Aggregation and Coordination Method for Flexible Resources Based on GNMTL-LSTM-Zonotope

Abstract

Demand-side flexible resources in building energy systems hold significant potential for enhancing grid reliability and operational efficiency. However, their effective coordination remains challenging due to the complexity of modeling and aggregating heterogeneous loads. To address this, this paper proposes a feasible region aggregation and coordination method for load aggregators based on a GNMTL-LSTM-Zonotope framework. A Gradient Normalized Multi-Task Learning Long Short-Term Memory (GNMTL-LSTM) model is developed to forecast the power trajectories of diverse flexible resources, including air-conditioning systems, energy storage units, and diesel generators. Using these predictions and associated uncertainty bounds, dynamic feasible regions for individual resources are constructed with Zonotope structures. To enable scalable aggregation, a Minkowski sum-based method is applied to merge the feasible regions of multiple resources efficiently. Additionally, a directionally weighted Zonotope refinement strategy is introduced, leveraging time-varying flexibility revenues from energy and reserve markets to enhance approximation accuracy during high-value periods. Case studies based on real-world office building data from Shandong Province validate the effectiveness, modeling precision, and economic responsiveness of the proposed method. The results demonstrate that the framework enables fine-grained coordination of flexible loads and enhances their adaptability to market signals. This study is the first to integrate GNMTL-LSTM forecasting with market-oriented Zonotope modeling for heterogeneous demand-side resources, enabling simultaneous improvements in dynamic accuracy, computational scalability, and economic responsiveness.

1. Introduction

With the increasing integration of renewable energy into modern power systems, the need for operational flexibility has become more critical [1]. Flexible loads, energy storage systems, electric vehicles, and other adjustable resources demonstrate great potential for balancing supply and demand, mitigating peak loads, and enhancing overall system resilience. Demand-side active participation in power scheduling has emerged as a key strategy to improve grid regulation capabilities and promote source-load synergy. In practical scheduling scenarios, grid operators typically require resources to report their adjustable power ranges, based on which the system constructs and solves an optimization model [2]. Therefore, accurately quantifying the flexibility of individual resources and efficiently aggregating them in a structured manner are essential prerequisites for enabling large-scale participation in grid regulation.

However, due to the diversity, scale, small individual capacity, and spatial distribution of demand-side resources, modeling and coordinating them one by one introduces substantial communication and computational burdens. It also hinders the development of a unified resource management framework for system-wide scheduling [3,4]. To address this, a widely adopted strategy in recent studies is “unified modeling + feasible region aggregation”, where individual resources are abstractly modeled and then aggregated into a collective pool for system-level participation. Some researchers have also introduced evaluation indices for flexibility aggregation to enhance adaptability and improve model transparency [5].

From a modeling perspective, two common approaches are the virtual generator model and the virtual energy storage model, which consider constraints such as power limits, energy states, ramp rates, and operational statuses [6,7]. For instance, heat pumps can be abstracted as virtual units with ramping constraints [8], while thermostatically controlled loads (TCLs) and electric vehicles with energy buffer characteristics are often modeled as virtual energy storage systems [9]. Additionally, some studies have constructed feasible regions directly using inequality constraints to capture power bounds, state transitions, and duration constraints, aiming to improve the realism and fidelity of the model [10].

In order to improve the efficiency of resource aggregation modeling, researchers have proposed a variety of approximate construction means. Convex polyhedron-based feasible domain expression is a class of classical methods that has good geometric interpretability in low-dimensional scenarios but suffers from high combinatorial complexity and unstable expression structure under high-dimensional conditions [11]. For this reason, region approximation by means of an external ellipsoid has been proposed in the literature [12], or resource aggregation approximation models are constructed based on polyhedral boundary contraction and projection mapping strategies [13]. Meanwhile, techniques such as virtual battery model [14], Gaussian kernel [15], and the inner-box approximation model [16] are also widely used for fast construction of flexible boundaries of resources. However, most of the above methods take the same kind of resources as the modeling object, and it is still difficult to meet the modeling demand of distributed heterogeneous resource co-regulation. In addition, some studies have also proposed the integration mechanism of fusion constraint space integration, which introduces the engineering regulation interface in the construction of response structure to effectively improve modeling ability [17].

Recently, Zonotope has attracted considerable attention due to its favorable mathematical properties, such as closure under Minkowski sum, central symmetry, and scalability [18]. These features allow efficient high-dimensional feasible region aggregation, making Zonotope a promising tool for flexibility modeling [19]. Prior studies have demonstrated that Zonotope aggregation achieves a good trade-off between modeling accuracy and computational efficiency. However, most existing Zonotope methods rely on idealized structures and fail to capture dynamic behavior or multi-temporal coupling in practical demand response applications [20]. Some improvements have been made via projection optimization and generator matrix reconfiguration, showing enhanced structural stability in specific scenarios [21].

Despite these advancements, traditional methods for flexible resource aggregation, such as virtual synchronous machine (VSM) models and Zonotope-based approaches, still exhibit several critical limitations:

• They are primarily designed for static constraint modeling and thus cannot effectively capture temporal couplings, state transitions, or ramping characteristics inherent in flexible resources.
• Their construction processes often require solving numerous linear programming problems, leading to increased computational burden and reduced scalability in large-scale applications.
• Most methods overlook the temporal value of flexibility, which may result in inefficient allocation of high-value flexibility resources and thereby compromise economic performance.

To overcome these limitations, this paper proposes a novel feasible region aggregation method based on a GNMTL-LSTM-Zonotope framework. The main purpose of this study is to develop a unified and efficient modeling framework that can dynamically capture the operational behaviors of heterogeneous flexible demand-side resources, accurately represent their high-dimensional constraints, and incorporate the economic value of flexibility over time. This framework aims to improve prediction accuracy and aggregation efficiency, while enabling market-oriented flexibility scheduling to enhance both the reliability and economic performance of power systems. Section 6 provides comparative results against common aggregation approaches to demonstrate the superiority of the proposed method. The main contributions are summarized as follows:

• A dynamic feasible region prediction framework is developed by integrating a GNMTL-LSTM network, enabling the capture of operational constraints over time and addressing the limitations of static Zonotope modeling.
• An approximate feasible region for individual resources is constructed using the GNMTL-LSTM-Zonotope approach. A market price–based weighting mechanism is introduced to reflect the economic value of flexibility across different time periods, and the Minkowski sum is applied for cluster-level aggregation.
• An optimization model is formulated to maximize the economic benefits of the load aggregator, which coordinates real-time control signals to dispatch flexible resources accordingly.
• Case studies involving diverse distributed energy resources demonstrate the advantages of the proposed method in terms of modeling accuracy, aggregation efficiency, and economic performance.

The novelty of this work lies in the first integration of GNMTL-LSTM forecasting with market-oriented Zonotope modeling for heterogeneous demand-side resources. This integration enables simultaneous improvements in dynamic modeling accuracy, computational scalability, and economic responsiveness. The main research object of this paper relates to medium-scale building clusters, and further research is needed to assess the scalability and adaptability of the approach in larger, more complex energy systems.

The structure of this paper is arranged as follows. Section 2 introduces the multi-type building load aggregation scenario and the aggregation–scheduling framework. Section 3 establishes feasible regional models for various types of resources. Section 4 proposes the GNMTL-LSTM-Zonotope-based feasible region characterization and aggregation method, including dynamic prediction, Zonotope construction, and market-oriented weighting strategies. Section 5 formulates the scheduling optimization model under the aggregated feasible region. Section 6 presents case studies based on real-world data to validate the effectiveness of the proposed method. Section 7 concludes the paper and outlines future research directions.

2. Flexible Resource Aggregation Regulatory Framework

This study aims to develop a unified modeling and control framework for heterogeneous flexible demand-side resources to enhance system reliability and economic performance through dynamic modeling and efficient aggregation.

Grounded in the scientific hypothesis that heterogeneous demand-side resources—when modeled dynamically and aggregated efficiently—can provide temporally adaptive flexibility, this research aligns with the United Nations Sustainable Development Goals (SDGs), particularly SDG 7 (Affordable and Clean Energy) and SDG 13 (Climate Action), aiming to improve renewable energy accommodation and reduce reliance on carbon-intensive generation.

Figure 1 illustrates the proposed flexible resource aggregation and control framework. This study employs an efficient feasible region aggregation technique based on GNMTL-LSTM-Zonotope modeling to aggregate multiple types of flexible resources within a load aggregator into a unified resource cluster. The load aggregator receives both external electricity market information and internal state data to perform coordinated optimization and control of its internal flexible resources. Upon completion of the optimization process, the aggregator dispatches control signals to the resource side to implement energy management of the internal resource cluster.

Moreover, the proposed GNMTL-LSTM-Zonotope-based framework is not limited to office buildings. Since the method does not depend on specific device types or control infrastructures, it can be extended to other flexible demand-side resources such as commercial complexes, hospitals, or district-level energy systems. This universality allows the method to serve as a foundational approach for large-scale coordination of diverse flexible resources.

The time step for aggregation and optimization control is set to 1 h, assuming constant device power output within each time interval. Communication latency between entities in the system framework is in the millisecond range and is therefore considered negligible in this study.

The detailed implementation process for flexible resource aggregation and control in the context of a load aggregator is as follows:

• Development of dynamic electrical models for various types of flexible resources within the aggregator to characterize their respective feasible control regions;
• Approximation of the original feasible region of individual flexible resources using the GNMTL-LSTM-Zonotope method;
• Considering demand-side scenarios with complex constraints and time-varying flexibility values, proposition of a correction strategy for the aggregated feasible region model and application the Minkowski sum to obtain the aggregated feasible region of the flexible resources;
• Formulation and solution of an optimization model aimed at maximizing the overall economic benefit of the aggregator, and issuing of control commands to flexible resources for day-ahead scheduling.

This research focuses on a load aggregator within a building energy system, targeting unified modeling and control strategies for representative flexible loads such as air conditioning, energy storage systems, and distributed generation units. To address challenges such as inconsistent resource responses and complex control couplings, a unified feasible region expression and optimal control method based on GNMTL-LSTM-Zonotope is proposed. An integrated framework encompassing modeling, aggregation, correction, and scheduling is established to support coordinated resource management and control.

3. Characterization of Feasible Regions for Aggregator Flexible Resources

To support flexible aggregation and coordinated control, it is essential to first characterize the feasible operating regions of distributed energy resources under the load aggregator. These regions, defined by physical and operational constraints, form the mathematical basis for subsequent Zonotope-based aggregation and optimization. This chapter develops linear models for three representative resource types using inequality-based formulations to enable scalable and unified approximation.

3.1. Air Conditioning Loads

As a typical distributed resource, air-conditioning systems exhibit considerable potential for controllability in temperature regulation. The dynamic thermal behavior between indoor and outdoor environments is commonly described using a first-order equivalent thermal parameter model for individual air conditioners, which is formulated as follows:

$${\displaystyle \frac{\mathrm{d}{T}^{\mathrm{in}}}{\mathrm{d}t}}={\displaystyle \frac{T^{\mathrm{out}}-T^{\mathrm{in}}}{R_{\mathrm{in}}{C}_{\mathrm{in}}}}-{\displaystyle \frac{Q_{\mathrm{ac}}}{C_{\mathrm{in}}}}$$

(1)

$$Q_{\mathrm{ac}}={\eta }_{\mathrm{ac}}{P}_{\mathrm{ac}}$$

(2)

where Tⁱⁿ denotes the indoor temperature, T^out represents the outdoor temperature, and Q_ac is the cooling capacity of the air-conditioning unit. R_in and C_in are thermodynamic parameters representing the thermal resistance of the indoor space and the thermal capacitance of the indoor air, respectively. η_ac is the coefficient of performance (COP) of the air conditioner, and P_ac is its electric power consumption.

Based on Equation (1), the dynamic electrical model of the air conditioner can be derived as follows:

$$T_{t+1}^{\mathrm{in}}=T_t^{\mathrm{in}}{\mathrm{e}}^{-{\displaystyle \frac{\Delta t}{R_{\mathrm{m}}{C}_{\mathrm{m}}}}}+({\eta }_{\mathrm{ac}}{R}_{\mathrm{in}}{P}_t^{\mathrm{ac}}+T_t^{\mathrm{out}})\left(1-{\mathrm{e}}^{-{\displaystyle \frac{\Delta t}{R_{\mathrm{m}}{C}_{\mathrm{m}}}}}\right)$$

(3)

$$T_{{\min }^{\mathrm{in}}}⩽T_t^{\mathrm{in}}⩽T_{{\max }^{\mathrm{in}}}$$

(4)

$$0⩽P_t^{ \mathrm{ac}}⩽P_{ \max }^{ \mathrm{ac}}$$

(5)

Let $T_t^{in}$ denote the indoor temperature at time step t, $T_{\mathrm{t}}^{\mathrm{out}}$ the outdoor temperature, and $P_{ac t}$ the power consumption of the air conditioner at time t. Δt is the time step duration, fixed to 1 h in this study. Tin max and Tin min represent the upper and lower bounds of the indoor thermal comfort range, respectively. Pac max is the rated maximum power of the air conditioner.

During the aggregation and regulation of air conditioners as flexible resources, parameters such as thermal characteristics, efficiency, and outdoor temperature are treated as constants. Thus, Equation (3) can be simplified as follows:

$$T_{t+1}^{\mathrm{in}}=k_1{T}_t^{\mathrm{in}}+(1-k_1)T_t^{\mathrm{out}}-k_2{P}_t^{\mathrm{ac}}$$

(6)

Taking a two-period example, all operational constraints of the air conditioner can be reformulated into a half-space representation of the form Ax ≤ b, where x is the vector of control variables, A is the constraint coefficient matrix, and b is the vector of constant terms. The corresponding feasible region is expressed as follows:

$$\left[\begin{array}{cc}1& 0\\ -1& 0\\ 0& 1\\ 0& -1\\ -k_2& 0\\ -k_1{k}_2& -k_2\\ k_1{k}_2& k_2\end{array}\right]\left[\begin{array}{l}{P}_1^{\mathrm{ac}}\\ P_2^{\mathrm{ac}}\end{array}\right]\le \left[\begin{array}{l}{P}_1^{\mathrm{ac}}\\ 0\\ P_{\max }^{\mathrm{ac}}\\ 0\\ T_{\max }^{\mathrm{in}}-k_1^1{T}_1^{\mathrm{in}}-{\displaystyle \sum _{i=1}^1{k}_1^{1-i}}(1-k_1)T_i^{\mathrm{out}}\\ -T_{\min }^{\mathrm{in}}+k_1^1{T}_1^{\mathrm{in}}+{\displaystyle \sum _{i=1}^1{k}_1^{1-i}}(1-k_1)T_i^{\mathrm{out}}\\ T_{\max }^{\mathrm{in}}-k_1^2{T}_1^{\mathrm{in}}-{\displaystyle \sum _{i=1}^2{k}_1^{2-i}}(1-k_1)T_i^{\mathrm{out}}\\ -T_{\min }^{\mathrm{in}}+k_1^2{T}_1^{\mathrm{in}}+{\displaystyle \sum _{i=1}^2{k}_1^{2-i}}(1-k_1)T_i^{\mathrm{out}}\end{array}\right]$$

(7)

The feasible region described in Equation (7) treats the air conditioner as a virtual energy storage system, where the difference between the baseline power and the actual operating power is interpreted as the virtual charging or discharging power. The baseline power refers to the power required to maintain the indoor temperature at a fixed reference level.

3.2. Energy Storage Systems

The charging and discharging processes of energy storage systems can be freely controlled and are not affected by environmental factors. The corresponding model for energy storage operation is formulated as follows.

• Charging and Discharging Power Constraints

$$0\le p_{\mathrm{es},m,t}^{\mathrm{c}}\le {\lambda }_{m,t}^{\mathrm{c}}{p}_{\mathrm{es},m,t}^{\mathrm{c},\max }$$

(8)

$$0\le p_{\mathrm{es},m,t}^{\mathrm{d}}\le {\lambda }_{m,t}^{\mathrm{d}}{p}_{\mathrm{es},m,t}^{\max }$$

(9)

$${\lambda }_{m,t}^{\mathrm{c}}+{\lambda }_{m,t}^{\mathrm{d}}\le 1$$

(10)

$$p_{m,t}^{\mathrm{d}}{p}_{m,t}^{\mathrm{c}}=0$$

(11)

where $p_{\mathrm{e}\mathrm{s},\mathrm{m},\mathrm{t}}^{\mathrm{c}}$ and $p_{\mathrm{e}\mathrm{s},\mathrm{m},\mathrm{t}}^{\mathrm{d}}$ denote the charging and discharging power of storage unit m at time t, respectively; $p_{\mathrm{e}\mathrm{s},\mathrm{m},\mathrm{t}}^{\mathrm{c},\mathrm{m}\mathrm{a}\mathrm{x}}$ and $p_{\mathrm{e}\mathrm{s},\mathrm{m},\mathrm{t}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ are the upper bounds on charging and discharging power; and ${\lambda }_{\mathrm{m},\mathrm{t}}^{\mathrm{c}}$ and ${\lambda }_{\mathrm{m},\mathrm{t}}^{\mathrm{d}}$ are binary variables indicating whether the unit is in charging or discharging mode at time t. Equation (11) enforces mutual exclusivity between charging and discharging states during any single time period.

2.

Energy Constraints

$$E_{\mathrm{es},m,t}=(1-{\gamma }_m)E_{\mathrm{es},m,t-1}+(p_{\mathrm{es},m,t}^{\mathrm{c}}{\eta }_m^{\mathrm{c}}-p_{\mathrm{es},m,t}^{\mathrm{d}}/{\eta }_m^{\mathrm{d}})\Delta t$$

(12)

$$E_{\mathrm{es},m}^{\min }\le E_{\mathrm{es},m,t}\le E_{\mathrm{es},m}^{\max }$$

(13)

where E_es,m,t−1 and E_es,m,t are the stored energy levels of storage unit m at time steps t − 1 and t, respectively; $E_{\mathrm{e}\mathrm{s},\mathrm{m}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $E_{\mathrm{e}\mathrm{s},\mathrm{m}}^{\mathrm{m}\mathrm{i}\mathrm{n}}$ represent the upper and lower bounds of energy storage; γ_m is the self-discharge rate; ${\eta }_{\mathrm{m}}^{\mathrm{c}}$ and ${\eta }_{\mathrm{m}}^{\mathrm{d}}$ are the charging and discharging efficiencies; and Δt is the duration of a single time period.

3.

Ramping Constraints

$$0\le \left|p_{\mathrm{es},m,t}^{\mathrm{c}}-p_{\mathrm{es},m,t-1}^{\mathrm{c}}\right|\le \Delta p_{\mathrm{es},m}^{\mathrm{c},\max }$$

(14)

$$0\le \left|p_{\mathrm{es},m,t}^{\mathrm{d}}-p_{\mathrm{es},m,t-1}^{\mathrm{d}}\right|\le \Delta p_{\mathrm{es},m}^{\mathrm{d},\max }$$

(15)

Here, Δ$p_{\mathrm{e}\mathrm{s},\mathrm{m}}^{\mathrm{c},\mathrm{m}\mathrm{a}\mathrm{x}}$ and Δ$p_{\mathrm{e}\mathrm{s},\mathrm{m}}^{\mathrm{d},\mathrm{m}\mathrm{a}\mathrm{x}}$ are the upper bounds on the ramping rate for charging and discharging power of storage unit mmm, respectively.

Taking a two-period case as an example, we unify the charging and discharging power as p_es,m,t, the charging/discharging efficiency as η_m, and the ramping constraint as Δ$p_{\mathrm{e}\mathrm{s},\mathrm{m}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$.

The energy levels of storage unit mmm over two time periods can then be expressed in terms of the charging/discharging power as follows:

$$E_{\mathrm{es},m,1}=(1-{\gamma }_m)E_{\mathrm{es},m,0}+{\eta }_m\Delta t{p}_{\mathrm{es},m,1}$$

(16)

$$\begin{array}{l}{E}_{\mathrm{es},m,2}=(1-{\gamma }_m)E_{\mathrm{es},m,1}+{\eta }_m\Delta t{p}_{\mathrm{es},m,2}\\ \hspace{1em}\hspace{1em} ={(1-{\gamma }_m)}^2{E}_{\mathrm{es},m,0}+(1-{\gamma }_m){\eta }_m\Delta t{p}_{\mathrm{es},m,1}+{\eta }_m\Delta t{p}_{\mathrm{es},m,2}\hfill \end{array}$$

(17)

All constraints can be reformulated into the standard form A_p ≤ b. Taking a two-period case as an example, the feasible operating region of the energy storage system can be characterized as

$$\left|\begin{array}{cc}1& 0\\ -1& 0\\ 0& 1\\ 0& -1\\ 1& -1\\ -1& 1\\ {\eta }_m\Delta t& 0\\ -{\eta }_m\Delta t& 0\\ (1-{\gamma }_m){\eta }_m\Delta t& {\eta }_m\Delta t\\ -(1-{\gamma }_m){\eta }_m\Delta t& -{\eta }_m\Delta t\end{array}\right|\left|\begin{array}{c}{p}_{es,m,1}\\ p_{es,m,2}\end{array}\right|\le \left|\begin{array}{c}{p}_{es,m,1}^{c,max}\\ \\ -p_{es,m,1}^{d,max}\\ \\ p_{es,m,2}^{max}\\ \\ -p_{es,m,2}^{d,max}\\ \\ \Delta p_{es,m}^{max}\\ \\ \Delta p_{es,m}^{max}\\ \\ E_{es,m}^{max}-(1-{\gamma }_m)E_{es,m,0}\\ \\ -E_{es,m}^{min}+(1-{\gamma }_m)E_{es,m,0}\\ \\ E_{es,m}^{max}-{(1-{\gamma }_m)}^2{E}_{es,m,0}\\ \\ -E_{es,m}^{max}+{(1-{\gamma }_m)}^2{E}_{es,m,0}\end{array}\right|$$

(18)

It can be observed that due to the need to account for energy layering and charging/discharging losses, the operational constraint coefficients of the devices become more complex. The proposed model incorporates loss factors and explores a generalized formulation method for Zonotope generators, enabling a more accurate representation of the geometric structure of complex feasible regions.

3.3. Diesel Generators

Although diesel generators typically serve as emergency backup power sources in conventional office buildings, they possess controllable start–stop flexibility and cost-effectiveness in the context of the large-scale building energy systems considered in this study. These units are powered by diesel combustion and exhibit operational characteristics that differ significantly from load-side resources such as air conditioners and energy storage systems. Specifically, their output is unidirectional (power generation only), and their operation must comply with ramping and start–stop constraints. The generator’s output characteristics are influenced by fuel calorific value fluctuations, combustion efficiency, and mechanical inertia. In practical operation, strict ramping limits must be enforced to ensure equipment safety, and the generator’s output range is constrained by the following equations:

$$P_{\min }^{\mathrm{dg}}⩽P_t^{\mathrm{dg}}⩽P_{\max }^{\mathrm{dg}}$$

(19)

$$\mathsf{\Delta }{P}_{\min }^{\mathrm{dg}}⩽P_t^{\mathrm{dg}}-P_{t-1}^{\mathrm{dg}}⩽\mathsf{\Delta }{P}_{\max }^{\mathrm{dg}}$$

(20)

Here, $P_{\mathrm{dg} t}$ denotes the output power of the diesel generator at time t; $P_{\max }^{\mathrm{dg}}$ and $P_{\min }^{\mathrm{dg}}$ represent the allowable upper and lower output limits, respectively; and Δ$P_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{d}\mathrm{g}}$ and Δ$P_{\mathrm{m}\mathrm{i}\mathrm{n}}^{\mathrm{d}\mathrm{g}}$ define the maximum ramp-up and ramp-down limits for a single time step.

In addition, the generation cost of the diesel unit consists of fuel procurement, equipment wear, and carbon emission costs. To account for the operational economics of diesel generators, the unit cost of power generation is modeled using a quadratic function. Based on this, the total operating cost of the system C₀ is expressed as

$$C_{\mathrm{o}}=\sum _{t=1}^T\left[a^{\mathrm{dg}}{\left(P_t^{\mathrm{dg}}\right)}^2+b^{\mathrm{dg}}{P}_t^{\mathrm{dg}}+c^{\mathrm{dg}}\right]$$

(21)

Here, T denotes the total number of time periods considered for system operation, typically set to 24 to represent a full-day scheduling cycle. The coefficients a^dg, b^dg and c^dg correspond to the fuel consumption nonlinearity, marginal cost per unit of power, and fixed start-up cost of a single diesel generator, respectively.

Based on the above analysis, under a two-period framework, the operational feasible region of the diesel generator can be characterized by the following mathematical expressions:

$$\left[\begin{array}{cc}\hfill 1& \hfill 0\\ \hfill -1& \hfill 0\\ \hfill 0& \hfill 1\\ \hfill 0& \hfill -1\\ \hfill -1& \hfill 1\\ \hfill 1& \hfill -1\end{array}\right]\left[\begin{array}{c}{P}_1^{\mathrm{dg}}\\ P_2^{\mathrm{dg}}\end{array}\right]⩽\left[\begin{array}{c}\hfill P_{\max }^{\mathrm{dg}}\\ \hfill -P_{\min }^{\mathrm{dg}}\\ \hfill P_{\max }^{{\mathrm{dg}}_2}\\ \hfill -P_{\min }^{\mathrm{dg}}\\ \hfill \mathsf{\Delta }{P}_{\max }\\ \hfill -\mathsf{\Delta }{P}_{\min }^{\mathrm{dg}}\end{array}\right]$$

(22)

3.4. Demand-Side Resources

Consider a demand-side device j with an operating power p_j,t at time period t. The flexibility feasible region of device j can be described by a combination of single-period constraints and multi-period coupling constraints, as follows.

• Power Constraint

Assuming the device operates at a constant power level within each time interval t, the power constraint can be written as

$$p_j^{\min }⩽p_{j,t}⩽p_j^{\max }\hspace{1em}t=1,2,\dots ,T$$

(23)

where p_j,t is the constant power of device j at time t; T is the set of all scheduling periods; and $p_{\mathrm{j}}^{\mathrm{m}\mathrm{i}\mathrm{n}}$ and $p_{\mathrm{j}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ denote the minimum and maximum allowable operating power of device j, respectively.

2.

Energy Constraint

$$e_j^{\min }⩽e_{j,t}⩽e_j^{\max }\hspace{1em}t=1,2,\dots ,T$$

(24)

where $e_{\mathrm{j}}^{\mathrm{m}\mathrm{i}\mathrm{n}}$ and $e_{\mathrm{j}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ denote the lower and upper bounds on the cumulative energy consumption of device j. Since the case t = 1 is already addressed in the power constraint, the energy constraint can be reformulated as

$$e_j^{\min }⩽{\displaystyle \sum _{i=1}^t{p}_{j,i}}⩽e_j^{\max }\hspace{1em}t=2,3,\dots ,T$$

(25)

3.

Ramping Constraint

In demand response applications, the rate of power change between time periods is an important operational consideration. The ramping constraint that governs the power variability of device j is expressed as

$$r_j^{\min }⩽p_{j,t}-p_{j,t-1}⩽r_j^{\max }\hspace{1em}t=2,3,\dots ,T$$

(26)

where $r_{\mathrm{j}}^{\mathrm{m}\mathrm{i}\mathrm{n}}$ and $r_{\mathrm{j}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ are the lower and upper bounds on the allowable power change between consecutive periods for device j.

Observing that constraints Equation (24) through Equation (27) are all linear in power variables, the feasible region of device j can be compactly represented as a set of linear inequalities:

$$P=\left\{p\in R^N:A_P⩽b\right\}$$

(27)

where P denotes the convex polytope representing the feasible operating region of the device, N is the number of decision variables (typically the number of time periods), and (A, b) encapsulate the linear constraint matrix and corresponding constant vector, respectively.

4.

State Constraint

The operational state of a flexible resource may vary with its power usage—for example, the state of charge of a battery or the internal temperature of a thermal system. In this study, the state evolution is assumed to follow a linear relationship with the current state variable x(t), an auxiliary variable u(t), and the power p(t), given by

$${\displaystyle \frac{\mathrm{d}x(t)}{\mathrm{d}t}}=ax(t)+bu(t)+cp(t)$$

(28)

where a, b, and c are dynamic system parameters. For example, in energy storage systems, a ≤ 0 indicates self-discharge, while b and c reflect the conversion efficiency of auxiliary variables and power into state transitions. After discretization, the state constraint can be written as

$$x_{k+1,\min }\le (a+1)x_k+b{u}_k+c{p}_k\le x_{k+1,\max },k=0,\dots ,N-1$$

(29)

To reduce the complexity of decision making at the system operator level, the load aggregator aggregates individual users’ feasible regions to form a collective feasible region for the user cluster. The purpose of the above modeling of distributed resources is to facilitate subsequent aggregation and centralized coordination. The aggregated feasible region represents the overall adjustable output range of all flexible resources under coordinated control. Mathematically, the feasible region aggregation problem is equivalent to performing a Minkowski sum (denoted ⊕) over the individual feasible regions, where P^J is the aggregated feasible region, and each p^j ∈ P^J denotes a feasible point within the feasible region of user j.

Figure 2 presents a schematic of the Minkowski sum over a two-period decision horizon involving two users. The individual feasible regions of User 1 and User 2 are denoted P¹ and P², respectively. Their aggregation yields a joint feasible region, illustrated as a trapezoid, whose geometric complexity increases compared to the individual regions. In essence, centralized coordination aims to identify an optimal point—i.e., a feasible power trajectory—within this aggregated region. In practice, load aggregation and control by aggregators are typically executed over a 24 h horizon, resulting in a high-dimensional feasible region (24 dimensions), which poses significant computational challenges for Minkowski summation.

In this chapter, to meet the regulation demands of real building energy systems, operational models are developed for three representative categories of flexible resources: thermal (air conditioners), electrochemical (energy storage systems), and demand-responsive (general controllable loads and diesel generators). Key operational characteristics such as power constraints, energy dynamics, and state transitions are abstracted and unified into a set of linear inequality representations. These original feasible regions are then aggregated using Minkowski summation to construct the operational boundary of the collective cluster. However, the exponential growth in computational complexity with increasing dimensionality necessitates the development of an efficient approximation method. The next chapter addresses this challenge by introducing an improved Zonotope-based modeling approach for feasible region aggregation with controllable structure and high computational efficiency.

4. Aggregation Method of Feasible Regions for Load Aggregators Based on GNMTL-LSTM-Zonotope

This chapter proposes a feasible region characterization and aggregation method for flexible resources, integrating GNMTL-LSTM prediction with Zonotope-based approximation modeling. By employing a multi-task learning LSTM model, the operational boundaries of air-conditioning systems, energy storage units, and diesel generators are predicted, enabling the construction of a dynamically driven Zonotope representation of the feasible region. On this basis, a generator optimization and precision control mechanism is designed to enhance both modeling efficiency and representational accuracy. Finally, a flexibility value-weighting strategy based on market prices is introduced to adjust the aggregation process. The Minkowski sum is applied to aggregate multiple resources, optimizing the approximation results in high-value time periods.

4.1. Overview of Traditional Zonotopes

A Zonotope is a class of centrally symmetric convex polytope characterized by favorable geometric properties and strong mathematical tractability. Compared to conventional polyhedral representations, the Zonotope employs a generator matrix-based parameterization (Z-representation), which enables efficient and unified modeling of feasible regions across multiple resources and time periods, while also supporting computationally efficient aggregation operations.

The Zonotope is particularly suited for flexible resource aggregation because it avoids the combinatorial complexity growth seen in high-dimensional polyhedral methods, allows direct incorporation of power, energy, and ramping constraints, and does not require explicit probabilistic assumptions. These advantages make it robust for heterogeneous resource modeling under practical engineering conditions. Unlike traditional polytopes expressed in half-space form (H-representation) or vertex form (V-representation), the Zonotope is a special type of polytope defined by a distinct mathematical structure known as the Z-representation. Its formal representation is given by [22]

$$Z=\left\{x\in {\mathrm{R}}^N:x=c+G\beta ,-\overline{\beta }⩽\beta ⩽\overline{\beta }\right\}$$

(30)

$$G=\left[g^{\left(1\right)},g^{\left(2\right)},\dots ,g^{\left(M\right)}\right]\in {\mathrm{R}}^{N\times M}$$

(31)

$$\parallel g^{(m)}{\parallel }_2=1\hspace{1em}m=1,2,\dots ,M$$

(32)

$$\beta ={\left[{\beta }_1,{\beta }_2,\dots ,{\beta }_M\right]}^{\mathrm{T}}$$

(33)

Here, N is the dimension of the space; c represents the center point of the polytope; G is the generator matrix composed of M generator vectors g^(m), which define the directions in which the polytope extends from its center; and β is the vector of extension lengths along each generator direction, bounded by their respective maximum values. $\parallel \cdot {\parallel }_2$ denotes the Euclidean (L2) norm.

In flexible resource modeling, Zonotopes can be employed to approximate the feasible output regions of resources such as air-conditioning systems, energy storage units, and generators under operational constraints, including power limits, energy constraints, and ramping capabilities. By representing the operating ranges of various resources at different time intervals as Zonotopes and leveraging their closure properties under Minkowski addition, feasible region aggregation can be effectively achieved.

When Zonotopes share the same generator matrix G, their Minkowski sum can be significantly simplified due to the alignment of their extension directions. Specifically, the aggregation process benefits from this shared structure, leading to a more computationally efficient summation procedure:

$$Z^{\mathrm{agg}}=Z_1\oplus Z_2\oplus \dots \oplus Z_n$$

(34)

$$c^{\mathrm{agg}}={\displaystyle \sum }_{s=1}^{N_c}{c}_s$$

(35)

$${\beta }_m^{\mathrm{agg}}={\displaystyle \sum }_{s=1}^{N_c}{\beta }_{m,s}$$

(36)

Here, Z^agg denotes the aggregated Zonotope; ⊕ represents the Minkowski sum operator; N_c is the number of individual feasible regions being aggregated, i.e., the number of flexible resources; c^agg is the center of the aggregated Zonotope; c_s is the center of the s individual Zonotope; β denotes the total extension length in the m generator direction for the aggregated Zonotope; and β_m,s represents the extension length of the s Zonotope in the m generator direction before aggregation.

4.2. Dynamic Feasible Region Characterization Method Based on GNMTL-LSTM-Zonotope

4.2.1. Construction of the GNMTL-LSTM Prediction Model

Considering the coupling among air-conditioning systems, energy storage units, and diesel generators, an MTL approach is adopted to enhance prediction accuracy and generalization capability. This is achieved by using shared layers to learn and extract auxiliary coupling information from related subtasks. The task learning process is implemented using LSTM neural networks, which possess memory capabilities.

LSTM networks incorporate gating mechanisms that allow the selective retention or forgetting of past information. Through iterative training and extraction of features from multiple hidden layers within the LSTM, the model is capable of capturing and learning the complex internal relationships within the input data. The structure of the proposed GNMTL-LSTM model is illustrated in Figure 3.

Due to the differing magnitudes of loss gradients across tasks during multi-task learning, tasks with smaller gradient magnitudes contribute less to the parameter updates in the shared layers during backpropagation, resulting in insufficient learning and underfitting of the shared representation. Conversely, tasks with larger gradient magnitudes tend to dominate the training process, potentially causing overfitting on those tasks and leading to reduced overall prediction accuracy for the aggregated load.

To address this issue, gradient normalization is introduced. The core idea is to penalize the network whenever the backpropagated loss gradient of any task becomes excessively large or small. This mechanism helps achieve a proper balance of loss gradients when all tasks train at a similar pace. If a task trains relatively faster, its loss coefficient is reduced, thereby allowing other tasks to exert greater influence during training.

The loss function of the prediction model is expressed as Equation (37):

$$L(t)={\displaystyle \sum w_i}(t)L_i(t)$$

(37)

Here, L(t) and L_i(t) represent the total loss of all tasks and the loss of task i at time t, respectively; w_i(t) denotes the loss weight of task i at time t.

To measure the magnitude of the loss gradients, the gradient norm G_Wi(t) for task i at time t and the average gradient norm $\overline{G_{\mathrm{W}}}(t)$ across all tasks are calculated, as shown in Equations (38) and (39). To reduce computational overhead, the weights from the last shared layer are used for these calculations.

$$G_{\mathrm{W}i}(t)=\sqrt{{\displaystyle \sum _j{\left[\frac{\partial (w_i(t)L_i(t))}{\partial W_j}\right]}^2}}$$

(38)

$$\overline{G_{\mathrm{W}}}(t)={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{G}_{\mathrm{W}i}}(t)$$

(39)

Here, W_j represents the parameters of the LSTM neural network; N denotes the total number of subtasks in the multi-task learning framework.

To evaluate the training rate of each task, the loss decay rate v_i(t) is calculated as the ratio of the loss L_i(t) of task i at time t to its initial loss L_i(0). Additionally, the relative change rate of the decay rate r_i(t) compared to the expected value is computed, as shown in Equations (40) and (41).

$$v_i(t)={\displaystyle \frac{L_i(t)}{L_i(0)}}$$

(40)

$$r_i(t)={\displaystyle \frac{v_i(t)}{E_{\mathrm{task}}[v_i(t)]}}$$

(41)

Here, Etask represents the average loss decay rate across all tasks.

A larger value of r_i(t) corresponds to a higher gradient magnitude, encouraging the task to train faster. Therefore, the expected gradient norm target for each task i is defined as follows:

$$G_{\mathrm{W}i}(t)\to \overline{G_{\mathrm{W}}}(t){\left[\begin{array}{c}{r}_i(t)\end{array}\right]}^a$$

(42)

Here, a is a hyperparameter that balances the training speeds of different tasks and represents the strength of the task’s resilience to returning to the normal training rate.

Based on the difference between the actual and target gradient norms, denoted GradLoss, the loss weights w(t) are dynamically adjusted to achieve balanced training across tasks, as shown in Equations (43) and (44).

$$G_{\mathrm{radLoss}}={\displaystyle \sum _i\mid }{G}_{\mathrm{W}i}(t)-\overline{G_{\mathrm{W}}}(t){[r_i(t)]}^a\mid$$

(43)

$$w_i(t+1)=w_i(t)+\lambda {\displaystyle \frac{\mathrm{d}{G}_{\mathrm{radLoss}}}{\mathrm{d}w(t)}}$$

(44)

Here, w_i(t + 1) represents the adjusted loss weight, and λ denotes the learning rate of the neural network.

4.2.2. Selection of Zonotope Generators

After modeling the load time-series data of flexible resources such as air conditioners, energy storage systems, and diesel generators, this study introduces the Zonotope as an approximate representation of the dynamic feasible operating boundaries of these resources.

Generators define the directions along which a Zonotope extends. For accurately approximating the original feasible regions of the resource models developed earlier, the selection of generator directions is critical to ensuring geometric fidelity.

In the case of air conditioning loads and energy storage systems, the corresponding models primarily involve power constraints and energy constraints. Power constraints impose bounds on the power variables at each time step, while energy constraints reflect the temporal coupling among those variables. The generator set designed to approximate the feasible regions of such resources is constructed as follows [23]:

$$g^{(m)}={\left[\begin{array}{c}\\ 0,\dots ,0,\underset{m}{\underbrace{1}},0,\dots ,0\end{array}\right]}^{\mathrm{T}}\hspace{1em}m=1,2,\dots ,N$$

(45)

$$g^{(N+m)}={\left[\begin{array}{c}\\ 0,\dots ,0,{\displaystyle \frac{-1}{\underset{m}{\underbrace{\sqrt{2}}}}},{\displaystyle \frac{1}{\underset{m+1}{\underbrace{\sqrt{2}}}}},0,\dots ,0\end{array}\right]}^{\mathrm{T}}m=1,2,\dots ,N-1$$

(46)

Here, N denotes the dimensionality of the scheduling horizon, and each generator is an N-dimensional vector.

Equation (45) includes N generators, each corresponding to the extension direction at a specific time period i, independent of other time steps. These generators thus represent the power constraints. Equation (46) includes N − 1 generators, each capturing the cumulative power over adjacent time steps, which makes them suitable for approximating energy constraints [24]. In total, there are 2N − 1 generators for modeling both constraint types.

In contrast to air-conditioning and storage systems, diesel generators must account for both power and ramping constraints. Therefore, the generators in Equation (47) are retained to approximate power constraints, while Equation (48) defines the generators associated with ramping constraints.

$$g^{(m)}={\left[\begin{array}{c}\\ 0,\dots ,0,\underset{m}{\underbrace{1}},0,\dots ,0\end{array}\right]}^{\mathrm{T}}\hspace{1em}m=1,2,\dots ,N$$

(47)

$$g^{(N+m)}={\left[\begin{array}{c}\\ 0,\dots ,0,{\displaystyle \frac{1}{\underset{m}{\underbrace{\sqrt{2}}}}},{\displaystyle \frac{1}{\underset{m+1}{\underbrace{\sqrt{2}}}}},0,\dots ,0\end{array}\right]}^{\mathrm{T}}m=1,2,\dots ,N-1$$

(48)

Equation (48) consists of N − 1 generators, each representing the difference in power between the current time step and the subsequent one. These generators are thus used to capture the ramping constraints of diesel generators.

4.2.3. Definition of Accuracy Metrics

To evaluate the geometric approximation accuracy of Zonotopes for distributed flexible resource feasible regions, a two-layer similarity index based on multi-dimensional vector projections is proposed. This metric jointly considers boundary coverage across various directions and the structural balance of the generator set.

As illustrated in Figure 4 using a two-dimensional example, the similarity between the Zonotope and the original feasible region is visualized, where P denotes the original feasible region and Z represents the approximated Zonotope. A set of unit normal vectors is selected to represent directional projections, and the width ratio of P and Z along each direction is computed to quantify the approximation quality.

The accuracy metric Λ is defined as follows:

$$\Lambda ={\omega }_1{\displaystyle \frac{1}{N_{\mathrm{f}}}}{\displaystyle \sum _{l=1}^{N_l}\frac{{\Delta }_{Z,l}}{{\Delta }_{P,l}}}+(1-{\omega }_1)\left[1-{\displaystyle \frac{{\displaystyle \sum _{m=1}^M{({\beta }_m-\tilde{\beta })}^2}}{M{\sigma }_{\max }}}\right]$$

(49)

Here, ω₁ is the weighting factor; N_f is the number of unit normal vectors; Δ_P,l and Δ_Z,l represent the widths of the original feasible region and the Zonotope along the l unit normal direction, respectively; β_m denotes the extension length of the m generator in that direction; β is the mean extension length across all generators; and σ_max is the maximum variance of the extension lengths, used for normalizing the second term in the index. In the definition of the accuracy metric Λ in Equation (49), the first term quantifies the approximation degree between the Zonotope and the original feasible region, while the second term measures the variation in extension lengths across all generator directions. The inclusion of the second term ensures that the Zonotope extends across all defined directions, promoting structural balance. The similarity index Λ ∈ [0, 1], and the trade-off between the two components is governed by the weighting factor ω₁ This metric captures both the boundary-matching fidelity and the directional coverage of the Zonotope, thereby enhancing the stability of the overall approximation.

4.2.4. Optimal Zonotope Search

In the previous section, a comprehensive accuracy metric Λ was formulated to assess the approximation quality of a Zonotope, capturing both boundary projection fidelity and generator structural balance. Given a predefined set of generator directions G, the final shape of a Zonotope is fully determined by its center position and the extension lengths β of its generators. Therefore, the next step is to formulate an optimization model that maximizes the Λ metric with respect to β in order to obtain the Zonotope structure that best fits the original feasible region.

• Computation of Parameters Related to the Accuracy Metric

When using a Zonotope to approximate the original feasible region P, the optimal Zonotope can be derived by solving an optimization problem based on the defined accuracy metric. A higher value of Λ, approaching 1, indicates a greater similarity between the Zonotope Z and the feasible region P.

The width ΔZ of the Zonotope in all unit normal directions can be computed through mathematical derivation as

$$\Delta Z=2|F^{\mathrm{T}}G|\beta$$

(50)

Here, F ∈ R^N×Nf is the matrix composed of the selected unit normal vectors.

Let ΔP denote the maximum distance between the two supporting hyperplanes of the original feasible region along each unit normal direction. This width can be determined via optimization. The original feasible region P is represented as a polytope in half-space form:

$$P=\{x\in R^N:Ax⩽b\}$$

(51)

For a given unit normal vector ƒι = [a₁, a₂, …, a_N], where x₁, x₂, …, x_N represent the coordinates along each dimension, and it is a constant.

Assume that

$$X_{\mathrm{P}}=[x_{1p},x_{2p},\dots ,x_{Np}]$$

(52)

$$X_{\mathrm{Q}}=[x_{1q},x_{2q},\dots ,x_{Nq}]$$

(53)

Here, X_P and X_Q represent a point outside and a point inside the hyperplane, respectively.

The directional width Δ_P,l of the original feasible region along the unit normal vector ƒι can be obtained by solving the following optimization problem:

$$\left\{\begin{array}{c}\begin{array}{c}\max {\Delta }_{P,l}=|a_1(x_{1p}-x_{1q})\\ +\dots +a_N(x_{Np}-x_{Nq})|\hfill \end{array}\hfill \\ \mathrm{s}.\mathrm{t}.\hspace{1em}A{X}_{\mathrm{p}}⩽b\hfill \\ \hspace{1em}A{X}_{\mathrm{Q}}⩽b\hfill & \hfill \end{array}\right.$$

(54)

2.

Optimization Model for Finding the Optimal Zonotope

This study formulates an optimization objective aimed at maximizing the accuracy metric Λ in order to identify the optimal Zonotope structure that best approximates the original feasible region.

$$\max {\omega }_1{\displaystyle \frac{2}{N_{\mathrm{f}}}}{\left({\displaystyle \frac{1}{\Delta P}}\right)}^{\mathrm{T}}|F^{\mathrm{T}}G{|}^{\mathrm{T}}\beta +(1-{\omega }_1)\left[1-{\displaystyle \sum _{m=1}^M\frac{{({\beta }_m-\tilde{\beta })}^2}{M{\sigma }_{\max }}}\right]$$

(55)

The optimization is subject to the following constraints:

$$Z(G,c,\beta )\subseteq P(A,b)$$

(56)

Based on mathematical derivations, these constraints can be reformulated as

$$\left\{\begin{array}{c}Ac+|AG|\beta ⩽b\hfill \\ \beta ⩾0\hfill \end{array}\right.$$

(57)

In summary, the aggregation process for distributed flexible resource feasible regions using Zonotope approximation is illustrated in Figure 5.

4.3. Flexibility–Revenue-Aware Directionally Weighted Zonotope Modeling Method

This paper proposes using Zonotopes to approximate the feasible operating region of a single user’s flexible resource, thereby enabling efficient aggregation of the feasible regions of resource clusters. However, due to geometric discrepancies between Zonotopes and the original convex polytopes, the proposed method sacrifices approximation accuracy for the sake of computational efficiency. To address this issue, a correction strategy for the objective weighting in the approximation model is introduced. This strategy prioritizes the retention of high-value segments of the feasible region, based on the differences in flexibility revenue across time periods. In the day-ahead spot market, settlement for demand-side flexible resources typically considers two components: the value of energy usage and the value of reserve capacity. The energy usage value is settled according to the actual dispatched energy within the feasible region, while the reserve value is based on the available upward or downward flexibility margin at a given time. Since the prices for both energy and reserve vary across time periods depending on market clearing outcomes, the flexibility provided by controllable loads yields different economic returns at different times. Therefore, when constructing a Zonotope to describe a user’s flexible operating region, greater approximation precision should be preserved for high-revenue periods, while the inevitable approximation error should be allocated to periods with lower economic value.

Assuming that the user participates in demand response under a price taker model, its flexibility revenue in the spot market can be expressed as

$${\max }_p{\mu }^{\mathrm{T}}(p_{\max }-p_{\min })+u(p)-{\lambda }^{\mathrm{T}}p$$

(58)

Here, λ denotes the clearing prices for energy across N time periods, and μ represents the reserve clearing prices. The function u(·) is the user’s electricity utility function. The electricity utility and energy cost are determined by the user’s optimal operating point, and this value remains unchanged as long as the optimal point is not excluded from the approximated feasible region. In contrast, reserve revenue depends on the maximum adjustable margin at each time period. Therefore, for directions corresponding to periods with high reserve clearing prices, the approximation accuracy of the feasible region should be preserved to maximize its economic value. Based on this insight, the objective function is modified to assign higher penalties for approximation errors in high-value time periods, thereby guiding the Zonotope approximation to allocate more flexibility in those periods. The specific process is as follows: the predicted reserve clearing prices are sorted in ascending order and divided into B segments. For each segment, a flexibility value weight is assigned as

$$w_n=\left\{\begin{array}{cc}1& {\overline{\mu }}_1⩽{\widehat{\mu }}_n⩽{\overline{\mu }}_{N/B}\\ 2& {\overline{\mu }}_{N/B+1}⩽{\widehat{\mu }}_n⩽{\overline{\mu }}_{2N/B}\\ \dots & n=1,\dots ,N\\ B& {\overline{\mu }}_{(B-1)N/B+1}⩽{\widehat{\mu }}_n⩽{\overline{\mu }}_N\end{array}\right.$$

(59)

Here, $\overline{\mu }$ denotes the sequence of reserve prices sorted in ascending order, and wⁿ is the weight corresponding to time period n.

The directional approximation accuracy for direction α^s is modified according to the computed weights as

$${\Lambda }_s^{\prime }={\Lambda }_s(w{({\alpha }^s)}^2)={\Lambda }_s\left({\displaystyle \sum _{n=1}^N{w}_n}{x}_n^2\right)$$

(60)

Let the normal vector ${\alpha }^s=(x_1,x_2,\dots ,x_N)$, where (⋅)² denotes the element-wise square of a matrix. Therefore, the term $\left({\displaystyle \sum _{n=1}^N{w}_n}{x}_n^2\right)$ represents the projection of the normal vector onto each coordinate axis, weighted by the economic value of reserve capacity in each direction. By assigning and summing the corresponding weights, a composite economic value weight for direction α^s is obtained. Utilizing this aggregated weight, the original similarity index Λ_s in the direction α^s is modified to a corrected value Λ_s′.

In summary, the Zonotope approximation problem considering flexibility revenue can be formulated as

$$\begin{array}{c}{\max }_{c,{\beta }_{\max }}{\displaystyle \frac{1}{S}}{\displaystyle \sum _{s=1}^S{\Lambda }_s^{\prime }}\hfill \\ \mathrm{s}.\mathrm{t}.Ac+|AG|{\beta }_{\max }⩽b\hfill \\ p^{*}=c+G\beta \hfill \\ -{\beta }_{\max }⩽\beta ⩽{\beta }_{\max }\hfill \end{array}$$

(61)

When the dimension is nnn, the computational complexity of the Minkowski sum for feasible regions represented by Zonotopes is only O(n). The construction process of the resource aggregation model based on Zonotopes is illustrated in Figure 6.

The overall computational process for feasible region aggregation is illustrated in Figure 7.

This chapter addresses the challenge of high-dimensional feasible region representation in the context of flexible resource aggregation and control and systematically develops a Zonotope-based approximation modeling framework. First, generator construction is investigated, and a set of directional structures is proposed to accommodate various constraint characteristics including power, energy, and ramping. Next, a similarity metric is designed to quantify the geometric alignment between the Zonotope and the original feasible region, followed by the formulation of an optimization model that maximizes this metric to obtain an optimal approximation structure. Subsequently, asymmetric constraints and energy loss models are introduced to enhance the modeling capability of Zonotopes for real-world devices. Finally, a revenue-aware correction mechanism is proposed, aligned with day-ahead market requirements, to prioritize approximation accuracy in high-value periods. The methods developed in this chapter lay a foundational basis for linear embedding in subsequent optimization routines and support premium management of flexible resources.

5. Aggregated Resource Cluster Control Method

Unlike traditional scheduling models that rely on detailed individual constraints, the Zonotope representation offers a compact and unified structure that facilitates direct integration into optimization frameworks. This chapter proposes a revenue-aware control framework that reformulates the aggregated Zonotope-based feasible regions into half-space representations, enabling seamless integration into optimization models for economically efficient scheduling of heterogeneous flexible resources.

5.1. Conversion from Zonotope to Half-Space Representation

In Section 3, this paper proposed a generator construction method for Zonotopes that accommodates complex linear constraints. This method expands the conventional generator design space—typically limited to “unit direction + adjacent differences”—and enables Zonotopes to be accurately inscribed within any original feasible region formed by linear inequalities. However, to facilitate the application of Zonotopes in scheduling models, it is necessary to convert their specialized structure into the standard half-space form Ax ≤ b, so that the Zonotope can be directly embedded as a constraint in resource scheduling formulations. The following presents a detailed derivation of the conversion method from Zonotope to half-space representation.

In fact, the special mathematical structure of a Zonotope admits an exact half-space formulation [25]. In the half-space form Ax ≤ b, each row of matrix A corresponds to the normal vector of a separating hyperplane, and the i-th element of vector bb represents the scalar product of any point on the i-th hyperplane with its associated normal vector. The absolute value of this scalar product denotes the distance from the origin to the corresponding half-space.

Let us define a matrix H as follows:

$$H=[h_1,h_2,\dots ,h_{n-1}]\in {ℝ}^{n\times n-1}$$

(62)

Here, h denotes an n-dimensional vector. Let $H^{[i]}\in {ℝ}^{n-1\times n-1}$ represent the matrix H with its i-th row removed. The cross product of the vectors in H is defined as

$$nX(H)={[\dots ,{(-1)}^{+1}\det (H^{[i]}),\dots ]}^{\mathrm{T}}$$

(63)

For an N-dimensional Zonotope with M generators, each non-parallel facet must be formed by selecting N−1 generators from the M generator set. Therefore, the total number of facets is $C_{N-1 M}$.

Let G^(γ,…,η) denote the generator matrix G with M − N + 1 generators removed. The corresponding half-space representation is then computed as follows:

$$\left\{\begin{array}{c}A=\left[\begin{array}{c}{A}^{+}\\ -A^{+}\end{array}\right]\hfill \\ b=\left[\begin{array}{c}{b}^{+}\\ b^{-}\end{array}\right]\hfill \end{array}\right.$$

(64)

$$A_i^{+}={\displaystyle \frac{nX{(G^{(\gamma ,\dots ,\eta )})}^{\mathrm{T}}}{\Vert nX(G^{(\gamma ,\dots ,\eta )}){\Vert }_2}}$$

(65)

$$\left\{\begin{array}{c}\hfill \\ b_i^{+}=A_i^{+}c+\Delta b_i\hfill \\ b_i^{-}=-A_i^{+}c+\Delta b_i\hfill \\ \Delta b_i={\displaystyle \sum _{m=1}^M|}{A}_i^{+}{g}^{(m)}|\hfill \end{array}\right.$$

(66)

Here, the index i ranges from 1 to $C_{N-1 M}$, corresponding to the different G^(γ,…,η) combinations being evaluated. In practical computation, $C_{N-1 M}$ represents the total number of iterations required.

5.2. Resource Cluster Optimization and Control Model

It is noteworthy that in Section 3, a revenue-aware accuracy metric was introduced to prioritize the preservation of economically valuable boundary directions during the construction of Zonotope-based feasible regions. This ensures that resources retain greater adjustability during critical time periods. To extend this modeling approach, a time-series weighting vector ww is introduced into the optimization objective function. This vector is constructed based on a weighted combination of reserve clearing prices and electricity prices [26].

The weighting vector guides the optimization model to allocate greater dispatchable power in high-revenue periods, thereby maximizing the economic utilization of flexible resources under revenue-driven conditions. With the goal of maximizing internal economic performance for the load aggregator, the following objective function is formulated for optimizing the control of the resource cluster:

$$\max \sum _{t=1}^T{w}_t\cdot \left(\begin{array}{l}(P_t^{\mathrm{BESS},\mathrm{agg}}+P_t^{\mathrm{AC},\mathrm{agg}}\\ -P_t^{\mathrm{AC},\mathrm{base}}-P_t^{\mathrm{DG},\mathrm{agg}}){\lambda }_t\mathsf{\Delta }t\end{array}\right)-C_o-\gamma \sum _{t=1}^T{P}_t^{\mathrm{BESS},\mathrm{agg}}$$

(67)

Here, P_t^BESS,agg epresents the operating power of the aggregated energy storage cluster at time t; P_t^AC,agg denotes the output of the aggregated air-conditioning cluster at time t; and w_t ∈ [0.5, 1.5] is the “flexibility weight factor” defined in Section 3.4 based on reserve price segmentation, where higher values indicate greater economic potential of boundary modulation and thus demand higher approximation precision. λ_t denotes the electricity price at time t; γ is the charging/discharging loss coefficient for the storage cluster; and C_o is the total operating cost of the diesel generator cluster. Incorporating C_o into the objective function ensures that the operation of diesel generators aligns with economically rational output ranges.

The constraint set is represented by the half-space form of the total Zonotope derived from each device cluster. As established in Section 5.1, the number of computations required for converting from Zonotope form to half-space representation is $C_{N-1 M}$. Given a 24 h scheduling horizon with hourly resolution, the resulting 24-dimensional space leads to substantial computational complexity. To address this, a two-stage aggregation and control framework is proposed, as illustrated in Figure 8.

The upper layer of Figure 8 represents the resource representation stage, where heterogeneous flexible resources are approximated and aggregated using Zonotopes. The middle layer corresponds to the structural transformation stage, where Zonotopes are mapped to half-space representations. The bottom layer is the control implementation stage, in which economic optimization scheduling is performed over segmented time intervals [27].

This paper adopts a staged implementation for the aggregation and control process to address the computational complexity arising from the transformation of Zonotope representations into half-space form. Lower-dimensional Zonotopes result in shorter conversion times. However, the optimization framework developed herein is based on price sensitivity; finer time segmentation reduces the length of intervals over which the model can capture price signals, thus limiting the economic performance of the load aggregator. Moreover, since the overall framework involves optimizing the control of resource clusters after aggregation, increasing the number of time segments results in more frequent repetitions of the aggregation and control procedures, which further contributes to computational overhead.

Therefore, balancing computational efficiency and the aggregator’s economic performance, the 24 h scheduling horizon is divided into two stages, each covering 12 h [28].

This chapter focuses on the modeling of resource aggregation and control, establishing a Zonotope-based optimization framework and proposing a method for converting its structure to half-space form. By incorporating the flexibility weighting mechanism introduced in Section 3, the objective function is guided by economic value to direct the dispatch of flexible resources. Additionally, the proposed two-stage control framework effectively mitigates the complexity of solving high-dimensional models. These methods not only preserve the structural advantages of Zonotope modeling but also enable temporal stratification of flexibility value in scheduling optimization, thereby providing a theoretical foundation for the engineering simulations presented in Section 5.

6. Simulation Validation

6.1. Modeling of Multi-Type Building Load Scenarios and Simulation Configuration

6.1.1. Feasible Region Modeling Schemes and Benchmark Methodologies

Since forecasting spot market clearing prices and determining aggregator bidding strategies are beyond the scope of this study, actual market electricity prices are adopted as exogenous inputs. This section focuses on evaluating the performance of feasible region modeling and aggregation methods in terms of accuracy, computational efficiency, and economic outcomes.

Three representative distributed energy storage devices with significantly different parameters are selected as flexible resource samples. These devices are modeled as ideal batteries (i.e., with no energy loss and 100% charge/discharge efficiency), and their technical specifications are listed in

Table 1. Distributed energy storage device parameters.

Device Model	Maximum Charging/Discharging Power/kW	Energy Storage Capacity/(kW·h)	Ramp Rate/(kW·h−1)	Initial SOC
A	17.2	100	8.6	[0.2, 0.8]
B	50	210	25	[0.2, 0.8]
C	14	27	7	[0.2, 0.8]

. The following four representative modeling and aggregation methods are used for comparison [29]:

M1 approximates the feasible region of flexible resources using the virtual synchronous machine model, followed by aggregation via Minkowski summation (M-Sum);

M2 approximates the feasible region using the Zonotope model, followed by aggregation via M-Sum;

M3 applies the proposed method in this paper to approximate the feasible region, followed by aggregation via M-Sum;

M4 applies the revenue-aware method proposed in this paper to approximate the feasible region, followed by aggregation via M-Sum.

6.1.2. Construction of Multi-Type Building Load Scenarios and Simulation Platform Setup

To validate the effectiveness of the proposed aggregation and control method for flexible resource feasible regions, this study selects the office building complex of the Shandong Electric Power Company as the case study. A typical building energy-use scenario is constructed, incorporating air-conditioning loads, energy storage systems, and controllable diesel generator units. This building features a variety of flexible loads capable of providing both real-time demand response and reserve support for aggregators.

The simulation adopts a 24 h scheduling horizon with a 1 h time step, capturing typical intra-day variations in building energy consumption under realistic operational conditions. The case study is based on a five-story office building of Shandong Electric Power Company, with a total floor area of approximately 5000 m², equipped with 100 air-conditioning units, 20 energy storage devices, and 5 diesel generators.

Equipment parameters are derived from actual measured data. To reflect device heterogeneity, stochastic perturbations are introduced: for air-conditioning units, the parameters k1 and k2 are randomly sampled within defined ranges to simulate environmental and operational variability, with indoor temperature maintained between 25 °C and 27.5 °C; for energy storage units, the initial energy states E0 are varied within 30–40 kWh to represent diverse starting conditions; for diesel generators, rated power ranges from 0 to 100 ± ε kW with a ramp rate constraint of ±30 kW. The rated power limits of all devices are also perturbed to emulate minor differences in equipment characteristics, thereby enhancing the realism of the aggregation and control process [30].

The building operates under local electricity market conditions, with the electricity price curve and outdoor temperature profile for the simulation day shown in Figure 9. These conditions provide the necessary environmental and economic context for verifying the effectiveness of the proposed method in a realistic demand-side resource aggregation scenario.

The simulation platform is built in Python 3.8 using the Spyder environment and executed on a machine equipped with an Intel Core i5-12400 2.50 GHz CPU and 8 GB RAM (The Intel Core i5-12400 CPU is manufactured by Intel Corporation in Leixlip, Ireland; the 8 GB RAM is sourced from Samsung Electronics in Icheon, Republic of Korea). During the conversion from Zonotope to half-space form, 8-core parallel computing is employed to accelerate computation and satisfy the processing demands of high-dimensional aggregation tasks.

6.2. Analysis of the GNMTL-LSTM Model

6.2.1. Construction of the GNMTL-LSTM Model

To obtain the optimal model architecture, the parameters during training are optimized using the AdamW algorithm. All hyperparameter settings are listed in

Table 2. Model hyperparameters.

Model Hyperparameters	Value
Shared Layer Neural Network	LSTM
Number of Shared Layers	3
Number of Neurons per Layer	256/128/64
Model Training Optimization Algorithm	AdamW
Initial Learning Rate	0.001
Learning Rate Decay Factor	0.5
Minimum Learning Rate	0.0001
Number of Iterations	600
Loss Weight Adjustment Function	MSE

. Among them, the number of hidden layers and the learning rate are key hyperparameters that significantly affect the model’s performance and generalization capability.

The impact of the number of LSTM layers on model prediction accuracy is shown in Figure 10a. As the number of LSTM layers increases, the R² score for each subtask initially improves, reaching its peak when the number of layers is three. However, when the number exceeds three, overfitting occurs, leading to a decline in R² scores and reduced model performance. Therefore, the optimal number of LSTM layers is set to three.

Figure 10b illustrates the relationship between the initial learning rate and the R² score of feasible region prediction. As the initial learning rate increases from 0.0001 to 0.1, the R² score for the feasible region of flexible resources rises rapidly at first and then gradually decreases. As a result, the optimal initial learning rate is chosen to be 0.001.

6.2.2. GNMTL-LSTM Prediction Analysis

To evaluate the prediction accuracy, the coefficient of determination (R²), root mean square error (RMSE), and mean absolute error (MAE) are selected as evaluation metrics. The prediction results of the proposed method are presented in

Table 3. Prediction results of the GNMTL-LSTM model.

Type	R2	RMSE	MAE
Air Conditioner	0.960	0.980	0.890
Energy Storage	0.912	1.009	0.950
Diesel Generator	0.922	1.027	1.121
Average	0.931	1.005	0.987

. The proposed approach demonstrates high prediction accuracy, with an average R² of 0.931, RMSE of 1.005, and MAE of 0.987 across multiple resource types. These results indicate that multi-task learning (MTL) can effectively capture the coupling relationships among various flexible resources, thereby improving the prediction accuracy of diverse loads.

At the same time, to validate the superiority of the GNMTL-LSTM framework, we compare it against several commonly used forecasting models, including standard LSTM, MTL-LSTM and multilayer perceptron (MLP). As shown in

Table 4. Comparison of prediction accuracy among different models.

Model	R2 (AC)	RMSE (AC)	MAE (AC)	R2 (ES)	RMSE (ES)	MAE (ES)	R2 (DG)	RMSE (DG)	MAE (DG)
LSTM	0.945	1.120	0.980	0.885	1.150	1.050	0.895	1.210	1.240
MTL-LSTM	0.953	1.050	0.930	0.900	1.080	0.990	0.910	1.120	1.170
GNMTL-LSTM	0.960	0.980	0.890	0.912	1.009	0.950	0.922	1.027	1.121

Note: AC = air conditioner, ES = energy storage, DG = diesel generator.

, GNMTL-LSTM outperforms all baseline models in terms of R², RMSE, and MAE across all resource types, indicating its effectiveness in capturing multi-dimensional temporal dynamics. For example, in air conditioner load forecasting, the GNMTL-LSTM achieved an R² of 0.960, outperforming both LSTM and MTL-LSTM, while both RMSE and MAE were reduced. Compared to LSTM (single-task), GNMTL-LSTM improved R² by 1.59% and reduced RMSE and MAE by 12.5% and 9.18%, respectively; compared to MTL-LSTM (multi-task), R² increased by 0.73%, while RMSE and MAE decreased by 6.67% and 4.30%, respectively.

Following the quantitative comparison of GNMTL-LSTM with the other two algorithms across the three evaluation metrics, we further analyze the residual plots for each model on the three resource types. The residuals, defined as the difference between observed and predicted load values at each time step, provide insight into forecast bias and error distribution beyond aggregate metrics.

As shown in Figure 11, the GNMTL-LSTM consistently exhibits residuals with the smallest magnitude and the tightest concentration around zero across all resources, indicating minimal systematic bias and improved prediction stability. The MTL-LSTM shows moderate residual variance, while the standard LSTM demonstrates the largest and most dispersed residuals, reflecting greater prediction uncertainty and bias.

To further validate the performance differences observed in the residual analysis, we conducted statistical significance tests using paired t-tests on the absolute prediction errors between models for each resource type.

Table 5. Statistical significance test results for model performance comparison across resources.

Resource	Model Comparison	t-Value	p-Value
Air Conditioner	GNMTL-LSTM vs. LSTM	−3.123	0.00478
	GNMTL-LSTM vs. MTL-LSTM	−1.954	0.06295
Energy Storage	GNMTL-LSTM vs. LSTM	−4.192	0.00035
	GNMTL-LSTM vs. MTL-LSTM	−0.545	0.59128
Diesel Generator	GNMTL-LSTM vs. LSTM	−2.756	0.01125
	GNMTL-LSTM vs. MTL-LSTM	−1.324	0.19843

shows paired t-test results comparing GNMTL-LSTM with LSTM and MTL-LSTM models for three resources. GNMTL-LSTM significantly outperforms traditional LSTM for air conditioners, energy storage, and diesel generators. However, the differences between GNMTL-LSTM and MTL-LSTM are not statistically significant, indicating similar performance. These results demonstrate that GNMTL-LSTM improves prediction accuracy over LSTM, while providing moderate gains beyond MTL-LSTM.

6.3. Validation of Single-Resource Approximation Accuracy and Control Effectiveness Based on GNMTL-LSTM-Zonotope

6.3.1. Approximation Strategies and Reserve Revenue Evaluation

This section builds upon the aggregation comparison framework established in Section 6.1.1 to further investigate the advantages of the proposed GNMTL-LSTM-Zonotope-based model in terms of feasible region approximation accuracy and reserve revenue representation. A 24 h scheduling period is defined, with an interval of 1 h between time slots. Three distributed energy storage units with significantly different parameters are selected for case studies, and four feasible region approximation methods (M1–M4) are employed for comparative analysis. Among them, M3 and M4 represent the proposed GNMTL-LSTM-Zonotope-based approaches, with the number of generators set to S = 100 and K = 50 unit normal vectors added for accuracy evaluation. Approximation accuracy is measured using precision indicators, as shown in Figure 12.

As shown in Figure 12, M1 yields the lowest average accuracy—only 22.19%—since it neglects temporal coupling and merely retains the “maximum inscribed cube” within the original feasible region. By incorporating state constraints, M2 improves the average accuracy to 41.50%. M3 further includes ramping constraints, raising the accuracy to 45.12%. M4, which additionally integrates flexibility-revenue-based weighting, achieves an average accuracy of 45.38%, which is nearly identical to M3 and shows no significant decline. These results indicate that while all models incur some degree of approximation error, the methods proposed in this paper maintain comparable accuracy while incorporating economic value weighting, thus offering superior overall performance. To further assess computational efficiency, Figure 13 presents a comparison of aggregation time across different models using 50 devices with heterogeneous initial states.

Figure 13 shows the following: M1 achieves the shortest runtime at only 62 s, as it omits coupling constraints and directly aggregates boundary points, making it the most computationally efficient. M2 requires 3685 s due to the need for facet-by-facet comparisons, with its computational burden increasing sharply with dimensionality. M3 introduces a normal vector search to simplify the objective function, reducing runtime to 1430 s, which is over 60% less than M2. M4 builds upon M3 by incorporating flexibility-based weighting, with a slight increase in runtime to 1577 s, yet it still significantly outperforms M2. Additionally, directly performing Minkowski sums on polytopes becomes computationally infeasible when the dimension exceeds three. To further evaluate economic performance,

Table 6. Comparison of standby revenue for each device (CNY).

Equipment Model	M1	M2	M3	M4
A	8.27	12.23	14.65	15.36
B	19.43	44.70	48.15	50.36
C	1.37	6.99	8.39	8.50

and Figure 14 compare the reserve market revenues of energy storage systems under the four models. The revenue differences stem from variations in the geometric shape of the approximated feasible regions.

As shown in Table 4 and Figure 14, the average reserve revenues for M1 through M4 are CNY 9.69, CNY 21.31, CNY 23.73, and CNY 24.74, respectively. M1, due to its substantial reduction of the feasible region, yields significantly lower reserve revenue compared to the other three methods. M3 achieves higher feasible region approximation accuracy than M2, resulting in reserve revenue improvements of 7.71% to 19.99% across the three device types. M4, which incorporates flexibility-revenue awareness, maintains comparable approximation accuracy to M3 while further increasing reserve revenue by 1.38% to 4.79%.

6.3.2. Feasible Region Approximation and Reserve Revenue Evaluation

Building upon the previous section, this part further verifies the adaptability and robustness of the proposed Zonotope model in representing complex resource behaviors. An electrochemical energy storage system with charging/discharging losses is selected as the study object. The corresponding feasible region model incorporates state recursion, energy loss, and efficiency constraints. A series of simulation analyses are conducted to evaluate the model’s approximation accuracy and aggregation efficiency. Key parameter settings are provided in

Table 7. Energy storage device parameters.

Energy Storage Parameters	Value
Maximum Storage Capacity/kWh	200
Minimum Storage Capacity/kWh	0
Maximum Charging Power/kW	19
Maximum Discharging Power/kW	24
Charge/Discharge Efficiency	0.9
Energy Loss Coefficient	0.1
Ramp Rate (/kW·h−1)	6.8
Initial State of Charge	[0.20, 0.75]

.

The Monte Carlo method is used to sample specific values for the initial state of charge of the storage units, generating a population of 100 energy storage devices. Each device’s feasible region is individually approximated using both the original Zonotope model (which does not account for losses) and the improved Zonotope model (which incorporates loss factors). The average similarity between the approximated and actual feasible regions is computed across a scheduling horizon of 2 to 8 h. The final approximation results are illustrated in Figure 15.

As shown in Figure 15, the geometric complexity of the feasible region increases in high-dimensional space as the scheduling horizon extends, resulting in a decline in the approximation accuracy for both the original and the improved Zonotope models. Additionally, variations in the initial state of charge of storage devices introduce fluctuations in approximation precision. In the figure, the red and blue regions represent the similarity ranges of the original and improved models, respectively. A comparison of their mean curves reveals that the improved model, which accounts for charging and discharging losses, achieves an overall improvement of approximately 4.6% in approximation accuracy compared to the original model. This enhancement confirms the model’s effectiveness in capturing complex operational constraints.

Furthermore, by applying the Minkowski sum to the individual feasible regions, aggregated feasible regions for the resource cluster are obtained. These include both controllable load users and storage devices under different scheduling durations. The following two case configurations are compared.

M1: There are 100 storage devices, with feasible regions characterized and aggregated using the original Zonotope model that ignores losses.

M2: There are 100 storage devices, with feasible regions characterized and aggregated using the improved Zonotope model that incorporates losses.

Table 8. Feasible region approximation and aggregation calculation time.

Method	Process	Time Period
		4	8	24
M1	Original Zonotope Approximation Time (without considering losses)/s	581.53	748.38	2038.16
	Aggregation Time/s	0.14	0.24	0.79
	Total Time/s	581.67	748.62	2038.95
	Average Approximation Accuracy	0.87	0.76	0.61
M2	For the Proposed Model in This Paper/s	606.56	789.46	2209.15
	Aggregation Time/s	0.17	0.25	1.03
	Total Time/s	606.73	789.71	2210.18
	Average Approximation Accuracy	0.91	0.82	0.72

presents the results for approximation accuracy and aggregation computation time under both case scenarios.

As shown in Table 8, the improved Zonotope model achieves average approximation accuracy improvements of 4.6%, 7.9%, and 18.0% over the original model across the three scheduling horizons, with all values exceeding 0.7, demonstrating strong approximation performance. Although the inclusion of charging and discharging loss constraints slightly increases computational effort, the aggregation process itself remains highly efficient due to the additive nature of Zonotope-based operations, with aggregation time accounting for less than 0.1% of the total runtime.

Moreover, the feasible region approximation can be performed in a day-ahead preprocessing phase, ensuring fast response during real-time scheduling. These results confirm that the proposed model effectively balances accuracy and computational efficiency.

6.4. Performance Analysis of the GNMTL-LSTM-Zonotope-Based Aggregation Strategy

6.4.1. Balancing Accuracy and Efficiency: Zonotope Parameter Sensitivity

This case study is based on the multi-type building load scenarios and simulation platform constructed in Section 6.1.1. The model parameters are derived from measured data of the Shandong Electric Power Company office building to ensure the evaluation framework’s realism and reproducibility. The aggregation process uses the aforementioned GNMTL-LSTM prediction results as inputs. Subsequently, a sensitivity analysis is conducted on the configuration of two types of weighting factors within the accuracy metrics. The Zonotope aggregation results of various resources under different weight settings are quantitatively compared (see

Table 9. Precision metrics of Zonotope approximation under different weights.

Resource Type	Precision Metrics Under Different Weights (12-Dimensional)
	ω = 0.4	ω = 0.6	ω = 0.8
Air Conditioning Load	0.8801	0.8419	0.8213
Energy Storage	0.8218	0.7863	0.7827
Diesel Generator	0.7868	0.7248	0.7049

), and the structural response characteristics are visually illustrated by the extension lengths along each generator direction (see Figure 16).

As the weight of the first component in the accuracy metric increases, the approximation accuracy of the Zonotope models for various devices declines accordingly. However, as shown in Figure 16, the extension lengths across generator directions become more balanced. Therefore, based on the results of the sensitivity analysis, the weighting factor ω is set to 0.6 in the subsequent resource cluster control simulations.

6.4.2. Comparative Analysis of the Zonotope Aggregation Method and Benchmark Models

This case study investigates the feasibility of the proposed Zonotope-based resource feasible region aggregation method across different spatial dimensions, as well as the accuracy with which Zonotopes approximate the original feasible regions. Furthermore, the proposed aggregation method is compared against other benchmark methods to evaluate their respective advantages and disadvantages in terms of aggregation accuracy. A comparison of accuracy metrics across multiple spatial dimensions is presented in

Table 10. Comparison of precision metrics for each aggregation method under different dimensions.

Method	Resource Type	Precision Metrics (ω = 0.6)
		6 Dimensional	12 Dimensional	24 Dimensional
GNMTL-LSTM-Zonotope	Air Conditioning Load	0.8657	0.8419	0.8367
	Energy Storage	0.8122	0.7863	0.7489
	Diesel Generator	0.7274	0.7248	/
Inner Box Approximation	Air Conditioning Load	0.8889	0.8602	0.8751
	Energy Storage	0.5769	0.4418	0.3749
	Diesel Generator	0.6319	0.6344	/
Zonotope (Randomly Selecting Generators)	Air Conditioning Load	0.6061	0.5211	0.4631
	Energy Storage	0.6102	0.5213	0.4405
	Diesel Generator	0.5029	0.4608	/

.

This analysis includes a comparison of various inner approximation aggregation methods, including the method proposed in this paper, a box-based inner approximation method, and a Zonotope aggregation method based on randomly selected generators. Across different spatial dimensions, the proposed Zonotope aggregation method consistently demonstrates higher approximation accuracy for aggregated feasible regions. This is primarily because the generator directions in the proposed method are chosen based on the structure of the original feasible region model, allowing for more comprehensive extension and coverage of the feasible region, thereby achieving better geometric fidelity.

For air conditioning load aggregation, however, the accuracy metric of the proposed method is slightly lower than that of the box-based approximation method. This is attributed to the fact that the generator selection for air conditioner clusters incorporates temporal coupling, reflecting the thermal dynamics of indoor environments. In reality, indoor temperature is more directly influenced by the air conditioner’s power at the current time step. Therefore, while the box-based approximation which disregards temporal coupling offers better accuracy in some cases, the generator structure used in this paper remains physically meaningful and necessary from a modeling standpoint.

6.4.3. Economic Evaluation of the Two-Stage Resource Control Strategy

To balance the computational complexity of feasible region structure transformation with control precision, a two-stage rolling control strategy is adopted for the optimization scheduling of the resource cluster. Specifically, the first 12 h of the day constitute Stage 1, during which preliminary control is executed, and the operational trajectories of various resources are obtained. These trajectories are then used to estimate the state parameters of resources at the beginning of hour 13, which serve as initial conditions for Stage 2 (the latter 12 h) of control.

To better reflect real-world operating scenarios, the initial energy states of storage units in Stage 2 are set within the range [29.8, 31.8] kWh, simulating the state dispersion resulting from performance differences after Stage 1 control. This strategy aims to improve the model’s representation of resource dynamics, thereby enhancing the effectiveness of economic dispatch.

The control results for each resource cluster across the two stages are illustrated in Figure 17, and the corresponding economic and computational performance metrics are summarized in

Table 11. Economic and computational efficiency metrics of two-stage regulation for resource clusters.

Stage	Resource Type	Aggregation Time Consumption/s	Conversion Time Consumption/s	Economic Benefits/CNY
Stage One (First 12 Hours)	Air Conditioning Load	1661.9	33.1	1030
	Energy Storage	331.5	29.6
	Diesel Generator	515.3	27.5
Stage Two (Last 12 Hours)	Air Conditioning Load	1732.3	32.3	1720
	Energy Storage	352.6	30.8
	Diesel Generator	479.8	28.6

. The total time consumed during the aggregation process is primarily influenced by the number of devices being aggregated. Since the Zonotope-based aggregation process is inherently a linear additive operation, with no coupling involved in the Minkowski summation among individual resources, the aggregation of each resource’s Zonotope can be computed independently. As such, this method is well suited for parallel computation or deployment across multiple machines, enabling scalability to larger test cases.

Furthermore, the coordinated control of the aggregated cluster offers notable economic benefits to the load aggregator, supporting profitability through improved scheduling efficiency.

Figure 17 illustrates the 24 h dispatch results of various devices, revealing a strong responsiveness to price signals. Diesel generators, due to their associated unit costs, primarily operate during high-price periods, with their output constrained by ramping limits. Energy storage systems exhibit arbitrage behavior by charging during low-price periods and discharging during high-price periods. Air conditioners, modeled as virtual storage units, demonstrate moderate flexibility under comfort constraints and also respond to price fluctuations.

Figure 18 illustrates the feasible region aggregated from the air-conditioning cluster and further compares its baseline and actual operating power. The results demonstrate that, while maintaining thermal comfort, part of the power consumption is effectively reduced, thereby enabling flexible demand response. Notably, the shape of the feasible region closely approximates the predefined Chvátal polytope, validating the structural integrity of the aggregation. However, due to the boundary contraction induced by the Zonotope approximation and the implementation of a two-stage control strategy, the resources are only exposed to localized price signals. This limitation partially diminishes the temporal coupling characteristics inherent to energy storage systems.

Overall, this case study validates the applicability of the Zonotope-based flexible resource aggregation and staged control method in practical load aggregator scenarios. While ensuring computational efficiency, the proposed approach significantly enhances both the economic performance and regulatory flexibility of system operations, demonstrating strong potential for real-world engineering applications.

7. Conclusions

This study presents a unified GNMTL-LSTM-Zonotope-based framework for the modeling and aggregation control of flexible resources under load aggregator coordination. The proposed approach addresses key challenges in feasible region construction, including dynamic operational behavior, high-dimensional constraints, and economic sensitivity. Response models are developed for various resource types—air-conditioning systems, energy storage, and diesel generators—and a multi-task LSTM structure is introduced to capture temporal dynamics, enabling the generation of feasible regions with embedded power, energy, and ramping constraints.

To enhance economic responsiveness, a directionally weighted approximation mechanism is designed based on flexibility revenue signals from energy and reserve markets, allowing the model to prioritize high-value time periods in flexibility representation. A Minkowski-sum-based aggregation strategy is then applied to efficiently combine individual feasible sets into a collective cluster-level model.

Simulation results based on real-world office building data demonstrate that the proposed method significantly improves feasible region accuracy, aggregation efficiency, and market-aligned flexibility allocation. These results validate the effectiveness of the approach in enabling intelligent, coordinated control of heterogeneous demand-side resources.

In future work, the proposed approach could be adapted for broader scenarios, such as electric vehicle (EV) fleet coordination, city-level virtual power plants (VPPs), or integrated demand-side management platforms. The transferable nature of the feasible region-based modeling makes it promising for large-scale applications involving heterogeneous and distributed energy resources. Additionally, incorporating historical operational data and real-time sensing can further enhance the adaptability of the model, supporting broader deployment in market-oriented flexibility scheduling. However, the current study primarily focuses on medium-scale building clusters, and further research is needed to assess the scalability and adaptability of the approach in larger, more complex energy systems. Moreover, the model assumes the availability of high-quality historical data, which may limit its applicability in data-scarce or noisy environments. Future work should explore more robust learning methods and real-time adaptive mechanisms to improve the generalizability and resilience of the framework.