Fixed Frequency Air Conditioners

• Fixed Frequency Air Conditioners (FFACs) are air-conditioning systems with fixed-speed compressors operating in a binary ON/OFF mode, crucial for modeling energy demand and control.
• Research employs thermal dynamics, hysteresis-based thermostat logic, and stochastic Markov models to capture cycling behavior, startup transients, and aggregate load flexibility.
• Studies illustrate FFACs’ dual role in demand-response services and even urban-scale moisture management, highlighting their impact on grid stability and atmospheric interactions.

Searching arXiv for recent and foundational papers on fixed-frequency air conditioners and related control/modeling topics. Using the arXiv search tool to verify and supplement the cited papers. Fixed Frequency Air Conditioners (FFACs) are air-conditioning systems whose compressor operates at fixed speed and whose electrical demand is therefore naturally represented as a binary switching process: ON, drawing rated power $P$, or OFF, drawing zero power. In the research literature summarized here, FFACs appear as residential air-to-air heat pumps, window-mount units, and broader air-conditioning loads within the class of thermostatically controlled loads (TCLs). Their significance lies in the conjunction of building thermodynamics, thermostat hysteresis, cycling constraints, and aggregate controllability: the same device class is studied as a building-energy component, as a demand-response resource for ancillary services and peak shaving, and, at urban scale, as a distributed dehumidification infrastructure with possible mesoscale weather impacts [(Garde et al., 2012); (Nandanoori et al., 2018); (Hiruma et al., 2019)].

1. Device class, thermal dynamics, and thermostat logic

In device-level models, FFACs are treated as switching loads governed by thermostat or hysteresis logic. A commonly used continuous-time thermal model for residential air-conditioners is

$$\dot{T}(t) = -\frac{(T(t) - T_a)}{C R} - \frac{\eta p(t)}{C},$$

where $T(t)$ is indoor temperature, $T_a$ is outside temperature, $C$ is thermal capacitance, $R$ is thermal resistance, $\eta$ is an efficiency parameter, and $p(t) \in \{0,P\}$ is the power state. The associated switching law is

$$p(t^+) = \begin{cases} 0, & \text{if } T(t) \leq T_{set} - \delta T/2 \ P, & \text{if } T(t) \geq T_{set} + \delta T/2 \ p(t), & \text{otherwise}, \end{cases}$$

with $T_{set}$ denoting the setpoint and $$\dot{T}(t) = -\frac{(T(t) - T_a)}{C R} - \frac{\eta p(t)}{C},$$0 the deadband (Nandanoori et al., 2018).

An equivalent formulation appears in control-oriented TCL models, where temperature $$\dot{T}(t) = -\frac{(T(t) - T_a)}{C R} - \frac{\eta p(t)}{C},$$1 and mode $$\dot{T}(t) = -\frac{(T(t) - T_a)}{C R} - \frac{\eta p(t)}{C},$$2 evolve according to

$$\dot{T}(t) = -\frac{(T(t) - T_a)}{C R} - \frac{\eta p(t)}{C},$$3

with mode changes determined by thermostat limits $$\dot{T}(t) = -\frac{(T(t) - T_a)}{C R} - \frac{\eta p(t)}{C},$$4 and $$\dot{T}(t) = -\frac{(T(t) - T_a)}{C R} - \frac{\eta p(t)}{C},$$5. A stochastic extension introduces diffusion through

$$\dot{T}(t) = -\frac{(T(t) - T_a)}{C R} - \frac{\eta p(t)}{C},$$6

so that exogenous randomness and heterogeneity are incorporated directly into ensemble models (Coffman et al., 2020).

These formulations establish the characteristic structure of FFAC dynamics. Electrical flexibility arises from discrete compressor switching, while service feasibility is constrained by thermal inertia, comfort bounds, and the deadband-mediated cycling process. A plausible implication is that any grid-service strategy for FFACs must reconcile sub-device binary actuation with population-level smoothing.

2. Dynamic modeling and experimental validation

A foundational modeling study distinguishes three levels of representation for residential air-to-air heat pumps suitable for FFACs: Model nº0, a simplified ideal-control model; Model nº1, a dynamic on/off model with a $$\dot{T}(t) = -\frac{(T(t) - T_a)}{C R} - \frac{\eta p(t)}{C},$$7 min time step and dead band; and Model nº2, a quasi-steady-state model with performance maps, coil bypass-factor calculations, and the same transient startup representation as Model nº1 (Garde et al., 2012).

Model	Main assumptions	Use and limits
Model nº0	Ideal control loop, constant COP, hourly cooling loads	Quick sizing; not suitable for transient or cycling analysis
Model nº1	On/off control, $$\dot{T}(t) = -\frac{(T(t) - T_a)}{C R} - \frac{\eta p(t)}{C},$$8 min time step, dead band, constant power when compressor is ON	Captures cycling and controller behavior
Model nº2	Performance maps versus outdoor temperature and entering air condition, bypass-factor coil model, startup transient	More accurate power and coil-process representation

The transient startup of the cooling capacity is modeled as a first-order lag,

$$\dot{T}(t) = -\frac{(T(t) - T_a)}{C R} - \frac{\eta p(t)}{C},$$9

with experimentally determined $T(t)$0 min. The same study reports that no significant power surge at compressor start was detected experimentally, supporting the constant-power assumption during ON periods for the tested FFAC. The experimentally observed controller dead band was $T(t)$1, and dynamic models with $T(t)$2 min resolution substantially improved agreement with measurements. Indoor air temperature matched within $T(t)$3 over daily cycles, and the best model matched electric consumption within $T(t)$4 of measured value (Garde et al., 2012).

The detailed performance model further expresses total and sensible cooling capacities as linear functions of entering air enthalpy, with coefficients dependent on outdoor temperature, and represents electrical power as

$T(t)$5

with fitted coefficients reported as $T(t)$6, $T(t)$7, and $T(t)$8 for the tested unit (Garde et al., 2012). This hierarchy of models formalizes a recurrent conclusion in FFAC research: short-timestep simulation is essential when on/off control and startup dynamics materially affect energy use and cycling losses.

3. Ensemble abstractions and control-oriented population models

When FFACs are deployed as an aggregate resource, device-level hybrid dynamics are lifted to probabilistic or distributional representations. One control-oriented framework models the ensemble through coupled Fokker-Planck equations for the ON and OFF subpopulations and discretizes them in temperature and time using a finite volume method. The resulting population model takes the Markov form

$T(t)$9

and admits the factorization

$T_a$0

Here, $T_a$1 represents the temperature evolution induced by device physics and weather, while $T_a$2 represents the balancing authority’s randomized switching policy. Aggregate power is extracted through

$T_a$3

A central result is the conditional-probability factorization

$T_a$4

which separates the effects of weather and control in the discrete ensemble dynamics (Coffman et al., 2020).

A later FFAC-specific study adopts a different stochastic abstraction: each unit is a discrete-time Markov process with binary state $T_a$5 and transition matrix

$T_a$6

where the transition probabilities depend only on indoor temperature, user-set temperature, and outdoor temperature. The transition functions are parameterized by sigmoid or logistic functions, and aggregate power is obtained by Monte Carlo sampling over a heterogeneous population:

$T_a$7

In that framework, the indoor-temperature update is written as

$T_a$8

This suggests that the FFAC literature has converged on two complementary ensemble views: density evolution over temperature-mode bins for control synthesis, and sampled binary-state Markov models for optimization and profit analysis (He et al., 14 Aug 2025).

4. Frequency response and ancillary-service provision

A prominent line of work treats FFACs as distributed resources for frequency response. In a hierarchical scheme for ensembles of switching loads, an aggregator assigns each committed device a frequency threshold $T_a$9 at the start of each control window, typically $C$0–$C$1 minutes. For under-frequency response, device $C$2 autonomously turns OFF when local frequency drops below its assigned threshold, provided that doing so does not violate local end-use or comfort constraints (Nandanoori et al., 2018).

The distinctive feature of this scheme is prioritized threshold allocation based on a device “fitness” metric,

$C$3

where availability represents the probability that the device is ON and operationally available when an event occurs, and quality can be expressed as

$C$4

with $C$5 the average actuation or switching delay. Devices are ordered by decreasing fitness, and the fittest are assigned thresholds closer to nominal frequency. The probability that all $C$6 selected devices succeed is modeled as

$C$7

Control accuracy is evaluated by the Reserve Margin Variability Target,

$C$8

Monte Carlo simulations on ensembles of $C$9 ACs with $R$0, $R$1, and power $R$2 showed that prioritized threshold allocation yielded response curves closely tracking the target response, with typical RMVT errors $R$3. Non-prioritized random allocation produced much higher errors, with mean RMVT in the $R$4–$R$5 range. The same study notes that sampling intervals should be short, for example $R$6 s, especially for events near nominal frequency where the frequency rate-of-change is high (Nandanoori et al., 2018).

These results place FFACs among the more tractable binary-load resources for fast, distributed response. At the same time, the literature does not treat their binary nature as costless: discretization error scales with device granularity, and actuation delays produce additional response error.

5. Switching constraints, market-based control, and robust optimization

Because FFACs are on/off machines, switching constraints are a first-order design requirement rather than an implementation detail. A market-based control framework for air-conditioning loads introduces explicit lockout states and a tunable parameter $R$7 to regulate switching frequency. Although that study does not mention FFACs by name, it focuses on air-conditioning loads with switching-frequency and lockout-time constraints that directly apply to fixed-frequency, single-stage devices (Yao et al., 2018).

In this framework, each unit submits a bid

$R$8

with bid price

$R$9

where $\eta$0 is the normalized state of air-temperature and

$\eta$1

The lockout offsets enforce minimum ON and OFF times, while $\eta$2 biases units to remain in their current state. The study reports that for mitigation of power fluctuation, switching frequency can be kept lower than with local thermostatic control if $\eta$3, and states that “participating in such ancillary services would not lead to more mechanical wear with proper control parameters.” For fast regulation services, however, switching frequency exceeded normal operation even at $\eta$4, leading the authors to conclude that new control methods are needed to protect the devices (Yao et al., 2018).

Experimental work on a window-mount air conditioner independently quantified the energetic “cost of control” under harmonic setpoint modulation. The imposed setpoint trajectory was

$\eta$5

and the excess power consumption was defined as

$\eta$6

For moderate modulation, exemplified by $\eta$7 and $\eta$8 min, the reported cost was about $\eta$9 W extra consumption for about $p(t) \in \{0,P\}$0 W of modulation amplitude. The same experiments showed a frequency response that peaks near the natural duty cycle, around $p(t) \in \{0,P\}$1 min, and rolls off at both faster and slower modulation periods; significant phase shifts and pronounced nonlinearity were observed because of dead band and on/off switching (Geller et al., 2016).

A more recent optimization study formulates FFAC demand-response participation as a robust mixed-integer linear programming problem under outdoor-temperature uncertainty. Individual units follow a Markov response model; aggregate baseline power is generated by Monte Carlo sampling; thermal comfort is enforced through bounds $p(t) \in \{0,P\}$2 and a discomfort budget; and uncertainty is represented by

$p(t) \in \{0,P\}$3

The objective maximizes aggregator profit, with binary scheduling variables $p(t) \in \{0,P\}$4, minimum ON/OFF-time constraints, and robustified comfort constraints obtained by dualization. Reported outcomes include peak-hour demand reduction by up to $p(t) \in \{0,P\}$5, pre- and post-cooling through load shifting, robust maintenance of indoor temperatures within the comfort envelope, and a $p(t) \in \{0,P\}$6 revenue improvement after comfort penalties (He et al., 14 Aug 2025). Together, these results indicate that FFAC flexibility is substantial, but it is shaped by mechanical wear limits, nonlinear response, and the economics of comfort compensation.

6. Urban-scale atmospheric interaction and storm modulation

Outside the conventional grid-services literature, FFACs have also been examined as a distributed means of modifying atmospheric moisture fields. A numerical study of the August 2014 Hiroshima storm training event proposed the strategic use of consumer air conditioners in dehumidification mode to remove near-surface moisture upstream of severe rainfall. The paper states that consumer air conditioners can typically remove about $p(t) \in \{0,P\}$7 kg of moisture from the air per hour in dehumidification mode, and notes that in Japan more than one air conditioner is installed per capita. Under that density, removal of half a kiloton of moisture within half an hour in a city with a population of one million is described as feasible (Hiruma et al., 2019).

The study introduces a control efficiency parameter,

$p(t) \in \{0,P\}$8

where $p(t) \in \{0,P\}$9 is the total rainfall reduction over the target region and $$p(t^+) = \begin{cases} 0, & \text{if } T(t) \leq T_{set} - \delta T/2 \ P, & \text{if } T(t) \geq T_{set} + \delta T/2 \ p(t), & \text{otherwise}, \end{cases}$$0 is the total moisture removed in the control volume. The reported efficiency is nonlinear: as total moisture removal increases, the efficiency per unit removed decreases approximately as a power law with exponent $$p(t^+) = \begin{cases} 0, & \text{if } T(t) \leq T_{set} - \delta T/2 \ P, & \text{if } T(t) \geq T_{set} + \delta T/2 \ p(t), & \text{otherwise}, \end{cases}$$1. The normalized efficiency depends on the area ratio $$p(t^+) = \begin{cases} 0, & \text{if } T(t) \leq T_{set} - \delta T/2 \ P, & \text{if } T(t) \geq T_{set} + \delta T/2 \ p(t), & \text{otherwise}, \end{cases}$$2, scaling with $$p(t^+) = \begin{cases} 0, & \text{if } T(t) \leq T_{set} - \delta T/2 \ P, & \text{if } T(t) \geq T_{set} + \delta T/2 \ p(t), & \text{otherwise}, \end{cases}$$3 for small targets and saturating for large ones (Hiruma et al., 2019).

Quantitatively, the Hiroshima simulations reported that $$p(t^+) = \begin{cases} 0, & \text{if } T(t) \leq T_{set} - \delta T/2 \ P, & \text{if } T(t) \geq T_{set} + \delta T/2 \ p(t), & \text{otherwise}, \end{cases}$$4 tons of moisture removal over a $$p(t^+) = \begin{cases} 0, & \text{if } T(t) \leq T_{set} - \delta T/2 \ P, & \text{if } T(t) \geq T_{set} + \delta T/2 \ p(t), & \text{otherwise}, \end{cases}$$5 target region reduced accumulated rainfall by about $$p(t^+) = \begin{cases} 0, & \text{if } T(t) \leq T_{set} - \delta T/2 \ P, & \text{if } T(t) \geq T_{set} + \delta T/2 \ p(t), & \text{otherwise}, \end{cases}$$6 mm, nearly $$p(t^+) = \begin{cases} 0, & \text{if } T(t) \leq T_{set} - \delta T/2 \ P, & \text{if } T(t) \geq T_{set} + \delta T/2 \ p(t), & \text{otherwise}, \end{cases}$$7 of an observed $$p(t^+) = \begin{cases} 0, & \text{if } T(t) \leq T_{set} - \delta T/2 \ P, & \text{if } T(t) \geq T_{set} + \delta T/2 \ p(t), & \text{otherwise}, \end{cases}$$8 mm. For a $$p(t^+) = \begin{cases} 0, & \text{if } T(t) \leq T_{set} - \delta T/2 \ P, & \text{if } T(t) \geq T_{set} + \delta T/2 \ p(t), & \text{otherwise}, \end{cases}$$9 target, the same removal yielded a $T_{set}$0 mm reduction, about $T_{set}$1 of $T_{set}$2 mm observed rainfall. The paper also identifies a converse effect: some summertime storms occurring inside or near a metropolis could be strengthened by the excess moisture discharged from large numbers of air conditioners used for cooling rooms. It further notes that small or unfavorably located moisture reductions may produce negative efficiency, meaning rainfall increases rather than decreases (Hiruma et al., 2019).

This body of work broadens the meaning of FFAC aggregation. In grid applications, aggregation is used to shape electrical demand while respecting local thermal constraints; in the storm-modulation study, aggregation shapes the urban moisture field. A plausible implication is that FFACs sit at the intersection of power systems, building thermodynamics, and atmospheric processes, with the relevant aggregate state depending on whether the controlled variable is electrical power, indoor comfort, or near-surface humidity.