$. $N_i$ represents the number of IACs for IAC group $i$. $\Delta T_{\mathrm{c},i}$ represents the UAT for IAC group $i$. $t_{\mathrm{s},i,l}$ and $t_{\mathrm{d},i,l}$ represent the start time and end time of regulation cycle $l$ for IAC group $i$, respectively. $t_{\mathrm{start}}$ and $t_{\mathrm{end}}$ represents the start time and end time of DR, respectively. Additionally, each IAC group needs to restore the target temperature to $T_1$.

After determining the parameters of participating IACs for each aggregated group, a dynamic priority weight, termed the impact factor $\sigma$, is incorporated into each decision cycle to enhance long-term fairness in regulation. This weight is inversely proportional to the IAC’s historical cumulative average impact. Specifically, an IAC that has been less affected in the past is assigned a higher dispatch priority in the current period. The impact factor is formulated as follows:

$${\sigma }_{i,j}={\displaystyle \sum _{l=1}^{l+1}(\frac{\Delta T_{\mathrm{set},i,j,l}\cdot l_{in}}{\Delta T_{\mathrm{c},i}\cdot (l+1)})}\begin{array}{cc}& \end{array}l\in \left[1,\mathrm{L}-1\right]$$

(22)

where $\Delta T_{\mathrm{set},i,j,l}$ represents temperature upward quantity for IAC $j$ in group $i$ during regulation cycle $l$. $\Delta T_{\mathrm{c},i}$ represents the UAT for IAC group $i$. ${\sigma }_{i,j}$ represents the impact factor for IAC $j$ in group $i$. $l_{in}$ represents the participated cycle length. Specifically, during each regulation interval, IACs are compared based on their impact factor. The smaller the impact factor, the higher the priority for an IAC to be selected for regulation.

4.4. Local Controller with Parameter Uncertainty

To simulate real-world robustness issues, the local controller operates based on estimated parameters rather than true parameters. This distinction is critical, as the parameter mismatch introduces disturbances that the control system must reject. The cooling capacity command is generated via a feedforward–feedback logic. The flowchart is illustrated in Figure 6.

$$Q_a\left(t\right)={\displaystyle \frac{T_o\left(t\right)-(T_{\mathrm{set},i,j,l}+\Delta T_{\mathrm{bias}}(t))}{\beta \cdot R_{\mathrm{est},i,j}}}+{\mathrm{K}}_{\mathrm{p}}{e}^{i,j}(t)+{\mathrm{K}}_{\mathrm{i}}{\displaystyle \sum e^{i,j}(t)}$$

(23)

where $\beta$ represents a coefficient adjusting the feedforward gain. ${\mathrm{K}}_{\mathrm{p}}$ and ${\mathrm{K}}_{\mathrm{i}}$ represent the proportional gain and integral gain coefficients of the feedback controller, respectively. $T_{\mathrm{set},i,j,l}$ represents temperature setpoint for IAC $j$ in group $i$ during regulation cycle $l$. $R_{\mathrm{est},i,j}$ represents the equivalent heat capacity for IAC $j$ in group $i$. $e^{i,j}(t)$ represents the local temperature tracking error for IAC $j$ in group $i$ during time $t$. $\Delta T_{\mathrm{bias}}(t)$ represents the dynamically global temperature bias. $e^{i,j}(t)$ and $\Delta T_{\mathrm{bias}}(t)$ are formulated as follows:

$$\left\{\begin{array}{l}{e}^{i,j}(t)=T_{\mathrm{in},i,j}(t)-T_{\mathrm{set},i,j,l}\\ \Delta T_{\mathrm{bias}}(t)={\mathrm{K}}_{\mathrm{g}}(P_{\mathrm{real},i}(t)-P_{\mathrm{set},i}(t))\end{array}\right.$$

(24)

where $T_{\mathrm{in},i,j}(t)$ represents the indoor temperature for IAC $j$ in group $i$ during time $t$. $P_{\mathrm{real},i}(t)$ represents the real electrical power for IAC in group $i$. ${\mathrm{K}}_{\mathrm{g}}$ represent the proportional gain coefficients of the feedback controller. To prevent specific units from being over-utilized, the dispatch logic replaces random selection with a priority-based sorting mechanism. This approach utilizes a Fairness Index, defined by the cumulative usage count $O_{\mathrm{usage},i}$, and a Safety Index $M_i(t)$, which is formulated as follows:

$$M_i(t)=T_{\mathrm{c},i,j}-T_{\mathrm{in},i,j}(t)$$

(25)

where $T_{\mathrm{c},i,j}$ represents the upward adjustable temperature for IAC $j$ in group $i$. The units are classified into a safe set and an unsafe set, which are formulated as follows:

$$\left\{\begin{array}{l}{\Omega }_{\mathrm{safe}}=\left\{i|M_i(t)>0.3\right\}\\ {\Omega }_{\mathrm{unsafe}}=\left\{i|M_i(t)\le 0.3\right\}\end{array}\right.$$

(26)

where the sorting rules are then applied as follows: units in ${\Omega }_{\mathrm{safe}}$ are sorted in ascending order of $O_{\mathrm{usage},i}$ to prioritize users with less prior participation, while units in ${\Omega }_{\mathrm{unsafe}}$ are sorted in descending order of $M_i(t)$ to utilize those with the largest safety margins as a backup. Finally, the top $O_{\mathrm{usage},i}$ units from the sorted list are activated for regulation.

To mitigate power surges at the onset of demand response $t_{\mathrm{start}}$, a buffer period $T_{\mathrm{buf}}$ is introduced. During this interval, the control signals are modulated by a sinusoidal S-curve, which is formulated as follows:

$$S(t)=\sin ({\displaystyle \frac{\pi }{2}}\cdot {\displaystyle \frac{t-(t_{\mathrm{start}}-T_{\mathrm{buf}})}{T_{\mathrm{buf}}}})\begin{array}{cc}& \end{array}t\in [t_{\mathrm{start}}-T_{\mathrm{buf}},t_{\mathrm{start}}]$$

(27)

The target reduction and setpoint shifts are scaled by $S(t)$, ensuring a smooth transition from zero to the nominal target, thereby utilizing the building’s thermal inertia to dampen grid impact.

5. Results

Assume a regional case comprising residential and office buildings with a total of 10,000 IACLs available for potential evaluation. Based on pre-collected user data, the potential evolution of the IAC cluster is conducted. It is important to note that this study assumes all aggregated users have pre-authorized participation for the entire duration of the demand response event. This assumption allows this study to focus on the validation of the dispatch precision and fairness mechanisms under defined capacity. To validate the effectiveness of the proposed method in preserving user comfort and to quantify the proportion of time users are exposed to high-temperature conditions, this paper employs the mean comfort factor and the max comfort factor as the evaluation metrics, which are formulated as follows:

$$\left\{\begin{array}{c}{\sigma }_{\mathrm{mean}}={\displaystyle \frac{{\displaystyle {\int }_{t_{\mathrm{start}}}^{t_{\mathrm{end}}}{I}_{\mathrm{PMV}}(t)dt}}{t_{\mathrm{start}}-t_{\mathrm{end}}}}\\ {\sigma }_{\max }=\max (I_{\mathrm{PMV}}(t))\end{array}\right.$$

(28)

where $I_{\mathrm{PMV}}(t)$ represents the PMV index for user at time $t$. ${\sigma }_{\mathrm{mean}}$ and ${\sigma }_{\max }$ represent the average value and max value of PMV index during the scheduling, respectively. To quantify the impact of temperature regulation on user comfort, a high-temperature influence factor ${\sigma }_{\mathrm{highmean}}$ is defined, which is formulated as follows:

$$\left\{\begin{array}{l}{\sigma }_{\mathrm{high},i,j}={\displaystyle \sum _{t=1}^Z{\left(\frac{T_{\mathrm{in},i,j}(t)-T_{1,i,j}}{\Delta T_{\mathrm{c},i}}\right)}^2}\\ {\sigma }_{\mathrm{highmean}}={\displaystyle \frac{{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{j=1}^{N_i}{\sigma }_{\mathrm{high},i,j}}}}{Z\cdot {\displaystyle \sum _{i=1}^I{N}_i}}}\end{array}\right.$$

(29)

where ${\sigma }_{\mathrm{high},i,j}$ represents the high-temperature influence for IAC $j$ in group $i$. $T_{\mathrm{in},i,j}(t)$ represents the indoor temperature for IAC $j$ in group $i$. $T_{1,i,j}$ represents the initial temperature for IAC $j$ in group $i$. $Z$ and $I$ represent the discrete simulation time and the number of IAC groups, respectively. We adopted this definition to strengthen the experimental validation for two reasons. First, the squared term amplifies the penalty for large deviations. This encourages the controller to exploit thermal inertia while preventing temperatures from lingering near critical bounds. Second, normalizing the deviation by the limit creates a dimensionless standard. This allows us to compare comfort loss fairly across users with different capacities, directly supporting the validation of our intra-group fairness strategy. The air conditioner and simulation parameters are listed in

Table 2. Simulation parameters of IAC.

IAC Type	Proportion (%)	PAC Rate (kW)	PAC Min (kW)	QAC Rate (kW)	QAC Min (kW)	Room Area (m2)	Normal Distribution Parameters
							Mean Value	Variance
1 HP	17.96	0.64	0.075	2.3	0.15	10–16	13	0.8
1.5 HP	37.45	0.86	0.1	3.5	0.2	14–22	18	0.6
2 HP	17.01	1.3	0.19	5.12	0.5	20–31	25	1.6
3 HP	27.58	2.1	0.25	7.2	0.9	28–45	35	1.6

.

5.1. IAC Cluster Potential Evaluation

To validate the effectiveness of incorporating differentiated user comfort in assessing the potential of IAC clusters, this study compares the adjustable capacities of the cluster under the following four methods. The meteorological data for the simulation, including outdoor temperature and solar radiation, were adopted from the work of Zhou [27]. The selected data represent a typical high-temperature summer day in Chengdu, China, with the ambient temperature peaking at 37 °C, ensuring a rigorous evaluation of the system’s peak-shaving performance. The aggregated power curve of the IAC cluster without control, the outdoor temperature and deviation value of potential evaluation are illustrated in Figure 7. The experimental results are illustrated in Figure 8. The methods are illustrated as follows:

Method 1 uses the AOSS method for evaluation, incorporating user comfort differentiation. Method 2 uses the ADC method for evaluation, considering user comfort but not its heterogeneity. Method 3 uses the ADC method for evaluation, incorporating user comfort differentiation.

As shown in Figure 7a, the aggregated power of the IAC cluster increases with rising outdoor temperature. As shown in Figure 7b, the deviation in the air conditioning potential evaluation remains within 1%, demonstrating excellent performance that meets the required control accuracy. This minor discrepancy primarily stems from inherent clustering errors and insufficient unit counts in certain groups, leading to capacity shortfalls.

Figure 8 compares the lower limit of IACL aggregated power for Method 1, Method 2, and Method 3. The lower bound of the aggregated power in Methods 1 and 2 is higher than that in Method 3, indicating that Methods 1 and 2 underestimate the adjustable potential of IACs. Incorporating differentiated user comfort further unlocks this potential compared to using a uniform comfort assumption. This is particularly prominent for IACLs with higher initial temperature setpoints because the limited adjustable range of Method 2 would lead to a significant underestimation of the cluster’s overall regulation capacity. Furthermore, the proposed ADC method, by setting an analog duty cycle within the IAC operating period, demonstrates a clear advantage over the AOSS method in exploiting peak-shaving capability.

It is acknowledged that user participation in DR inevitably affects thermal comfort. A bigger analog duty cycle $D_{\mathrm{c},i}$ enables deeper exploitation of IAC potential but results in a higher average impact on user’s comfort. During potential evaluation, as the duty cycle gradually increases, a proportion of users’ indoor temperature reaches the steady state before the end of the PSS increases, leading to a rise in average comfort impact. If the duty cycle continues to increase, the model degenerates to the method described in reference [26], where IACs remain in a continuous PSS without a TCS. Conversely, as the duty cycle decreases, the proportion of users not reaching a steady state by the end of the support period increases, and the average comfort impact decreases. If the duty cycle decreases to 1:1, the model degenerates to the AOSS method. The ADC method balances adjustable potential and user comfort. It achieves significant potential exploitation while enabling timely callback to mitigate user impact. A detailed comparative analysis of comfort performance is presented in Section 5.2.

5.2. Subsection

To validate the effectiveness of the proposed method in optimal scheduling, this section conducts a comparative case study using a downward regulation scenario. The performance of the IAC cluster under the ADC method is evaluated against other methods, focusing on aggregate power tracking deviation and the impact on user comfort. The scenario assumes the control center requires a sustained load reduction of 200 kW, 400 kW and 600 kW from IACLs in the region during the peak temperature period 12:00–15:00. During this peak-shaving event, the cluster is regulated using Method 1, the proposed Method 3, and the method in reference [26], which named Method 4, respectively. The results of power tracking deviations are summarized in

Table 3. Power reduction deviations for Methods 1, 3 and 4.

Except Reduction Power (kW)	Deviations (kW)
	Method 1	Method 3	Method 4
200	2.27	1.76	0.78
400	3.24	3.08	1.54
600	/	3.98	2.58

. The expected load curtailment curve and the actual load curtailment curve are illustrated in Figure 8.

As shown in Figure 9 and Table 3, Method 1 could not participate in the 600 kW load curtailment task due to insufficient adjustable potential, and thus its tracking deviation could not be quantified. Among the applicable methods, the accuracy with the preset target power curve is ranked from highest to lowest as follows: Method 4, Method 3, and Method 1. The superior accuracy of Method 4 stems from its simpler regulation strategy, which enables more precise tracking of the target trajectory. The advantage of Method 3 lies in its more favorable consideration of user comfort. Nevertheless, all methods maintained relatively small deviations during the regulation process. The mean value of all IACLs’ high-temperature influence factor is shown in

Table 4. The mean value of IACLs’ high-temperature influence factors for Method 1, 3 and 4.

Except Reduction Power (kW)	Mean Value of High-Temperature Influence Factor
	Method 1	Method 3	Method 4
200	0.4082	0.2896	0.4633
400	0.5319	0.2897	0.5845
600	/	0.2903	0.7451

. The impact on user comfort is illustrated in Figure 10, which shows the average and maximum comfort factors.

As shown in Table 4, when the reduction power is 400 kW, the high-temperature influence factor of Method 3 is reduced by 45% and 50%, respectively, compared with Method 1 and Method 4. Methods 1 and 4 exhibit larger high-temperature influence factors, though for distinct reasons. Method 1 operates near its upper potential limit, necessitating a significant increase in the IAC temperature setpoint, which makes the indoor temperature more prone to reaching high levels. Method 4 maintains the IAC at low electrical power for extended periods without reversion, causing the indoor temperature to remain high for a prolonged duration. This shows that the ADC method can balance the temperature setpoint and reversion time, significantly reducing the user’s time at high temperatures.

As shown in Figure 10, the elements of the violin diagram are defined as follows: The outer shape represents the kernel density estimation of the data distribution. Wider sections indicate higher frequency. Inside each violin, the central white dot denotes the median value; the thick black bar indicates the interquartile range (IQR), spanning from the 25th to the 75th percentiles; and the thin vertical line represents the rest of the distribution range, extending to the adjacent values (1.5 IQR). The distributions of the average comfort factor are ranked from best to worst as Method 4, Method 3, and Method 1. Method 4 exhibits higher maximum and mean values of the average comfort factor, indicating that some IACs remained in the PSS for extended periods without timely callback. Although user comfort limits were not exceeded, this resulted in users being subjected to the upper temperature adjustment limit for prolonged durations, which is unfavorable for long-term DR participation. In contrast, Methods 1 and 3 show lower maximum and mean values of the average comfort factor. It demonstrates that timely thermal restoration after IAC participation can reduce the time users spend at the upper temperature limit, thereby improving overall comfort. As shown in Figure 10b, for Methods 1, 3, and 4, there are cases where the maximum comfort factor is zero, indicating that some air conditioners did not participate in regulation. This highlights the need to introduce an impact factor to enhance long-term regulatory fairness. It set to prevent situations where some IACs are dispatched repeatedly while others participate minimally or not at all. The mean value of the standard deviation for final impact with the IACL groups is illustrated in

Table 5. The standard deviation of final impact.

Except Reduction Power (kW)	Standard Deviation
	with Impact Factor	Without Impact Factor
200	5.46	18.09
400	5.79	18.20
600	9.52	29.06

.

As shown in Table 5, final impact refers to the final value of the average impact factor. As power curtailment increases, the standard deviation rises progressively. This trend is driven by larger temperature adjustments, which widen the disparity in the impact on individual AC units. Furthermore, incorporating the impact factor reduces the mean standard deviation by 67% compared to without impact factor. Without this factor, priority ranking causes some IACs to remain in DR for extended periods while others remain idle, severely compromising the fairness of the regulation among users. This concentration reduces the disparity in the level of impact experienced by IACs. Concurrently, this method effectively address the issue where some IACs previously avoided participation. This enhancement ensures greater fairness in user engagement and improves the practicality of the control strategy. Furthermore, during regulation, some users are not subjected to prolonged periods at the upper comfort limit, which makes the strategy more suitable and sustainable for implementation under high outdoor temperature conditions.

5.3. Model Robustness Analysis

To validate the robustness of the proposed strategy under model uncertainty, comparative experiments were conducted introducing parameter estimation errors ranging from 0% to 20%. The proposed closed-loop strategy was benchmarked against a traditional open-loop control method (Direct Load Control). The primary objective was to evaluate the system’s ability to maintain high-precision grid power tracking while simultaneously safeguarding user thermal comfort under varying degrees of model mismatch. A critical disparity between the two methods is revealed when analyzing user comfort, as detailed in

Table 6. Mean violation rate, violation number and deviation of 20% errors for open-loop and proposed methods.

Except Reduction Power (kW)	Time Violation Rate (%)		Number Violation Rate (%)		Deviations (kW)
	Open-Loop	Proposed Method	Open-Loop	Proposed Method	Open-Loop	Proposed Method
200	16.0	0.2	35.09	0.98	5.746	6.426
400	20.4	0.3	41.23	2.60	11.281	11.214
600	25.6	0.8	44.41	3.41	15.757	15.366

.

As shown in Table 6, both strategies maintain the tracking deviations at a lower level regardless of the parameter mismatch. This indicates that the proposed strategy successfully integrates complex fairness constraints without compromising response precision. The introduction of the feedback mechanism and fairness dispatch does not degrade the aggregate tracking performance, demonstrating the method’s capability to meet strict grid regulation requirements. The time violation rate represents the total violation time of all air conditioners divided by the total adjustment time. The number violation rate represents the number of IAC exceeding the limit divided by the total number of IACs. In the open-loop mode, the comfort violation rate and violation number surge significantly as the parameter error increases, reaching 25.6% and 4441, respectively, at a 20% mismatch level. This degradation occurs because the open-loop controller, lacking real-time state awareness, fails to account for model deviations. Consequently, it erroneously prolongs the off-time of units with low thermal inertia, which have smaller actual values, causing their indoor temperatures to drift beyond acceptable limits. This result underscores that the “high precision” of open-loop control is achieved at the expense of user experience. In contrast, although some air conditioners still exceed the limit, the violation number has been greatly reduced tenfold compared to open-loop control, with the highest rate of limit violations being only 3.4%, and the duration of these violations not exceeding 0.8% of the total control time. The proposed method maintains the violation rate at an exceptionally low level, even under severe parameter uncertainty.

6. Conclusions

Starting from the objectives of reducing the duration of high indoor temperatures and finely exploiting the peak-shaving potential within IAC operating periods, this paper proposes an ADC method that comprehensively accounts for differentiated user comfort. The main conclusions of this study are as follows:

(1) To address the issue in existing methods where prolonged IAC participation in DR leads to sustained high indoor temperatures, this paper proposes the ADC method. This approach represents an advancement over AOSS control and fills a gap in the existing research on frequency control for IACs. Through grouped rotation control, it establishes a continuous and sustainable peak-shaving capacity without excessively compromising user comfort. Compared to the AOSS method, the ADC method demonstrates significantly greater adjustable potential and better facilitates long-duration user participation in demand response under high-temperature conditions.

(2) IACLs are classified according to their UAT. Based on IACL parameters and the varying tolerance levels of different users, the allowable temperature adjustment range for each category is determined. Regulating according to these differentiated adjustment ranges effectively prevents the thermal comfort index $I_{\mathrm{PMV}}$ from exceeding the predefined limit. Compared to existing methods, the proposed method significantly reduces the duration of user exposure to high temperatures, lowering the high-temperature influence factor by 45–50%. The proposed model aligns more closely with real conditions, thereby enabling a more accurate evaluation and utilization of the IACs’ regulation potential.

(3) The proposed impact factor has demonstrated considerable necessity within the rotation control method. Compared to existing methods, the proposed method lowers the standard deviation of user impact by 67%. Its implementation leads to a marked improvement in thermal comfort conditions compared to the scenario without it, resulting in a more uniform distribution of user comfort. This enhancement significantly promotes fairness among users during the regulation process.

(4) The proposed framework greatly enhances system robustness for practical engineering implementation. The results demonstrate that the proposed local controller effectively mitigates user comfort violations under parameter uncertainty. Specifically, the total violation time is limited to merely 0.8% of the duration, and the proportion of air conditioning units experiencing violations is drastically reduced from 44% to 3.4%.

The proposed method employs a group rotation control strategy that fully exploits the regulation potential of IACLs while accounting for differentiated user comfort and ensuring fairness during regulation events. However, this study primarily focuses on maximizing long-term potential excavation and does not fully address the cost considerations inherent in practical dispatch operations. Moreover, we acknowledge that in practical applications, user willingness to participate is often time-varying and stochastic. The proposed hierarchical framework is designed with the flexibility to address this by integrating a time-varying availability constraint vector into the dispatch queue. Furthermore, the research is centered on day-ahead peak shaving services, and the feasibility of applying this method to other DR services, such as frequency regulation, remains unexplored. Future work will focus on incorporating these dynamic user availability profiles to conduct a comprehensive potential evaluation, assessing how stochastic user participation impacts the aggregate regulation capacity and system reliability and concentrating on integrating cost optimization and extending the method’s application to frequency regulation and other ancillary services.