Data-Driven MPC with Multi-Layer ReLU Networks for HVAC Optimization Under Iraq’s Time-of-Use Electricity Pricing

Abstract

Enhancing the energy management capabilities of modern smart buildings is essential for energy conservation, which is valuable for modern power networks maintaining a tight power balance under high renewable penetration. This study introduces a data-driven control strategy based on the model predictive control (MPC) for HVAC (heating, ventilation, and air conditioning) systems considering the time-of-use (ToU) electricity rates in Iraq. A multi-layer neural network is first constructed using time-delayed embedding for the modeling of building thermal dynamics, where the rectified linear unit (ReLU) is used as the activation function for the hidden layers. Based on such piecewise affine approximation, an optimization model is developed within the receding horizon control framework, which incorporates the data-driven model and is transformed into a mixed-integer linear programming facilitating efficient problem solving. To validate the efficiency of the proposed approach, a simulation model of the building’s thermal network is constructed using Simscape considering several thermal effects among the building components. Simulation results demonstrate that the proposed approach improves the economic performance of the building while maintaining thermal comfort levels within acceptable range.

1. Introduction

According to the International Energy Agency (IEA), buildings account for approximately 40% of global energy consumption, with even higher proportions in industrialized and urbanized regions [1]. In Baghdad, Iraq, the building sector accounts for 48% of residential energy consumption, with cooling and heating being the primary contributors to energy demand, surpassing lighting and other purposes. In Baghdad households, annual energy use shows distinct seasonal patterns: cooling accounts for 42.43% of total consumption during summer months, while heating represents 26.56% in winter. When aggregated annually, these climate control systems collectively comprise approximately 69% of total household energy consumption, reflecting Iraq’s extreme temperature variations between seasons [2]. Since 2003, Iraq has grappled with a severe energy crisis, with a predominant focus to increase the supply rather than improving energy efficiency on the demand side. The technical and commercial losses surpassing 50% of generated energy highlight the urgent need for effective energy conservation and management. Enhancing building energy efficiency in Iraq, particularly in cooling and heating systems, presents a significant opportunity to address the country’s power crisis, especially given the extreme environmental temperatures in Iraq’s dry and semi-arid climate. During the summer months, temperatures frequently rise above 40 °C and can sometimes exceed 50 °C [3]. Addressing the efficiency of building energy systems not only mitigates the adverse effects of this extreme heat on daily life but also reduces the overall energy consumption and environmental impact of the country [1] (

Table 1. List of nomenclature.

MPC	Model predictive control
HVAC	Heating, ventilation, and air conditioning
ReLU	Rectified linear unit
ToU	Time-of-use (electricity pricing)
MILP	Mixed-integer linear programming
COP	The energy efficiency coefficient
$T_{in}{T}_{out}$	Indoor/outdoor temperature at time t (°C)
$T_t^{wall},T_t^{roof}$ $T_t^{window}$, $T_t^{out}$	Wall/roof/window/outdoor temperature at time t (°C)
$T_{min}^{in}$, $T_{max}^{in}$	Lower and upper limits of the indoor temperature
$Q_t^{AC}$	Thermal input from HVAC at time t (W)
$Q_t^{dist}$	Thermal disturbance input
$f$	Thermodynamic mapping relationship
$z_t^{AC}$	Binary HVAC status [0: off, 1: on] at time t
$P_t^{AC}$	The electrical power of HVAC
$C_p$	Specific heat capacity of air (J/kg·K)
$m^{AC}$	The air flow rate
$T^{set}$	Temperature set point of the supplied air
$t$	Time step index
$H$	Prediction horizon
$M$	Historical time window for delayed embedding
C in, C wall	Thermal capacitances (J/°C)
Rout, R wall	Thermal resistances (°C/W)

).

The Iraqi government is actively tackling energy challenges, emphasizing efficiency and sustainability. A key initiative involves urging large industrial and commercial facilities to adopt energy-efficient technologies and flexible load regulation practices. This incentivizes dynamic adjustments in electricity consumption based on grid conditions and pricing. However, significant challenges persist in the power sector, with frequent power outages affecting many households. Efforts to alleviate shortages include enhancing grid flexibility, reducing network losses, strengthening regional interconnections, optimizing gas usage, and boosting renewable energy integration [4]. Addressing these challenges necessitates a comprehensive approach involving demand reduction, capacity enhancement, infrastructure improvement, and encouraging non-essential demand to shift away from peak hours.

Traditional on–off control logic for HVAC systems is inadequate for advanced building energy management applications. In contrast, optimization-based approaches like Model Predictive Control (MPC) provide superior performance by anticipating future system behavior through predictive algorithms. This advanced control methodology enables optimal decision-making that simultaneously addresses complex system dynamics, operational constraints, and multiple performance objectives. Specifically, MPC’s capability to solve constrained multi-objective optimization problems makes it particularly suitable for modern HVAC control challenges, where energy efficiency must be balanced with thermal comfort requirements [5]. MPC offers several advantages, including the ability to manage constraints related to temperature and humidity for comfort, leverage weather, and occupancy forecasts for improved performance. Current research delves into diverse aspects of MPC for HVAC controls, examining thermal dynamics in various contexts [6], exploring time-sharing control schemes [7], and investigating heuristic algorithms to achieve optimal solutions. By using a model of the system, MPC can forecast a system’s future behavior and adjust control inputs in advance. This enables the optimization of energy utilization in HVAC systems [8,9], resulting in better energy utilization and lower costs [10], like Midea’s iBUILDING and Siemens’ Xcelerator [11]. By adjusting HVAC power based on electricity prices or external commands, buildings can maintain comfort while reducing consumption, offering valuable flexible resources for grids. Participation in ancillary services markets can lead to over 20% electricity cost savings for large buildings [12]. References [8,13,14] investigated cooperative planning of household energy storage and HVAC systems using the Energy Plus simulation software (version 8.8). They addressed system uncertainty, improved predictive control in different scenarios, and conducted field tests. Reference [15] optimized household AC and battery storage systems to enhance energy efficiency, demonstrating significant energy and cost savings through coordinated operation, supporting sustainable residential energy use. References [16,17] proposed an active distribution network optimization model integrating smart buildings for room temperature control and grid loss reduction. Reference [17] suggested a robust HVAC optimization strategy for grid peak shaving backup. Reference [6] proposed building thermal dynamics and analyzed economic operation under various building parameters.

Existing studies primarily use traditional thermal models, autoregressive models, or simulations for accurate building thermodynamic modeling. Accurately modeling thermal dynamics in real buildings is challenging due to various factors [8,9,18]. The traditional MPC requires an accurate mathematical model of the system, which can be challenging to obtain, especially for complex systems like HVAC [19,20]. Data-driven modeling approaches, such as neural networks, can learn the system dynamics from available data without the need for explicit mathematical modeling. This makes data-driven MPC particularly suitable for systems where first-principles models are difficult to derive [19,21]. While traditional MPC for HVAC systems relies on physics-based white-box models of buildings and equipment, real-world complexity often makes accurate parameter identification challenging [22].

Recent advancements in artificial intelligence (AI) have revolutionized the HVAC field. Data-driven strategies, such as those covered in references [23,24,25], take advantage of artificial intelligence’s capacity to handle intricate variables. For instance, researchers can model a heating, ventilation, and air conditioning (HVAC) system using data-driven black-box models like neural networks. These models learn complex relationships from historical data, easily adapt to system changes, and enable efficient real-time control applications. Reference [23] employed data-driven techniques to create a multi-region building aggregation model for energy management. Reference [24] introduced a thermodynamic model using deep learning, while reference [25] demonstrated that recurrent neural networks outperform white-box and grey-box models in temperature prediction accuracy. However, integrating these deep learning models into the optimization frameworks presents challenges due to their nonlinear nature, often requiring linearization or heuristic methods for effective solution [26,27], and thus, they are not suitable for real-time control.

Rectified Linear Unit (ReLU) activation functions, defined as f(x) = max (0, x), are employed in our neural network architecture for three key reasons relevant to HVAC control. First, their piecewise linear nature enables the exact transformation of the neural network into an equivalent mixed-integer linear program, which is crucial for efficient MPC optimization [28]. Second, while ReLU networks avoid the vanishing gradient problem common in sigmoid/tanh functions during backpropagation, we mitigate potential ‘dying ReLU’ issues through careful initialization and batch normalization [22]. Third, the sparse activations induced by ReLU improve computational efficiency while maintaining the network’s ability to approximate building thermal dynamics [21]. This combination of mathematical tractability and empirical performance makes ReLU particularly suitable for our data-driven MPC framework.

This paper proposes a data-driven HVAC optimization strategy that integrates a multi-layer ReLU neural network within a model predictive control (MPC) framework. The thermal dynamics of a building are modeled using a time-delay embedded neural network, where rectified linear units (ReLU) serve as the activation functions in the hidden layers. Leveraging this piecewise affine representation, an optimization model is formulated within a receding horizon framework. The resulting problem is reformulated as a mixed-integer linear program (MILP), enabling efficient and tractable optimization. To assess the performance of the proposed approach, a detailed simulation of a building’s thermal behavior is developed in Simscape, capturing various thermal interactions among building components. A key contribution of this work is the exact MILP reformulation of the ReLU-based neural network, which allows for real-time optimization with guaranteed global optimality, addressing a major limitation of traditional nonlinear neural networks that often rely on approximations or heuristics. The control framework also incorporates time-of-use (TOU) electricity pricing and HVAC operational constraints, such as start–stop limits. Validated using 50 days of real summer temperature data from Baghdad (July and August), the method demonstrates strong performance under extreme weather conditions. Compared to standard on/off control strategies, the proposed approach significantly reduces energy costs while maintaining indoor thermal comfort, making it a viable solution for energy limited environments.

The paper is structured into sections. Section II presents the typical building thermal network simulation model, serving as the basis for data collection and algorithm deployment detailed in Section III. Section IV is the optimization of the control models and analyses simulation results in Section V, with the paper concluding in Section VI.

2. Simulation Model of a Typical Building Thermal Network

Simscape is a MATLAB/Simulink R2023a physics model library with hybrid multiphysics simulation capabilities that provide modeling capabilities for thermal insulation and thermal exchange effects. In this paper, we first construct a thermal network simulation model of a typical building, including an air-conditioning system and sub-models of the room heating network, where the room structure is composed of internal air, walls, windows, and roofs. Figure 1 shows a schematic diagram of the room heat network subsystem, showing that the internal air exchanges heat with the external ambient air through walls, windows, and roofs, respectively, with each heat exchange channel considering heat convection, heat conduction, and heat storage effects. The simulation model is mathematically expressed as:

$${\left[T_{t+1}^{in},T_{t+1}^{wall},T_{t+1}^{roof},T_{t+1}^{window}\right]}^T=f{\left(T_t^{in},T_t^{wall},T_t^{roof},T_t^{window},Q_t^{AC},Q_t^{dist}\right)}^T$$

(1)

where $T_{t+1}^{in},T_{t+1}^{wall},T_{t+1}^{roof},T_{t+1}^{window}$ represents the temperatures of indoor air, walls, the roof, and windows at time t+1 (next timestep), $T_t^{out}$ represents the outdoor air temperature, $Q_t^{AC}$ represents the thermal input from the HVAC system, $Q_t^{dist}$ is the thermal disturbance input, and f represents the thermodynamic mapping relationship.

Considering a simplified HVAC system setup, it is assumed that the supply air temperature and the flow rate are constant, and the heat generated is expressed as

$$Q_t^{AC}=C_p{m}^{AC}\left(T^{set}-T_t^{in}\right)z_t^{AC}$$

(2)

In the equation, $m^{AC}$ and $T^{set}$ represent the air flow rate and temperature setpoint of the supplied air, respectively, $C_p$ is the specific heat capacity of air, and $z_t^{AC}$ represents the switch status of HVAC at time t. Its total electricity cost is

$${Cost}_t=\underset{0}{\overset{t}{\int }}{p}_t{P}_t^{AC}dt={\int }_0^t{p}_t\frac{Q_t^{AC}}{COP}dt$$

(3)

where $P_t^{AC}$ is the electrical power of HVAC, $COP$ is the energy efficiency coefficient, and $p_t$ represents the time of use (ToU) electricity price. Figure 2 is a straightforward chart that illustrates the TOU electricity price in Iraq-Baghdad.

Table 2. Time-of-use electricity prices in Baghdad-Iraq.

	Time-of-Use Electricity	Price in Baghdad Province
Super-Peak period	20:00–22:00	Basic electricity price × 180%
Peak period	9:00–15:00	Basic electricity price × 149%
Shoulder period	7:00–9:00, 15: 00–20:00, 22:00–23:00	Basic electricity price
Off-peak period	23:00–7:00	Basic electricity price × 48%

lists the designated peak, off-peak, and valley times as well as the associated electricity prices. The base rate for electricity is 19.375 Iraqi Dinar per kWh.

3. Data-Driven Modeling of Building Thermal Dynamics

The thermal dynamics of a building include heat exchange between multiple different media, which is difficult to obtain from traditional mechanism models with unknown parameters for accurate multi-step temperature prediction. In this paper, a multi-layer neural network is used to model the thermal dynamics of the building. The structure of the neural network is shown in Figure 3, where the linear rectification function is used as the activation function for hidden layer neurons. Compared with the common activation functions such as Sigmoid and Tanh, ReLU networks are easy to train and generally more robust [25]. The output layer uses the Fully Connected (FC) layer to linearly map the output of the hidden layer to a specified dimension.

To realize the multi-step prediction of the room temperature of the building in the future H-step, time-delayed embedding is used to construct the input features, and the input and output of the ReLU network are respectively expressed as

$${\mathit{X}}_t={\left[\begin{array}{c}{\mathit{x}}_{t-M},{\mathit{x}}_{t-M+1},{\dots \mathit{x}}_t,T_{t+1}^{out},T_{t+2}^{out},T_{t+H-1}^{out},\\ z_{t-M}^{AC},z_{t-M+1}^{AC},\dots .,z_{t+M-1}^{AC}\end{array}\right]}^T$$

(4)

$$Y_t={\left[T_{t+1}^{in},T_{t+2}^{in},\dots ,T_{t+H}^{in}\right]}^T$$

(5)

where M is the delayed time step, and

$${\mathit{x}}_t={\left[T_t^{in},t^{cos},t^{sin}\right]}^T$$

(6)

In addition to the indoor and outdoor temperatures and the 0–1 state of the air conditioner, the time features are introduced into the feature vector to capture the periodic changes in the system via

$$t^{sin}=\mathit{sin}\left({\displaystyle \frac{2\pi t}{8760}}\right)$$

(7)

$$t^{cos}=\mathit{cos}\left({\displaystyle \frac{2\pi t}{8760}}\right)$$

(8)

The output of the ReLU network is the future indoor temperature and does not include the other signals (e.g., wall temperature and window temperature) that are not under regulation. The ReLU network is a piecewise linear affine system, which can be expressed as

$${\mathit{O}}_0{=\mathit{X}}_t$$

(9)

$${\mathit{O}}_{i+1}=\mathit{max}\left({\mathit{W}}_i{\mathit{O}}_i+{\mathit{b}}_i,0\right) \forall i=0,\dots .N_H-1$$

(10)

$${\mathit{O}}_t^{FC}={\mathit{W}}_{FC}{\mathit{O}}_n+{\mathit{b}}_{FC}$$

(11)

where (${\mathit{W}}_{\mathit{i}}$, ${\mathit{b}}_{\mathit{i}}$) is the weight and offset of the hidden layer and (${\mathit{W}}_{\mathit{F}\mathit{C}}$, ${\mathit{b}}_{\mathit{F}\mathit{C}}$) is the weight and offset of the fully connected layer.

The Mean Squared Error (MSE) is selected as the loss function

$${\mathrm{M}}_{SE}={\displaystyle \frac{1}{N}}{\sum }_{n=1}^N{‖{\mathit{O}}_{\mathit{n}}^{\mathit{F}\mathit{C}}-{\mathit{Y}}_{\mathit{n}}‖}_2^2$$

(12)

where N is the number of samples, ${\mathit{Y}}_{\mathit{n}}$ is the actual output of the system, and the loss function represents the error between the output of the ReLU network and the actual output of the system. The training of the ReLU network uses the ADAM algorithm, which is suitable for large amounts of data and shows better performance in optimization problems with high feature dimensions [26,29].

To highlight the distinctions between the proposed ReLU–MPC approach and existing data-driven and traditional RC-based MPC methods,

Table 3. Tabular comparison: ReLU–MPC vs. typical data-driven MPC.

Criterion	Existing Data-Driven MPC	Autoregressive/RC Models	Our ReLU–MPC Method
Modeling Technique	Black-box NNs (e.g., RNNs, LSTMs)	Grey-box (first-order RC circuits)	ReLu-based piecewise affine NN
Activation Function	Sigmoid/Tanh (nonlinear, smooth)	Not applicable (linear ODEs)	ReLU (enables exact MILP reformulation)
Optimization Solver	Nonlinear/heuristic solvers	MILP/LP (after linearization)	MILP (Gurobi/Mosek) with certified optimality
Computational Cost	High (nonconvex, iterative solvers)	Moderate (depends on model structure)	Low (linear constraints enable fast MILP solving)
Multi-Step Prediction	Prone to error accumulation (especially with RNNs)	Sensitive to parameter mismatch and drift	Stable (via time-delay embedding in PWA structure)

presents a comparative summary based on key modeling and control criteria.

While autoregressive and RC models have been widely used for building thermal modeling, they present fundamental limitations that our ReLU–MPC method systematically addresses. Autoregressive models exhibit exponential error accumulation in multi-step predictions; our experiments show a 200% increase in MSE for 30 min horizons compared to single-step forecasts. While these models can technically incorporate constraints through post-processing (e.g., projection methods), this decoupled approach often degrades solution quality and requires additional stabilization heuristics.

RC models face different challenges: they approximate building thermodynamics through first-order equivalents, neglecting crucial multi-media heat transfer dynamics (e.g., simultaneous wall–roof–air interactions). Our tests reveal this simplification leads to 15% higher prediction errors compared to our data-driven approach. The models’ dependence on manually calibrated R/C parameters makes them both building-specific and noise-sensitive, particularly problematic for Iraq’s extreme temperature variations.

4. MPC Problem Formulation

We first give the schematic diagram of the proposed data-driven MPC framework for the air conditioning operations as shown in Figure 4. The data-driven thermal dynamics model provides the mapping between system states and control inputs in a piece affine fashion. The MPC optimization problem incorporates this model as a mixed-integer programming that can be solved by off-the-shelf solvers. The historical data continuously improve the accuracy of the ReLU network.

4.1. Optimization Model

This article chooses the control scenario within a summer cooling setting. In the typical control mode, when the room temperature reaches the upper limit, $z_t^{AC}$ is set to 1, indicating that the air conditioner is turned on, and when the lower limit is reached, it is set to 0. In this mode, the HVAC load is not managed across time, and it is difficult to take advantage of the ToU price structure. To clarify how the MPC control model works, consider a simple scenario: Suppose the current room temperature is 26 °C, the comfort range is set between 18–24 °C, and the electricity price is expected to spike in the next 30 min. The MPC controller predicts that if the air conditioner is turned on now, the temperature will reach 22 °C before the peak price period begins. Therefore, the controller decides to precool the room early, so that it can turn off the AC during the high-price hours while keeping the temperature within the comfort range. This decision is made by solving the optimization problem that minimizes electricity costs while satisfying thermal comfort and system constraints over the prediction horizon. The optimization problem model is constructed under the rolling horizon control framework, and its objective function is expressed as

$$\underset{Z_k^{\mathit{AC}},T_k^{\mathit{in}}}{\mathit{min}}{\sum }_{K=1}^H{p}_k{\displaystyle \frac{C_P{m}^{AC}(T^{set}-T_K^{in})z_k^{AC}}{COP}}\Delta t$$

(13)

Expressing the total cost of electricity in steps from the current moment t to the future H step. $\Delta t$ is the control cycle length, and k represents the k step in the future from the current time t. Constraints for the optimization model include temperature comfort, HVAC start-stop time, and thermodynamic model constraints. First, the room temperature needs to be controlled within the temperature comfort range:

$$T_{min}^{in}{\le T}_k^{in}{\le T}_{max}^{in} k=\mathrm{1,2},..,H$$

(14)

where $T_{min}^{in}$ and $T_{max}^{in}$ represent the lower and upper limits of the indoor temperature. In addition, the frequent start-up of HVAC equipment may seriously affect the service life of its internal compressors and other components, so the start-stop time constraint is added to the optimization model as

$${\mathit{z}}_{\left[K,K+S-1\right]}^{AC}\ge z_K^{AC}-z_{K-1}^{AC} K=1\dots .H$$

(15)

$${\mathit{z}}_{\left[K,K+S-1\right]}^{AC}\le 1-\left(z_{K-1}^{AC}-z_K^{AC}\right)$$

(16)

$$z_K^{AC}\in \left\{\mathrm{0,1}\right\}$$

(17)

In the formula ${\mathit{z}}_{\left[K,K+S-1\right]}^{AC}={\left[z_K^{AC},z_{K+1}^{AC},\dots ,z_{K+S-1}^{AC}\right]}^T$, which represents the column vector of HVAC start-stop state in the time domain shown by a subscript, S represents the minimum start-stop time of HVAC; Equation (15) means that, when the air conditioner is turned on at k time, the start-stop state of the air conditioner in the minimum start-stop time range is 1, and the value on the right side of the inequality is 0 or −1 in other cases, and the inequality is naturally established. Equation (16) similarly limits the HVAC control instruction under the shutdown state, and $z_0^{AC}$ represents the air-conditioning start-stop state at the previous moment.

Finally, the trained ReLU network gives the dynamic constraints, which are expressed as follows:

$${\mathit{O}}_{i+1}^t=\mathit{max}\left({\mathit{W}}_i{\mathit{O}}_i^t+{\mathit{b}}_i,0\right),i=0,\dots ,N-1$$

(18)

$${\mathit{T}}_{\left[1,H-1\right]}^{in}={\mathit{W}}_{FC}{\mathit{O}}_N^t+{\mathit{b}}_{FC}$$

(19)

$${\mathit{O}}_0^t=[{\mathit{X}}_{\left[t-M,t\right]}^T,{\mathit{z}}_{\left[t-M,t\right]}^{ACT},{\mathit{T}}_{\left[t,t+M-1\right]}^{outT},{\mathit{z}}_{\left[1,H-1\right]}^{ACT}{]}^T$$

(20)

where ${\mathit{X}}_{\left[t-M,t\right]}^T,{\mathit{z}}_{\left[t-M,t\right]}^{AC T}$ indicates the historical trajectory of the system, which introduces a feedback mechanism in the optimization control; ${\mathit{z}}_{\left[1,H-1\right]}^{AC T}$ is the sequence of decision variables at time t; and ${\mathit{T}}_{\left[t,t+M-1\right]}^{out T}$ is the predicted value of outdoor temperature in the future.

4.2. Model Reformulation

The above optimization model contains a bilinear term in the objective function and a max operator in the constraints, which is difficult to solve directly, so it is reconstructed as follows:

$${min}_{Z_K^{AC},T_K^{in},X_K}={\sum }_{K=1}^H{p}_K{\displaystyle \frac{C_p{m}^{AC}\left(T^{set}{z}_K^{AC}-X_K\right)}{COP}}$$

(21)

subject to (14), (15), (19), (20)

(22)

$$Z_K^{AC}{T}_{min}^{in}\le X_K\le Z_K^{AC}{T}_{max}^{in}$$

(23)

$$X_K\ge T_K^{in}+(Z_K^{AC}-1)T_{max}^{in}$$

(24)

$$X_K\le T_K^{in}+(Z_K^{AC}-1)T_{min}^{in}$$

(25)

$$O_{i+1}^t\ge {\mathit{W}}_i{\mathit{O}}_i^t+{\mathit{b}}_i$$

(26)

$$O_{i+1}^t\le {\mathit{W}}_i{\mathit{O}}_i^t+{\mathit{b}}_i+M\left(1-{\mathit{d}}_i^t\right)$$

(27)

$$0\le O_i^t\le M{\mathit{d}}_i^t, i=1, \dots , N-1$$

(28)

The variable $X_K$ is introduced to replace the bilinear term in the objective function; Equations (23) and (25) represent the McCormick envelope, and the nonlinear term is convex according to the upper and lower limits of $z_K^{AC} \mathrm{a}\mathrm{n}\mathrm{d} T_K^{in}$; Equations (26) and (27) are the reconstruction of the max operator; the column vector $d_i^t\in {\left\{\mathrm{0,1}\right\}}^{n_i}$ represents the activation state of all neurons in the i-th hidden layer—when an element is 1, it means that the neuron is activated; and M is the big-M constant. The reconstructed optimization model has the form of mixed-integer linear programming, which can be solved efficiently using existing optimization solvers, e.g., Gurobi. Under the rolling horizon control framework, the optimization model is solved, and $z_1^{AC}$ is applied to the system. The execution procedure is repeated for each control cycle.

5. Case Studies

5.1. Basic Data

All simulation examples in the paper were conducted on a Core i5 computer with 16 GB RAM. The optimization was established and solved using YALMIP and calling the Mosek solver [27]. The simulation model of the building thermal network was built based on the nominal parameters from the example in MATLAB/Simulink [25]. The room dimensions were assumed to be 30 m × 10 m × 4 m, with 6 windows measuring 1 m in length and width and 0.01 m in thickness. The roof had a slope angle of 60 degrees and a thickness of 0.2 m. Air properties included a specific heat capacity of 1005.4 J/kg/K, a density of 1.225 kg/m³, and indoor air quality calculated at 1496 kg based on room volume. Mass and heat transfer coefficients for roofs, walls, and windows were determined as shown in

Table 4. Mass and thermal dynamic parameters of the building.

	Roof	Wall	Window
Mass (kg)	3846.9	122,880	162
Specific heat capacity (J/kg/K)	835	835	840
Thermal conductivity (W/m/K)	0.038	0.038	0.78
Thermal convection coefficient (W/m2/K)	12/38	24/34	25/32

, with the final row indicating thermal convection coefficients for each component with indoor/outdoor air. The simulation examples explored the impact of human-generated thermal disturbances on control effectiveness. Assuming a normal distribution of people in the room, each contributing 100 W of heat, the ReLU–MPC control strategy successfully maintained the room temperature within the desired range despite external disturbances without needing to predict room occupancy. This success was due to the feedback mechanism inherent in the MPC approach, which allowed the controller to dynamically adapt to the changing thermal conditions in the room.

It was assumed that HVAC operated in the cooling mode with a supply air temperature of 18 °C, flow rate of 1 kg/s, COP of 0.95, and room temperature control range of 18–24 °C. The MPC control period was 1 min, the predicted time domain was 30 min, and the minimum start-stop time for HVAC was 3 min.

Table 5. Simulation parameters.

Parameter	Value/Range	Description
Control Horizon (∆t)	1 min	MPC execution interval
Prediction Horizon	30 steps (30 min)	Optimization window
Supply Temp ($T^{set}$)	18 °C	Fixed cooling setpoint
Flow Rate ($m^{AC}$)	1 kg/s	Constant air mass flow
COP	0.95	Typical for Iraqi residential units
Comfort Range	18–24 °C	Iraqi building standards

summarizes simulation parameters (control horizon, prediction horizon, AC specs).

Historical outdoor ambient temperature data for Baghdad, Iraq, spanning 50 days in July and August, are plotted in Figure 5.

5.2. The Testbench

To evaluate the performance of the proposed ReLU–MPC framework, a hybrid testbench was developed that integrated empirical weather data, physics-based modeling, and optimization. The test environment leveraged the same single-zone building structure defined in Section 5.1 and extended it into a dynamic simulation using a customized Simscape “House Heating System” in MATLAB/Simulink. The HVAC system operated in cooling mode with a fixed 18 °C supply air temperature, 1 kg/s airflow, and a COP of 0.95. A time-of-use (TOU) electricity pricing scheme from Hubei Province was adopted for cost optimization. Real outdoor temperature data collected in Baghdad, Iraq over a 51 day period in July and August 2024 was integrated into the simulation via an API and aligned with the MPC’s 1-min control interval. Human-induced thermal disturbances were modeled as stochastic internal heat gains (100 W/person) without requiring occupancy prediction, relying instead on the feedback nature of MPC to handle variability.

The ReLU neural network, trained on historical data using the Adam optimizer, yielded a multi-step prediction MSE of 0.0317. The MPC optimization problem was implemented in YALMIP and solved using Mosek, with a 30-min prediction horizon and a 3-min HVAC cycling constraint.

Two test scenarios were assessed: one with a constant outdoor temperature and another with full-day variations in both weather and electricity price. The results confirmed accurate temperature regulation and energy cost savings, demonstrating the testbench’s ability to blend data-driven control with physical realism. Figure 6 and Figure 7 show the modular Simulink setup used to simulate thermal dynamics and implement the ReLU–MPC controller.

Figure 6 illustrates the high-level modular structure of the house cooling system used in the testbench. It included major functional blocks such as the Heater Switch, Thermostat, Heater, House Thermal Network, and Daily Temperature Variation. The Heater Switch enabled manual control of the system for testing purposes. The Thermostat received the measured room temperature (T room) and compared it to predefined thresholds to determine whether the air conditioning system should be turned On or Off. Based on this signal, the Heater block either supplied cooling power or remained inactive, and it also tracked energy usage and cost. The Daily Temperature Variation block provided time-varying ambient temperature data (Tatm) using realistic weather inputs. The House Thermal Network block represented the building’s thermal structure and simulated heat transfer between the indoor air and the environment. Finally, the Heating Results block collected and displayed key simulation outputs, including temperature and cost.

Figure 7 provides the detailed internal layout of the House Thermal Network block from Figure 6. It modeled the thermal interactions between the indoor air and the building envelope components: roof, walls, and windows. Each component included thermal mass elements (e.g., Thermal Mass Roof, Thermal Mass Wall, Thermal Mass Window) connected by conductive and convective paths. For instance, heat transfer occured via Air–Roof Convection, Half Roof–Air Conduction, and Atm–Roof Convection chains. Similarly, heat exchange through the walls and windows was modeled with parallel structures. A Room Temperature Sensor measured the indoor air temperature (Troom) based on the Thermal Mass Air, and this signal was used by the thermostat for control decisions. The external ambient temperature (Tatm) was introduced at the boundary of the thermal network, enabling the dynamic simulation of environmental effects. Together, Figure 6 and Figure 7 form a comprehensive testbench, linking control logic with physically realistic thermal behavior to evaluate the performance of the ReLU–MPC framework under real-world conditions.

5.3. ReLU Network Modeling Accuracy Analysis

The outdoor temperature historical data was used to generate 51 system trajectories, with 50 trajectories serving as the training set for the ReLU network. After ablation studies comparing four-hidden-layer configurations ([100], [10-10], [20-20-20], [10-10-10]), the three-hidden-layer [10-10-10] architecture was selected as optimal, balancing prediction accuracy (MSE = 0.0317 for 30 min multi-step forecasts) and computational efficiency. The network was trained using the Adam optimizer with an initial learning rate of 0.01 that decayed by 0.5 every 100 iterations over 1000 epochs alongside L2 regularization (λ = 1 × 10⁻⁴) and a mini-batch size of 3. Dropout was omitted due to the model’s compact size (30 neurons) and the moderate dataset size, which inherently prevented overfitting.

ReLU was chosen over alternatives such as Leaky ReLU and Swish due to its piecewise linearity, which enables exact MILP reformulation: a critical requirement for MPC integration. Additionally, ReLU demonstrates superior gradient stability in shallow networks, and empirical tests revealed a 15% accuracy improvement over Leaky ReLU for thermal modeling, aligning with findings in similar low-dimensional systems. While alternatives like Swish can mitigate dying ReLU issues, their nonlinearity complicates MILP formulation, and their benefits proved negligible in this architecture. Beyond its modeling advantages, ReLU’s inherent sparsity enhances inference efficiency, making it particularly well-suited for real-time control applications.

The performance of the ReLU neural network was benchmarked against a traditional RC (resistor–capacitor) model, whose parameters of thermal resistance (R = 0.0036) and heat capacity (C = 1.577 × 10⁶) were identified by solving an optimization problem aimed at minimizing the prediction error across all temperature trajectories. While the RC model achieved high single-step prediction accuracy (MSE = 0.001), it suffered from substantial bias in multi-step forecasting with a significantly higher error (MSE = 0.2207), as illustrated in Figure 8.

In contrast, the ReLU network maintained robust performance over longer prediction horizons, yielding a considerably lower multi-step MSE of 0.0317. These results highlight the advantage of data-driven thermodynamic modeling in capturing system dynamics more accurately over extended time periods.

The Resistance–Capacitance (RC) model is commonly used for HVAC system modeling, which is used as the mathematical form of the thermal dynamics that is embedded into the optimization model. The RC model integrates thermal resistance (R) and capacitance (C) to represent heat conduction and storage within the system. Thermal resistance characterizes the resistance of transferring heat within a zone, while thermal capacity represents the ability to store heat. An A larger than C indicates that the system can absorb more heat, resulting in gradual response to variations in heat input.

We wrote the RC model in the discrete form as:

$$T_{t+1}=(1-{\displaystyle \frac{\Delta t}{RC}})T_t+{\displaystyle \frac{\Delta t}{RC}}{T}_{out}-{\displaystyle \frac{\Delta t}{C}}{\displaystyle \frac{S_t{P}_h}{COP}}$$

(29)

where $S_t$ represents the AC switch state, with $S_t=1$ being on and $S_t=0$ being off, $T_t$ represents the current indoor temperature, and $T_{out}$ is the supply air temperature.

We obtained the RC parameters of the target building using a regression approach considering the objective function in (30), which minimizes the fitting error between the predicted and actual temperatures. The objective function is the sum of squared differences between the predicted and actual values for different time steps

$$\underset{R,C}{min}{\sum }_{i=0}^N(T_i-T_i^m{)}^2$$

(30)

where $T_i^m$ is the measured temperature from the simulations, and $T_i$ is the prediction provided by (29). The constraints of optimization problems are simply

$$R\ge 0,C\ge 0$$

(31)

To estimate the optimal thermal resistance R and thermal capacitance C, we minimized the squared error between the measured indoor temperatures and the predicted values over 50 daily trajectories. The optimization problem was formulated as:

$$\underset{R>0,C>0}{\mathit{min}}\sum _{j=1}^{50}\sum _{t=1}^{24\ast 60}{\left\{T_{j,t+1}^{in}-\left[{\displaystyle \frac{∆t}{RC}}{T}_{j,t}^{out}+\left(1-{\displaystyle \frac{∆t}{RC}}\right)T_{j,t}^{in}+{\displaystyle \frac{C_P{m}^{AC}}{C}}\left(T^{set}-T_{j,t}^{in}\right)z_{j,t}^{AC}\right]\right\}}^2$$

(32)

where:

• $T_{j,t}^{in}$ and $T_{j,t}^{out}$ are the indoor and outdoor temperatures for the j-th day at minute t;
• $∆t$ = 60 is the sampling interval;
• $C_P$ is the specific heat capacity of air;
• $m^{AC}$ is the supply air mass flow rate;
• $T^{set}$ is the AC setpoint temperature;
• $z_{j,t}^{AC}$ ∈ {0,1} is the on/off status of the air conditioner.

This objective function captures both passive heat exchange through the building envelope and active cooling from the HVAC system. The parameters R and C were identified by solving this problem using numerical optimization tools in MATLAB_R2023a.

In addition to the RC model comparison shown in Figure 6, the ReLU network was tested across multiple trajectories under variable weather and load conditions. These additional tests consistently showed that ReLU maintained stable multi-step accuracy, especially during rapid temperature transitions, which are known to challenge traditional models. To offer a more complete evaluation,

Table 6. Comparison of the performance of our ReLU network with several commonly used modeling techniques.

Model	MAE (°C)	MSE (°C2)	RMSE (°C)	MAPE (%)	R2 Score
ReLU Network	0.128	0.0317	0.178	1.65	0.982
RC Model	0.305	0.2207	0.470	4.82	0.881
LSTM (Baseline)	0.196	0.0843	0.290	2.77	0.940
ARX Model	0.264	0.1510	0.389	3.92	0.904

compares the performance of our ReLU network with several commonly used modeling techniques, including the RC model, ARX model, and a basic LSTM architecture. Performance was assessed using five standard metrics over 30 min prediction horizons. The results confirmed that the ReLU model achieved the best overall performance across all metrics, with significantly lower errors and higher predictive reliability.

Although this study focused on a representative building with fixed dimensions and historical temperature data for Baghdad, the proposed ReLU–MPC framework is not limited to these conditions. Specifically, we used 50 days of outdoor temperature data from Baghdad during July and August, two of the hottest months of the year, to test the robustness of the model under extreme weather conditions. Preliminary tests on varied building sizes (e.g., smaller rooms with lower thermal mass and larger halls with high insulation) and synthetic weather scenarios (e.g., extreme heatwaves or milder coastal climates) have shown that the controller adapts well by retraining the neural network on new data. These results suggest that the framework can generalize to a broad range of building profiles and climatic conditions with minimal retraining effort. Extending this to a broader set of buildings is a promising direction for future work.

6. Analysis of Control Performance

6.1. Scenario 1: Constant Outside Temperature

This study compares the ReLU–MPC control method with the conventional 0–1 control strategy in terms of room temperature regulation and economic operation in buildings. Scenario 1 examined control effectiveness over a short period, assuming a constant outdoor temperature of 30 °C, as shown in Figure 9. At around 4:00, the price signal changed to test the system response under ReLU–MPC, showing that MPC maintained indoor temperature near the lower limit to minimize power use without exceeding limits, unlike the benchmark control which saw temperature fluctuate, failing to achieve energy savings. When incorporating time-of-use tariffs, the MPC strategy optimally activated air conditioning to avoid high electricity price hours, yielding the lowest total electricity cost. Additionally, the MPC ensured that the on–off cycling of HVAC met state retention limits, preventing frequent cycling that could shorten equipment life.

6.2. Scenario 2: Outdoor Temperature Changes over a Wide Range

Scenario 2 involved a real outdoor temperature-driven simulation experiment with actual electricity prices drawn from Table 2 for 24 h. Figure 10 illustrates changes in indoor/outdoor temperatures, electricity bills, and prices. Despite significant outdoor temperature fluctuations (blue line), data-driven MPC maintained room temperature within range while reducing HVAC operation costs. Time-of-use electricity pricing had minimal impact, as results were mainly influenced by electricity price jumps, outdoor temperature, and system constraints. Comparing MPC and MPC TOU effects during a price signal jump between 20:00 and 22:00 revealed that MPC TOU reduced HVAC startup time during flat pricing (20:30–21:00) and activated equipment during low-price periods after 21:00, ensuring effective indoor temperature regulation.

Table 7. Comparison of the levels of building economic operation in the two scenarios.

Scenario	Control Method	Savings (%)	IQD/kWh	95% CI (IQD/KWh)	Robustness Range *
1	0–1 Control	_	_	_	-
	ReLU–MPC (TOU)	31	14.2	13.6–14.8	12.8–15.1
2	0–1 Control	_	_	_	_
	ReLU–MPC (TOU)	64	29.3	28.2–30.4	27.5–30.9

* Range maintained across forecast errors, sensor noise, and occupancy variations.

provides a summary and comparison of the building’s economic operation under two different scenarios, using the electricity cost from traditional 0–1 control as a baseline. The results in the table show that the 0–1 control method lacked predictive energy management capabilities, leading to higher electricity costs across all scenarios. In contrast, ReLU–MPC maintained the indoor temperature at the lower limit, optimizing power savings within the constraints, and, consequently, its electricity costs were significantly lower than those of the traditional 0–1 control. When time-of-use (TOU) electricity pricing was considered, MPC-TOU could further reduce building electricity costs by pre-cooling indoor air before peak electricity prices took effect, with the most noticeable impact observed in Scenario 1.

The results clearly demonstrate the economic benefits of the proposed ReLU–MPC control strategy. In Scenario 1 (constant outdoor temperature), ReLU–MPC with time-of-use pricing reduced electricity costs by 31%, achieving an average price of 14.2 IQD/kWh compared to the baseline method. In Scenario 2 (variable outdoor temperatures), the savings were even more substantial, with ReLU–MPC reducing costs by 64%, achieving an average price of 29.3 IQD/kWh. These savings reflect the controller’s ability to pre-cool the building during lower-price periods and avoid operation during peak rates a key improvement over traditional 0–1 control, which lacks any price-awareness or foresight.

7. Conclusions

In this paper, we proposed a data-driven model predictive control (MPC) strategy for building HVAC systems using a multi-layer ReLU neural network. The network enabled the accurate multi-step prediction of indoor temperatures, and its piecewise affine structure allowed integration into a mixed-integer linear programming (MILP) framework for optimal control. By embedding the learned thermal dynamics into the MPC problem and incorporating time of use (TOU) electricity pricing, the controller effectively minimized energy costs while maintaining thermal comfort. Simulation results using a Simscape thermal model confirmed that the ReLU–MPC approach adapted well to varying outdoor temperatures and achieved significant electricity savings of up to 64% compared to traditional control methods. This demonstrates the practical potential of the proposed method for improving building energy efficiency in hot climates like Iraq.

8. Policy Implications

Our study demonstrates how MPC-enabled HVAC systems can support Iraq’s energy transition through three key pathways. First, for consumers, targeted tax incentives in neighborhoods equipped with Advanced Metering Infrastructure (AMI), which enables two-way communication between utilities and customers, could encourage adoption. Public education campaigns delivered through the Mawani App, Iraq’s official digital service platform for government and utility services, can highlight the real-world cost savings of 14–29 Iraqi dinars (IQD) per kilowatt-hour (kWh). These savings exceed the monetary value of current demand response incentives, making MPC systems more attractive and accessible. Second, utilities can immediately integrate our approach with existing programs like the Basra Peak Pricing Pilot, potentially doubling participant savings while leveraging installed smart meters. Third, policymakers should incorporate MPC standards into Electricity Law No. 42 revisions, following Iraq’s National Energy Strategy timeline: commercial certification by 2025, residential rollout with AMI deployment (2026–27), and nationwide implementation post-2028.

Crucially, our system’s proven reliability above 50 °C directly addresses Iraq’s extreme climate challenges. Future implementation should focus on the following: (1) deploying temperature sensor networks alongside AMI infrastructure, and (2) refining models using Baghdad’s historical climate data. This comprehensive approach creates a tailored solution for Iraq’s grid modernization needs while maximizing energy savings across all stakeholder groups.