Single-level MILP incentive pricing for customer directrix load-based demand response in smart buildings

Abstract

Introduction:

With the high penetration of renewable energy, tapping the regulation potential of smart buildings in distribution networks plays a pivotal role in enabling orderly renewable energy accommodation. Customer directrix load-based demand response serves as an effective mechanism for guiding smart building interaction. However, its incentive price, derived as the equilibrium point of a leader-follower game between the operator and load entities, relies on a corresponding bilevel optimization model that suffers from low computational efficiency.

Methods:

To address this limitation, an efficient incentive pricing method is proposed. The proposed linear similarity metric and incentive revenue formulation ensure the linearity of the lower-level optimization problem. Subsequently, based on primal-dual feasibility, an aggregated strong-duality equality, and a discrete-price reformulation, the original bilevel model is accurately reformulated into a single-level mixed-integer linear programming problem, enabling a one-shot determination of the incentive price.

Results:

Simulation results demonstrate that the proposed linear metric is highly consistent with traditional nonlinear metrics in terms of ranking monotonicity. Compared with time-of-use-only demand response and fixed incentive pricing strategies, the proposed method can identify an economically balanced incentive price that effectively guides smart building energy consumption behavior, improves renewable energy accommodation, and maintains favorable economic performance for both the distribution system operator and smart buildings.

Discussion:

These findings support the use of the proposed single-level MILP framework for balancing renewable energy accommodation with the economic interests of the distribution system operator and smart buildings.

1 Introduction

China’s “dual carbon” goals are accelerating the transition to a new power system centered on Renewable Energy (RE) (Han et al., 2021). This transition, however, introduces significant real-time power balance challenges, as the system is strained by dual pressures: high volatility and uncertainty from large-scale RE integration on the generation side, and increasingly complex user behaviors on the load side. With the traditional generation-follows-load unidirectional regulation mode proving unsustainable (Xiao and Zheng, 2022), it has become essential to unlock load-side regulation potential via Demand Response (DR) to enhance the power system’s flexible regulation capability (Panda et al., 2022).

National policies, such as the Action Plan for Carbon Dioxide Peaking Before 2030 (State Council of the People’s Republic of China, 2021) and the 14th Five-Year Plan (People’s Daily, 2021), are driving a profound transition for urban buildings. They are evolving from passive electricity consumers into “prosumers”—smart buildings with flexible capabilities that integrate assets like distributed Photovoltaics (PV), energy storage (ES), and bidirectional electric vehicles (EVs) (Buckman et al., 2014; Sharma et al., 2024). This transformation establishes them as key entities for tapping regulation potential in distribution networks (Yao et al., 2025). Consequently, an urgent need has emerged to establish efficient grid-friendly interaction mechanisms. These mechanisms, leveraging communication and control technologies (Hao et al., 2017), are essential to guide building participation in load-side management and provide support for system peak shaving, frequency regulation, RE accommodation, and emergency load reserves (Ouammi, 2021).

Existing DR mechanisms are primarily classified into Price-based DR (PBDR) (Wang et al., 2025) and Incentive-based DR (IBDR) (González-Cabrera and Gutiérrez-Alcaraz, 2017). PBDR guides voluntary user adjustments through dynamic price signals, such as time-of-use (TOU), peak, and real-time pricing. However, this approach faces significant challenges, including uncertainties in response quantity and effectiveness, difficulties in dynamic price determination, and the risk of exposing users to price volatility. IBDR, in contrast, provides economic compensation based on pre-established agreements for actions like load curtailment or direct load control. This mechanism’s core challenge stems from its reliance on a customer baseline load to assess contributions. This reliance not only presents prediction challenges for operators (regarding accuracy and efficiency) but also introduces the moral hazard of users artificially inflating their baselines during non-response periods (Wu et al., 2024).

To address the shortcomings of PBDR and IBDR, Customer Directrix Load (CDL)-based DR has recently emerged as a promising mechanism (Fan et al., 2020). This approach is centered on the CDL, which is defined from the grid’s perspective as an ideal load profile capable of smoothing the fluctuations caused by non-dispatchable generation and inflexible loads (Meng et al., 2026). CDL-DR fundamentally shifts the user’s objective: instead of merely responding to dynamic prices or curtailing load, users are actively incentivized to regulate their consumption behavior to track the CDL. Their incentive revenue is then directly linked to this tracking similarity and a corresponding incentive price. Furthermore, this mechanism introduces user utility to quantify the response cost, enabling a fairer and more precise assessment and compensation for the regulation value users provide to the grid.

Determining the incentive price is the core challenge in designing the CDL-DR mechanism (Ma et al., 2024). This price embodies a crucial trade-off: it must be high enough to effectively guide user behavior, yet low enough to remain financially sustainable for the operator. Existing methods for resolving this trade-off and establishing the optimal price generally fall into two main categories:

The first category is the game-theoretic approach, which models the operator-user interaction as a leader-follower game (Xu et al., 2022). This method, often implemented using models like dual-loop Stackelberg games (Meng et al., 2023) or Nash bargaining (Xu et al., 2023), relies on an iterative process to find an equilibrium between the operator’s price and the users’ consumption. While this approach is mechanistically intuitive, its fundamental reliance on a repetitive solution process is a major drawback, resulting in significant computational inefficiency.

The second category is the KKT-based reformulation approach. This approach replaces the lower-level problem with its optimality conditions, including stationarity, primal feasibility, dual feasibility, and complementary slackness conditions. Although this method effectively converts the bilevel problem into a single-level model, the transformed formulation is usually nonconvex and nonlinear, and thus remains difficult to solve directly. For such a transformed single-level model, one possible solution route is to retain the nonconvex structure and apply global optimization methods; for example, Li et al. (2023) developed a powerful global algorithm for indefinite separable quadratic knapsack problems with box constraints, but this type of method requires the problem to possess a specific structure. Alternatively, complementarity conditions can be reformulated through Big-M or SOS1 techniques, leading to MIQP or MINLP models (Zhang et al., 2025; Yan et al., 2026; Hong et al., 2023), which can be handled by decomposition methods (Geoffrion, 1972) or spatial branch-and-bound algorithms (Belotti et al., 2013) for nonconvex quadratic and bilinear formulations. However, these approaches remain computationally demanding due to the presence of the per-constraint complementary slackness conditions and the nonlinear constraints introduced by conventional similarity metrics, such as Euclidean distance. Specifically, each lower-level inequality constraint introduces one complementary slackness condition, causing the model size and combinatorial complexity to grow rapidly as the number of buildings increases.

This paper proposes an efficient incentive pricing method for smart buildings within the CDL-DR mechanism, specifically targeting the RE accommodation scenario. We first formulate the pricing problem as a bilevel optimization model, accounting for the external characteristics of smart buildings (including bidirectional resources like ES and EVs). To ensure the computational tractability of the reformulation, we introduce a linear similarity metric and a corresponding linear incentive revenue formulation, which guarantee that the lower-level optimization problem is a Linear Programming (LP) problem. This linearity allows the original bilevel model to be accurately reformulated into a single-level Mixed-Integer Linear Programming (MILP) problem using primal-dual feasibility, an aggregated strong-duality equality, and a discrete-price reformulation. Unlike the KKT-based reformulation approach, which explicitly linearizes each complementary slackness condition, the proposed reformulation embeds the optimality of all building-level LPs without introducing binary variables for each lower-level inequality constraint. The remaining bilinear term is then handled by a discrete-price MILP representation. The resulting MILP is computationally tractable and can be efficiently solved to a global optimum over the selected price grid by commercial solvers.

The rest of this paper is organized as follows. Section 2 introduces the proposed grid interaction mechanism for smart buildings. Section 3 details the single-level MILP model for incentive pricing. Section 4 presents the case studies and analyzes the simulation results. Finally, Section 5 concludes the paper.

2 Smart building–grid interaction mechanism

2.1 Customer directrix load

The CDL is defined from the grid’s perspective as the ideal energy consumption profile. It is formulated in Equation 1 as the ratio of the ideal consumption curve to the total energy consumption:where is the CDL (ranging from 0 to 1), is the ideal energy consumption curve, and is the total energy consumption over the scheduling horizon .

In scenarios characterized by high RE penetration, the ideal response profile theoretically mirrors the inverse of the system’s net load curve. This implies that adjustable loads should ramp up consumption during periods of surplus generation (e.g., midday PV peaks) and ramp down during system load peaks, thereby maintaining power balance. Crucially, the CDL framework is versatile; its specific definition is not static but can be tailored to diverse grid requirements, ranging from peak shaving and fluctuation mitigation to emergency reserve provision. Focusing specifically on the RE accommodation scenario, this paper adopts the target CDL profile illustrated in Figure 1. Maximizing the alignment between a smart building’s actual consumption profile and this target CDL directly maximizes the effectiveness of RE accommodation.

FIGURE 1

2.2 Interaction framework

Smart buildings serve as critical flexible resources for grid-friendly interactions. To fully unlock their regulation potential, this paper employs a CDL-based interaction mechanism (illustrated in Figure 2) between a Distribution System Operator (DSO) and smart buildings. In this framework, the DSO broadcasts two key signals: (1) the CDL, representing the ideal consumption profile derived from the net load curve for this RE accommodation scenario; and (2) the incentive price, defined as the unit revenue awarded when a building’s profile aligns perfectly with the CDL.

FIGURE 2

In this framework, smart buildings function as aggregators of flexible resources. Distinct from general controllable loads, which are typically single-type devices, a smart building integrates heterogeneous assets, including PV, ES, EVs, and Heating, Ventilation, and Air Conditioning (HVAC) systems. Consequently, its regulation capability extends beyond simple unidirectional power curtailment to include energy-constrained bidirectional load shaping. Furthermore, to ensure user privacy and computational scalability, the DSO interacts only with the aggregated external characteristics of the building rather than directly controlling internal devices. Driven by the similarity-based pricing, smart buildings actively optimize their internal scheduling to reshape their net load profiles toward the CDL. This process allows them to maximize incentive revenue and maintain occupant comfort while facilitating orderly RE accommodation.

The determination of the incentive price is fundamentally a leader-follower game driven by the conflicting interests of the stakeholders. Smart buildings, aiming to maximize their interaction revenue, naturally favor higher prices. Conversely, the DSO faces a strategic trade-off: while it seeks to minimize operational costs by limiting incentives, it must simultaneously offer sufficient compensation to elicit the magnitude of demand response necessary for orderly RE accommodation. Therefore, finding the equilibrium price that reconciles the DSO’s cost-efficiency with the buildings’ willingness to participate constitutes the pivotal challenge of this mechanism.

2.3 Analytical CDL formulation

While defining the CDL as a simple inverse of the net load curve is intuitive for RE accommodation, it can benefit from a more systematic mathematical derivation. This section proposes a generalized formulation method, demonstrating that the aforementioned inverse-curve approach is merely a special case of this broader framework.

First, we construct an optimization model aimed at minimizing RE curtailment. The objective function incorporates a quadratic penalty on the curtailment volume to mitigate large-scale RE curtailment:where denotes the accommodated RE, is the day-ahead forecasted maximum RE output, and is the CDL to be determined.

Assuming the DSO’s target power purchase is and the conventional load is , the system must satisfy the following energy balance constraint:

Additionally, the CDL must be normalized:

As detailed in Supplementary Appendix, by solving the KKT conditions for the optimization problem Equations 2–4, the analytical optimal solution for the CDL is derived as:

where is defined in Equation 6 as:

Physically, represents the instantaneous ideal load demand, while the second term in Equation 5 captures the mean deviation between the aggregate ideal demand and the building’s total energy capacity . The prefactor outside the brackets normalizes the CDL, ensuring compliance with Equation 4. Consequently, the optimal formulation strategy revealed by Equation 5 dictates that the limited total energy should be preferentially allocated to periods exhibiting higher ideal load demand. The specific magnitude of this allocation is determined by the extent to which the demand in a given period exceeds the average demand across all periods. This mechanism underscores the regulation role of the CDL in shaping the electricity consumption behavior of smart buildings.

In the best case where the building’s total energy capacity exactly matches the cumulative ideal demand, RE is fully accommodated (see Supplementary Equation S9 in Supplementary Appendix). Under this perfect equilibrium, the optimal CDL solution degenerates to the form given in Equation 7:

This result mathematically validates the “inverse net load” strategy as a special case of our generalized formulation. However, such perfect synchronization is virtually nonexistent in real-world operations due to the inherent mismatch between volatile RE output and finite building consumption capabilities.

3 Single-level MILP model for incentive pricing

The incentive pricing mechanism embodies the economic trade-off between the DSO and smart buildings, a relationship mathematically formulated as a bilevel optimization problem. In the upper-level (leader) model, the DSO determines the incentive price to minimize total operational costs, treating the buildings’ response as a dependent variable. Conversely, in the lower-level (follower) model, smart buildings treat this price as an exogenous parameter and optimize their consumption profiles to maximize incentive revenue. This structure captures the sequential decision-making process where the operator anticipates the users’ optimal response to pricing signals.

3.1 Linear definition of similarity and incentive revenue

Let denote the incentive price to be determined, represent the CDL similarity, and be the reported day-ahead total energy consumption. The incentive revenue is defined as:

In existing literature, CDL similarity is conventionally characterized using Euclidean distance. This approach quantifies the deviation using Equation 9 and the resulting similarity metric using Equation 10:where denotes the power consumption of the -th smart building at time . Here, Equation 10 maps the geometric distance into a normalized similarity metric.

To quantify the deviation between the actual consumption profile and the CDL, we employ the Manhattan distance:

Based on the inverse relationship that a smaller distance indicates higher alignment, we define the linear CDL similarity as:

Assuming the building load profile is nonnegative and normalized, i.e., , the Manhattan distance satisfies , and thus Equation 12 ensures , satisfying the normalization criterion stated above.

The absolute value in Equation 11 introduces nonlinearity. However, since the lower-level problem aims to maximize incentive revenue, the distance term enters the objective function with a negative coefficient. This ensures a tight linearization. Consequently, Equation 11 can be exactly reformulated by introducing auxiliary variables :

The non-negativity of the similarity is imposed by Equation 15:

This formulation establishes a linear similarity metric. The validity of this metric, especially its ranking monotonicity compared with the traditional Euclidean metric, is quantitatively analyzed in Section 4.

The initial revenue formulation in Equation 8 focuses solely on the magnitude and profile of the load response, neglecting the smart building’s physical regulation capabilities, such as maximum response duration, response frequency, actuation limits, and ramping rates. We introduce the response quality coefficient in Equation 16 to quantify this “quality of response”:where denotes the number of response quality indicators, and denotes the normalized score of the -th indicator. A general normalization rule is given in Equation 17:where is the evaluated capability of building for indicator , and is a reference value specified by the DSO according to program requirements, historical performance records, or representative benchmarks for similar buildings. In this paper, is treated as an exogenous interface parameter obtained from the building response quality evaluation, and its inclusion does not change the structure of the proposed pricing model.

Integrating the response quality coefficient and the linearized similarity metric into the base formula yields the specific incentive revenue formulation for the lower-level problem:Within the lower-level context, the incentive price functions as a fixed parameter, while is the decision variable. This structural distinction ensures that the lower-level objective remains linear.

In the upper-level problem, the role of the incentive price is reversed: becomes the decision variable, while the load profile is treated as a fixed input. Consequently, the DSO’s total incentive cost is expressed as a linear function of :

3.2 Upper-level problem: DSO pricing model

As previously stated, the DSO’s objective is to minimize operational costs. The objective function is specifically formulated as:where is the electricity purchased by the DSO from the upstream grid during period , is the unit penalty cost for RE curtailment, represents the curtailed power of RE unit , and and are given by the lower-level optimal response of smart building .

The net electricity trading cost of the DSO is defined in Equation 21 as:where and are the upstream purchase price and the retail selling price charged by the DSO, respectively, and represents the predictable electricity consumption of rigid loads within the DSO service area.

To ensure electricity supply, the DSO must satisfy the energy balance constraint for each period:where is the maximum available output from the -th RE unit, and represents the predicted electricity consumption of other non-participating loads.

Finally, the decision variables are constrained by their respective operational limits:where and define the permissible range for the incentive price, and limits the maximum electricity purchased from the upstream grid during period . This completes the formulation of the upper-level problem, defined by (Equations 19–23a).

3.3 Lower-level problem: smart building response model

Modeling the external characteristics of the -th smart building requires balancing model accuracy, privacy preservation, and computational tractability. A smart building may contain heterogeneous resources, including rigid loads, distributed generation, shiftable loads, HVAC systems, ES, and EV fleets. Directly exposing device-level models to the DSO would not only increase the dimensionality of the pricing problem but also reveal private operational information of the building. Therefore, this paper adopts a resource-level aggregated model to represent the net consumption behavior of the smart building. This model distinguishes major flexibility categories to reliably capture the essential physical boundaries and response capabilities, while preserving a linear lower-level structure so that the subsequent single-level reformulation remains tractable.

The aggregated net power consumption of building at time is formulated as:Here, and denote the rigid load demand and building-side distributed generation output, respectively; and represent the shiftable load and equivalent HVAC power consumption; and and are the net charging power of the equivalent ES device and aggregated EV fleet, respectively. For these bidirectional resources, positive values indicate charging and negative values indicate discharging.

Shiftable loads represent flexible electricity demands whose operation can be temporally redistributed within the scheduling horizon. The operating power of the shiftable load is bounded by:where and denote the lower and upper bounds of the shiftable load, respectively. The total energy requirement of the shiftable load is enforced as:where is the required energy consumption of the shiftable load over the scheduling horizon.

b. HVAC system:

The HVAC system is modeled through an aggregated thermal dynamic equation. Let denote the indoor temperature of building at time . The indoor temperature evolves as:where is the outdoor temperature, is the thermal inertia coefficient, represents exogenous thermal gains such as solar radiation and internal heat gains, is the HVAC regulation coefficient, and is the product of the operating-mode sign and the efficiency coefficient. Specifically, corresponds to cooling, while corresponds to heating.

The HVAC power is bounded by:where is the maximum HVAC power. The indoor temperature is constrained by the comfort range and terminal condition as follows:where and denote the lower and upper indoor comfort limits, respectively.

c. Energy storage:

Assuming negligible conversion losses, the operational constraints for ES are formulated as follows:where and represent ES charging and discharging, respectively; and denote the maximum charging and discharging powers; is the state of charge; and are the minimum and maximum allowable SOC levels; and represents the ES energy capacity.

d. EV fleet:

Assuming negligible charging and discharging losses, the aggregated EV fleet is modeled with time-varying power and energy boundaries, which capture vehicle arrivals, departures, and mobility requirements. The operational constraints are formulated as follows:where and represent EV fleet charging and discharging, respectively; and denote the time-varying maximum charging and discharging powers; is the aggregated energy state of the EV fleet; and represent the energy brought by arriving EVs and the required energy taken by departing EVs; and and denote the time-varying lower and upper energy boundaries, respectively.

Notably, the net electricity cost inherently captures arbitrage opportunities from the equivalent ES and EV fleet, as discharging during peak price periods naturally reduces the cost term.

Accordingly, the objective function for the lower-level problem is formulated as:where is the incentive revenue (calculated via Equation 18), and represents the retail electricity price. The functions and denote the utility of shiftable-load usage and thermal comfort, respectively.

To maintain model tractability, the two utility functions are represented by concave piecewise-linear approximations. Introducing auxiliary variables and , the linearized utility constraints are given by:where and denote the slope and intercept of the -th linear segment for the shiftable-load utility, and and denote those for the thermal comfort utility.

This completes the formulation of the lower-level optimization problem, defined by constraints Equations 13, 14 and Equations 24–35b. It is noted that the overall system model comprises such structurally identical lower-level problems, one for each participating smart building.

3.4 Single-level reformulation

After applying the linear similarity metric and the piecewise linear approximation of the utility functions, the lower-level problem of each smart building becomes an LP. Therefore, its optimality can be equivalently characterized by primal feasibility, dual feasibility, and strong duality.

For building , let collect all lower-level decision variables, and all inequality constraints are collected as shown in Equation 36:while all equality constraints are collected as shown in Equation 37:where dual variables and is unrestricted.

Given the incentive price , the lower-level problem of building can be compactly written as:Here, denotes the non-incentive part of the lower-level objective, while represents the incentive revenue in Equation 18.

For embedding the lower-level optimality conditions into the upper-level problem, the incentive price parameter is replaced by the upper-level decision variable . Accordingly, the dual problem associated with Equations 38a–c is given by Equations 39a–c

According to strong duality for LPs, the lower-level response of building is optimal if and only if the primal problem is feasible, the dual problem is feasible, and the primal and dual objective values are equal. Therefore, the strong-duality condition is:

The term contains a bilinear product between continuous variables. Since is shared by the bilinear terms of all buildings, these terms can be aggregated. Define the aggregate quantity Φ in Equation 41 as:and introduce the auxiliary variable:

Then, instead of imposing Equation 40 for each building separately, the strong-duality conditions can be aggregated as

Under primal and dual feasibility, each building-level duality gap is nonnegative; thus, forcing their sum to zero ensures zero gap and optimality for every building.

The resulting single-level model minimizes the upper-level DSO objective Equation 20, where the lower-level response parameters are replaced by the corresponding endogenous variables and the aggregate incentive payment is represented by . The feasible region consists of the upper-level DSO constraints Equation 19 and Equations 22, 23, the lower-level primal feasibility constraints Equations 13, 14 and Equations 24–35b, the lower-level dual feasibility constraints Equations 39b, 39c, and the aggregated strong-duality equality Equation 43. The only remaining nonlinear term is in Equation 42.

3.5 Linearization of bilinear terms

To linearize , this paper discretizes the feasible interval of the incentive price into candidate price levels, as illustrated in Figure 3. This treatment is consistent with practical incentive schemes, where prices are usually announced with a finite price resolution. The candidate prices are defined as:

FIGURE 3

where indicates that the -th candidate price is selected. Substituting the discrete price expression into Equation 42 gives Equation 45:

Since , the aggregate incentive quantity satisfies Equation 46:

Thus, can be exactly represented by the following Big-M constraints in Equation 47:

The system-level bilinear term can then be expressed linearly in Equation 48 as:

Together with the primal-dual strong-duality reformulation above, this discrete-price representation completes the single-level MILP reformulation.

The maximum discretization error in the bilinear term is bounded by Equation 49:

Hence, increasing reduces and improves the approximation of the incentive price.

4 Case study

4.1 Comparison of similarity metrics

To validate the rationale behind using the proposed linear Manhattan metric, we benchmark it against established nonlinear alternatives. Specifically, we consider two variants of Euclidean similarity and and the widely used Cosine similarity , as defined in Equations 50a–c:where the parameters are set to and , respectively.

To empirically evaluate the consistency across these metrics, we generated a synthetic dataset comprising 1,000 random -dimensional vectors (elements uniformly distributed in [0, 1]). The core objective is to assess monotonic consistency. To visualize this, we performed a dual-sorting analysis: the sample vectors were sorted in ascending order based on the Euclidean similarity and the Manhattan similarity , respectively. Under each sorting criterion, the similarity scores defined by the alternative metrics were synchronously rearranged to match the sorted index. The resulting trends from these two perspectives are illustrated in Figure 4 (displaying raw, unsmoothed data).

FIGURE 4

Visual inspection of Figure 4 reveals that the linear Manhattan metric demonstrates strong consistency with the nonlinear Euclidean benchmark , exhibiting only minor local fluctuations. To quantify this alignment, we utilize Spearman’s rank correlation coefficient, where a value of one implies identical ranking logic. The analysis yields a coefficient of 0.9643 between and (and 0.9893 for Cosine similarity). This high correlation strongly suggests that provides a reliable approximation of the monotonic ranking properties of the Euclidean metric. Consequently, a sample that yields a high similarity score under the Euclidean definition will correspondingly yield a high score under the Manhattan definition, with the distinct advantage that the latter preserves mathematical linearity.

4.2 Optimal CDL under different RE outputs

Figure 5 illustrates the deviations from the benchmark. In the RE-surplus scenario, the optimal CDL profile is significantly sharpened, becoming further elevated during RE peaks and suppressed during periods of scarcity. This shift demonstrates that when the total adjustable load is insufficient, the limited consumption capacity must be prioritized to absorb RE during peak output hours. Conversely, the RE-shortage scenario exhibits a flatter trend. This difference arises because the ample adjustable load capacity provides sufficient overall accommodation capability, alleviating the need for aggressive load reshaping to meet RE targets.

FIGURE 5

4.3 Smart building response analysis

To validate the proposed model, we construct a simulation case study comprising smart buildings. These are classified into ten distinct types (B1 to B10), each containing three homogeneous buildings with parameters detailed in Table 1. The simulation spans a daily horizon of hours with . Figure 5 illustrates the forecasted maximum RE output.

In terms of economic parameters, the RE curtailment penalty is set to 5 CNY/kWh. The grid electricity price follows a TOU scheme, as depicted in Figure 6, which highlights the correlation between price signals and the system net load. For the incentive mechanism, the incentive price bounds are set as and CNY/kWh. Regarding algorithmic settings, the shiftable-load usage utility and thermal comfort utility are both approximated by 10 linear segments, and candidate prices are used for the incentive price to facilitate the reformulation. The resulting single-level MILP model is solved using the commercial solver Gurobi with a 0.01% optimality gap.

FIGURE 6

Under the defined simulation parameters, the model yields an optimal incentive price of 0.2265 CNY/kWh. This value aligns in magnitude with the DSO’s grid purchase price, indicating that the economic game between the upper-level DSO (cost minimization) and lower-level buildings (utility maximization) successfully converges to an equilibrium. Regarding computational performance, the proposed method demonstrates significant tractability. Despite the binary variables introduced by the discrete-price reformulation, the resulting single-level MILP problem maintains a moderate scale and converges within just 4.39 s.

Figure 7 visualizes the changes in the aggregated consumption profiles across all smart buildings compared to the TOU-only baseline. The target CDL features a “dual-peak, dual-valley” profile, necessitating load ramping during high-RE periods and shedding during system peaks. In Case 1, baseline consumption is regulated solely by the TOU price; buildings naturally shift usage toward low-price periods, yielding a low CDL similarity of 0.7632. Following the introduction of CDL incentives in Cases 2–4, the load profiles exhibit varying degrees of reshaping. In Case 2, because the fixed price is too low to significantly disrupt the TOU-driven pattern, the consumption profile remains largely unchanged from Case 1, marginally increasing the similarity to 0.7820. Conversely, applying the upper-bound incentive price in Case 3 drives substantial shifts that closely track the CDL, particularly during – and –. The similarity accordingly jumps to 0.9037, confirming that adequate incentives effectively align building consumption with grid objectives. Under the optimal incentive price in Case 4, the resulting profile shows a similar, albeit slightly less aggressive, reshaping pattern, achieving a similarity of 0.9007.

FIGURE 7

Table 2 further compares the economic outcomes across the four cases. Under the sole regulation of the TOU price, Case 1 concurrently yields the lowest DSO sales revenue and building electricity costs; the maximum RE curtailment penalty, particularly during the high-PV window of –, as illustrated in Figure 8, where TOU pricing imperfectly matches RE accommodation requirements; and the lowest building net benefit due to the absence of incentive revenue.

FIGURE 8

Conversely, adopting a fixed high incentive price in Case 3 maximizes DSO sales revenue, RE accommodation, and building net benefits; however, this aggressive payment significantly inflates incentive costs, reducing DSO net revenue by 3.3% compared to Case 1. Case four resolves this by striking an optimal economic balance. While Case 4 accepts a marginal drop in CDL similarity and a negligible RE curtailment penalty compared to Case 3, it successfully boosts DSO net revenue by 7.7%. This demonstrates that the optimized pricing mitigates the financial burden of manually specified high prices with almost no compromise to load-guidance efficacy. Driven by the upper-level objective that prioritizes DSO interests, the proposed model identifies a price that slightly reduces building-side profits relative to Case 3, but secures superior DSO profitability alongside near-perfect RE accommodation. Consequently, both DSO net revenue and building net benefits are substantially improved relative to Cases 1 and 2.

Figure 9 contrasts the consumption profiles of building types B2 and B10, highlighting the impacts of resource capacity and response quality. Constrained by a lower response quality coefficient and limited ES and EV capacities, Building B2 fails to maintain consistent tracking throughout the scheduling horizon, yielding a similarity of 0.7597. Conversely, Building B10 leverages a higher response quality coefficient and superior storage capacity to enhance its load-shaping capability, closely matching the CDL across most intervals with a higher similarity of 0.9098.

FIGURE 9

Figure 10 illustrates the aggregate charging/discharging power and energy trajectory of the ES and EV resources within the representative building B10. This operational pattern is jointly driven by the TOU electricity price and the CDL-tracking incentive. During –, when the power purchase cost is high and the CDL is not at its peak, these flexible energy resources tend to discharge to reduce the building’s electricity procurement costs while avoiding excessive deviation from the CDL. During –, the electricity price drops to its valley level while the CDL rises rapidly toward its peak. In this period, the economic arbitrage signal aligns perfectly with the CDL-tracking signal, inducing intensive charging and a substantial increase in stored energy. During –, the electricity price peaks again while the CDL decreases sharply; therefore, the ES and EV resources switch to discharging, which concurrently lowers electricity costs and suppresses the building’s net consumption in accordance with the CDL trend. In the final intervals, the charging/discharging actions are primarily adjusted to satisfy the remaining energy and terminal operational requirements. Overall, the trajectory demonstrates that the smart buildings utilize their inherent storage flexibility to coordinate price arbitrage with CDL tracking, thereby enabling effective adjustment of their net power profiles to track the CDL.

FIGURE 10

4.4 Sensitivity analysis

To evaluate the model’s performance under varying RE penetration levels, we introduce the RE output coefficient as a scaling multiplier applied to the baseline RE profile. The sensitivity results, visualized in Figure 11, demonstrate that this coefficient exerts a non-monotonic influence on both the optimal incentive price and the CDL similarity. When the RE output coefficient is small, the need for CDL-based load guidance is limited, resulting in relatively low incentive prices and similarity scores. As the coefficient grows, the marginal value of steering building consumption toward high-RE periods increases, driving up both indicators. Within the coefficient range of –, the optimal incentive price attains its upper bound while the similarity stays at a high plateau, indicating that the available building flexibility is fully leveraged. However, once the RE output becomes excessive, the CDL requirements surpass the physical regulation limits of the building cluster. In this saturation region, maintaining a high incentive price fails to further enhance tracking alignment but significantly inflates the DSO’s financial burden. Consequently, the optimal incentive price decreases, and the tracking similarity drops accordingly.

FIGURE 11

To investigate the impact of the price discretization resolution on the reformulation accuracy and computational efficiency, a sensitivity analysis regarding the number of candidate prices is conducted. As illustrated in Figure 12, when is relatively small, the objective error drops sharply, whereas the runtime grows only marginally. As continues to increase, the decrease in the objective error slows down and gradually plateaus below 0.3%, while the runtime escalates exponentially. Simultaneously, Figure 13 depicts the evolution of the optimal incentive price along with its discretization error bound. The optimal price initially decreases and eventually stabilizes around CNY/kWh as expands. Therefore, selecting strikes an excellent trade-off between solution accuracy and computational efficiency, securing an exceptionally low objective error (under 0.3%) and a tight price deviation (within CNY/kWh) while maintaining a highly efficient solution speed (within 5 s).

FIGURE 12

FIGURE 13

Figure 14 shows the sensitivity of the proposed model to the number of piecewise linear segments used for utility-function approximation. As the number of segments increases, the objective value first changes noticeably and then gradually becomes stable, indicating that a finer approximation improves the accuracy of the utility representation. However, the runtime increases significantly when too many segments are used, especially after the segment number exceeds 50. Therefore, increasing the number of segments beyond a moderate level brings limited improvement in the objective value but substantially increases the computational burden. Based on this accuracy–efficiency trade-off, 10 segments are adopted.

FIGURE 14

4.5 Computational comparison of different reformulations

To further demonstrate the computational advantage of the proposed reformulation, three solution strategies are compared in this subsection. M1 denotes the direct bilinear formulation, which solves the single-level model with the remaining bilinear term in (54) and is used as a high-accuracy benchmark. M2 denotes the conventional KKT-complementarity MILP reformulation, where the optimality of the lower-level LP is represented by stationarity, primal feasibility, dual feasibility, and complementary slackness conditions. M3 denotes the proposed MILP reformulation, in which the lower-level complementarity conditions are replaced by primal-dual feasibility and an aggregated strong-duality equality.

As shown in Table 3, because the continuous bilinear product is retained, the objective value and the optimal incentive price obtained by M1 are globally optimal. However, directly solving this bilinear term increases the computational burden, requiring 29.16 s to resolve the case.

M2 fails to converge within the 1-h time limit. This computational intractability stems from the complementary slackness conditions introduced by each lower-level inequality. When linearized via Big-M or SOS1 methods, the number of binary variables scales heavily with the number of buildings, time periods, and piecewise-linear utility segments, causing a substantial branch-and-bound burden. Moreover, the Big-M relaxation often yields loose continuous bounds, further hindering the solver’s search process.

By contrast, the proposed M3 converges in only 4.39 s, reducing the runtime by 84.9% compared to M1. The obtained objective value exhibits a relative deviation of only 0.296% from the direct bilinear benchmark. Similarly, the optimal incentive price yields a tight absolute deviation of only CNY/kWh. These results demonstrate that the proposed reformulation preserves high pricing accuracy while achieving a substantial reduction in computational time. As for the contrast with the conventional KKT method (M2), this computational breakthrough primarily stems from the elimination of per-constraint complementary slackness. The proposed reformulation instead aggregates lower-level optimality into a single primal-dual strong-duality equality. Consequently, the formulation introduces only binary variables for candidate price selection regardless of the number of smart buildings, substantially reducing the dependence of the combinatorial complexity on the lower-level inequality scale and demonstrating superior scalability for large-scale pricing problems.

5 Conclusion

Future research will extend this work in two key directions. First, we aim to address the heterogeneity in response capabilities across diverse smart buildings by exploring differentiated pricing mechanisms, thereby achieving more refined incentive allocation. Second, we plan to investigate distributed solution algorithms tailored for large-scale building clusters, specifically to further enhance the computational efficiency and scalability of the pricing framework.

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