Towards Realistic Virtual Power Plant Operation: Behavioral Uncertainty Modeling and Robust Dispatch Through Prospect Theory and Social Network-Driven Scenario Design

Abstract

The growing complexity of distribution-level virtual power plants (VPPs) demands a rethinking of how flexible demand is modeled, aggregated, and dispatched under uncertainty. Traditional optimization frameworks often rely on deterministic or homogeneous assumptions about end-user behavior, thereby overestimating controllability and underestimating risk. In this paper, we propose a behavior-aware, two-stage stochastic dispatch framework for VPPs that explicitly models heterogeneous user participation via integrated behavioral economics and social interaction structures. At the behavioral layer, user responses to demand response (DR) incentives are captured using a Prospect Theory-based utility function, parameterized by loss aversion, nonlinear gain perception, and subjective probability weighting. In parallel, social influence dynamics are modeled using a peer interaction network that modulates individual participation probabilities through local contagion effects. These two mechanisms are combined to produce a high-dimensional, time-varying participation map across user classes, including residential, commercial, and industrial actors. This probabilistic behavioral landscape is embedded within a scenario-based two-stage stochastic optimization model. The first stage determines pre-committed dispatch quantities across flexible loads, electric vehicles, and distributed storage systems, while the second stage executes real-time recourse based on realized participation trajectories. The dispatch model includes physical constraints (e.g., energy balance, network limits), behavioral fatigue, and the intertemporal coupling of flexible resources. A scenario reduction technique and the Conditional Value-at-Risk (CVaR) metric are used to ensure computational tractability and robustness against extreme behavior deviations.

1. Introduction

As power systems worldwide transition toward distributed, decarbonized, and digitized architectures, the role of virtual power plants (VPPs) has shifted from a supplementary aggregation tool to a foundational pillar for managing decentralized flexibility [1]. At the distribution level in particular, where the spatial heterogeneity of load profiles, the temporal volatility of distributed energy resources (DERs), and the diversity of consumer participation behaviors intersect, VPPs serve as critical intermediaries between end-users and grid operators [2]. Their function is no longer confined to aggregating rooftop solar or coordinating small battery fleets; instead, they are expected to orchestrate complex multi-energy portfolios under uncertainty, aligning user behaviors with system-level dispatch, stability, and market objectives [3]. Traditional optimization frameworks for VPP scheduling have made substantial contributions in handling the physical and economic aspects of distributed systems [4]. They account for energy balance, network constraints, storage cycling, and cost minimization under uncertain renewables through models such as two-stage stochastic programming, robust optimization, and chance-constrained formulations [5]. However, these formulations tend to abstract away the most volatile and least predictable component of a VPP: the end-user. In most existing models, user-side flexibility is either treated as a deterministic quantity or as a probabilistic input with fixed distributions derived from historical participation [6]. These simplifications fall short of capturing the psychological, social, and contextual factors that drive demand-side response (DSR) behaviors in the real world [7].

Indeed, decades of empirical research on demand response pilots, behavioral trials, and residential energy interventions have consistently demonstrated that user participation is neither static nor fully rational. Households may reject lucrative DR offers due to comfort concerns or habit inertia, while small commercial users may respond inconsistently based on daily operational uncertainty [8]. Moreover, behavior is not only individualized but socially embedded: Users respond not just to incentives but also to peers, norms, and narratives [9]. Existing VPP models that assume uniform or price-elastic response mechanisms fail to represent these dynamics, resulting in dispatch schedules that are either infeasible in practice or suboptimal in terms of realized flexibility [10].

A small but growing body of literature has started to acknowledge this modeling gap. Some researchers have explored the incorporation of probabilistic user models into energy scheduling, representing participation likelihoods via Bernoulli processes or Gaussian mixture models. Others have attempted to characterize behavioral response using statistical regressions fitted to participation datasets [5,11]. However, most of these efforts stop at scenario generation and do not propagate user behavior uncertainty into the core of the dispatch optimization process. Furthermore, they rarely distinguish between the behavioral typologies of different user classes—residential versus commercial versus industrial—or model intra-class heterogeneity arising from income, comfort preferences, device type, or temporal availability [12]. This paper advances the state of the art by proposing a novel behavior-aware aggregation and dispatch strategy for distribution-level VPPs that explicitly models heterogeneous and uncertain user behavior and integrates this modeling into a tractable, robust optimization process [13]. At its core, our framework addresses the fundamental disconnect between the behavioral richness of end-users and the optimization-centric logic of current VPP scheduling [14]. We argue that behavioral diversity—far from being a noise to average out—is a structural feature of decentralized energy systems that must be directly modeled, simulated, and incorporated into dispatch strategies. To this end, we develop a two-layer methodological architecture [15]. The first layer constructs a behavioral modeling engine grounded in behavioral economics and social network theory. Rather than relying on abstract or empirically undefined probability distributions, we use Prospect Theory to model individual users’ decision-making under uncertainty. Prospect Theory, introduced in cognitive psychology and now widely applied in behavioral finance and marketing, provides a utility function that is nonlinear, asymmetric, and reference-dependent [16]. Users are more sensitive to losses than gains, weigh probabilities subjectively, and evaluate outcomes not in absolute terms but relative to their expectations. We parameterize this model using survey-based elicitation of user preferences, aligned with previous work in experimental energy economics that has validated such techniques. Additionally, we incorporate Social Influence Theory by modeling how peer decisions affect individual participation rates. Drawing on real-world evidence that DR participation can increase when neighboring users participate, we build a recursive peer-influence structure where user response is a convex combination of individual willingness and the behavior of socially proximate users [17].

These psychological and sociological components are synthesized into a probabilistic participation map across all users in the VPP. Each user is associated with a time-varying likelihood of responding to a dispatch signal, contingent upon the offered incentive, social exposure, behavioral inertia, and their own fatigue from recent past events. This map is not static; it evolves over time through an online learning mechanism that incorporates observed behaviors into real-time parameter updates. Such adaptivity ensures that the model is responsive to long-term behavioral drift and short-term context fluctuations, both of which are well-documented in empirical demand response programs.

The second layer of our framework is a robust dispatch optimization model that integrates the behavioral uncertainty defined above into a two-stage stochastic program. In the first stage, the VPP operator determines pre-commitment schedules for energy resources—flexible loads, storage units, and demand curtailment plans—under uncertainty about actual user participation. The second stage then adjusts dispatch actions after behavioral outcomes are realized, modeled through a scenario tree derived from the probabilistic behavioral map. Each scenario represents a possible realization of user participation across time and across nodes in the distribution network. These scenarios are generated not only through sampling but also filtered and clustered using K-medoids and weighted by similarity-based softmax distributions to ensure both diversity and computational tractability.

The dispatch formulation incorporates a wide set of operational constraints, including nodal energy balance, storage charge-discharge dynamics, voltage deviation limits, and time-coupling constraints for devices such as electric vehicles and thermostatically controlled loads. Crucially, it also includes novel behavioral constraints such as participation fatigue—users are less likely to respond again immediately after a recent event—and social saturation, where the marginal benefit of peer influence decays with repeated exposure. Furthermore, to safeguard system performance under low participation scenarios, we incorporate a Conditional Value-at-Risk (CVaR) constraint on the objective function, allowing the operator to limit expected costs or dispatch errors under the worst 1$-\beta$1$-\beta$1$-\beta$% of behavioral scenarios.

Figure 1 illustrates the overall architecture of the DC/AC hybrid microgrid under study. Renewable energy sources, including photovoltaic (PV) arrays and wind turbines, are placed on the generation side, while battery storage and hydrogen storage/fuel cell systems are integrated as energy buffers. These components are connected to the DC bus, which links to the AC bus through an inverter. On the demand side, both critical and non-critical loads are supplied, highlighting the system’s ability to prioritize essential demand. At the bottom, the Energy Management System (EMS), designed with fuzzy logic and JAYA-based controllers, coordinates the operation of all components. The accompanying dataset mapping specifies the input data for each subsystem, including meteorological data, load profiles, storage specifications, converter characteristics, and control parameters, thereby ensuring the reproducibility and clarity of the study.

To demonstrate the efficacy of our approach, we test it on a distribution-level feeder populated with 500 residential users, 30 small commercial buildings, and five industrial facilities, each assigned realistic device configurations including HVAC systems, water heaters, refrigeration loads, EVs, and stationary battery systems. Behavioral parameters are drawn from both synthesized datasets based on field surveys and participation traces from DR pilot studies. The results show that our behavior-aware model outperforms deterministic baselines on multiple metrics: It improves VPP scheduling accuracy by 22%, reduces real-time deviation from planned dispatch by 31%, and increases user engagement rates by 18%. Sensitivity analysis further reveals that the value of social modeling grows with user density and correlation among load profiles, suggesting that as energy systems become more decentralized and interconnected, social behavior modeling becomes not only relevant but essential.

This work pushes the frontier of VPP operation toward a truly socio-technical optimization framework. It demonstrates that by combining behavioral economics, probabilistic modeling, and robust optimization, one can build systems that are not only technically feasible but also behaviorally plausible and operationally resilient. It moves beyond the dichotomy of techno-economic versus human-centric design and instead proposes a synthesis: a model that is computationally rigorous, behaviorally grounded, and practically deployable. The proposed framework is flexible and extensible. It can be adapted to include gamification-based incentive design, privacy-aware coordination mechanisms, or the integration of behavioral feedback into retail electricity tariffs. It is a platform for future work in participatory energy systems, where user engagement is not merely assumed but actively modeled, predicted, and optimized. In conclusion, as VPPs evolve into a core operational mechanism for distributed power systems, their success will increasingly depend on their ability to align with human behaviors—diverse, context-sensitive, and socially mediated. This paper provides the foundation for that alignment, offering a scalable, data-driven, and theoretically robust approach to behavior-aware VPP aggregation and dispatch that integrates the full complexity of the human element into the machinery of modern grid optimization. Prospect Theory, first introduced by Kahneman and Tversky in behavioral economics, is a descriptive model of decision-making under risk and uncertainty. Unlike classical expected utility theory, which assumes that individuals behave rationally and evaluate outcomes in terms of absolute gains, Prospect Theory recognizes that real-world decision-makers exhibit systematic cognitive biases. Specifically, people tend to evaluate outcomes relative to a reference point rather than absolute values, display loss aversion (losses are perceived more severely than equivalent gains), and apply nonlinear probability weighting, where small probabilities are overweighted and large probabilities are underweighted. The resulting utility function is typically concave for gains, convex for losses, and steeper in the loss domain, which reflects the psychological reality that the pain of losing outweighs the pleasure of gaining. By embedding this theory into demand response modeling, the proposed framework captures heterogeneous and bounded-rational user decisions that cannot be explained by conventional linear or risk-neutral models.

2. Mathematical Modeling

Table 1. Summary of main symbols, variables, and parameters used in Section 2 and Section 3.

Symbol	Name	Description	Unit
Sets and indices
$\mathcal{U}$	Users	Set of end-users (residential, commercial, industrial)	–
$\mathcal{K}$	User classes	User types; ${\mathcal{U}}_{\kappa }\subset \mathcal{U}$	–
$\mathcal{N}$	Nodes	Distribution network nodes	–
$\mathcal{T}$	Time periods	Discrete intervals (e.g., 15 min)	–
$\Omega$	Scenarios	Participation scenarios; weights ${\varrho }_{\omega }$	–
Decision variables
${\Pi }_{n,\tau ,\omega }^{\mathrm{gen}}$	Flexible generation	Dispatch of flexible gen at node n	kW
${\Lambda }_{n,\tau ,\omega }^{\mathrm{chg}}$	Storage charge	Charging power	kW
${\Lambda }_{n,\tau ,\omega }^{\mathrm{dis}}$	Storage discharge	Discharging power	kW
${{\rm Y}}_{u,\tau ,\omega }$	DR response	Realized demand response of user u	kW
${\Delta }_{n,\tau ,\omega }^{+/-}$	Mismatch	Positive/negative deviation from plan	kW
$V_{n,\tau ,\omega }$	Voltage	Node voltage magnitude	p.u.
${\Sigma }_{n,\tau ,\omega }^{\mathrm{imb}}$	Imbalance slack	Residual imbalance at node n	kW
${\sigma }_{u,\tau }^{\mathrm{EV}}$	EV SoC	EV battery state-of-charge	kWh
Operational/network parameters
${\alpha }_n^{\mathrm{gen}}$	Gen. cost coeff.	Marginal cost of flexible generation	$/kWh
${\beta }_n^{\mathrm{stor}}$	Storage cost	Charge/discharge cost (or wear)	$/kWh
${\chi }_n$	Imbalance penalty	Penalty on imbalance $\Sigma$	$/kW
${\lambda }_n,{\mu }_n$	Mismatch weights	Quadratic penalty on ${\Delta }^{+/-}$	$/kW2
${\overline{\Pi }}_n^{\mathrm{gen}}$	Gen. cap.	Maximum flexible generation	kW
${\overline{\Lambda }}_n^{\mathrm{chg}/\mathrm{dis}}$	Storage limits	Max charge/discharge power	kW
$Y_{n,m}$	Admittance	Linearized admittance matrix	–
${\underline{V}}_n,{\overline{V}}_n$	Voltage bounds	Lower/upper voltage limits	p.u.
${\eta }_u^{\mathrm{dis}}$	EV efficiency	Discharge efficiency	–
${\Xi }_{n,\tau }^{\mathrm{baseline}}$	Baseline load	Typical demand at node n	kW
Behavioral/social parameters
${\psi }_{\kappa }(·)$	Value function	Prospect Theory value: concave gains ${\alpha }_{\kappa }$, convex losses ${\beta }_{\kappa }$	–
${\lambda }_{\kappa }$	Loss aversion	Factor $>1$ weighting losses vs. gains	–
$w_{\kappa }(·)$	Prob. weighting	Subjective overweight/underweight of p	–
${\vartheta }_{\kappa }$	Logit sensitivity	Steepness of participation logit	–
${\delta }_{\kappa }^{\mathrm{ref}}$	Ref. threshold	Incentive threshold in logit	$/kWh
${\rho }_{\kappa }^{\mathrm{soc}}$	Social sensitivity	Weight of peer influence	–
${\omega }_{uv}$	Influence matrix	Normalized influence from v to u	–
${\kappa }_{\kappa }^{\mathrm{fatigue}}$	Fatigue rate	Exponential decay of repeated response	–
${\sigma }^{\mathrm{crit}}$	Participation threshold	Viability filter for aggregate response	kW
Scenario/risk/optimization parameters
${\varrho }_{\omega }$	Scenario weight	Probability/weight of scenario $\omega$	–
$\gamma$	Softmax temp.	Concentration in scenario reweighting	–
$\beta$	CVaR level	Tail probability in ${\mathrm{CVaR}}_{\beta }$	–
${\varpi }_{1,2,3}$	Trade-off weights	Balance cost, utility, mismatch in objective	–
${ϵ}^{\mathrm{tol}}$	Stop tol.	Tolerance for objective improvement	–
${\delta }^{\mathrm{step}}$	Step tol.	Tolerance for decision change	–
${\eta }^{\left(k\right)}$	Step size	Iterative learning rate in solver	–
${\eta }_{\kappa }^{\mathrm{learn}}$	Online learn rate	Update gain for behavioral parameters	–

provides a consolidated overview of all the mathematical notations employed in the modeling and methodology sections. The symbols are grouped into five categories for clarity. The first category, sets and indices, defines the fundamental elements of the model such as users, nodes, time intervals, and behavioral scenarios. The second category, decision variables, includes dispatch quantities for flexible generation, storage charging and discharging, demand response participation, mismatches, and state variables such as voltage and electric vehicle state-of-charge. The third category, operational and network parameters, summarizes the system-level coefficients and constraints, including generation costs, storage costs, imbalance penalties, voltage limits, and baseline loads. The fourth category, behavioral and social parameters, highlights the Prospect Theory-based constructs, such as value functions, loss aversion, probability weighting, logit sensitivity, and peer influence weights, which capture bounded-rational user behavior. Finally, scenario and risk optimization parameters are included, covering scenario weights, softmax scaling parameters, CVaR risk levels, and solver tolerances.

This single integrated table is designed to enhance readability by allowing readers to quickly locate the meaning, role, and unit of each symbol, avoiding the need to repeatedly search through the text in Section 2 and Section 3.

To rigorously capture the interaction between user-side behavioral uncertainty and system-level operational decisions in the context of distribution-level virtual power plants (VPPs), we construct a two-layer mathematical modeling framework that integrates behavioral response modeling with a scenario-based stochastic dispatch optimization. The behavioral layer formalizes user participation decisions using Prospect Theory to reflect cognitive biases such as loss aversion, diminishing sensitivity, and nonlinear probability weighting. In parallel, social influence dynamics are encoded through a weighted peer network, wherein each user’s likelihood of responding to a demand response (DR) signal is modulated by the historical behavior of their neighbors. These individual- and network-level response mechanisms yield a high-dimensional, time-varying participation probability distribution across the user population, from which a representative set of behavioral scenarios is sampled and reduced.

Figure 2 illustrates the overall structure of the proposed behavior-aware optimization framework for virtual power plant operation. The process begins with data input consisting of observations and external signals, combined with the decision environment that provides system context. These inputs feed into the constraints and objectives block, where technical and economic requirements are specified. A central optimization engine then determines scheduling and dispatch decisions, guided by external control strategies, policies, and incentives. The behavioral modeling module, incorporating Prospect Theory and social influence, provides bounded-rational user participation responses that directly shape the optimization outcomes. The resulting decisions generate outputs and feedback, such as performance metrics, which are iteratively returned to update both the behavioral model and the decision environment. This loop ensures that the framework adaptively aligns technical operation with human and policy-driven factors under uncertainty.

The operational layer formulates a two-stage stochastic program. In the first stage, the VPP operator commits to a dispatch schedule for flexible resources—including responsive loads, electric vehicle (EV) charging, and distributed storage—based on the statistical structure of user behavior and system forecasts. In the second stage, real-time adjustments are implemented once participation outcomes are realized, with the objective of minimizing total operating cost, dispatch mismatch penalties, and risk exposure, while satisfying technical constraints such as energy balance, resource limits, network flow constraints, and behavioral feasibility. Behavioral fatigue, temporal coupling (e.g., inter-temporal availability of storage and EVs), and social feedback loops are embedded into the constraint set to ensure realism in both behavioral dynamics and operational feasibility.

Let $\mathcal{U}$, $\mathcal{T}$, and $\Omega$ denote the sets of users, time intervals, and behavioral scenarios, respectively. We introduce a comprehensive set of decision variables to represent both first-stage commitments and second-stage recourse actions. The model is solved over a reduced scenario tree derived from probabilistic behavioral simulations, with additional robustness enforced through a Conditional Value-at-Risk (CVaR) formulation. The following subsections formally define the objective function and constraints governing the behavior-aware VPP dispatch optimization.

$$\begin{array}{cc}\hfill \underset{\Pi ,\Lambda ,\Sigma }{min}& \sum _{\tau =1}^{\mathcal{T}}\sum _{\omega \in \Omega }{\varrho }_{\omega }·\left(\sum _{n\in \mathcal{N}}\left[{\alpha }_n^{\mathrm{gen}}(\tau ,\omega )·{\Pi }_{n,\tau ,\omega }^{\mathrm{gen}}+{\beta }_n^{\mathrm{stor}}\left(\tau \right)·{\Lambda }_{n,\tau ,\omega }^{\mathrm{chg}}+{\chi }_n·{\Sigma }_{n,\tau ,\omega }^{\mathrm{imb}}\right]\right)\hfill \end{array}$$

(1)

This formulation represents the expected cost across all participation scenarios $\omega \in \Omega$, weighted by their scenario probabilities ${\varrho }_{\omega }$, for each time period $\tau$. The term ${\alpha }_n^{\mathrm{gen}}$ represents the marginal cost of flexible generation dispatch at node n, while ${\Pi }_{n,\tau ,\omega }^{\mathrm{gen}}$ is the power scheduled under scenario $\omega$. The storage charging cost is captured by ${\beta }_n^{\mathrm{stor}}$ multiplied by ${\Lambda }_{n,\tau ,\omega }^{\mathrm{chg}}$, and ${\chi }_n$ penalizes any imbalance ${\Sigma }_{n,\tau ,\omega }^{\mathrm{imb}}$ arising from deviations between expected and realized user participation. The objective thus minimizes the comprehensive expected operational expenditure of the VPP, taking into account uncertainty from behavior-driven flexibility response.

$$\begin{array}{cc}\hfill \underset{{\rm Y}}{max}& \sum _{\tau =1}^{\mathcal{T}}\sum _{\kappa \in \mathcal{K}}\sum _{u\in {\mathcal{U}}_{\kappa }}\left[{\theta }_{\kappa }^{\mathrm{risk}}·\left({\int }_0^{{{\rm Y}}_{u,\tau }}{\psi }_{\kappa }\left(\zeta \right)d\zeta \right)+{\varphi }_{\kappa }^{\mathrm{social}}·log\left(1+\sum _{v\in {\mathcal{N}}_u}{{\rm Y}}_{v,\tau }\right)\right]\hfill \end{array}$$

(2)

This utility function characterizes the behavioral satisfaction derived from voluntary load curtailment ${{\rm Y}}_{u,\tau }$ by end-user u of type $\kappa \in \mathcal{K}$. The first integral represents the value function under Prospect Theory with risk attitude ${\theta }_{\kappa }^{\mathrm{risk}}$, using a nonlinear utility kernel ${\psi }_{\kappa }(·)$ that reflects loss aversion. The second term captures the effect of social reinforcement, where ${\varphi }_{\kappa }^{\mathrm{social}}$ governs the strength of peer influence via logarithmic aggregation over neighboring participants ${\mathcal{N}}_u$. Together, the expression quantifies the heterogeneous and bounded-rational utility received by all agents from participating in demand response, allowing the model to internalize complex psychological and sociological effects.

$$\begin{array}{cc}\hfill \underset{\Delta }{min}& \sum _{\tau =1}^{\mathcal{T}}\sum _{\omega \in \Omega }{\varrho }_{\omega }·\sum _{n\in \mathcal{N}}\left[{\lambda }_n·{\left({\Delta }_{n,\tau ,\omega }^{+}\right)}^2+{\mu }_n·{\left({\Delta }_{n,\tau ,\omega }^{-}\right)}^2\right]\hfill \end{array}$$

(3)

Here, the model penalizes asymmetric real-time mismatches between scheduled dispatch and actual user engagement. The positive and negative deviations are denoted ${\Delta }_{n,\tau ,\omega }^{+}$ and ${\Delta }_{n,\tau ,\omega }^{-}$ respectively, with quadratic costs weighted by ${\lambda }_n$ and ${\mu }_n$. This term reflects behavioral uncertainty in aggregate VPP participation and enforces dispatch reliability. Higher values of ${\lambda }_n$ and ${\mu }_n$ can reflect nodes with tighter stability margins or critical load sensitivity, allowing the system operator to enforce more conservative schedules where necessary. The quadratic structure penalizes large deviations more harshly and encourages probabilistic balance planning under user behavior variability.

$$\begin{array}{cc}\hfill min& {\varpi }_1·\left(\sum _{\tau ,\omega ,n}{\varrho }_{\omega }·{\alpha }_n^{\mathrm{gen}}·{\Pi }_{n,\tau ,\omega }^{\mathrm{gen}}\right)\hfill \\ & -{\varpi }_2·\left(\sum _{\tau ,\kappa ,u}{\theta }_{\kappa }^{\mathrm{risk}}·{\int }_0^{{{\rm Y}}_{u,\tau }}{\psi }_{\kappa }\left(\zeta \right)d\zeta \right)\hfill \\ & +{\varpi }_3·\left(\sum _{\tau ,\omega ,n}{\varrho }_{\omega }·\left({\lambda }_n·{\left({\Delta }_{n,\tau ,\omega }^{+}\right)}^2+{\mu }_n·{\left({\Delta }_{n,\tau ,\omega }^{-}\right)}^2\right)\right)\hfill \end{array}$$

(4)

This weighted multi-objective formulation combines all previous elements into a single scalarized optimization target. The term weighted by ${\varpi }_1$ captures the expected generation cost, the negative term with ${\varpi }_2$ captures aggregated behavioral utility from demand response (thus making it a benefit to be maximized), and the term with ${\varpi }_3$ penalizes behavioral-induced dispatch mismatches. This trade-off formulation allows a system planner to tune between economic efficiency, behavioral incentives, and operational resilience. By adjusting ${\varpi }_1,{\varpi }_2,{\varpi }_3$, decision-makers can design strategies aligned with policy priorities such as cost minimization, user empowerment, or dispatch reliability under uncertainty.

$$\begin{array}{c}\hfill \sum _{n\in \mathcal{N}}\left({\Pi }_{n,\tau ,\omega }^{\mathrm{gen}}+{\Lambda }_{n,\tau ,\omega }^{\mathrm{dis}}-{\Lambda }_{n,\tau ,\omega }^{\mathrm{chg}}+\sum _{u\in {\mathcal{U}}_n}{{\rm Y}}_{u,\tau ,\omega }\right)=\sum _{n\in \mathcal{N}}\left({\Xi }_{n,\tau ,\omega }^{\mathrm{load}}+{\Sigma }_{n,\tau ,\omega }^{\mathrm{imb}}\right)\end{array}$$

(5)

This is the core nodal energy balance constraint across all distribution nodes $n\in \mathcal{N}$ for time $\tau$ under scenario $\omega$. It ensures that net generation (including discharging storage and behavioral response ${\rm Y}$) minus net consumption (including charging and base load) must match the total imbalance $\Sigma$. This is the operational glue that maintains energy consistency in every VPP dispatch scenario.

$$\begin{array}{c}\hfill \left|\sum _{m\in \mathcal{N}}{Y}_{n,m}·\left(V_{n,\tau ,\omega }-V_{m,\tau ,\omega }\right)\right|\le {\nu }_n^{\max },\forall n\in \mathcal{N}\end{array}$$

(6)

Here, we enforce linearized nodal voltage constraints using the admittance matrix $Y_{n,m}$, ensuring voltage differences across neighboring nodes remain within the allowable deviation limit ${\nu }_n^{\max }$. This keeps the power quality within regulatory limits even under uncertain behavioral dispatches.

$$\begin{array}{c}\hfill {\underline{V}}_n\le V_{n,\tau ,\omega }\le {\overline{V}}_n,\forall n\in \mathcal{N},\forall \tau ,\forall \omega \end{array}$$

(7)

Each node’s voltage magnitude must lie within bounds defined by ${\underline{V}}_n$ and ${\overline{V}}_n$. This is critical for low-voltage feeders, where fluctuations in user behavior can lead to instability if not properly bounded.

$$\begin{array}{c}\hfill 0\le {\Pi }_{n,\tau ,\omega }^{\mathrm{gen}}\le {\overline{\Pi }}_n^{\mathrm{gen}},0\le {\Lambda }_{n,\tau ,\omega }^{\mathrm{chg}}\le {\overline{\Lambda }}_n^{\mathrm{chg}},0\le {\Lambda }_{n,\tau ,\omega }^{\mathrm{dis}}\le {\overline{\Lambda }}_n^{\mathrm{dis}}\end{array}$$

(8)

These inequalities define the operating bounds of generation and storage systems at node n. They reflect physical limitations on how much power can be generated, charged into storage, or discharged at any time $\tau$ under any scenario $\omega$.

$$\begin{array}{c}\hfill {\Xi }_{n,\tau }^{\mathrm{baseline}}=\sum _{\kappa \in \mathcal{K}}\sum _{u\in {\mathcal{U}}_{n,\kappa }}{\rho }_{\kappa }·{\zeta }_{u,\tau }^{\mathrm{typ}},\forall n,\forall \tau \end{array}$$

(9)

This aggregates the typical baseline consumption at node n, computed from user-type dependent load profiles ${\zeta }_{u,\tau }^{\mathrm{typ}}$, scaled by a behavioral alignment coefficient ${\rho }_{\kappa }$. It acts as the reference demand before any flexible participation occurs.

$$\begin{array}{c}\hfill \mathbb{P}\left({{\rm Y}}_{u,\tau }>\eta \right)={\displaystyle \frac{1}{1+exp\left(-{\vartheta }_{\kappa }·\left({\iota }_{u,\tau }^{\mathrm{inc}}-{\delta }_{\kappa }^{\mathrm{ref}}\right)\right)}},\forall u,\forall \tau \end{array}$$

(10)

This logistic function estimates the probability of user u of type $\kappa$ participating in demand response at time $\tau$, based on the offered incentive ${\iota }_{u,\tau }^{\mathrm{inc}}$ and a reference threshold ${\delta }_{\kappa }^{\mathrm{ref}}$. ${\vartheta }_{\kappa }$ governs how sharply users transition from rejection to participation.

$$\begin{array}{c}\hfill {\psi }_{\kappa }\left(\zeta \right)=\left\{\begin{array}{cc}{\zeta }^{{\alpha }_{\kappa }},\hfill & \mathrm{if}\zeta \ge 0\hfill \\ -{\lambda }_{\kappa }·{(-\zeta )}^{{\beta }_{\kappa }},\hfill & \mathrm{if}\zeta <0\hfill \end{array}\right.\end{array}$$

(11)

This is the Prospect Theory value function used to model user perception of gains and losses. It is asymmetric, concave for gains, convex for losses, and loss-averse by factor ${\lambda }_{\kappa }>1$. This nonlinearity makes users more sensitive to perceived loss in utility than equivalent gains.

$$\begin{array}{c}\hfill {{\rm Y}}_{u,\tau }={\widetilde{{\rm Y}}}_{u,\tau }^{\mathrm{self}}+{\rho }_{\kappa }^{\mathrm{soc}}·\sum _{v\in {\mathcal{N}}_u}{\omega }_{uv}·{{\rm Y}}_{v,\tau },\forall u,\forall \tau \end{array}$$

(12)

This recursive structure models social contagion, where user u’s response ${{\rm Y}}_{u,\tau }$ is a mix of personal willingness ${\widetilde{{\rm Y}}}_{u,\tau }^{\mathrm{self}}$ and weighted influence from their peers $v\in {\mathcal{N}}_u$, with influence strength ${\omega }_{uv}$. ${\rho }_{\kappa }^{\mathrm{soc}}$ is the social sensitivity coefficient for user type $\kappa$.

$$\begin{array}{c}\hfill {\Omega }_{\tau }^{\mathrm{eff}}=\sum _{\omega \in \Omega }{\varrho }_{\omega }·\mathbb{I}\left[\sum _{u\in \mathcal{U}}{{\rm Y}}_{u,\tau ,\omega }>{\sigma }^{\mathrm{crit}}\right],\forall \tau \end{array}$$

(13)

This constraint defines an effective scenario filter, where only participation realizations that exceed a critical aggregated threshold ${\sigma }^{\mathrm{crit}}$ are considered operationally viable. This keeps the scenario set sparse and computationally tractable.

$$\begin{array}{c}\hfill {{\rm Y}}_{u,\tau }\le {{\rm Y}}_{u,\tau -1}·exp(-{\kappa }_{\kappa }^{\mathrm{fatigue}}),\forall u,\forall \tau >1\end{array}$$

(14)

Participation fatigue is modeled via exponential decay, ensuring users who responded in the previous period $\tau -1$ are less likely to respond again at $\tau$. This captures human behavioral inertia and rest requirements, governed by decay rate ${\kappa }_{\kappa }^{\mathrm{fatigue}}$.

$$\begin{array}{c}\hfill \mathrm{If}{{\rm Y}}_{u,\tau }^{\mathrm{EV}}>0\Rightarrow {\sigma }_{u,\tau +1}^{\mathrm{EV}}={\sigma }_{u,\tau }^{\mathrm{EV}}-{\eta }_u^{\mathrm{dis}}·{{\rm Y}}_{u,\tau }^{\mathrm{EV}},\forall u\in {\mathcal{U}}^{\mathrm{EV}},\forall \tau \end{array}$$

(15)

Finally, for users with electric vehicles, the battery state-of-charge ${\sigma }_{u,\tau }^{\mathrm{EV}}$ evolves dynamically according to the discharged energy ${{\rm Y}}_{u,\tau }^{\mathrm{EV}}$ and efficiency ${\eta }_u^{\mathrm{dis}}$. This constraint guarantees realistic flexibility modeling by enforcing battery physics over time.

3. The Proposed Method

$$\begin{array}{c}\hfill {\mathcal{C}}_{\kappa }^{\mathrm{cluster}}=arg\underset{{\mathcal{C}}_1,\dots ,{\mathcal{C}}_L}{min}\sum _{l=1}^L\sum _{u\in {\mathcal{C}}_l}{∥\left({\nu }_u^{\mathrm{load}},{\xi }_u^{\mathrm{resp}},{\psi }_u^{\mathrm{time}}\right)-{\mu }_l∥}_2^2\end{array}$$

(16)

We initiate the methodology with a clustering operation to categorize users into L behavioral types by minimizing intra-cluster variance across three key dimensions: load signature ${\nu }_u^{\mathrm{load}}$, historical response pattern ${\xi }_u^{\mathrm{resp}}$, and temporal regularity ${\psi }_u^{\mathrm{time}}$. The centers ${\mu }_l$ represent cluster centroids in behavioral space, and this step grounds the subsequent parameterization steps.

$$\begin{array}{c}\hfill {\vartheta }_{\kappa }=arg\underset{\vartheta }{min}\sum _{u\in {\mathcal{U}}_{\kappa }}\sum _{\tau =1}^{\mathcal{T}}{\left(\mathbb{I}\left[{{\rm Y}}_{u,\tau }^{\mathrm{obs}}>0\right]-{\displaystyle \frac{1}{1+exp\left(-\vartheta ·\left({\iota }_{u,\tau }^{\mathrm{inc}}-{\delta }_{\kappa }^{\mathrm{ref}}\right)\right)}}\right)}^2\end{array}$$

(17)

We estimate the behavioral sensitivity coefficient ${\vartheta }_{\kappa }$ by fitting a logistic regression model to observed response events ${{\rm Y}}_{u,\tau }^{\mathrm{obs}}$, aiming to minimize the squared prediction error across all users of class $\kappa$. This parameter controls how sharply users react to incentives, forming a cornerstone of the probabilistic response map.

$$\begin{array}{c}\hfill {\lambda }_{\kappa },{\alpha }_{\kappa },{\beta }_{\kappa }=arg\underset{\lambda ,\alpha ,\beta }{min}\sum _{u\in {\mathcal{U}}_{\kappa }}\sum _{\tau }\left|V_{u,\tau }^{\mathrm{perceived}}-{\psi }_{\kappa }\left({\zeta }_{u,\tau }\right)\right|\end{array}$$

(18)

To encode Prospect Theory behavior, we fit the value function parameters—loss aversion ${\lambda }_{\kappa }$, gain curvature ${\alpha }_{\kappa }$, and loss curvature ${\beta }_{\kappa }$—to minimize the absolute error between reported perceived values $V_{u,\tau }^{\mathrm{perceived}}$ and the theoretical function ${\psi }_{\kappa }(·)$. This process tailors subjective utility modeling to each user type.

$$\begin{array}{c}\hfill {\omega }_{uv}={\displaystyle \frac{f_{uv}^{\mathrm{comm}}·f_{uv}^{\mathrm{geo}}}{{\sum }_{w\in {\mathcal{N}}_u}{f}_{uw}^{\mathrm{comm}}·f_{uw}^{\mathrm{geo}}}},\forall u,v\in \mathcal{U}\end{array}$$

(19)

Social influence weights ${\omega }_{uv}$ are computed using normalized products of communication frequency $f_{uv}^{\mathrm{comm}}$ and spatial proximity $f_{uv}^{\mathrm{geo}}$, capturing both digital and physical interactions. This ensures that peer effects are grounded in empirical user relationships.

$$\begin{array}{c}\hfill {\mathbb{S}}^{\mathrm{scen}}=\left\{{\mathbf{s}}^{\left(1\right)},\dots ,{\mathbf{s}}^{\left(S\right)}\right\}=\mathrm{Sample}\left(\mathcal{B}\left[{{\rm Y}}_{u,\tau }\sim {\mathbb{P}}_{\kappa }\right]\times \mathcal{W}\left({\Xi }_{n,\tau }^{\mathrm{weather}}\right)\times \mathcal{L}\left({\zeta }_{u,\tau }^{\mathrm{latent}}\right)\right)\end{array}$$

(20)

Scenario generation involves sampling from a joint distribution over user behavioral realizations ${{\rm Y}}_{u,\tau }$, weather-modulated load profiles ${\Xi }_{n,\tau }^{\mathrm{weather}}$, and latent behavioral drivers ${\zeta }_{u,\tau }^{\mathrm{latent}}$. The resulting ensemble ${\mathbb{S}}^{\mathrm{scen}}$ spans plausible operational futures for robust planning.

$$\begin{array}{c}\hfill \underset{\mathit{x},\mathit{y}}{min}{\mathbb{E}}_{\omega \in {\mathbb{S}}^{\mathrm{scen}}}\left[{\mathcal{L}}_1\left(\mathit{x}\right)+{\mathcal{L}}_2({\mathit{y}}_{\omega },\omega )+\mathcal{R}(\mathit{x},{\mathit{y}}_{\omega })\right]\end{array}$$

(21)

This compact stochastic optimization formulation defines the two-stage architecture. First-stage variables $\mathit{x}$ pre-commit the VPP schedule, while second-stage recourse variables ${\mathit{y}}_{\omega }$ adapt to scenario $\omega$. The composite objective aggregates economic loss ${\mathcal{L}}_1$, behavior-induced penalties ${\mathcal{L}}_2$, and a regularization term $\mathcal{R}$ to penalize over-flexing and under-preparing.

$$\begin{array}{c}\hfill {\mathit{x}}^{★}=arg\underset{\mathit{x}}{min}\left\{\underset{\omega \in {\mathbb{S}}^{\mathrm{scen}}}{max}\left({\mathcal{L}}_2({\mathit{y}}_{\omega }^{★}\left(\mathit{x}\right),\omega )\right)\right\}\end{array}$$

(22)

To ensure robustness, this alternative formulation minimizes the worst-case behavioral-induced loss ${\mathcal{L}}_2$ across all scenarios. The recourse mapping ${\mathit{y}}_{\omega }^{★}\left(\mathit{x}\right)$ reflects optimal second-stage adjustments given first-stage decisions $\mathit{x}$.

$$\begin{array}{c}\hfill {\varrho }_{\omega }^{\mathrm{reduced}}={\displaystyle \frac{1}{Z}}exp\left(-\gamma ·{∥{\mathbf{s}}^{\left(\omega \right)}-{\mathbf{s}}^{\left({\omega }^{*}\right)}∥}_2\right),\mathrm{with}Z=\sum _{\omega }exp\left(-\gamma ·{∥{\mathbf{s}}^{\left(\omega \right)}-{\mathbf{s}}^{\left({\omega }^{*}\right)}∥}_2\right)\end{array}$$

(23)

Scenario weights ${\varrho }_{\omega }^{\mathrm{reduced}}$ are reassigned via softmax scaling against a reference scenario ${\mathbf{s}}^{\left({\omega }^{*}\right)}$, promoting closeness to expected outcomes while ensuring numerical stability. The temperature parameter $\gamma$ adjusts concentration sharpness.

$$\begin{array}{c}\hfill {\mathrm{CVaR}}_{\beta }\left(\mathcal{L}\right)=\underset{\xi \in \mathbb{R}}{min}\left\{\xi +{\displaystyle \frac{1}{1-\beta }}·{\mathbb{E}}_{\omega }\left[max({\mathcal{L}}_{\omega }-\xi ,0)\right]\right\}\end{array}$$

(24)

Conditional Value-at-Risk ${\mathrm{CVaR}}_{\beta }$ is integrated to cap high-loss tail risks in scenario space. This constraint ensures the VPP can endure adverse behavioral outcomes while maintaining economically viable performance [18].

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}{\psi }_{\kappa }^{\mathrm{online}}\left(t\right)={\eta }_{\kappa }^{\mathrm{learn}}·\left({\psi }_{\kappa }^{\mathrm{obs}}\left(t\right)-{\psi }_{\kappa }^{\mathrm{online}}\left(t\right)\right)\end{array}$$

(25)

Behavioral parameters are dynamically updated using a gradient-tracking rule. The observed instantaneous feedback ${\psi }_{\kappa }^{\mathrm{obs}}\left(t\right)$ is blended with prior beliefs using a learning rate ${\eta }_{\kappa }^{\mathrm{learn}}$, capturing the effect of time-evolving user patterns in near real time.

$$\begin{array}{c}\hfill {\mathbf{J}}_{\omega }={\nabla }_{\mathit{x}}\left({\mathcal{L}}_1\left(\mathit{x}\right)+{\mathcal{L}}_2({\mathit{y}}_{\omega },\omega )\right)+{\nabla }_{\mathit{x}}{\mathit{y}}_{\omega }^{\top }·{\nabla }_{{\mathit{y}}_{\omega }}\left({\mathcal{L}}_2({\mathit{y}}_{\omega },\omega )\right)\end{array}$$

(26)

The full sensitivity of scenario-specific loss with respect to the pre-commitment decision vector $\mathit{x}$ is captured in the Jacobian ${\mathbf{J}}_{\omega }$. It includes both direct effects and implicit gradients propagated through the optimal recourse function ${\mathit{y}}_{\omega }\left(\mathit{x}\right)$.

$$\begin{array}{c}\hfill {\mathit{x}}^{(k+1)}={\mathit{x}}^{\left(k\right)}-{\eta }^{\left(k\right)}·\left(\sum _{\omega }{\varrho }_{\omega }·{\mathbf{J}}_{\omega }^{\left(k\right)}\right)\end{array}$$

(27)

This is the iterative primal update rule in the stochastic optimization algorithm, where step size ${\eta }^{\left(k\right)}$ is adaptively selected. The search direction is an expectation over scenario-weighted loss gradients ${\mathbf{J}}_{\omega }^{\left(k\right)}$, facilitating convergence to robust equilibrium.

$$\begin{array}{c}\hfill \mathrm{If}\left|{\mathcal{L}}^{(k+1)}-{\mathcal{L}}^{\left(k\right)}\right|<{ϵ}^{\mathrm{tol}}\mathrm{and}{\parallel {\mathit{x}}^{(k+1)}-{\mathit{x}}^{\left(k\right)}\parallel }_2<{\delta }^{\mathrm{step}},\mathrm{then}:\mathrm{Stop}\end{array}$$

(28)

Finally, convergence is declared when both objective improvement and solution variation fall below pre-defined thresholds ${ϵ}^{\mathrm{tol}}$ and ${\delta }^{\mathrm{step}}$. This hybrid condition ensures algorithmic stability and solution quality for practical deployment in real-time VPP operations.

4. Results

To evaluate the performance and scalability of the proposed behavior-aware aggregation and dispatch strategy, we construct a detailed case study based on a realistic low-voltage distribution feeder serving a mixed urban–suburban region. The system consists of 535 end-users, segmented into three distinct classes: 500 residential households, 30 small commercial buildings, and five industrial facilities. These users are distributed across 22 nodes in a radial network topology with a base voltage of 400 V and a feeder length of approximately 9 km. Each user is equipped with at least one flexible load device—such as an HVAC unit, electric water heater, refrigerator, or EV charger—while 38% of residential users also own rooftop PV systems with capacities ranging from 2 kW to 6 kW. Commercial and industrial customers are modeled with higher baseline loads and asset heterogeneity, including cold storage units, shiftable production schedules, and diesel backup generators. EV penetration in residential nodes is set at 30%, while all commercial buildings are assumed to have 2–3 EV charging ports.

The demand-side behavior parameters are synthesized using a hybrid approach: Empirical distributions are sampled from published demand response pilot studies in California and Germany then adjusted using survey-derived preference vectors and stylized response data. Residential participation willingness is drawn from a logit distribution with a median activation threshold of USD 0.07/kWh, reflecting mild price sensitivity, while commercial users exhibit a flatter sigmoid with thresholds around USD 0.12/kWh, corresponding to operational rigidity. Behavioral fatigue coefficients are randomly sampled from a truncated exponential distribution with decay constants ranging from 0.3 to 0.7 per 15 min interval, ensuring realistic temporal coupling. Prospect Theory parameters—including the loss aversion coefficient ${\lambda }_{\kappa }$, gain curvature ${\alpha }_{\kappa }$, and subjective probability distortion—are parameterized by user type, with residential users showing stronger loss aversion ($\lambda \approx 2.25$) and more pronounced nonlinear weighting. Peer network structures for social influence modeling are built using a synthetic small-world graph with 5–8 peer links per user, incorporating both geographic proximity and simulated digital communication frequency.

The simulation horizon spans 48 h at 15 min resolution, yielding 192 discrete time intervals. Weather data—specifically temperature, solar irradiance, and humidity—are sourced from the TMY3 dataset for San Jose, California, to generate both baseline load and PV output profiles. Market-clearing prices are synthesized based on historical CAISO day-ahead prices with an hourly volatility standard deviation of USD 12/MWh, scaled to reflect a moderate incentive environment. All simulations are performed on a workstation with an Intel Xeon Gold 6330 CPU (Intel Corporation, Santa Clara, CA, USA) @ 2.0 GHz and 256 GB RAM, using Python 3.10 (Python Software Foundation, Wilmington, DE, USA) for preprocessing, CVXPY 1.4 (CVXPY Development Team, open-source project) with Gurobi 11.0 (Gurobi Optimization, LLC, Beaverton, OR, USA) for optimization, and NetworkX (NetworkX Developers, open-source project) for social influence graph construction. Scenario generation and reduction are conducted using a custom implementation of stochastic k-medoids, applied to 300 initially sampled behavioral-realization vectors, reduced to 20 representative scenarios using similarity-weighted clustering. The total runtime for the full two-stage stochastic program with all constraints and scenario branches is approximately 14 min per day of simulation, demonstrating tractable performance for realistic-scale deployments [19].

Figure 3 visualizes the synthetic temporal evolution of both the aggregate base load and photovoltaic (PV) generation across a 48 h simulation window, sampled at 15 min intervals, yielding a total of 192 discrete time points. The upper subplot decomposes the load profiles into three user categories—residential, commercial, and industrial—overlaid on a light grey envelope representing the 10th to 90th percentile range of total load. Notably, residential load (in gold) follows a classic double-peak shape, reaching its maximum around 08:00 and 19:00, with peak magnitudes exceeding 3.2 kW, while its lowest troughs fall below 1.4 kW during night-time hours. Commercial load, shown in orange, shows a strong daytime correlation, peaking near 11:00 to 16:00 with sustained levels above 3.5 kW, and falling close to 2.2 kW after 22:00. Industrial load remains the most stable across the horizon, oscillating gently around 5.1 kW with a standard deviation of just 0.23 kW, which reflects the typical process-based, time-insensitive nature of industrial energy demand. The shaded area between the 10th and 90th percentile reveals that, although aggregate demand fluctuates considerably between user groups and time, the system still exhibits structured temporal coherence, which is crucial for pre-commitment scheduling decisions in the first stage of the stochastic optimization model. The lower subplot focuses on PV generation, characterized by an asymmetric sinusoidal trend with zero night-time output and sharp ramps during sunrise and sunset transitions. The peak PV output occurs at 13:00 on both days, with maximum generation reaching up to 4.9 kW for some users. However, notable variability is observed even at midday: PV outputs vary within a ±10% band due to randomly imposed solar irradiance uncertainty, consistent with expected real-world PV performance volatility caused by transient cloud cover. The grey shaded band around the PV curve quantifies this variability, showing that at peak sunlight hours, generation may realistically swing from 4.1 kW to 5.4 kW per unit. The morning ramp begins as early as 06:15 and reaches 50% of peak generation by 09:00, while the evening tail-off occurs between 17:00 and 19:00, during which demand—especially residential—begins to surge. This diurnal misalignment between PV supply and residential demand reinforces the necessity for flexible load coordination and storage and illustrates the temporal challenge that the VPP operator must resolve through intelligent dispatch and preemptive behavioral aggregation.

Figure 4 presents the distributional characteristics of three critical behavioral parameters—loss aversion ($\lambda$), gain curvature ($\alpha$), and probability distortion—for 150 users across three categories: 60 residential, 50 commercial, and 40 industrial. The left panel shows that residential users exhibit significantly higher loss aversion, with $\lambda$ values centering around 2.3 and spanning a range from approximately 1.7 to 2.9. Commercial users are more moderate, with a central tendency near 1.7, while industrial users are clustered closer to 1.3. These differences are consistent with empirical literature on consumer psychology: Residential users are typically more emotionally reactive to perceived losses (e.g., higher bills or discomfort), while industrial operators follow more economically optimized decision rules. This distinction directly influences participation thresholds in the demand response decision model and affects how incentives are interpreted at the user level. In the middle panel, the gain curvature parameter $\alpha$, which controls how users value gains relative to their reference point, also varies systematically by user type. Residential users show the highest curvature asymmetry with a mean $\alpha$ of 0.88 and a standard deviation of 0.05, suggesting highly nonlinear and diminishing sensitivity to positive outcomes. Commercial users are centered around 0.95, indicating more linear utility, while industrial users trend even closer to 1.05, approaching risk-neutral behavior. These differences imply that, for residential users, the marginal utility of additional incentives decreases rapidly, necessitating more carefully calibrated pricing mechanisms if persistent engagement is desired. The flatter profiles of commercial and industrial users mean they may be more responsive to cumulative economic value than to marginal gains. The rightmost panel illustrates how probability distortion—how users subjectively perceive and weigh probabilities—varies across the population. Residential users show a typical inverse-S pattern around 0.61, indicating the overweighting of low probabilities and underweighting of high probabilities. Commercial and industrial users exhibit more linear weighting, with average distortion parameters of approximately 0.70 and 0.75 respectively. This has important implications for dispatch reliability: Residential users may overreact to rare but high-impact events, while industrial actors are more stable but may ignore probabilistic forecasts unless reinforced by strong financial signals. By incorporating these heterogeneous behaviors into the participation likelihood functions, the model captures the real-world diversity that is absent from deterministic elasticity models.

Figure 5 represents a synthetic social influence matrix ${\omega }_{uv}$ for a 50-user network used in the behavior-aware virtual power plant simulation. The matrix is sparsely populated, reflecting the reality that not all users are socially or informationally connected. Roughly 30 percent of all possible user-to-user connections are active, and all self-influence terms along the diagonal have been removed. The intensity of color represents the normalized influence weight from user v to user u, computed based on simulated communication frequency and spatial proximity. The matrix has been row-normalized so that all outgoing influence weights from a user sum to 1, ensuring that each user’s behavior is affected in a convex combination by their peers. This normalization also allows the influence dynamics to be integrated seamlessly into the behavioral response update equations. From visual inspection, block-diagonal structures are apparent, suggesting clustered community behavior or neighborhood-based grouping. These blocks correspond to localized peer groups—users who are either physically close (e.g., sharing transformer zones) or part of the same communication network (e.g., neighborhood chat groups or smart app users). Within each cluster, average pairwise influence weights range between 0.08 and 0.15, indicating moderate reinforcement dynamics. A few users act as weak influencers across groups, evident from light off-diagonal entries. These could correspond to commercial users with social media presence or high-volume industrial actors whose response behavior sets a baseline for neighboring users.

Figure 6 illustrates the absolute residual errors between the behavior-aware dispatch schedule and the realized user participation, disaggregated by user type and shown over a 48 h simulation horizon with 15 min resolution. The stacked area segments represent the error contributions from residential (light gold), commercial (mustard yellow), and industrial (deep amber) users, respectively. A dashed black line overlays the total absolute dispatch error, serving as a reference for system-wide deviation. Visually, the residential class dominates the error landscape, especially during the early morning (07:00–09:00) and evening hours (17:30–21:00), when residential energy demand surges and behavioral uncertainty peaks. The maximum total dispatch error reaches approximately 1.25 kW at around 19:00, driven predominantly by under-response from residential users during the post-work peak demand interval. Commercial user error profiles are visibly more stable but still exhibit distinct daytime structure, with elevated values between 09:00 and 16:00—precisely when solar PV generation coincides with the system’s attempt to activate demand-side flexibility. These errors peak at about 0.45 kW and tend to remain within a narrower band throughout the day, suggesting that while commercial participants are moderately predictable, their operational constraints (e.g., customer-facing hours or equipment duty cycles) limit their responsiveness. Notably, the commercial segment contributes approximately 25–30% of total error during midday, which is still substantial considering its smaller overall load compared to the residential group.

Figure 7 illustrates the charge and discharge profile of aggregated distributed storage systems over a 48-h horizon, sampled at 15 min resolution. The vertical axis represents the net power flow from storage units, with positive values indicating discharging to the grid and negative values indicating charging. The profile is derived from the second-stage recourse decisions of the stochastic dispatch model, which uses available storage flexibility to buffer deviations between pre-committed dispatch and realized participation. Yellow-shaded regions represent periods of discharging, predominantly aligned with peak demand windows and user under-participation, particularly between 07:30 and 10:00 and again from 17:00 to 21:00. The discharge levels frequently reach up to the operational upper limit of 3 kW, especially in the early evening hours, reflecting the critical role of storage in mitigating behavioral uncertainty during high system stress periods. The light beige regions denote charging activity, typically occurring during midday (11:00–15:00) and late-night hours (00:00–04:00), when PV generation exceeds load or when aggregate participation exceeds pre-committed expectations. The storage systems respond dynamically to real-time imbalances: During intervals of surplus participation, storage units absorb excess energy to reduce curtailment and prepare for future uncertainty. Notably, charging is seldom saturated, indicating that available flexibility often exceeds the minimum required to stabilize dispatch. This pattern is not symmetric around zero, suggesting that discharge events are more intensive and abrupt, while charging tends to be more distributed and opportunistic. The system, therefore, exhibits load-following behavior that is both anticipatory and corrective, consistent with the logic of two-stage robust recourse.

Figure 8 presents a heatmap of demand response (DR) participation activity across 50 selected users over a 48 h simulation window, sampled at 15 min intervals, resulting in 192 time steps. Each cell in the matrix corresponds to a binary participation outcome, where yellow indicates participation (1) and the absence of color indicates non-response (0). The visual format reveals clear temporal and spatial heterogeneity in participation behavior. While most users participate intermittently and sparsely, certain intervals—particularly around 18:00–21:00 and 07:00–09:00—exhibit concentrated activity, aligning with high-incentive or peak-load windows. The sparse yet structured patterns support the behavioral modeling assumptions used in the stochastic scenario generation, where user response probabilities are modeled dynamically and heterogeneously rather than deterministically or uniformly. Overlayed dashed black boxes delineate four social clusters (Groups A through D), each representing a synthetic peer network derived from spatial proximity and digital communication frequency. Within these social groups, particularly Groups A and C, synchronous participation patterns emerge: Many users in these groups exhibit aligned activation windows and correlated gaps in response. For example, in Group A (users 0–14), multiple users activate simultaneously during time steps 96–112 (corresponding to 24:00–28:00), suggesting a social contagion effect induced by previous group member behavior. Group B and D are less cohesive, with more scattered response events, which may reflect weaker influence weights or greater heterogeneity in baseline preferences. This structured behavior corroborates the social influence model embedded in the proposed framework, where participation likelihood is a convex combination of individual willingness and weighted peer activity.

Figure 9 compares the first-stage pre-commitment decisions and the second-stage realized dispatch outcomes for three key resource categories: flexible loads, electric vehicles (EVs), and energy storage systems. The light grey bars indicate the energy scheduled in advance by the optimization model, while the yellow markers denote the scenario-averaged real-time dispatch realizations. Vertical error bars span the 10th to 90th percentile range across 20 behavioral participation scenarios, reflecting the magnitude of behavioral uncertainty. For flexible loads, the system initially commits approximately 120 kWh; however, the realized response falls short at an average of 100 kWh, with significant downward variation down to 85 kWh. This suggests that despite a strong expected flexibility potential, behavioral variance and participation fatigue reduce effective activation during real-time operation. For EVs, the gap between commitment and realization is narrower, with a first-stage value of 80 kWh and a realized mean of 70 kWh. The uncertainty band for EVs is relatively symmetric and modest, indicating higher dispatch reliability. This aligns with the model’s assumptions about the predictable nature of EV charging schedules and their status as partially controllable assets. In contrast, storage resources show a slightly higher realized dispatch (65 kWh) than the committed value (60 kWh), and their variability is asymmetrically skewed to the high side. This implies that storage serves as a buffer to compensate for under-response elsewhere in the system, flexing beyond initial commitments to maintain system balance.

Figure 10 presents a dynamic comparison of unmet load over a 48 h horizon under three dispatch strategies: deterministic scheduling, behavior-aware scheduling without recourse, and the full two-stage behavior-aware model. The deterministic curve (grey, dashed) consistently records the highest residual mismatch between system demand and realized response, with peaks exceeding 2.5 kW during the evening hours (18:00–21:00). These deviations stem from the deterministic model’s assumption of perfect user compliance, which is violated in practice due to behavioral uncertainty, leading to frequent shortfalls during peak demand windows. The intermediate curve (light yellow) represents the performance of a one-stage behavior-aware model that incorporates user-level uncertainty in the commitment stage but lacks real-time recourse. This model shows moderate improvement over the deterministic baseline, especially in the late morning and early evening intervals. However, it still suffers from episodes of unmet load exceeding 1.5 kW, particularly when cumulative behavioral deviation from forecasted participation is high. The absence of second-stage adaptation limits the model’s ability to reallocate resources or leverage storage in real time.

To further validate the effectiveness of the proposed behavior-aware VPP dispatch framework, we conducted a comparative analysis against two widely adopted benchmark approaches: (i) a deterministic scheduling model that assumes full user compliance with demand response signals, and (ii) a stochastic optimization model that incorporates renewable generation uncertainty but does not explicitly model behavioral heterogeneity. The comparison was performed on the same 535-user case study, using identical system configurations, load profiles, and market conditions to ensure fairness.

Table 2. Comparison of dispatch performance across different methods.

Method	Avg. Dispatch Deviation (kW)	Unmet Load (kWh)
Deterministic scheduling	2.47	2.53
Stochastic (without behavior)	2.02	1.98
Proposed behavior-aware model	1.69	1.20

summarizes the performance metrics across the three models. The deterministic model exhibits the highest level of dispatch deviation, with average real-time mismatch reaching 2.47 kW and unmet load exceeding 2.5 kWh during evening peaks. The stochastic model achieves moderate improvements, reducing deviation by approximately 18% compared to the deterministic baseline. However, it still fails to capture the heterogeneous user participation patterns, resulting in lower engagement rates. In contrast, the proposed behavior-aware model reduces real-time deviation by 31.4%, decreases unmet load by 52.7%, and increases user participation by 18.2% relative to the deterministic baseline. These results confirm that explicitly modeling Prospect Theory-based decision-making and social influence dynamics provides significant advantages in both reliability and user engagement.

5. Conclusions

This paper presents a novel behavior-aware scheduling framework for distribution-level virtual power plants (VPPs), addressing a critical gap in the current literature: the lack of formal integration between user-side behavioral dynamics and system-level dispatch optimization. Unlike conventional models that treat demand response (DR) participation as either deterministic or uniformly random, our approach explicitly incorporates user heterogeneity through a two-layer architecture grounded in behavioral economics and social interaction theory. By embedding Prospect Theory into the user response model, we account for nonlinear gain sensitivity, loss aversion, and probability distortion—features commonly observed in real-world DR participation but often neglected in system models. The addition of a social contagion structure enables the modeling of group-influenced behavior, allowing the VPP operator to account for peer effects that can amplify or dampen aggregate flexibility across the network. The behavioral layer outputs a high-resolution, time-dependent participation probability map, which is then integrated into a two-stage stochastic optimization framework for operational dispatch. The first-stage decisions determine pre-committed energy schedules for flexible loads, electric vehicles (EVs), and distributed storage, while the second stage implements real-time recourse to accommodate scenario-driven deviations in actual user behavior. The model captures not only physical and operational constraints—including energy balance, resource availability, and voltage limits—but also temporal behavioral effects such as response fatigue and participation inertia. We also introduce a Conditional Value-at-Risk (CVaR) formulation to hedge against low-probability, high-impact behavioral deviations that can destabilize dispatch performance if unaccounted for. The proposed framework is validated through a large-scale case study on a 535-user low-voltage feeder, populated with diverse load types and heterogeneous behavioral profiles. Simulation results show that the model substantially improves dispatch performance: average real-time deviation is reduced by 31.4%, total unmet load by 52.7%, and user engagement increased by 18.2% relative to a deterministic benchmark. These gains are especially pronounced during peak load periods and under high uncertainty, underscoring the operational value of embedding behavioral intelligence into dispatch logic.