Bilevel Carbon-Aware Dispatch and Market Coordination in Power Networks Under Distributional Uncertainty

Abstract

The accelerating transition toward carbon neutrality necessitates the synergistic integration of power and hydrogen systems to mitigate renewable intermittency; however, coordinating regulatory policies with the operational flexibility of these coupled systems remains a critical challenge under deep uncertainty. Motivated by this gap, this study develops a bilevel carbon price-coupled optimization framework for integrated power–hydrogen systems, aiming to coordinate environmental policy design with operational scheduling under deep uncertainty. The upper-level model represents the decision-making of a market regulator that determines the optimal carbon price and emission allowances to maximize overall social welfare, while the lower-level model captures the coordinated operation of electricity and hydrogen subsystems that minimize total dispatch cost, including renewable utilization, electrolyzer conversion, and fuel-cell recovery.To address stochastic variations in renewable generation and load demand, a Distributionally Robust Optimization (DRO) formulation is introduced using Wasserstein ambiguity sets, ensuring decision feasibility against worst-case probability distributions. The bilevel structure is efficiently solved via a Benders–Column-and-Constraint Generation (CCG) algorithm, which decomposes policy and operation layers into tractable subproblems with provable convergence. Case studies on a 33-bus integrated power–hydrogen network demonstrate that the proposed framework effectively balances economic efficiency and carbon reduction. Results show that the optimal carbon price of approximately 45 $/tCO₂ achieves a 27% emission reduction with only a 9% cost increase, revealing a near-optimal social welfare equilibrium. Hydrogen subsystems operate flexibly, with electrolyzer utilization increasing by 30% and storage cycling deepening by 15%, enabling enhanced renewable absorption. Sensitivity analyses confirm that the DRO layer reduces operational risk by 4% compared with stochastic optimization, validating robustness against distributional shifts. The study provides a rigorous and computationally efficient paradigm for policy-coordinated decarbonization, highlighting the synergistic role of carbon pricing and cross-energy scheduling in the next generation of resilient low-carbon energy systems.

1. Introduction

The accelerating decarbonization of modern energy systems has transformed power networks into deeply interconnected socio-technical infrastructures, where renewable variability, sector-coupling technologies, and carbon-market regulations interact in increasingly complex ways. Within this evolving landscape, hydrogen has emerged as a central enabler of cross-sectoral flexibility. Acting as both a clean energy carrier and a long-duration storage medium, hydrogen provides a means of absorbing surplus renewable generation through electrolysis and converting it back to electricity via fuel cells when system conditions tighten [1,2]. This dual capability creates an unprecedented opportunity to mitigate renewable intermittency, enhance operational resilience, and support long-term emission reduction goals. At the same time, the integration of hydrogen into electricity markets introduces new operational couplings, technological constraints, and strategic decision layers that require careful modeling to ensure economically and environmentally sound system performance.

Despite these opportunities, the coordination of hydrogen-enabled multi-energy systems under decarbonization policies remains an open challenge. As carbon pricing mechanisms such as cap-and-trade programs and dynamic emission penalties evolve, system operators must respond to policy signals while managing renewable uncertainty, multi-energy conversion processes, and emission compliance requirements. The resulting interactions form a multi-layered feedback loop: carbon prices influence dispatch and hydrogen utilization; operational decisions reshape system emissions; and emission outcomes feed back into regulatory objectives. Capturing this bidirectional linkage between policy and system operation requires an analytical framework capable of jointly representing regulatory intent, cross-energy coordination, and uncertainty-driven variability [3,4,5]. Traditional models—whether deterministic, stochastic, or multi-stage—typically treat carbon prices as exogenous parameters and do not reflect how policy-making and real-time operation co-evolve within an integrated energy ecosystem. This absence of policy–operation coupling limits their capacity to evaluate system behavior under realistic market conditions and undermines their relevance for long-term carbon-neutral planning.

Motivated by these gaps, this study develops a unified framework that models the co-evolution of carbon-market regulation and hydrogen-coupled system operation. At a conceptual level, the work views integrated energy scheduling as a dynamic interaction between the upper-level regulator and the lower-level system operator, reflecting the real-world structure of emission governance. Carbon caps and prices determined at the regulatory level shape operational decisions, while dispatch outcomes and emission realizations feed directly into welfare-driven policy objectives. At a methodological level, the framework embeds a Wasserstein-based distributionally robust formulation to address renewable and load uncertainties, ensuring resilience against distributional deviations without relying on precise probabilistic assumptions [6,7]. The resulting bilevel architecture captures the full spectrum of dependencies—policy-driven, techno-economic, and uncertainty-based—that govern hydrogen-enabled decarbonized systems. To address these challenges, this study aims to establish a unified optimization framework that bridges the gap between macro-level carbon policy design and micro-level power–hydrogen system dispatch. Specifically, this paper seeks to answer the following three key research questions:

• RQ1: How can carbon pricing signals be endogenously optimized to align regulatory social welfare goals with the operational cost minimization of integrated energy systems?
• RQ2: How can the system effectively hedge against the distributional ambiguity of renewable generation and load demand without relying on possibly inaccurate probability assumptions?
• RQ3: What algorithmic strategy can efficiently solve the resulting bilevel, non-convex, and distributionally robust optimization problem within a reasonable timeframe?

The contributions of this work are fourfold. First, it establishes a bilevel optimization model that explicitly links carbon price determination with multi-energy system operation, thereby bridging regulatory design and operational behavior within a single analytical structure. Second, it integrates hydrogen production, storage, and conversion dynamics into the lower-level dispatch model, revealing how hydrogen flexibility reshapes emission pathways and carbon price equilibria. Third, it incorporates a Wasserstein distributionally robust formulation that safeguards operational feasibility under uncertainty, offering a tractable yet rigorous alternative to classical stochastic frameworks. Fourth, it develops a Benders-type decomposition scheme tailored for the proposed bilevel–robust structure, enabling scalable solution of high-dimensional and cross-sectoral energy coordination problems. Together, these innovations provide a cohesive foundation for analyzing the interplay between carbon policies, renewable variability, and hydrogen-enabled flexibility, offering actionable insights for the design of carbon-neutral and uncertainty-resilient energy systems. The remainder of this paper is organized as follows. Section 2 reviews the relevant literature on power system optimization, carbon pricing mechanisms, and integrated hydrogen energy systems. Section 3 details the theoretical formulation of the bilevel optimization model, the distributionally robust framework, and the proposed Benders–Column-and-Constraint Generation (CCG) solution algorithm. Section 4 describes the data sources and system parameters used in the empirical analysis. Section 5 presents the numerical results, including case studies, sensitivity analyses, and comparisons with benchmark methods. Finally, Section 6 concludes the paper and summarizes the key findings and policy implications. In the context of the existing literature, the theoretical contribution of this work lies in formalizing the coupling between regulatory carbon pricing and distributionally robust operational scheduling, addressing the lack of policy-endogenous models in prior studies. Empirically, the study contributes by quantifying the economic and environmental benefits of hydrogen flexibility under worst-case renewable uncertainty, offering a validated benchmark for policy impact assessment.

The remainder of this paper is organized as follows. Section 2 reviews the relevant literature on power system optimization, carbon pricing mechanisms, and integrated hydrogen energy systems. Section 3 details the theoretical formulation of the bilevel optimization model, the distributionally robust framework, and the proposed Benders–Column-and-Constraint Generation (CCG) solution algorithm. Section 4 describes the data sources and system parameters used in the empirical analysis. Section 5 presents the numerical results, including case studies, sensitivity analyses, and comparisons with benchmark methods. Finally, Section 6 concludes the paper and summarizes the key findings and policy implications.

2. Literature Review

Research on power system optimization has evolved significantly over the past two decades, beginning with deterministic unit commitment and economic dispatch models that assume perfect foresight and stable system conditions [8,9,10]. These classical formulations provided analytical tractability but proved insufficient as renewable penetration increased sharply and system variability became intrinsic. To address the limitations of deterministic assumptions, stochastic programming frameworks were introduced to incorporate uncertainty through scenario generation, probability distributions, and multi-stage decision structures [11,12]. While effective in representing probabilistic environments, stochastic models require accurate distributional information and large scenario sets, challenges that often render them computationally burdensome and vulnerable to data sparsity. This methodological evolution laid the foundation for more robust uncertainty-handling techniques but left unresolved the question of how to maintain system reliability when renewable and load distributions deviate significantly from historical data [13,14,15].

The emergence of distributionally robust optimization (DRO) addressed many of the shortcomings inherent in conventional stochastic methods by optimizing system performance over a set of plausible probability distributions rather than assuming a single true distribution [16,17,18]. Using ambiguity sets such as Wasserstein balls, moment-based bounds, or $\Phi$-divergence measures, DRO enables system operators to hedge against worst-case realizations while retaining computational tractability. DRO has been applied to renewable bidding, reserve scheduling, and investment planning, demonstrating strong resilience under data uncertainty. However, existing studies primarily focus on electricity-only systems and seldom consider multi-energy interactions, particularly the uncertainty implications introduced by hydrogen production, storage, and reconversion. As a result, the literature still lacks robust frameworks capable of simultaneously addressing uncertainty in cross-sectoral energy flows and policy-driven operational adjustments.

Parallel to these methodological developments, extensive research has examined carbon pricing mechanisms such as cap-and-trade programs, carbon taxes, and hybrid regulatory schemes that directly influence emission outcomes and dispatch behavior [19,20,21]. Bilevel and Stackelberg-type formulations have been widely used to model the hierarchical interaction between regulators and power system operators, capturing how emission caps, allowances, and penalty prices shape market operations. These studies reveal that regulatory decisions significantly reshape generation portfolios and dispatch trajectories [22,23]. Yet, most existing works treat the electricity sector in isolation and do not integrate emerging flexibility technologies such as hydrogen electrolysis, storage, and conversion. The absence of hydrogen within carbon-market-aware optimization frameworks overlooks the pivotal role electrolyzers can play in shifting emission patterns by absorbing renewable surplus and displacing carbon-intensive generation in later periods. Thus, the policy–technology feedback loop remains underexplored.

In the domain of integrated power–hydrogen systems, recent studies have expanded the representation of hydrogen technologies beyond simple controllable loads. Early research modeled electrolysis as a flexible consumption device embedded in power dispatch models, while later efforts introduced full hydrogen networks including storage tanks, fuel cells, compressors, and pipelines [24,25,26]. These works demonstrated the operational benefits of hydrogen in enhancing flexibility and mitigating renewable curtailment. Despite these advances, most formulations adopt fixed or exogenous carbon prices and therefore do not capture the endogenous dependence of hydrogen utilization on carbon policy. Moreover, the majority of existing studies focus on techno-economic coordination and do not incorporate uncertainty-driven behavioral or policy feedback into joint optimization. This gap prevents current models from fully assessing the broader system impacts of hydrogen under evolving regulatory conditions.

A final stream of literature concerns decomposition techniques for large-scale optimization, such as Benders decomposition, column-and-constraint generation, and cutting-plane algorithms [27,28,29]. These methods have enabled tractable solutions for multi-stage, nonlinear, and high-dimensional decision problems in power systems. When combined with robust optimization or stochastic formulations, decomposition techniques facilitate the iterative refinement of uncertainty sets and operational decisions. However, their applications to bilevel, cross-sectoral energy systems remain limited, particularly in settings where carbon pricing interacts directly with hydrogen system flexibility and renewable uncertainty. The integration of decomposition-based solution strategies into bilevel DRO formulations therefore represents a promising but underdeveloped avenue in existing research [30].

In summary, the existing literature provides important foundations in uncertainty modeling, carbon-market regulation, hydrogen–electricity coordination, and large-scale optimization. Nonetheless, several critical gaps persist: carbon prices are rarely endogenized within hydrogen-enabled systems; uncertainty is typically handled in either power or hydrogen domains but not jointly; and decomposition approaches have not been fully exploited for scalable bilevel formulations that integrate regulatory and operational decisions. These gaps motivate the development of a unified framework that simultaneously captures hydrogen flexibility, renewable uncertainty, and policy–operation co-evolution within a mathematically rigorous and computationally tractable optimization structure.

3. Theoretical Formulation and Algorithmic Realization of the Carbon Price-Coupled Power–Hydrogen Optimization Model

The following section constructs the theoretical foundation and computational realization of the proposed carbon price-coupled bilevel optimization framework. The formulation begins with a rigorous representation of hierarchical decision-making between regulatory and operational layers, where the upper-level entity determines carbon prices and emission quotas to maximize social welfare, while the lower-level energy operator minimizes system-wide dispatch costs subject to carbon and operational constraints. To embed resilience under renewable and demand uncertainty, the model incorporates a Wasserstein-based Distributionally Robust Optimization (DRO) reformulation, enabling the solution to remain valid under statistically perturbed probability distributions. Beyond the formulation, the section presents a stepwise decomposition strategy based on Benders and Column-and-Constraint Generation (CCG) principles, translating the complex bilevel structure into a sequence of tractable subproblems. The dual transformation and subgradient-based optimality cuts jointly ensure convergence with provable robustness. Together, these mathematical and algorithmic layers embody a unified philosophy: that sustainable energy coordination emerges not only from optimal economic design but also from resilient computational structure.

Figure 1 provides a theory-grounded representation of the regulatory and operational architecture underlying the proposed carbon-regulated power–hydrogen coordination framework. At the upper layer, the government or market operator issues carbon prices and allocates emission allowances, consistent with established principles of carbon-market economics in which policy instruments guide downstream operational behavior. This regulatory signal propagates into the integrated energy system, where the central block reflects the coupled power–hydrogen configuration widely discussed in the literature on multi-energy systems and sector coupling. The power system supplies electricity for hydrogen production through electrolyzers, while the hydrogen subsystem offers a flexible sink for renewable surplus and a dispatchable source via fuel-cell conversion, forming a bidirectional coordination pathway highlighted in cross-vector optimization research. External drivers such as variable wind and solar generation and evolving market demand enter the system as exogenous uncertainties, echoing prior studies emphasizing the need for flexibility under renewable intermittency and load variability. Collectively, the diagram synthesizes theoretical insights from regulatory economics, integrated energy system modeling, and hydrogen–electricity sector coupling to justify the bilevel structure of the study, in which upper-level policy decisions shape lower-level dispatch outcomes and the resulting system behavior feeds back to determine environmental and economic performance.

Table 1. Nomenclature.

Symbol	Description	Unit
$p_{g,t}$	Electric power output of generator g at time t	MW
$p_t^{\mathrm{ren}}$	Available renewable generation at time t	MW
$d_t$	Electric load demand at time t	MW
$u_{g,t}$	Commitment status of thermal generator g	–
$P_g^{max}$	Maximum capacity of generator g	MW
$E_t$	State-of-charge of energy storage system at time t	MWh
$E^{max}$	Maximum storage energy capacity	MWh
$p_t^{\mathrm{ch}}$	Charging power of storage at time t	MW
$p_t^{\mathrm{dis}}$	Discharging power of storage at time t	MW
${\eta }^{\mathrm{ch}}$	Charging efficiency of storage	–
$h_t^{\mathrm{ely}}$	Hydrogen produced by electrolyzer at time t	kg
$h_t^{\mathrm{fc}}$	Hydrogen consumed by fuel cell at time t	kg
$H_t$	Hydrogen storage level at time t	kg
$H^{max}$	Maximum hydrogen storage capacity	kg
${\eta }^{\mathrm{ely}}$	Conversion efficiency of electrolyzer	–
${\eta }^{\mathrm{fc}}$	Conversion efficiency of fuel cell	–
$e_g$	Emission factor of generator g	tCO2/MWh
${\lambda }_t$	Carbon price or emission penalty at time t	$/tCO2
${\pi }_t$	Market clearing electricity price	$/MWh
$C^{\mathrm{op}}$	Total operational cost of the system	$
$C^{\mathrm{H}2}$	Cost associated with hydrogen production or conversion	$
x	Vector of lower-level operational decision variables	–
y	Upper-level decision vector (policy or carbon regulation)	–
$r_t$	Renewable/load uncertainty realization at time t	–
$\mathbb{Q}$	Ambiguity set of probability distributions in DRO	–
$ϵ$	Wasserstein radius defining robustness level	–
$W(·)$	Wasserstein distance metric used in DRO	–
$\theta$	Dual variable associated with DRO reformulation	–
$f(·)$	Objective or cost function used in optimization	–
$g(·)$	System constraints representing technical limits	–

provides a consolidated list of the main symbols used throughout the manuscript. The notation spans power-system operation, hydrogen conversion processes, carbon-market interactions, and uncertainty modeling, and the table offers a clear reference point to support readers in navigating the mathematical formulation.

Figure 2 provides a high-level representation of the hierarchical structure underlying the proposed bilevel optimization framework and clarifies how policy decisions and operational responses are interconnected across the two levels. The upper-level block depicts the regulator’s role in determining the carbon price and emission allowances, which serve as strategic levers for shaping economic incentives and guiding system-wide emission outcomes. These policy variables flow downward into the operational layer, where they directly influence dispatch decisions by modifying generation costs, shifting the relative competitiveness of hydrogen technologies, and altering the economic conditions under which renewable energy is utilized. At the lower level, the system operator responds to these policy signals through a coordinated scheduling process that jointly optimizes electricity generation, hydrogen production and consumption, storage operations, and renewable integration under uncertainty. The embedded distributionally robust optimization component illustrates that the operator’s decisions account not only for expected conditions but also for adverse deviations in renewable and demand patterns, thereby enhancing system resilience. The upward arrows highlight the feedback loop inherent in bilevel structures: operational outcomes—such as total emissions, renewable utilization, and system cost—become inputs to the regulator’s welfare assessment. This closed-loop interaction emphasizes that policy formulation and operational behavior are mutually dependent, with each layer shaping the feasible choices and economic consequences of the other. The diagram therefore offers an intuitive and accessible visualization of the uncertainty–policy–operation interface captured in the mathematical formulation, helping readers understand the dynamic coupling between carbon-market instruments and multi-energy system dispatch.

$$\begin{array}{cc}\hfill \underset{\begin{array}{c}{P}_{g,t},P_{el,t},P_{fc,t},\\ P_{st,t}^{ch},P_{st,t}^{dis},S_t,H_t\end{array}}{min}\left\{\sum _{t\in \mathcal{T}}\right[& \sum _{g\in \mathcal{G}}\left({\alpha }_g^{\mathrm{op}}{P}_{g,t}+{\beta }_g^{\mathrm{quad}}{P}_{g,t}^2+{\pi }_c{\kappa }_g^{\mathrm{em}}{P}_{g,t}\right)\hfill \\ & +\sum _{\nu \in \mathcal{H}}\left({\varphi }_{\nu }^{\mathrm{el}}{P}_{el,\nu ,t}+{\psi }_{\nu }^{\mathrm{fc}}{P}_{fc,\nu ,t}+{\rho }_{\nu }^{\mathrm{st}}(P_{st,\nu ,t}^{ch}+P_{st,\nu ,t}^{dis})\right)\hfill \\ & +\sum _{\omega \in \Omega }{\lambda }_{\omega }^{\mathrm{imb}}\left|D_{\omega ,t}-\sum _{g\in \mathcal{G}}{P}_{g,t}-\sum _{\nu \in \mathcal{H}}(P_{fc,\nu ,t}-P_{el,\nu ,t})\right|\hfill \\ & +{\vartheta }^{\mathrm{dro}}\underset{Q\in {\mathcal{B}}_{ϵ}\left(\widehat{Q}\right)}{sup}{\mathbb{E}}_Q\left[\sum _{g,\nu }{\Gamma }_{g,\nu ,t}{({\Theta }_t^{\mathrm{wind}}+{\Xi }_t^{\mathrm{solar}}-{\delta }_t^{\mathrm{load}})}^2\right]\left]\right\}\hfill \end{array}$$

(1)

Equation (1) defines the comprehensive system-level objective function for the carbon price-coupled, distributionally robust integrated power–hydrogen system. The first summation over $g\in \mathcal{G}$ captures the thermal generation cost, comprising the linear fuel term ${\alpha }_g^{\mathrm{op}}{P}_{g,t}$, the quadratic production cost ${\beta }_g^{\mathrm{quad}}{P}_{g,t}^2$, and the carbon-emission component ${\pi }_c{\kappa }_g^{\mathrm{em}}{P}_{g,t}$ weighted by the carbon price ${\pi }_c$. The second summation models hydrogen subsystem operation, including the electrolyzer energy consumption ${\varphi }_{\nu }^{\mathrm{el}}{P}_{el,\nu ,t}$, the fuel cell conversion ${\psi }_{\nu }^{\mathrm{fc}}{P}_{fc,\nu ,t}$, and hydrogen storage cycling losses ${\rho }_{\nu }^{\mathrm{st}}(P_{st,\nu ,t}^{ch}+P_{st,\nu ,t}^{dis})$, establishing cross-domain coupling between electricity and hydrogen vectors. The third summation penalizes imbalance costs under uncertainty scenarios $\omega \in \Omega$, where deviations between total generation and stochastic demand $D_{\omega ,t}$ are scaled by imbalance factor ${\lambda }_{\omega }^{\mathrm{imb}}$. Finally, the last robust term integrates a Wasserstein-based distributionally robust adjustment, where the ambiguity set ${\mathcal{B}}_{ϵ}\left(\widehat{Q}\right)$ centered at empirical distribution $\widehat{Q}$ bounds possible stochastic variations. The inner expectation quantifies squared deviations among wind generation ${\Theta }_t^{\mathrm{wind}}$, solar generation ${\Xi }_t^{\mathrm{solar}}$, and demand fluctuation ${\delta }_t^{\mathrm{load}}$, weighted by the sensitivity matrix ${\Gamma }_{g,\nu ,t}$. This objective thereby unifies cost minimization, carbon regulation, hydrogen–electricity synergy, and resilience against renewable uncertainty within a single integrated optimization framework.

$$\begin{array}{cc}\hfill \sum _{g\in \mathcal{G}}{P}_{g,t}+\sum _{\nu \in \mathcal{H}}{P}_{fc,\nu ,t}+P_{st,t}^{\mathrm{dis}}& =\sum _{\nu \in \mathcal{H}}{P}_{el,\nu ,t}+P_{st,t}^{\mathrm{ch}}+D_t^{\mathrm{load}}+{\eta }_t^{\mathrm{loss}},\forall t\in \mathcal{T}\hfill \end{array}$$

(2)

The relationship above ensures the nodal power equilibrium at every time period. Electricity generated by thermal units $P_{g,t}$, fuel cells $P_{fc,\nu ,t}$, and discharged energy storage $P_{st,t}^{\mathrm{dis}}$ must precisely match the sum of system consumption, consisting of electrolyzer input $P_{el,\nu ,t}$, charging power $P_{st,t}^{\mathrm{ch}}$, net demand $D_t^{\mathrm{load}}$, and conversion losses ${\eta }_t^{\mathrm{loss}}$. This coupling secures the instantaneous energy balance that underlies all feasible operation trajectories.

$$\begin{array}{cc}\hfill P_{el,\nu ,t}{\eta }_{\nu }^{\mathrm{el}}& =P_{h2,\nu ,t}^{\mathrm{in}},P_{fc,\nu ,t}=P_{h2,\nu ,t}^{\mathrm{out}}{\eta }_{\nu }^{\mathrm{fc}},\forall \nu \in \mathcal{H},t\in \mathcal{T}\hfill \end{array}$$

(3)

Here the electrochemical conversion interface between the electric and hydrogen subsystems is formalized. On the left, the electrical input of an electrolyzer $P_{el,\nu ,t}$ is translated into hydrogen inflow $P_{h2,\nu ,t}^{\mathrm{in}}$ through its efficiency ${\eta }_{\nu }^{\mathrm{el}}$; on the right, the fuel-cell process reverses this pathway, generating electrical power $P_{fc,\nu ,t}$ from hydrogen outflow $P_{h2,\nu ,t}^{\mathrm{out}}$ scaled by ${\eta }_{\nu }^{\mathrm{fc}}$. Together these linear equalities enforce strict mass–energy conversion consistency across the two domains.

$$\begin{array}{cc}\hfill H_{t+1}& =H_t+\sum _{\nu \in \mathcal{H}}\left(P_{h2,\nu ,t}^{\mathrm{in}}{\eta }_{\nu }^{\mathrm{st},\mathrm{in}}-{\displaystyle \frac{P_{h2,\nu ,t}^{\mathrm{out}}}{{\eta }_{\nu }^{\mathrm{st},\mathrm{out}}}}\right)-{\chi }_t^{\mathrm{leak}},\forall t\in \mathcal{T}\hfill \end{array}$$

(4)

The temporal evolution of hydrogen inventory is described above. The storage state $H_t$ is augmented by inflows $P_{h2,\nu ,t}^{\mathrm{in}}$ weighted by charging efficiency ${\eta }_{\nu }^{\mathrm{st},\mathrm{in}}$ and depleted by outflows $P_{h2,\nu ,t}^{\mathrm{out}}$ corrected for discharging efficiency ${\eta }_{\nu }^{\mathrm{st},\mathrm{out}}$; a small leakage term ${\chi }_t^{\mathrm{leak}}$ reflects natural loss in containment. This constraint establishes dynamic continuity of stored hydrogen and links the system’s present and future states, forming the backbone of intertemporal coordination between production, storage, and utilization processes.

$$\begin{array}{c}\hfill P_g^{min}{\delta }_{g,t}\le P_{g,t}\le P_g^{max}{\delta }_{g,t},\forall g\in \mathcal{G},t\in \mathcal{T}\end{array}$$

(5)

Thermal generation units operate only within their permissible limits. The binary variable ${\delta }_{g,t}$ denotes the on/off commitment state of unit g at time t. When a generator is active, its output $P_{g,t}$ must remain between its minimum and maximum operating thresholds $P_g^{min}$ and $P_g^{max}$; otherwise, both bounds collapse to zero. This constraint guarantees feasible production ranges for all generating units while implicitly embedding unit commitment behavior in the scheduling framework.

$$\begin{array}{c}\hfill 0\le P_{el,\nu ,t}\le P_{el,\nu }^{max},0\le P_{fc,\nu ,t}\le P_{fc,\nu }^{max},\forall \nu \in \mathcal{H},t\in \mathcal{T}\end{array}$$

(6)

The feasible operating ranges of electrolyzers and fuel cells are described above. The first inequality bounds the electrical consumption of each electrolyzer $P_{el,\nu ,t}$ by its rated capacity $P_{el,\nu }^{max}$, while the second limits the fuel-cell output $P_{fc,\nu ,t}$ to its maximum permissible generation $P_{fc,\nu }^{max}$. Together, these ensure technical integrity of the hydrogen subsystem, preventing overloading and enabling consistent energy conversion between domains.

$$\begin{array}{c}\hfill E_t=\sum _{g\in \mathcal{G}}{\alpha }_g^{{\mathrm{CO}}_2}{P}_{g,t},\forall t\in \mathcal{T}\end{array}$$

(7)

Carbon emissions associated with electricity generation are quantified through the above linear relation. For each time interval, the total emission $E_t$ equals the sum of all generator outputs $P_{g,t}$ weighted by their specific emission factors ${\alpha }_g^{{\mathrm{CO}}_2}$. This expression provides a transparent and tractable mapping between economic dispatch decisions and environmental impact, allowing direct coupling with carbon pricing and policy constraints in the upper-level market formulation.

$$\begin{array}{cc}\hfill E_t& \le {\Omega }_c,\forall t\in \mathcal{T}\hfill \end{array}$$

(8)

A carbon-cap condition limits total emissions in each period. The quantity $E_t$ computed from generation outputs must not exceed the allowance ${\Omega }_c$ determined by the upper-level market. This constraint directly connects system operation with environmental policy, embedding emission regulation within the lower-level scheduling model.

$$\begin{array}{cc}\hfill {\Xi }_t& ={\left[{\Theta }_t^{\mathrm{wind}},{\Xi }_t^{\mathrm{solar}},D_t^{\mathrm{load}}\right]}^{\top },{\tilde{\Xi }}_t\sim \mathbb{P}\in {\mathcal{U}}_{ϵ}\hfill \end{array}$$

(9)

Uncertainty enters the system through renewable and load vectors. Each random vector ${\Xi }_t$ comprises wind generation ${\Theta }_t^{\mathrm{wind}}$, solar generation ${\Xi }_t^{\mathrm{solar}}$, and stochastic demand $D_t^{\mathrm{load}}$. The tilde denotes random realization drawn from an unknown distribution $\mathbb{P}$ lying within a Wasserstein ambiguity set ${\mathcal{U}}_{ϵ}$, whose radius $ϵ$ quantifies the decision-maker’s robustness preference.

$$\begin{array}{cc}\hfill {\mathcal{U}}_{ϵ}& =\left\{\mathbb{P}\in \mathcal{M}(\Xi ):W_p(\mathbb{P},\widehat{\mathbb{P}})\le ϵ\right\},W_p(\mathbb{P},\widehat{\mathbb{P}})=\underset{\Pi \in \Gamma (\mathbb{P},\widehat{\mathbb{P}})}{inf}{\left({\mathbb{E}}_{\Pi }\left[{\parallel \Xi -\widehat{\Xi }\parallel }_p^p\right]\right)}^{1/p}\hfill \end{array}$$

(10)

The ambiguity set above defines all plausible probability distributions $\mathbb{P}$ whose p-Wasserstein distance from the empirical sample distribution $\widehat{\mathbb{P}}$ does not exceed the tolerance $ϵ$. The transport plan $\Pi \in \Gamma (\mathbb{P},\widehat{\mathbb{P}})$ maps mass between the true and empirical distributions, and the cost of this mapping measured by ${\parallel \Xi -\widehat{\Xi }\parallel }_p^p$ establishes the degree of deviation allowed. By adopting this metric, the model formally controls sensitivity to sample uncertainty while remaining computationally tractable in the reformulated optimization.

$$\begin{array}{cc}\hfill {\vartheta }^{\mathrm{dro}}\underset{\mathbb{P}\in {\mathcal{U}}_{ϵ}}{sup}{\mathbb{E}}_{\mathbb{P}}\left[L(\mathit{x},\Xi )\right]& =\underset{\lambda \ge 0}{inf}\left\{\lambda {ϵ}^p+{\displaystyle \frac{1}{N}}\sum _{n=1}^N\underset{\xi }{sup}\left(L(\mathit{x},\xi )-\lambda {\parallel \xi -{\widehat{\xi }}^{\left(n\right)}\parallel }_p^p\right)\right\}\hfill \end{array}$$

(11)

The dual representation above converts the intractable worst-case expectation into a convex minimization with respect to $\lambda$. Here, $L(\mathit{x},\xi )$ denotes the scenario-dependent cost function, and ${\widehat{\xi }}^{\left(n\right)}$ represents the n-th empirical sample. The reformulation preserves the full distributional rigor of the Wasserstein DRO while providing computational tractability via linear–conic solvers. Parameter $\lambda$ operates as a Lagrange multiplier regulating the trade-off between conservatism ($ϵ$) and cost realism.

$$\begin{array}{cc}\hfill {\mathcal{C}}_{\mathrm{elec}}\left(\mathit{x}\right)& =0,{\mathcal{C}}_{\mathrm{hyd}}\left(\mathit{x}\right)=0,{\mathcal{C}}_{\mathrm{env}}\left(\mathit{x}\right)\le 0\hfill \end{array}$$

(12)

Feasibility of the lower-level system is enforced through grouped constraint functions. ${\mathcal{C}}_{\mathrm{elec}}\left(\mathit{x}\right)$ represents all electrical power-flow and capacity relations, ${\mathcal{C}}_{\mathrm{hyd}}\left(\mathit{x}\right)$ captures hydrogen energy balances and conversion couplings, and ${\mathcal{C}}_{\mathrm{env}}\left(\mathit{x}\right)$ embeds emission caps and sustainability regulations. These compact formulations ease analytical derivation and allow KKT-based reformulation.

$$\begin{array}{cc}\hfill {\nabla }_{\mathit{x}}\mathcal{L}(\mathit{x},\mathit{\mu },\mathit{\nu })& ={\nabla }_{\mathit{x}}f\left(\mathit{x}\right)+{\mathit{J}}_{{\mathcal{C}}_{\mathrm{eq}}}^{\top }\mathit{\mu }+{\mathit{J}}_{{\mathcal{C}}_{\mathrm{ineq}}}^{\top }\mathit{\nu }=0\hfill \end{array}$$

(13)

Stationarity of the lower-level optimization problem is expressed by the above KKT condition, where $\mathit{\mu }$ and $\mathit{\nu }$ are dual multipliers corresponding to equality and inequality constraints, respectively. The Jacobian matrices ${\mathit{J}}_{{\mathcal{C}}_{\mathrm{eq}}}$ and ${\mathit{J}}_{{\mathcal{C}}_{\mathrm{ineq}}}$ linearize all constraint sets around the current solution, ensuring gradient orthogonality between the feasible manifold and the cost function $f\left(\mathit{x}\right)$.

$$\begin{array}{c}\hfill 0\le \mathit{\nu }\perp {\mathcal{C}}_{\mathrm{ineq}}\left(\mathit{x}\right)\le 0\end{array}$$

(14)

Complementarity conditions link the primal and dual spaces. The notation $a\perp b$ denotes orthogonality $a^{\top }b=0$ with both non-negative, enforcing that an inequality constraint becomes active only when its corresponding multiplier is positive. This structure naturally emerges in MPEC (mathematical programs with equilibrium constraints) representations of hierarchical optimization problems.

$$\begin{array}{cc}\hfill \mathcal{L}(\mathit{x},\mathit{\mu },\mathit{\nu })& =f\left(\mathit{x}\right)+{\mathit{\mu }}^{\top }{\mathcal{C}}_{\mathrm{eq}}\left(\mathit{x}\right)+{\mathit{\nu }}^{\top }{\mathcal{C}}_{\mathrm{ineq}}\left(\mathit{x}\right)\hfill \end{array}$$

(15)

The Lagrangian function aggregates the primal objective $f\left(\mathit{x}\right)$ and constraint contributions weighted by their respective multipliers. It serves as a unifying bridge between primal feasibility and dual sensitivity, enabling efficient single-level transformation once embedded into the upper-level problem.

$$\begin{array}{cc}\hfill \underset{{\pi }_c,{\Omega }_c}{max}\{W({\pi }_c,{\Omega }_c)& =\sum _{t\in \mathcal{T}}\left[U\left(D_t\right)-C_{\mathrm{sys},t}\left({\pi }_c\right)-{\pi }_c{E}_t\right]\}\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& E_t\le {\Omega }_c,0\le {\pi }_c\le {\pi }_c^{max},{\Omega }_c^{min}\le {\Omega }_c\le {\Omega }_c^{max}\hfill \end{array}$$

(16)

The upper-level carbon-market optimization maximizes overall social welfare $W({\pi }_c,{\Omega }_c)$. It balances the consumer utility $U\left(D_t\right)$ against the system’s operational cost $C_{\mathrm{sys},t}$ and carbon payments ${\pi }_c{E}_t$. The associated constraints maintain emission compliance $E_t\le {\Omega }_c$ and bound the policy variables ${\pi }_c$ and ${\Omega }_c$ within regulatory limits. Through this layer, carbon pricing feedback and allowance design co-evolve with the lower-level scheduling model, completing the bilevel interaction between environmental policy and operational optimization.

$$\begin{array}{cc}\hfill \underset{\mathit{x}}{min}f\left(\mathit{x}\right)& =C_{\mathrm{sys}}({\pi }_c,\mathit{x})+{\mathcal{I}}_{\mathrm{KKT}}\left(\mathit{x}\right)\hfill \end{array}$$

(17)

The hierarchical optimization problem can be equivalently represented as a single-level mathematical program with equilibrium constraints (MPEC). Here, the primal decision vector $\mathit{x}$ consolidates all lower-level dispatch variables, while the indicator term ${\mathcal{I}}_{\mathrm{KKT}}\left(\mathit{x}\right)$ embeds the complementarity and feasibility conditions described in (13) and (14). This transformation collapses the bilevel structure into one tractable yet nonconvex formulation, enabling the use of decomposition-based solvers and dual reformulations.

$$\begin{array}{cc}\hfill \underset{\mathit{x},\theta }{min}\theta \mathrm{s}.\mathrm{t}.\theta & \ge C_{\mathrm{sys}}({\pi }_c,{\mathit{x}}^{\left(k\right)})+{\nabla }_{\mathit{x}}{C}_{\mathrm{sys}}{({\pi }_c,{\mathit{x}}^{\left(k\right)})}^{\top }(\mathit{x}-{\mathit{x}}^{\left(k\right)}),\forall k\in \mathcal{K}\hfill \end{array}$$

(18)

A Benders-type master formulation approximates the nonlinear recourse cost by iterative linearization. Each iteration k yields a supporting hyperplane cut for the master problem through the linearized first-order expansion of $C_{\mathrm{sys}}$. The auxiliary variable $\theta$ accumulates the current upper bound of the total operational cost, providing a continuously tightening approximation of the lower-level response surface.

$$\begin{array}{cc}\hfill {\Phi }^{\left(k\right)}& =\underset{\mathit{y}\in \mathcal{Y}}{min}\left\{f({\mathit{x}}^{\left(k\right)},\mathit{y}):\mathcal{A}\mathit{y}\le \mathit{b}\left({\mathit{x}}^{\left(k\right)}\right)\right\},\mathit{y}\equiv \mathrm{dispatch}\mathrm{and}\mathrm{hydrogen}\mathrm{variables}.\hfill \end{array}$$

(19)

Within each iteration, the subproblem solves for optimal operational responses to a fixed upper-level decision ${\mathit{x}}^{\left(k\right)}$. This stage computes the minimal dispatch cost subject to the linearized feasibility region $\mathcal{A}\mathit{y}\le \mathit{b}\left({\mathit{x}}^{\left(k\right)}\right)$, effectively acting as the operational oracle that informs the master-level welfare optimization.

$$\begin{array}{c}\hfill |z_{\mathrm{UB}}^{\left(k\right)}-z_{\mathrm{LB}}^{\left(k\right)}|\le {\epsilon }_{\mathrm{tol}}\end{array}$$

(20)

Convergence of the decomposition scheme is monitored through the optimality gap between upper and lower bounds of the system cost, denoted $z_{\mathrm{UB}}^{\left(k\right)}$ and $z_{\mathrm{LB}}^{\left(k\right)}$. When the gap falls below the tolerance ${\epsilon }_{\mathrm{tol}}$, the iterative process terminates, guaranteeing near-optimal performance of the bilevel carbon price-coupled operation.

$$\begin{array}{cc}\hfill {\pi }_c^{(r+1)}& =(1-{\rho }_r){\pi }_c^{\left(r\right)}+{\rho }_r{\pi }_c^{{*}_{\mathrm{adv}}},{\Omega }_c^{(r+1)}=(1-{\rho }_r){\Omega }_c^{\left(r\right)}+{\rho }_r{\Omega }_{c_{\mathrm{adj}}}^{*}\hfill \end{array}$$

(21)

Policy parameters are updated adaptively through convex combinations of historical and adversarial realizations. At iteration r, the carbon price ${\pi }_c^{\left(r\right)}$ and quota ${\Omega }_c^{\left(r\right)}$ are blended with the latest optimal responses ${\pi }_{c_{\mathrm{adv}}}^{*}$ and ${\Omega }_{c_{\mathrm{adj}}}^{*}$ using a learning rate ${\rho }_r$. This dynamic feedback rule stabilizes the regulatory–operational interaction and promotes progressive convergence of policy signals toward equilibrium.

$$\begin{array}{cc}\hfill {\mathit{x}}^{(r+1)}& =arg\underset{\mathit{x}}{min}\left\{f\left(\mathit{x}\right)+{\vartheta }_r{\parallel \mathit{x}-{\mathit{x}}^{\left(r\right)}\parallel }_2^2\right\}\hfill \end{array}$$

(22)

A proximal regularization step refines primal decisions and mitigates oscillations in continuous–discrete variable interactions. The quadratic penalty ${\parallel \mathit{x}-{\mathit{x}}^{\left(r\right)}\parallel }_2^2$ stabilizes iterative updates, ensuring convergence even under strong nonconvexity and high-dimensional coupling among energy carriers.

$$\begin{array}{c}\hfill \mathrm{If}|{\Phi }^{(r+1)}-{\Phi }^{\left(r\right)}|\le {\delta }_{\mathrm{stop}},\mathrm{then}\mathrm{terminate}\mathrm{optimization}.\end{array}$$

(23)

Finally, the algorithm halts when successive recourse-cost values differ by less than the stopping threshold ${\delta }_{\mathrm{stop}}$. This final condition confirms the stabilization of both policy and operational decisions, yielding the globally consistent solution for carbon price-coupled, distributionally robust coordination of the integrated power–hydrogen system.

$$\begin{array}{cc}\hfill \underset{\mathbb{P}\in {\mathcal{U}}_{ϵ}}{sup}{\mathbb{E}}_{\mathbb{P}}\left[L(\mathit{x},\Xi )\right]& \approx \underset{\omega \in \Omega }{max}\left\{L(\mathit{x},{\Xi }_{\omega }):W_p({\mathbb{P}}_{\omega },\widehat{\mathbb{P}})\le ϵ\right\}\hfill \end{array}$$

(24)

The inner worst-case search within the distributionally robust layer is replaced by a discrete approximation. Each scenario ${\Xi }_{\omega }$ represents a specific realization of renewable generation and demand, and the Wasserstein distance $W_p({\mathbb{P}}_{\omega },\widehat{\mathbb{P}})$ quantifies how far the scenario deviates from the empirical baseline. This formulation forms the computational core of the column-and-constraint generation (CCG) procedure, identifying adversarial conditions that maximize system loss and enriching the uncertainty set dynamically.

$$\begin{array}{cc}\hfill L^{(r+1)}\left(\mathit{x}\right)& =(1-{\rho }_r)L^{\left(r\right)}\left(\mathit{x}\right)+{\rho }_r{L}_{\mathrm{adv}}^{*}\left(\mathit{x}\right),{\rho }_r\in (0,1)\hfill \end{array}$$

(25)

Loss aggregation evolves adaptively as the algorithm iterates. At each step r, the updated expected loss $L^{(r+1)}$ is expressed as a convex combination of the previous loss estimate $L^{\left(r\right)}$ and the adversarial realization $L_{\mathrm{adv}}^{*}$ discovered through (24). The relaxation parameter ${\rho }_r$ controls the stability and speed of convergence, ensuring that the final estimate represents a balanced reflection of both empirical and worst-case observations.

$$\begin{array}{cc}\hfill {\lambda }^{(r+1)}& =arg\underset{\lambda \ge 0}{min}\left\{\lambda {ϵ}^p+{\displaystyle \frac{1}{N}}\sum _{n=1}^N\underset{\xi }{sup}\left(L({\mathit{x}}^{\left(r\right)},\xi )-\lambda {\parallel \xi -{\widehat{\xi }}^{\left(n\right)}\parallel }_p^p\right)\right\}\hfill \end{array}$$

(26)

Updating the dual variable $\lambda$ provides a self-correcting mechanism for distributional robustness. By solving the inner convex minimization, the algorithm automatically calibrates the trade-off between conservatism and empirical fit for iteration $r+1$. Smaller values of $\lambda$ imply heavier penalization of large deviations, while larger values reduce robustness in favor of empirical accuracy. This step maintains stability in the Wasserstein-DRO structure and ensures consistency of risk adjustment across iterations.

$$\begin{array}{cc}\hfill {\Psi }^{(r+1)}& ={\Phi }^{(r+1)}-{\pi }_c^{(r+1)}{E}^{(r+1)}+U\left(D_t\right)\hfill \end{array}$$

(27)

The welfare functional ${\Psi }^{(r+1)}$ synthesizes the total benefit of the integrated power–hydrogen system at iteration $r+1$. It combines the minimized system operation cost ${\Phi }^{(r+1)}$ with the carbon payment term $-{\pi }_c^{(r+1)}{E}^{(r+1)}$ and consumer utility $U\left(D_t\right)$. This aggregate indicator captures the combined effects of market decisions, emission regulation, and end-user satisfaction—serving as the final performance measure for the bilevel carbon price-coupled optimization.

$$\begin{array}{c}\hfill \mathrm{If}|{\Psi }^{(r+1)}-{\Psi }^{\left(r\right)}|\le {\zeta }_{\mathrm{stop}},\mathrm{terminate}\mathrm{iteration}.\end{array}$$

(28)

Termination occurs when welfare variation between successive iterations becomes negligible. The tolerance ${\zeta }_{\mathrm{stop}}$ governs this stopping condition. Once the improvement of $\Psi$ stabilizes, the entire hierarchy—from the upper-level carbon policy to the lower-level operational scheduling—is considered converged, yielding a distributionally robust equilibrium for the integrated power–hydrogen system under carbon market coupling.

Table 2. Computational performance of the Benders–CCG algorithm.

System Configuration	Iterations	Benders Cuts	CCG Columns	Total Runtime
	(Master + Subproblem)	Generated	Added	(Seconds)
Base Case (33-bus)	21 + 178	54	37	142.6
High Uncertainty Scenario	24 + 231	67	49	198.4
High Hydrogen Penetration	19 + 164	48	34	127.3
EV-Dominant Scenario	22 + 205	61	45	176.9
Stochastic Benchmark (for comparison)	–	–	–	389.5

summarizes the computational performance of the Benders–CCG algorithm and provides insight into the scalability and numerical stability of the proposed solution framework across several representative scenarios. The iteration counts reported for both the master problem and the subproblems show that the decomposition structure converges reliably even when the operating environment becomes more complex due to higher uncertainty, increased hydrogen participation, or intensified EV charging dynamics. The number of Benders cuts generated reflects the extent to which the algorithm refines the feasible region of the master problem, while the number of CCG columns added indicates the progressive enrichment of the uncertainty representation within the DRO formulation. Together, these metrics demonstrate that the algorithm systematically constructs a tighter and more informative approximation of the underlying decision space as the solution progresses. The total runtime remains within a tractable range for all scenarios, highlighting the practical feasibility of applying Benders–CCG to integrated electricity–hydrogen systems of realistic size. Even in uncertainty-heavy or behaviorally complex cases, the algorithm completes within a few hundred seconds, suggesting that the inherent iterative structure does not impose excessive computational burdens. The comparison with the stochastic benchmark underscores this advantage: the deterministic-equivalent stochastic formulation requires substantially longer solution times, reinforcing the computational benefits of the proposed distributionally robust and decomposition-based approach. Overall, the performance results demonstrate that the Benders–CCG methodology offers a balanced combination of robustness, efficiency, and scalability, making it suitable for both research-oriented studies and potential real-time or near-real-time planning applications in multi-energy systems.

Compared with conventional deterministic or stochastic optimization frameworks, the proposed model provides a more resilient representation of uncertainty and policy–operation interactions. Deterministic bilevel approaches assume fixed parameter values and therefore cannot capture variability in renewable output or demand behavior. Scenario-based stochastic models incorporate uncertainty but rely on the accuracy and completeness of scenario sets, which may lead to sensitivity under distributional shifts. In contrast, the Wasserstein-based distributionally robust formulation used in this study constructs an ambiguity set around empirical samples and evaluates worst-case realizations within that set, enabling a more reliable assessment of system behavior under uncertain conditions. This structure strengthens the analytical capability of the bilevel carbon pricing model by integrating uncertainty handling, operational flexibility, and policy design within a unified optimization framework.

The choice of the CS–ARDL model is informed by the empirical properties of the dataset and the interconnected structure of the system being analyzed. The underlying variables display substantial cross-sectional dependence driven by shared technological transitions, coordinated carbon pricing signals, and common socio-economic influences across units. Such dependence introduces unobserved common factors that render first-generation panel estimators, including conventional ARDL, FMOLS, and DOLS approaches, biased and inconsistent. The CS–ARDL methodology addresses this challenge by incorporating cross-section averages of both dependent and explanatory variables, thereby correcting for latent global shocks and accommodating co-movements induced by system-wide drivers. Moreover, CS–ARDL is capable of simultaneously capturing short-run dynamic adjustments and long-run equilibrium relationships without imposing restrictive homogeneity assumptions or requiring extensive pre-testing procedures. Its flexibility allows heterogeneous adjustment speeds across units while maintaining robustness to correlated disturbances. These characteristics make the CS–ARDL model particularly suitable for empirical environments influenced by jointly evolving policy interventions, coupled energy infrastructures, and multi-region behavioral dynamics, offering an econometrically reliable basis for quantifying both transient responses and structural linkages within the integrated system.

4. Data

A transparent and well-structured presentation of the empirical foundations is essential for ensuring the interpretability, reproducibility, and methodological credibility of integrated energy system studies. Given that the proposed carbon price-coupled power–hydrogen optimization model operates at the intersection of renewable energy variability, prosumer behavior, mobility-driven demand fluctuations, and cyber-physical uncertainty, the empirical datasets supporting the analysis must be sufficiently rich, granular, and representative of real-world operating conditions. The modeling framework requires data capable of characterizing not only the physical operation of the electricity and hydrogen subsystems but also the behavioral and stochastic components that shape multi-energy coordination. To satisfy these requirements, the study incorporates datasets from well-established empirical repositories, validated engineering benchmarks, and widely accepted behavioral modeling sources. These datasets jointly enable a holistic reproduction of the operational environment in which the proposed bilevel, distributionally robust optimization model is evaluated.

Table 3. Key datasets and system parameters used in the analysis.

Category	Data Used in the Study	Unit
PV Generation Profile	15-min irradiance and PV output for a semi-urban California location (37.77° N, 122.42° W), covering a 24-h summer day.	kW, kWh
PV Capacity per Node	Uniform distribution between 50–100 kW across 12 community nodes.	kW
System Peak Load	Aggregated peak demand of 3.2 MW after stochastic diversification.	MW
Load Profile	Residential + commercial profile with 15-min resolution, diversified across 12 nodes.	kW
Number of Households	360 households (30 per community node).	–
EV Penetration	One EV per household; total 360 EVs.	–
EV Battery Capacity	40–80 kWh per EV.	kWh
EV Charging Power Limit	Maximum 7 kW AC charging.	kW
Mobility Patterns	Arrival/departure windows based on CHTS survey distributions.	–
Behavioral Parameters	Inertia   Beta(2,5); peer-conformity   Beta(3,4); self-efficacy distributed across agents.	–
Cyber Attack Model [31]	FDI probability per bus: truncated Gaussian ($\mu$ = 0.3, $\sigma$ = 0.1, clipped to [0, 1]).	–
Network Topology	Modified IEEE 33-bus system with 12 prosumer nodes.	–
Hydrogen System Data	Electrolyzer and fuel cell efficiencies, tank capacity, converter ratings.	–
Temporal Resolution	96 intervals per day (15-min time steps).	–

summarizes the datasets utilized in the empirical analysis of the integrated power–hydrogen coordination framework. The study employs multiple data sources spanning renewable generation, load behavior, mobility patterns, cyber-physical disturbance modeling, and system-level configuration parameters. The datasets collectively enable a realistic representation of multi-energy interactions, behavioral variability, and uncertainty propagation across the electricity–hydrogen system.

The selection of these datasets follows established principles in integrated energy system analysis, ensuring that the computational experiments reflect operational realism and variability consistent with real-world conditions. High-resolution renewable datasets are essential for capturing solar intermittency and its influence on electrolyzer operation. Empirical demand and EV mobility data provide heterogeneous consumption and charging patterns necessary for representing distribution-level behavioral dynamics. Behavioral parameters allow the incorporation of socially influenced decision mechanisms, enabling a more realistic modeling of demand-response interactions. Cyber-attack probability distributions support the examination of resilience under data perturbations, a factor increasingly emphasized in modern cyber-physical system research. Finally, standardized system configuration data from the IEEE 33-bus network ensures compatibility with widely used benchmarks in distribution system studies. Collectively, these datasets form a coherent and representative basis for evaluating the proposed optimization framework.

5. Results

To empirically evaluate the effectiveness of the proposed socially-aware, distributionally robust optimization framework, we conduct a detailed case study using a modified IEEE 33-bus radial distribution system. The test system is enriched with prosumer participation, human behavioral patterns, and cyber-physical interaction layers. Specifically, 12 of the 33 nodes are designated as mixed-use community nodes, each equipped with rooftop photovoltaic (PV) generation, electric vehicle (EV) fleets, and shared thermal storage units. PV generation profiles are constructed using high-resolution solar irradiance data from NREL’s NSRDB dataset, corresponding to a semi-urban location in California (37.77° N, 122.42° W) over a 24-h summer day, with temporal granularity of 15 min. For each community node, a typical rooftop PV system capacity is assumed to be 50–100 kW, following a uniform distribution. Demand-side load profiles are synthesized using residential and small commercial demand patterns obtained from the Pecan Street database, further diversified using stochastic perturbations and scaled to yield a total system peak load of 3.2 MW. The social behavior of agents—reflecting EV charging preferences, load-shifting willingness, and demand response compliance—is modeled using a parametrized Social Force Model (SFM) combined with time-evolving Social Cognitive Theory (SCT) features. A virtual community of 360 households (30 households per community node) is simulated, where individual decision inertia, peer influence, and self-efficacy are encoded as agent-level parameters drawn from Beta distributions (e.g., inertia $\sim \mathrm{Beta}$(2,5), peer-conformity $\sim \mathrm{Beta}$(3,4)). Each household is randomly assigned an EV with a battery capacity ranging from 40 to 80 kWh and a charging power cap of 7 kW. The mobility model uses departure and arrival time windows based on empirical travel survey data from the California Household Travel Survey (CHTS), ensuring that EV availability constraints are consistent with realistic patterns. Cyber disruptions are modeled as false data injection (FDI) events targeting local renewable generation measurements and demand forecasts, with attack probabilities following a truncated Gaussian distribution across buses $(\mu =0.3,\sigma =0.1,\mathrm{clipped}\mathrm{to}[0,1\left]\right)$.

The socially aware component in the results reflects the way in which behavioral dynamics influence the load patterns that enter the lower-level optimization. The Social Force Model and Social Cognitive Theory elements do not modify the mathematical structure of the bilevel formulation directly; instead, they shape the temporal and spatial characteristics of electric vehicle charging behavior. These behavioral rules determine arrival distributions, parking durations, and charging propensities, which are then aggregated into the exogenous load coefficients used in the dispatch problem. In this sense, the connection between the behavioral layer and the optimization layer is conceptual rather than algebraically embedded: behavioral interactions generate the demand profiles, and the optimization framework subsequently determines the cost-minimizing operational response to those profiles. This clarification highlights that socially aware influences enter the model through the construction of data-driven demand inputs rather than through explicit coupling terms within the mathematical formulation.

Figure 3 provides a comprehensive system-level illustration of the energy-balance pathways within the integrated power–hydrogen framework, offering an intuitive interpretation of how electrical energy, hydrogen conversion processes, and storage dynamics jointly influence overall operational behavior. The figure depicts renewable electricity, primarily generated by photovoltaic units, as the fundamental upstream energy source. This renewable inflow is distributed through two principal pathways: direct delivery to the power distribution network to satisfy real-time electrical demand, and diversion to the electrolyzer, where electrical energy is converted into hydrogen via electrochemical processes. This conversion establishes a critical coupling between the electrical and chemical energy domains and enables surplus renewable generation to be transformed into a storable energy carrier rather than curtailed. The generated hydrogen is routed into the hydrogen storage system, which plays a central role in enabling temporal decoupling between energy production and consumption. By accumulating chemical energy during periods of high renewable availability and releasing it during periods of scarcity or increased demand, the storage unit provides long-term flexibility that is not achievable through electrical storage alone. When the system experiences reduced renewable output or elevated load requirements, the stored hydrogen is supplied to the fuel cell, where it is reconverted into electricity and injected back into the power network. This bidirectional interaction highlights hydrogen’s dual functionality as both an energy sink and a dispatchable energy source, thereby enhancing system resilience and operational robustness. In addition, the figure illustrates how electricity flows from the distribution network toward downstream loads, including electric vehicle charging demand and other consumer sectors, emphasizing the interaction between supply-side decisions and demand-side requirements. The Sankey-style visualization explicitly conveys the relative magnitudes of these energy pathways and facilitates an intuitive understanding of how energy is redistributed across conversion, storage, and consumption stages. Importantly, the diagram also reflects the influence of carbon price signals on operational decisions. Higher carbon costs implicitly shift energy flows toward low-emission pathways by incentivizing renewable utilization, increasing electrolyzer operation, and promoting deeper engagement of hydrogen storage and fuel cell generation. Through this visual abstraction, Figure 3 bridges the gap between the mathematical optimization model and its physical interpretation, clearly demonstrating how multi-energy coupling, storage utilization, and carbon-aware dispatch collectively shape the system’s operational equilibrium.

Figure 4 depicts how the temporal variation of community load interacts with photovoltaic (PV) generation across 12 distributed nodes over 96 time intervals of 15 min each, forming a dense 12 × 96 matrix that captures the diurnal energy balance dynamics. The x-axis corresponds to time, while the y-axis enumerates the community nodes. Each cell’s color encodes load intensity from 80 kW (deep blue) to 170 kW (bright yellow), providing a precise visual metric of spatial demand heterogeneity. The white contour lines overlay PV generation, peaking around intervals 30–60 (corresponding to 7:30–15:00) when irradiance levels reach 900 W/m², while load profiles simultaneously show partial reduction due to self-consumption. This creates a clearly observable complementarity between mid-day renewable abundance and evening demand surges, confirming the intrinsic temporal mismatch that drives the need for hydrogen and storage coupling. From a system coordination viewpoint, the figure exposes how inter-node variation amplifies the overall uncertainty envelope. Node 3 and Node 7 exhibit synchronized high loads with late-afternoon peaks exceeding 160 kW, whereas Node 9 experiences morning-biased consumption linked to industrial activity. The PV contours flatten in low-irradiance nodes (standard deviation ≈ 12 kW), illustrating location-dependent renewable resource diversity. These localized discrepancies directly translate into uneven dispatch requirements and motivate adaptive power–hydrogen conversion scheduling in the lower-level optimization. The color gradients, especially in the 0–6 h and 18–24 h windows, quantify curtailment vulnerability, as load demand remains unbalanced with PV output, suggesting curtailment ratios of roughly 25–35% during off-peak hours.

Figure 5 visualizes the construction of the uncertainty set in the renewable generation feature space, capturing the essence of the Distributionally Robust Optimization (DRO) framework applied in this study. Each point in light blue represents an empirical renewable scenario derived from historical data, characterized by two normalized renewable attributes—“Renewable Feature 1” and “Renewable Feature 2.” These dimensions can correspond to correlated stochastic drivers such as normalized solar irradiance and wind capacity factor. The teal-colored points denote the adversarial samples identified through the inner maximization process of the DRO formulation, where the algorithm seeks the worst-case realizations within the Wasserstein ambiguity ball (depicted as the dashed circle boundary). The spatial distribution of these adversarial samples is deliberately skewed toward the regions of higher operational sensitivity—where small deviations in renewable generation may lead to disproportionately large cost or emission impacts. The circular dashed boundary represents the Wasserstein radius $\epsilon$, defining the allowable distance between the empirical distribution and any perturbed adversarial distribution. This geometric interpretation conveys how the DRO mechanism balances conservatism and adaptiveness: smaller $\epsilon$ values limit the search to near-empirical variations, producing optimistic but potentially fragile solutions, while larger $\epsilon$ values expand the uncertainty region to cover rare, high-impact scenarios. In this visualization, the adversarial samples remain mostly concentrated within the upper-right quadrant, suggesting that the power–hydrogen system exhibits higher vulnerability under simultaneous increases in both renewable features—such as wind gusts coinciding with strong solar peaks—conditions that can cause congestion or over-generation in integrated systems.

Figure 6 visualizes the complex relationship between electrolyzer operational intensity and the evolution of hydrogen storage levels under the coordinated electricity–hydrogen scheduling framework. The horizontal axis represents the instantaneous power input to the electrolyzer, ranging from 0 to 100 kW, while the vertical axis shows hydrogen inventory levels spanning from roughly 180 kg to 300 kg over a 24-h cycle divided into 15-min intervals. The density distribution, rendered in a Parula gradient, reveals two distinct clusters: a primary linear regime between 35 kW and 75 kW where hydrogen accumulation follows an approximately proportional trajectory (slope ≈ 0.8 kg/kW, R² = 0.87), and a secondary saturation regime beyond 85 kW where marginal storage growth plateaus due to system efficiency decline. Nearly 62% of the samples fall in the mid-range band (40–80 kW), indicating that electrolyzers typically operate under partial load conditions to balance renewable variability and carbon price-induced economic signals. The high-density region near 280 kg reflects storage saturation points, aligning with the storage tank’s nominal capacity constraint, while the dispersed low-power region (<30 kW) corresponds to periods of reduced renewable availability. The overall correlation coefficient between electrolyzer power and stored hydrogen reaches 0.83, confirming the model’s accurate representation of short-term conversion dynamics.

Figure 7 illustrates the behavioral variability in electric vehicle activity across 360 households participating in the community microgrid. The horizontal axis represents local time (0–24 h), while the vertical scale denotes the number of vehicles within each hourly bin. Two major peaks are observed: a departure peak centered around 8.1 h with a standard deviation of 1.2 h and an arrival peak centered around 18.3 h with a standard deviation of 1.5 h. Approximately 70% of vehicles depart between 7:00 and 9:30, while 74% return between 17:00 and 20:30, defining a typical unavailability window of about 9.5 h for grid interaction. This pattern indicates that EVs are primarily idle during nighttime, a period with lower grid demand and thus greater flexibility for smart charging or discharging operations. Around 8–10% of vehicles deviate from this pattern, leaving before 6:30 or returning after 21:00, reflecting user heterogeneity modeled through stochastic behavioral parameters. The histogram’s bimodal distribution directly influences the system’s temporal flexibility structure. During peak midday PV production, the majority of EVs are away from the grid, causing renewable surplus to be redirected toward hydrogen production rather than charging. Conversely, the convergence of arrival times with the evening load peak produces a coinciding demand surge of roughly 220 kW system-wide if uncontrolled. The variance asymmetry between departure and arrival distributions (1.2² vs. 1.5²) implies that evening charging uncertainty is more significant, aligning with the model’s motivation to apply Distributionally Robust Optimization (DRO) for demand-response scheduling. The empirical spread confirms that social behavior must be embedded as a constraint set within the operational layer. By quantifying daily behavioral rhythms and statistical variability, this figure transforms human mobility patterns into operational parameters, connecting the social-cognitive dimension of energy use to system-level optimization.

Table 4. Comparative performance of different carbon price and optimization scenarios.

Scenario	Total Cost (103 USD)	Emission (tCO2)	Renewable Share (%)	Welfare Index
Baseline	982.5	42.7	68.4	0.73
Low Carbon Price	995.2	39.1	72.3	0.78
Moderate Carbon Price	1013.8	34.5	76.9	0.82
High Carbon Price	1048.6	29.8	82.1	0.88
DRO-Optimized	1002.4	33.2	80.4	0.91

presents a comprehensive comparison of system performance under different carbon price regimes and optimization paradigms, highlighting the fundamental trade-offs between economic and environmental outcomes. At a baseline state, the system operates with a total cost of 982.5 × 10³ USD and emits 42.7 tCO₂, corresponding to a renewable energy penetration of 68.4% and a welfare index of 0.73. As carbon prices increase, the emission levels decline progressively—from 39.1 tCO₂ under low carbon prices to 29.8 tCO₂ under high carbon prices—indicating effective decarbonization. However, this improvement in environmental performance comes at a moderate increase in cost, reaching 1048.6 × 10³ USD at the highest carbon price. The welfare index rises from 0.73 to 0.88, signifying that carbon regulation not only mitigates emissions but also enhances the overall societal benefit when economic externalities are internalized. The DRO-optimized scenario provides a notable balance: with 1002.4 × 10³ USD in cost and 33.2 tCO₂ emissions, it achieves 80.4% renewable integration and the highest welfare score (0.91). This demonstrates the capability of the proposed bilevel DRO framework to achieve both operational efficiency and resilience under uncertainty. The nonlinear response across scenarios also reveals the diminishing marginal benefit of carbon pricing beyond the moderate level, offering insights for regulatory calibration.

Table 5. Operational characteristics of major components in the integrated power–hydrogen system.

System Component	Avg Cap Usage (%)	Hydrogen Eff. (%)	Emission Factor (kgCO2/MWh)	Marginal Cost (USD/MWh)
Thermal Unit A	76.5	–	470.3	58.6
Thermal Unit B	71.2	–	512.8	61.4
Electrolyzer	83.1	78.6	0.0	52.9
Fuel Cell	64.7	62.3	32.5	55.1
Battery Storage	48.5	–	0.0	46.2

details the operational characteristics of individual system components, emphasizing technological differentiation in efficiency, emission intensity, and marginal cost behavior. Thermal Unit A and B exhibit average capacity utilization rates of 76.5% and 71.2%, with emission intensities of 470.3 and 512.8 kgCO₂/MWh respectively, reflecting their critical yet carbon-intensive role in ensuring reliability. Electrolyzers and fuel cells, in contrast, demonstrate much lower emission factors (0.0 and 32.5 kgCO₂/MWh) and higher energy conversion efficiencies of 78.6% and 62.3%. The electrolyzer’s high utilization rate of 83.1% highlights its importance in flexible hydrogen production, particularly when surplus renewable power is available. Battery storage operates at 48.5% utilization, primarily functioning as a short-term balancing mechanism to mitigate renewable intermittency rather than a constant dispatch source. The marginal costs range from 46.2 USD/MWh for storage to 61.4 USD/MWh for thermal generation, illustrating the relative competitiveness of cleaner technologies. The observed operational structure reveals a dynamic interplay between cost minimization and emission abatement, where higher-efficiency units compensate for the variability of renewables, achieving an optimized equilibrium through the integrated power–hydrogen dispatch model.

Table 6. Impact of uncertainty levels on system performance and emission outcomes.

Uncertainty Level (%)	Expected Cost (103 USD)	Cost Std. Dev. (USD)	Hydrogen Curtailment (%)	Emission Reduction (%)
0 (Deterministic)	974.6	0.0	3.5	0.0
10 (Mild)	987.8	8.7	4.2	2.4
20 (Moderate)	1008.3	16.9	5.3	5.6
30 (Severe)	1045.9	23.8	6.7	8.1
DRO (Adaptive)	1016.5	10.2	4.1	9.8

explores the system’s sensitivity to uncertainty, quantifying how stochastic variability in renewable output and demand influences total cost, curtailment, and emission reduction. Under deterministic conditions, the system exhibits minimal variability, with an expected cost of 974.6 × 10³ USD and a stable hydrogen curtailment ratio of 3.5%. As uncertainty intensifies from 10% to 30%, both cost and standard deviation increase significantly—from 987.8 × 10³ USD to 1045.9 × 10³ USD and from 8.7 to 23.8 USD—reflecting elevated balancing and reserve requirements. Concurrently, hydrogen curtailment grows from 4.2% to 6.7%, indicating the reduced efficiency of energy utilization when uncertainty is unmanaged. Emission reduction improves moderately (from 0.0% to 8.1%), yet this comes at the expense of economic stability. The DRO-based adaptive approach achieves a balanced outcome: a moderate expected cost of 1016.5 × 10³ USD, a low deviation of 10.2 USD, and a reduced curtailment of 4.1%, while achieving the highest emission reduction of 9.8%. This result underlines the intrinsic value of distributionally robust modeling, which minimizes the risk of extreme scenarios without overcompensating through excessive reserves or conservative operation. The adaptive model effectively mitigates the cost–risk–emission triad by dynamically responding to uncertain realizations, demonstrating that robustness and efficiency can coexist under well-calibrated optimization. Overall, this table validates the robustness, stability, and environmental superiority of the proposed bilevel carbon price-coupled optimization strategy.

Figure 8 portrays the probabilistic structure of uncertainty across ten representative stochastic realizations of renewable generation and load demand. The horizontal axis measures standardized deviation (−2.5 to +2.5), while the vertical dimension represents scenario index, each ridge depicting the probability density of one sampled scenario. The Parula-colored ridges transition from dark blue (low variance) at the bottom to yellow-green (high variance) at the top, visually encoding increasing uncertainty levels. The lower ridges (Scenarios 1–4) exhibit near-Gaussian distributions with standard deviations around 0.45, representing typical renewable fluctuations under stable weather. The upper ridges (Scenarios 8–10) widen substantially, with standard deviations reaching up to 0.85, corresponding to extreme high-load or low-generation cases. The ensemble envelope width grows by approximately 70% across all layers, mirroring the Wasserstein ambiguity radius $ϵ$ = 0.3 used for DRO calibration. The inter-scenario overlap region, covering roughly 65% of cumulative probability mass, defines the nominal operation confidence zone, providing a clear visual analogue of the ambiguity set’s geometry in the optimization framework. This ridge-based visualization translates the mathematical concept of distributional uncertainty into an intuitive topological representation. The lower, narrower ridges correspond to base-case operation close to expected forecasts, while the higher, broader ridges simulate risk-averse configurations responding to worst-case distributions. Such progression clarifies how the optimization process shifts from expectation-based decisions toward robust, conservatively feasible ones as uncertainty grows. The smooth transition between ridges ensures statistical continuity essential for the iterative convergence of the Benders–CCG algorithm. Quantitatively, the ridge ensemble reproduces the empirical error distribution derived from 1000 Monte Carlo samples of renewable and demand data, maintaining a 95% confidence coverage. This figure thus serves as both a methodological visualization and a validation of uncertainty calibration, reinforcing that the case study’s DRO layer is not abstractly assumed but statistically grounded in realistic data behavior.

Figure 9 presents the quantitative interaction between environmental policy pressure and operational performance of the integrated power–hydrogen system. The horizontal axis represents carbon price levels ranging from 30 $/tCO₂ to 60 $/tCO₂, while the left and right vertical axes denote total operating cost and total carbon emissions, respectively. As the carbon price increases, the system’s cost curve (blue solid line) exhibits a nearly linear upward trend from approximately 1.02 × 10⁶$ to 1.31 × 10⁶$, with an average gradient of about +9.4% per 10 $/tCO₂. Simultaneously, the emission curve (grey line) declines inversely from 51 tCO₂ to 37 tCO₂ (−27%), yielding an elasticity of −0.29 tCO₂ per$ increment. The shaded blue region surrounding the cost curve represents the uncertainty band derived from the Distributionally Robust Optimization (DRO) ambiguity set ($ϵ$ = 0.3), spanning roughly ±3.2% around the expected trajectory. This figure reveals the system’s policy sensitivity: moderate carbon pricing (≈45 $/tCO₂) achieves near-optimal environmental efficiency without disproportionate cost escalation. Below this threshold, emission reduction remains marginal because dispatch decisions favor conventional units due to their lower short-term marginal cost. Beyond 50 $/tCO₂, cost rises steeply as hydrogen conversion and storage units are forced to operate at higher marginal efficiencies, leading to diminishing returns in emission mitigation. The inflection between 40 and 50 $/tCO₂ corresponds to a significant shift in marginal abatement cost (MAC), calculated at approximately 47 $/tCO₂ for the base scenario. Operationally, this regime transition coincides with a 30% increase in electrolyzer utilization and a 15% rise in renewable curtailment absorption, confirming the coupled mechanism between upper-level carbon regulation and lower-level scheduling behavior. The smoothness and convexity of both curves indicate stable bilevel coordination—an essential validation that the DRO-Benders framework properly internalizes policy variability and economic equilibrium.

Figure 10 provides a macroeconomic interpretation of how carbon market regulation redistributes benefits across social stakeholders. The x-axis denotes the carbon price levels (30, 40, 50, 60 $/tCO₂), and the y-axis represents total social welfare (in 10³$). The stacked bars decompose welfare into three interacting components: consumer surplus (light blue), producer surplus (blue), and carbon revenue (grey). At 30 $/tCO₂, consumer surplus dominates (≈500 × 10³$), reflecting affordable electricity prices and moderate generation margins, while carbon revenue remains negligible. As the price rises to 40 $/tCO₂, consumer surplus declines (−6%), producer surplus slightly increases (+7%), and carbon revenue doubles to ≈100 × 10³$. By 50 $/tCO₂, producer surplus peaks at 340 × 10³$, and carbon revenue reaches 150 × 10³$, partially compensating the 15% reduction in consumer surplus. At 60 $/tCO₂, total welfare reaches 950 × 10³$, slightly lower than its maximum, signaling that excessive carbon taxation starts imposing efficiency losses. The decomposition clearly outlines a policy equilibrium zone near 45 $/tCO₂, where total welfare attains its local maximum while maintaining a balanced benefit distribution—consumer share ≈ 44%, producer share ≈38%, and carbon revenue ≈18%. Beyond this point, further price escalation primarily transfers welfare from consumers to government carbon funds without proportionate emission benefits (see Figure 2), confirming the theoretical optimality of moderate carbon pricing derived from the bilevel welfare maximization framework. The increasing slope of the grey bars also quantifies the fiscal potential of carbon markets: a +10 $/tCO₂ increment yields roughly +50 × 10³$ of public revenue, which could fund renewable subsidies or grid reinforcement, reinforcing the policy feedback loop.

Table 7. Comparison of modeling approaches.

Feature	Deterministic Bilevel	Stochastic Optimization	Wasserstein DRO (Proposed)
Uncertainty Treatment	Assumes fixed known parameters; no uncertainty modeling.	Uses scenario sets derived from historical data.	Models distributional deviations around empirical data.
Robustness to Data Shifts	Low; sensitive to parameter variation.	Moderate; depends on scenario representativeness.	High; protects against unseen but plausible outcomes.
Computational Complexity	Low; single optimization problem.	High; grows with number of scenarios.	Moderate; solved via Benders–CCG decomposition.
Data Requirements	Minimal; point estimates only.	Requires large scenario sets.	Requires only samples; does not depend on full distributions.
Policy Sensitivity	Rigid; limited adaptability to volatility.	Captures expected response to uncertain inputs.	Captures worst-case policy–operation interactions.
Suitability for Carbon Pricing	Limited; ignores uncertainty in renewable profiles.	Good; reflects expected market outcomes.	Excellent; accounts for risk under regulatory uncertainty.
System Flexibility Representation	Basic; lacks stochastic or robust dynamics.	Scenario-driven flexibility.	Scenario-free and distributionally flexible.

provides a concise comparison between deterministic bilevel optimization, scenario-based stochastic optimization, and the Wasserstein distributionally robust approach adopted in this study. The deterministic framework offers computational simplicity but cannot represent variability in renewable generation, load fluctuations, or behavioral uncertainties, which limits its effectiveness in policy-sensitive multi-energy systems. The stochastic paradigm introduces uncertainty modeling through scenario sets and captures expected outcomes, yet its performance depends heavily on the representativeness and size of the scenario library. Large scenario sets improve accuracy but substantially increase computational cost, and scenario truncation may omit rare but operationally significant deviations. The proposed Wasserstein DRO formulation addresses these limitations by constructing an ambiguity set around empirical samples and optimizing against the worst-case distribution within that set. This allows the model to incorporate uncertainty without relying on extensive scenario enumeration while maintaining robustness to unseen but plausible system conditions. The comparison also highlights key differences in data requirements and policy responsiveness: deterministic models require minimal data but exhibit limited adaptability, whereas stochastic models depend on significant historical data and may underperform under distributional shifts. In contrast, the DRO approach balances robustness and tractability by leveraging sample-based information and providing guaranteed performance under uncertain policy–operation interactions. From a computational standpoint, deterministic models are the lightest, stochastic models become increasingly demanding with scenario growth, and the proposed method achieves a moderate and manageable complexity through Benders–CCG decomposition. Overall, the table underscores the methodological advantages of the DRO-based bilevel formulation, particularly in settings where uncertainty, carbon pricing, and hydrogen flexibility interact in ways that traditional models cannot fully capture.

Discussion of Practical Implications

The empirical findings presented herein bear significant practical implications for the design and operation of next-generation low-carbon energy systems, particularly when viewed in light of the existing literature. First, regarding regulatory policy, the identification of a welfare-maximizing carbon price equilibrium (approximately 45 $/tCO₂) challenges the efficacy of static, exogenous pricing schemes often assumed in earlier deterministic studies [19,20]. Our results suggest that regulators should adopt adaptive pricing mechanisms that account for the operational flexibility of cross-sectoral technologies. Practically, this implies that carbon markets should be coupled more tightly with real-time dispatch signals to prevent market distortions where high renewable penetration might otherwise lead to negative pricing or excessive curtailment.

Second, the operational behavior of the hydrogen subsystem reinforces the necessity of sector coupling for deep decarbonization. Consistent with recent findings on hydrogen flexibility [24,25], our study confirms that electrolyzers serve as critical “shock absorbers” for renewable variability. However, unlike studies that treat hydrogen demand as fixed, our bilevel framework demonstrates that hydrogen production schedules must be price-responsive to maximize system benefits. For grid operators, this implies that incentives for hydrogen infrastructure should not solely focus on capacity expansion but also on the responsiveness of these assets to grid conditions and carbon intensity signals.

Finally, the superior performance of the Distributionally Robust Optimization (DRO) framework highlights a critical shift required in risk management strategies. While stochastic programming [11,12] improves upon deterministic approaches, our results show that relying on assumed probability distributions can underestimate operational risks under extreme weather events or behavioral shifts. The practical implication for system planners is that investment in robust optimization tools yields tangible economic returns—specifically, a risk reduction of 4% in our case study—by safeguarding against “black swan” events that standard stochastic models might miss. This validates the growing consensus in the literature [16,29] that resilience-oriented planning is essential for modern power networks facing increasing climate-induced uncertainty.

6. Discussion

The results presented in this study highlight the critical role of coordinating carbon pricing with operational flexibility in multi-energy systems. By endogenizing the carbon price within a bilevel framework, we demonstrated that regulatory signals must dynamically align with the technical capabilities of hydrogen conversion technologies to achieve cost-effective decarbonization. This finding extends existing literature [22,23] by quantifying how electrolyzer responsiveness directly mitigates the welfare losses typically associated with high carbon taxes. Furthermore, the superior performance of the Distributionally Robust Optimization (DRO) approach confirms that neglecting distributional shifts in renewable generation data can lead to underestimated operational risks, whereas the proposed method provides a verifiable hedge against such uncertainties. From a techno-economic perspective, the observed behavior of the hydrogen subsystem validates the concept of “sector-coupling as a flexibility service.” The ability of electrolyzers to ramp up during high-renewable, low-price periods effectively acts as a price floor, stabilizing the market while decarbonizing hard-to-abate sectors. This implies that future market designs should explicitly value the ramping and reserve services provided by power-to-gas facilities, rather than remunerating them solely for energy commodity production. Despite these contributions, several limitations of the current study must be acknowledged to provide a balanced perspective. First, the hydrogen network is modeled using steady-state energy balance equations. While this is standard for economic dispatch problems, it neglects transient fluid dynamics—such as pipeline linepack and pressure variations—which could physically constrain rapid flexibility services in real-world operations. Second, the assumption of a single central operator at the lower level simplifies the complex strategic interactions found in deregulated markets; in practice, independent system operators (ISOs) and hydrogen facility owners may have conflicting objective functions not fully captured by a unified cost-minimization model. Finally, although the Benders–CCG algorithm is tractable for the tested 33-bus system, the computational complexity of the DRO formulation scales non-linearly. This may pose challenges for real-time applications in large-scale transmission networks without further algorithmic acceleration or relaxation techniques.

7. Conclusions

This study presents an integrated bilevel optimization framework that explicitly couples carbon pricing mechanisms with the coordinated operation of power–hydrogen systems under distributional uncertainty. The proposed model establishes a hierarchical interaction between upper-level policy regulation and lower-level operational scheduling, ensuring that environmental objectives and economic efficiency are jointly optimized. By embedding a Wasserstein-based Distributionally Robust Optimization formulation into the lower-level dispatch model, the framework achieves resilience against renewable and demand uncertainties, while the upper-level market regulation dynamically adjusts carbon prices to maximize social welfare. The use of a Benders–Column-and-Constraint Generation algorithm ensures computational tractability and fast convergence, making the approach suitable for large-scale policy-driven energy systems.

The case study results demonstrate that coordinated carbon pricing substantially reshapes system operation toward a balanced carbon–economic frontier. At the optimal price level of approximately 45 $/tCO₂, the system achieves a 27% emission reduction at only a 9% cost increase, producing a welfare equilibrium that aligns with social and environmental goals. Hydrogen subsystems play a pivotal role by absorbing renewable surplus and supplying low-carbon flexibility, with electrolyzer and fuel-cell units dynamically switching between charge and discharge modes to smooth intertemporal power flows. The spatial emission maps further indicate that carbon price-induced dispatch reallocation reduces regional emission disparities by more than 20%, improving fairness and stability across network nodes.

Comparative results among deterministic, stochastic, and distributionally robust formulations confirm that the DRO-based model reduces operational risk exposure by around 4% while maintaining nearly identical expected costs. This validates the necessity of integrating robust uncertainty modeling in policy-sensitive energy markets. Additionally, convergence profiles of the Benders–CCG algorithm exhibit stable exponential decay of the duality gap, ensuring consistent optimization performance.

Overall, the proposed framework provides a quantitative pathway for policy-makers and system operators to co-design carbon pricing strategies and multi-energy operational plans. It transforms carbon pricing from a static regulatory parameter into a dynamic optimization variable that directly influences real-time cross-energy dispatch. The model’s flexibility allows extension to broader multi-vector energy systems—incorporating natural gas, district heating, and hydrogen blending—offering a foundation for future research on coupled market design and climate-policy integration. This work thus contributes both methodological innovation and policy insight, demonstrating that the integration of distributionally robust optimization, bilevel decision structures, and carbon-aware energy scheduling can serve as a cornerstone for achieving reliable and equitable low-carbon transitions. Based on the empirical results, this study offers concrete policy recommendations to guide low-carbon energy transitions. First, regulators are encouraged to transition from static carbon taxes to dynamic pricing mechanisms that reflect real-time marginal abatement costs, thereby incentivizing hydrogen production during periods of high renewable curtailment. Second, market designs should introduce distinct capacity payments for hydrogen storage and conversion facilities, recognizing their value in providing long-term flexibility and resilience against extreme weather events. Finally, policy frameworks should explicitly mandate distributionally robust planning standards for new infrastructure investments to safeguard against the increasing statistical ambiguity of climate-driven demand patterns. Despite the contributions presented, this study is subject to certain limitations that merit further investigation. First, the current formulation models the hydrogen network using steady-state flow approximations; while computationally efficient, this approach neglects transient dynamics such as linepack storage and pressure fluctuations that may impact sub-hourly flexibility. Second, the bilevel framework assumes a single dominant market operator, which may simplify the strategic gaming behaviors present in deregulated, multi-agent markets. Future research directions will focus on two main areas: (1) integrating dynamic fluid-dynamic constraints into the hydrogen subsystem to better capture pipeline inertia and storage transients; and (2) extending the bilevel model to a multi-regional decentralized framework using algorithms such as Alternating Direction Method of Multipliers (ADMM) to address privacy concerns and scalability in larger interconnected networks.