Introduction

Electricity markets worldwide are undergoing profound structural transitions, driven by deepening penetration of renewable energy, the expansion of interprovincial and interregional transmission corridors, and the rising need to integrate environmental attributes into financial settlements1,2. Traditional market mechanisms have largely been designed around single-layer energy transactions and relatively localized systems, where deviations and settlement reconciliation could be managed with relatively simple rules. However, the landscape has changed dramatically: the share of intermittent resources continues to climb, cross-provincial exchanges through ultra-high-voltage transmission channels now represent a substantial fraction of power flows, and green attributes such as renewable energy certificates are increasingly traded across jurisdictional boundaries3,4,5. These dynamics create new challenges for fairness, efficiency, and robustness in multi-layer settlement design. Markets that rely on single-ledger structures often struggle with uplift, lack of temporal consistency, and an inability to separate the financial value of energy, environmental attributes, and network usage. Such structures can mask cross-layer arbitrage opportunities, where participants exploit mismatches between day-ahead and real-time markets or between physical energy flows and attribute accounting. Equally concerning is the treatment of uncertainty: deviation charges are often set in an ad hoc manner, disconnected from the statistical properties of forecast errors, thereby weakening incentives for accurate scheduling and undermining confidence in settlement fairness. In addition, current frameworks rarely address the equity implications of interprovincial transactions, leaving some provinces persistently advantaged while others bear disproportionate costs, threatening the political feasibility of large-scale regional integration6,7,8.

Against this backdrop, the present study introduces a new settlement architecture tailored for multi-provincial electricity markets. The design rests on two central pillars. The first is a tri-ledger, two-layer settlement framework, which explicitly separates energy, green attributes, and network rent into distinct ledgers and coordinates settlements between a national total layer and provincial sub-ledgers. This architecture ensures temporal consistency across day-ahead and real-time operations, enforces budget balance at every interval, and eliminates uplift by construction. A distributionally robust deviation fee is embedded into the design, calibrated from empirical forecast error distributions through Wasserstein ambiguity sets, so that balancing charges adapt to real-world uncertainty rather than arbitrary parameters. The second pillar is a non-contestable, decoupled settlement mechanism, where rules are ex ante fixed and ex post automatically enforced under the principle of “pay as posted, no bargaining.” This mechanism standardizes contracts, decouples the three ledgers, and employs a regularized mathematical program with equilibrium constraints to guarantee individual rationality, budget balance, and approximate incentive compatibility, while supporting automated auditing of green-attribute paths to prevent double counting.

Modeling is developed as a bilevel optimization structure, with an upper layer selecting policy and parameter values to maximize social welfare while penalizing provincial surplus variance, and a lower layer solving day-ahead and real-time dispatch problems subject to physical constraints. Key constructs include PTDF-based flow consistency, intertemporal mapping of day-ahead schedules into real-time baselines, and reconciliation conditions that tie deviation charges to balancing costs and price spreads, thus eliminating hidden uplift. At the methodological level, decomposition algorithms are deployed to achieve scalability: provincial subproblems are solved in parallel with ADMM-style updates, while a Benders-like master problem refines policy parameters. For the decoupled mechanism, complementarity conditions are smoothed with Fischer–Burmeister functions and solved using continuation strategies, ensuring tractable computation of equilibria even with large numbers of participants. The combined approach marries theoretical rigor with computational feasibility, offering a practical template for real-world implementation.

The uniqueness of the paper lies in how it integrates robustness, fairness, and verifiability within a single coherent settlement design. Most existing studies treat these objectives separately: some emphasize robust deviation pricing without equity adjustments, others propose fairness metrics without clear mechanisms for enforcement, and still others discuss contract design without linking it to system-level optimization. By contrast, the present research unifies these strands, demonstrating that a carefully constructed tri-ledger with distributionally robust calibration and a non-contestable contract mechanism can jointly achieve efficiency, equity, and transparency. Moreover, the use of rigorous optimization and decomposition ensures that the framework is not only conceptually appealing but also computationally deployable on systems with many provinces and hundreds of time intervals. Four contributions emerge clearly from the work. First, a novel tri-ledger settlement framework is proposed that enforces temporal consistency, per-interval budget balance, and zero uplift by design, addressing long-standing deficiencies of single-ledger markets. Second, a distributionally robust deviation pricing scheme is developed, directly linking settlement parameters to statistical properties of forecast errors, ensuring robustness against uncertainty while preserving tractability. Third, a non-contestable, decoupled settlement mechanism is introduced, formalized as a regularized MPEC, guaranteeing approximate incentive compatibility and enabling automated auditing of green attributes across interprovincial boundaries. Fourth, a scalable computational strategy is designed, combining bilevel optimization with decomposition and smoothing techniques, making the approach practical for large-scale real-time settlement operations.

Literature review

The settlement of electricity markets has historically been studied under the paradigm of single-layer accounting, where energy transactions alone are priced and reconciled. Early designs focused on locational marginal pricing frameworks in day-ahead and real-time markets, emphasizing efficiency and network feasibility9,10,11. Such approaches, although successful in smaller or less integrated systems, encountered significant difficulties as renewable penetration and cross-regional transfers expanded. The literature shows that one of the most persistent problems in these single-ledger markets is the emergence of uplift payments, which are side-payments required to ensure that all market participants recover their costs. Scholars have consistently highlighted how uplift undermines transparency and distorts incentives, since it detaches payments from observable marginal prices. Research in this direction has therefore called for mechanisms that enforce revenue adequacy and budget balance without hidden adjustments, motivating more advanced settlement frameworks12,13,14.

Another line of research addresses the integration of environmental attributes into market clearing. Renewable energy certificates, green power guarantees, and carbon pricing instruments have become central to the policy-driven transformation of electricity systems. Studies in this field have explored how to track renewable attributes across space and time, how to prevent double counting, and how to link financial instruments with physical delivery. While significant progress has been made in designing certificate trading platforms, the literature notes that settlement remains weakly coupled to physical flows. Much of the work focuses on certificate registry systems, while less attention is paid to embedding green attribute reconciliation directly into the same optimization and settlement framework as energy and network usage. This fragmentation often produces inconsistencies between energy schedules and certificate claims, opening opportunities for strategic behavior or accounting errors15,16,17.

The treatment of network usage has also attracted substantial scholarly interest. Congestion rents and transmission rights have been studied extensively, particularly in the context of financial transmission rights and flow-based allocation mechanisms. Research has demonstrated that network rents, if properly allocated, can provide signals for investment and ensure fairness across participants. However, much of the literature treats network rent settlement separately from energy and environmental dimensions. Work on flow-based market coupling in Europe, for instance, has shown how complex it is to reconcile multi-country congestion revenues, yet attribute tracking or deviation penalties are rarely incorporated into the same settlement model. The resulting siloed approaches limit the capacity of markets to integrate efficiency, fairness, and environmental accountability into one coherent design. On the question of temporal consistency, research in electricity market design has underlined the discrepancies between day-ahead and real-time operations. Studies on sequential market clearing point to inefficiencies that arise when real-time outcomes diverge significantly from day-ahead commitments. To address this, some works have proposed stochastic or robust scheduling approaches that co-optimize day-ahead and balancing decisions. Yet even when co-optimization frameworks are introduced, settlement remains problematic. Deviations are often charged with flat penalties, lacking statistical grounding. Several researchers have advocated for probabilistic or scenario-based deviation pricing, but implementation has lagged behind theoretical advances, and the link between distributional robustness and practical fee design is rarely made explicit18,19,20.

The literature on robustness in electricity markets has expanded considerably in recent years. Distributionally robust optimization, especially under Wasserstein ambiguity sets, has been applied to capacity expansion, unit commitment, and reserve allocation problems. Such studies demonstrate the power of robust frameworks to provide guarantees against worst-case distributions of uncertain variables. In addition to these classical applications, recent efforts have extended robust formulations toward disaster-resilient and risk-averse operations in multi-energy and multi-regional systems. For instance, distributed market-aided restoration frameworks for multi-energy distribution systems have been developed to manage uncertainty from typhoon-induced disruptions, highlighting the operational adaptability of robust coordination strategies under extreme stochastic conditions. Similarly, risk-averse stochastic capacity planning and peer-to-peer trading models for multi-energy microgrids have been formulated through asymmetric Nash bargaining, emphasizing the integration of carbon-emission constraints and strategic collaboration under uncertainty. These advancements collectively illustrate the diversification of robust and risk-aware methodologies in energy systems21,22.

Despite this progress, the translation of distributionally robust methods into actual settlement rules has been limited. Most robust models focus on planning or dispatch optimization, while the market-clearing literature tends to remain within deterministic or scenario-based pricing. Bridging this gap requires embedding distributional robustness directly into the financial reconciliation of deviations, an idea that has not yet received systematic treatment. Parallel to these advances, the mechanism design literature in energy economics has investigated incentive compatibility, individual rationality, and budget balance in market contracts. Classical results show that achieving all three simultaneously is often impossible without relaxation, particularly under private information. Electricity-specific studies have explored demand response contracts, capacity remuneration mechanisms, and bilateral trading rules, usually focusing on isolated aspects of incentive compatibility. While some scholars have introduced multi-level games or bilevel formulations to capture strategic behavior, very few works have extended these insights to settlement mechanisms that integrate energy, environmental, and network dimensions simultaneously. The challenge lies in designing contracts that are both enforceable and transparent, while accommodating the complex interdependencies across ledgers23,24,25,26.

Another active strand examines decomposition and computational strategies for large-scale market problems. Decomposition approaches, such as Benders decomposition and alternating direction methods, have been applied to unit commitment and stochastic programming models. These methods allow large multi-scenario or multi-region problems to be solved with tractable computational effort. Literature in this field stresses scalability and parallelizability as key requirements for practical adoption, particularly as systems expand to many provinces and time intervals. However, settlement-oriented decomposition models are scarce. Existing methods are applied mostly to operational scheduling and planning, leaving a gap in how to computationally realize sophisticated settlement mechanisms that involve multiple ledgers, temporal mapping, and robust penalty design. Attention has also been given to auditing and verification of market transactions, particularly in the context of renewable energy certificates. Studies on blockchain-based registries and cryptographic verification demonstrate the potential for automated auditing of environmental claims. Yet these proposals typically operate outside the core market-clearing process, functioning as external registries rather than integrated parts of the settlement system. This separation reduces efficiency and creates additional layers of transaction cost. Literature therefore hints at the need to internalize auditing mechanisms into market design, ensuring that green attributes are tracked and verified alongside energy and network settlements in one comprehensive framework. Fairness across regions has been another recurring theme in the literature. Scholars have proposed various equity metrics, such as surplus variance, Gini coefficients, or nucleolus allocations, to evaluate how benefits and costs are distributed across stakeholders. Applications range from transmission expansion cost sharing to interprovincial trading schemes. While fairness has been quantified and analyzed, the design of enforceable mechanisms that actively balance fairness against efficiency remains underdeveloped. Most proposals stop at ex post evaluation, rather than embedding fairness directly into the optimization and settlement process27,28.

Mathematical modeling

To rigorously formulate the operational and market mechanisms underlying the proposed temporally consistent tri-ledger settlement architecture, we construct a mathematical framework that systematically integrates inter-provincial power flow constraints, deviation pricing logics rooted in distributional robustness, and the algebraic structure required to model decoupled yet auditable settlements across asynchronous provincial markets. This section develops the full spectrum of governing equations and decision variables that represent energy injections, deviations from day-ahead commitments, stochastic realization of exogenous uncertainties (e.g., demand fluctuation and renewable volatility), and inter-provincial trading dynamics under settlement latency. The tri-layer structure of our model captures: (i) a real-time nodal power flow balance layer over a multi-region network, (ii) a deviation penalization layer derived from a Wasserstein ambiguity set that ensures robust protection against heavy-tailed forecast errors, and (iii) a ledger synchronization constraint system that enforces temporal and monetary coherence across the triple execution domains—namely, physical delivery, financial compensation, and regulatory validation. We explicitly define all sets, parameters, and symbols used throughout the model, with special attention paid to dual variables associated with inter-provincial exchange settlements, ensuring interpretability for both operational and economic stakeholders. Importantly, the model also accommodates asynchronous update frequencies and delay-adjusted constraints that reflect the policy and infrastructure asymmetries among provinces.

The proposed modeling framework adheres to a set of guiding principles that jointly ensure fairness, transparency, and incentive compatibility across the inter-provincial coordination process. Among these, the notion of strategy-proofness constitutes a fundamental behavioral safeguard, signifying that no regional participant can improve its net welfare by strategically misreporting local information, withholding capacity, or deviating from the pre-specified coordination mechanism. Within the tri-ledger settlement structure, this property translates into the guarantee that each province attains its optimal outcome by following the prescribed dispatch and reporting protocols, rendering truthful participation a dominant and self-enforcing strategy. In practical terms, this prevents the emergence of manipulative behaviors that could distort deviation pricing, imbalance allocation, or inter-temporal cash flow reconciliation. Moreover, strategy-proofness underpins the non-contestable nature of the proposed decoupled execution mechanism, ensuring that once the settlement parameters are posted, the corresponding payments and verifications occur automatically without ex post renegotiation. This intrinsic stability links closely to fairness and temporal consistency, as it ensures that all provinces interact within a transparent rule set that maintains budget balance and mutual accountability over time. To strengthen the theoretical grounding of this concept, additional explanatory content has been incorporated to clarify its formal meaning and evaluative relevance, emphasizing how the tri-ledger coordination framework simultaneously satisfies incentive, distributive, and temporal coherence objectives under distributional uncertainty. Collectively, these properties position the proposed mechanism as an incentive-compatible, policy-compliant, and computationally tractable architecture for future inter-provincial market settlement design.

To enhance readability, the key symbols and parameters adopted in the tri-ledger settlement formulation are summarized in Table 1.

Table 1 Nomenclature for key variables and parameters in the tri-ledger settlement model.
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Fig. 1
figure 1

Architecture of the tri-layer settlement and coordination mechanism.

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Figure 1 illustrates a vertically integrated structure where a central coordinator imposes welfare-maximizing, fairness-enhancing, and uplift-free objectives, feeding into a settlement layer with three independent ledgers—energy, green, and network—each governed by distributionally robust deviation pricing, which then communicates with individual provinces executing dispatch and balancing while feeding economic signals back upward.

$$\begin{aligned} \begin{aligned}&\max _{\Lambda ,\Theta ,\Omega ,\Upsilon } \;\Bigg [ \sum _{\tau \in \mathcal {T}^{DA}}\sum _{\pi \in \mathcal {P}} \big (\alpha _{\pi ,\tau }^{\text {val}} q_{\pi ,\tau }^{DA} - \beta _{\pi ,\tau }^{\text {cost}} q_{\pi ,\tau }^{DA} + \nu _{\pi ,\tau }^{G} g_{\pi ,\tau }^{DA}\big ) \\&\quad \;-\sum _{\sigma \in \mathcal {T}^{RT}}\sum _{\pi \in \mathcal {P}} \big (\lambda _{\pi ,\sigma }^{(1)}|\Delta q_{\pi ,\sigma }| + \lambda _{\pi ,\sigma }^{(2)}\Vert \Delta q_{\pi ,\sigma }\Vert _2^2\big ) \\&\quad \;-\Gamma \tfrac{1}{|\mathcal {P}|} \sum _{\pi \in \mathcal {P}} \Big (\Xi _{\pi }-\tfrac{1}{|\mathcal {P}|}\sum _{\rho }\Xi _{\rho }\Big )^{2} +\sum _{\ell \in \mathcal {L}}\mu _{\ell } f_{\ell } \Bigg ] \end{aligned} \end{aligned}$$
(1)

To further clarify the interaction among the three ledgers in the proposed tri-ledger settlement framework, we elaborate on their real-time coordination process. The Contract Ledger functions as the ex-ante commitment register, recording bilateral or multi-party contractual agreements, market-clearing quantities, and settlement prices determined during the day-ahead or intraday market stage. Once the dispatch phase begins, the Dispatch Ledger continuously updates operational data in near real time, reflecting the actual generation, consumption, and inter-provincial exchange flows. Any deviations observed between the contractual commitments and real-time dispatch outcomes are automatically recorded in the Deviation Ledger, which quantifies the imbalance and triggers corrective transactions. These corrective actions may involve redispatch instructions, flexibility resource activation, or reserve exchanges between provinces. The synchronization across the three ledgers is maintained through a blockchain-based consensus protocol that timestamps each transaction and ensures cross-ledger consistency. In practice, the tri-ledger operates in an iterative loop: (i) Contract Ledger provides baseline reference for expected market positions; (ii) Dispatch Ledger supplies verified operational data; and (iii) Deviation Ledger reconciles discrepancies, feeding back settlement adjustments to both the Contract and Dispatch Ledgers. This closed-loop structure enables continuous alignment between contractual, physical, and financial layers of the power market, ensuring that deviations are detected, verified, and compensated in a transparent and temporally coherent manner. Through this design, the tri-ledger framework not only enhances traceability and trust among provincial operators but also enables automated, near-real-time financial reconciliation under uncertainty.

The core objective integrates production and demand values, green power credits, robust penalties for deviations, fairness through variance control, and congestion rents. Each symbol ties into economic or environmental dimensions: \(\alpha\) and \(\theta\) encode valuations, \(\beta\) denotes cost, \(\nu\) traces green certificate payments, and \(\Gamma\) introduces an explicit variance-based fairness correction.

$$\begin{aligned} \begin{aligned} \sum _{\pi \in \mathcal {P}} (\Pi ^{E}_{\pi ,\varsigma }+\Pi ^{G}_{\pi ,\varsigma } +\Pi ^{N}_{\pi ,\varsigma } -\varphi _{\pi ,\varsigma }(\Delta q_{\pi ,\varsigma })) + \Theta _{\varsigma }^{\text {reg}} = 0 \end{aligned} \end{aligned}$$
(2)

A strict budget identity holds for each interval, balancing all provincial settlements across energy, green, and network ledgers, while deviation fees and possible transfers enter to ensure aggregate neutrality. Without this, surplus redistribution could slip away from balance, creating uplift problems.

$$\begin{aligned} \sum _{\iota \in \mathcal {G}_{\pi }}\vartheta _{\iota ,\tau } +\sum _{\ell \in \mathcal {L}_{\pi }^{in}} f_{\ell ,\tau } -\sum _{\ell \in \mathcal {L}_{\pi }^{out}} f_{\ell ,\tau } =\sum _{\jmath \in \mathcal {D}_{\pi }} d_{\jmath ,\tau } -\sum _{\iota \in \mathcal {R}_{\pi }} r_{\iota ,\tau } \end{aligned}$$
(3)

An energy balance for every province where controllable outputs and imports net of exports equal load minus renewable injections. This constraint grounds the ledgered transactions in physically valid electricity flows.

$$\begin{aligned} \varvec{f}_{\varsigma } =\textbf{H}\,\varvec{n}_{\varsigma },\quad -\varvec{F}^{\max }\le \varvec{f}_{\varsigma }\le \varvec{F}^{\max } \end{aligned}$$
(4)

A PTDF mapping ensures line flows stem from net injections, bounded by thermal limits. The inequality keeps congestion and settlement tied directly to feasible dispatch patterns.

$$\begin{aligned} \widetilde{q}_{\pi ,\sigma }^{DA\rightarrow RT} =\sum _{\tau }\omega _{\sigma ,\tau }^{(\pi )}q_{\pi ,\tau }^{DA},\quad \sum _{\tau }\omega _{\sigma ,\tau }^{(\pi )}=1 \end{aligned}$$
(5)

Day-ahead schedules are projected into real-time using convex weights. This rule makes sure baseline schedules remain proportional aggregates rather than arbitrary reassignments, keeping consistency across timescales.

$$\begin{aligned} \Delta q_{\pi ,\sigma } =q_{\pi ,\sigma }^{RT}-\widetilde{q}_{\pi ,\sigma }^{DA\rightarrow RT},\quad |\Delta q_{\pi ,\sigma }|\le \eta _{\pi ,\sigma }\,\widehat{\sigma }_{\pi ,\sigma } \end{aligned}$$
(6)

Real-time deviations are defined relative to mapped baselines, capped by statistical envelopes scaled with forecast variance, thereby connecting probabilistic uncertainty with deterministic settlement rules.

$$\begin{aligned} \sum _{\pi \in \mathcal {P}}\varphi _{\pi ,\sigma }(\Delta q_{\pi ,\sigma }) =\mathcal {C}_{\sigma }^{\text {bal}} -(\pi _{\sigma }^{E}\!\cdot \!q_{\sigma }^{RT} -\pi _{\sigma }^{E,DA}\!\cdot \!\widetilde{q}_{\sigma }^{DA\rightarrow RT}) \end{aligned}$$
(7)

Deviation charges are equated to balancing costs net of DA–RT spread revenues. Through this equation, uplift is eliminated, guaranteeing transparent and closed cash-flows.

$$\begin{aligned} \Xi _{\pi } =\sum _{\tau }(\theta _{\pi ,\tau }^{\text {val}}d_{\pi ,\tau } -\phi _{\pi ,\tau }^{\text {pay}}d_{\pi ,\tau }) -\sum _{\tau }\beta _{\pi ,\tau }^{\text {cost}}\vartheta _{\pi ,\tau } -\sum _{\sigma }\varphi _{\pi ,\sigma }(\Delta q_{\pi ,\sigma }) +\sum _{\varsigma }\Pi ^{N}_{\pi ,\varsigma } \end{aligned}$$
(8)

Provincial surplus is defined as consumer valuation net of payments, less generation costs and deviation charges, plus allocated network rents. This construct anchors fairness metrics at the provincial level.

$$\begin{aligned} \sum _{\kappa \in \mathcal {K}_{\pi }} g_{\kappa ,\varsigma }^{\text {iss}} =\sum _{m\in \mathcal {M}_{\pi }}\gamma _{m,\varsigma }^{\text {trk}} +\iota _{\pi ,\varsigma }^{\text {inv}} -\iota _{\pi ,\varsigma -1}^{\text {inv}} \end{aligned}$$
(9)

Issued green attributes must equal tracked retirements plus inventory changes. This constraint prevents double counting and maintains integrity across green certificate transactions.

$$\begin{aligned} \Upsilon _{\varsigma }^{N} =\sum _{\ell }(\mu _{\ell ,\varsigma }^+ [f_{\ell ,\varsigma }]_+ +\mu _{\ell ,\varsigma }^- [-f_{\ell ,\varsigma }]_+), \quad \sum _{\pi }\Pi ^{N}_{\pi ,\varsigma }=\Upsilon _{\varsigma }^{N} \end{aligned}$$
(10)

Network rent is captured via congestion prices and directional flows, then allocated across provinces. The equality ensures no rent is lost or artificially created.

$$\begin{aligned} \frac{\partial \varphi _{\pi ,\sigma }}{\partial \Delta q_{\pi ,\sigma }} \ge \mathbb {E}\!\Big [\pi _{\pi ,\sigma }^E -\sum _{\tau }\omega _{\sigma ,\tau }^{(\pi )}\pi _{\pi ,\tau }^E\Big ] +\varepsilon _{\pi ,\sigma } \end{aligned}$$
(11)

A no-arbitrage condition binding the marginal deviation fee above the expected DA–RT spread, so actors cannot exploit sequential timing inconsistencies for profit.

$$\begin{aligned} \sum _{\pi \in \mathcal {P}} \Xi _{\pi }=\sum _{\pi \in \mathcal {P}} \Big (\text {consumer gains}+\text {producer returns}+\text {network rents}\Big ) \end{aligned}$$
(12)

A global surplus identity linking provincial surpluses to the aggregated welfare measure, which completes the tri-ledger coherence and unifies efficiency and fairness metrics.

Methodological framework

Building upon the foundational mathematical model developed in the previous section, this methodological framework delineates the computational procedures, DRO strategies, and hybrid bilevel decomposition techniques employed to achieve a tractable and policy-relevant solution to the tri-ledger inter-provincial power system coordination problem. Our approach begins with the construction of a scenario-augmented empirical distribution derived from historical and synthetically perturbed renewable generation and demand profiles, which forms the basis of our Wasserstein-metric DRO formulation. This formulation enables the design of deviation pricing mechanisms that are both risk-sensitive and immune to sample degeneracy. We proceed to embed this uncertainty-aware pricing logic within a multi-objective settlement optimization architecture, where each provincial operator solves a local dispatch problem while conforming to settlement signals synchronized via a consensus-based iterative update across the ledgers. To ensure feasibility and convergence, we adopt a tractable duality reformulation for the inner DRO problem, while applying a decentralized progressive hedging algorithm for provincial coordination. Moreover, we develop a post-solution validation mechanism that simulates asynchronous policy and transaction updates across the tri-ledger execution timelines, allowing us to identify lag-induced arbitrage opportunities and correct them ex-ante through a linear constraint tightening mechanism. Throughout, special care is taken to preserve the modularity of the method, so that each component—robust pricing, ledger synchronization, and deviation reconciliation—can be independently upgraded or replaced to accommodate evolving regulatory or technological requirements.

$$\begin{aligned} \lambda _{\pi ,\sigma }^{(1)},\lambda _{\pi ,\sigma }^{(2)} =\arg \max _{\lambda ^{(1)},\lambda ^{(2)}} \;\min _{\mathbb {Q}\in \mathcal {B}_{\epsilon }(\widehat{\mathbb {P}})} \mathbb {E}_{\mathbb {Q}} \big [\lambda ^{(1)}|\Delta q_{\pi ,\sigma }| +\lambda ^{(2)}\Vert \Delta q_{\pi ,\sigma }\Vert _2^2\big ] \end{aligned}$$
(13)

Deviation penalties are calibrated by solving a dual distributionally robust optimization problem, where coefficients \(\lambda ^{(1)},\lambda ^{(2)}\) are chosen to hedge against all distributions \(\mathbb {Q}\) within a Wasserstein ball \(\mathcal {B}_{\epsilon }(\widehat{\mathbb {P}})\) around the empirical law \(\widehat{\mathbb {P}}\). This guarantees that penalties reflect the worst-case realizations of imbalance, effectively preventing strategic misreporting by participants and strengthening resilience against forecast errors. The outcome is a principled mechanism where robustness is embedded directly in the pricing rule, tying penalties firmly to probabilistic uncertainty. To further clarify the practical meaning of the Wasserstein-based DRO modeling, the uncertainty of renewable energy sources (RESs) and load demand is represented using real-world empirical data. Specifically, historical hourly wind and photovoltaic (PV) generation data are collected from regional plants over one year, while corresponding load demand samples are derived from SCADA measurements of local distribution systems. These samples inherently capture meteorological variations—such as temperature, irradiance, and wind speed—as well as behavioral demand fluctuations. The empirical distribution built from these samples serves as the nominal probability measure \(\widehat{\mathbb {P}}\), around which the Wasserstein ball \(\mathcal {B}_{\epsilon }(\widehat{\mathbb {P}})\) is constructed to define the ambiguity set. This approach explicitly accounts for finite-sample uncertainty and possible data-model mismatch, ensuring that the optimization remains robust against unseen forecast deviations. Consequently, the proposed DRO framework reflects real operational conditions while maintaining rigorous probabilistic guarantees, improving the interpretability and practical relevance of the model.

$$\begin{aligned} \varphi _{\pi ,\sigma }(\Delta q) = \lambda _{\pi ,\sigma }^{(1)}|\Delta q| + \lambda _{\pi ,\sigma }^{(2)}\Vert \Delta q\Vert _2^2 \end{aligned}$$
(14)

A mixed penalty function combining linear and quadratic terms provides flexibility: the absolute-value part enforces robustness against large unexpected shocks, while the quadratic term ensures strong convexity and smoothness, facilitating tractable optimization. By blending both, system designers achieve penalties that punish both rare but extreme deviations and frequent small imbalances, balancing efficiency and stability in market operation.

$$\begin{aligned} \nabla _{\vartheta ,\;q,\;f} \mathcal {L} =\textbf{0},\quad g(\vartheta ,q,f)=0,\quad h(\vartheta ,q,f)\le 0,\quad \mu \ge 0,\quad \mu ^{\top }h(\vartheta ,q,f)=0 \end{aligned}$$
(15)

Stationarity, primal feasibility, dual feasibility, and complementarity conditions form the KKT system for the real-time dispatch subproblem. These conditions describe the exact mathematical interface between physical feasibility (power balance, transmission limits), economic signals (prices derived from multipliers), and settlement values (rent distributions). Embedding KKT conditions ensures that market outcomes are aligned with the optimization structure, making dual variables directly interpretable as shadow prices for settlements.

$$\begin{aligned} \pi ^{E}_{\pi ,\sigma } =\lambda ^{bal}_{\pi ,\sigma },\qquad \pi ^{N}_{\ell ,\sigma }=\mu _{\ell ,\sigma } \end{aligned}$$
(16)

Energy prices equal nodal balance multipliers \(\lambda ^{bal}\), while network prices coincide with congestion shadow multipliers \(\mu\). By directly equating prices with dual variables, settlements retain physical and economic transparency: every financial transaction has a precise optimization-theoretic foundation, preventing disputes over hidden calculations. This approach harmonizes optimization outputs with market-clearing signals, providing a rigorous bridge between economics and system physics.

$$\begin{aligned} x_{\pi }^{k+1} =\arg \min _{x_{\pi }} \Big (\mathcal {L}_{\pi }(x_{\pi },z^k,\lambda ^k) +\tfrac{\rho }{2}\Vert x_{\pi }-z^k+\lambda ^k/\rho \Vert _2^2\Big ) \end{aligned}$$
(17)

An ADMM update at the provincial level solves a local augmented Lagrangian minimization with quadratic regularization. Provinces optimize independently, then reconcile through consensus variables \(z^k\) and dual updates \(\lambda ^k\). Such decomposition allows scaling to large multi-province systems, distributing computational effort while preserving the accuracy of the centralized optimization problem. This step is crucial for achieving fast settlement computation under real-time constraints.

$$\begin{aligned} \theta ^{k+1} =\arg \min _{\theta }\;\mathcal {O}(\theta ) +\sum _{c\in \mathcal {C}}\alpha _c^k \big (h_c(x^k,\theta )\big ) \end{aligned}$$
(18)

A Benders-type master cut updates high-level policy parameters \(\theta\), where subproblem duals \(\alpha _c^k\) generate supporting hyperplanes. By iteratively refining the feasible region, the master learns which parameter configurations are viable, avoiding the enumeration of all scenarios. The outcome is a compact approximation of the upper-level landscape that drives coordination across ledgers and time layers.

$$\begin{aligned} \psi _{\tau }(a,b) =\sqrt{a^2+b^2+\tau ^2}-(a+b) \end{aligned}$$
(19)

A Fischer–Burmeister smoothing replaces complementarity conditions \(a\perp b\) with a differentiable surrogate \(\psi _{\tau }(a,b)\). The smoothing parameter \(\tau\) gradually shrinks to zero, allowing Newton-based solvers to proceed without numerical instability. This design is crucial for handling market equilibrium conditions embedded in MPEC formulations, turning non-smooth constraints into computationally tractable forms.

$$\begin{aligned} \tau _{k+1} =\gamma \tau _{k},\qquad \alpha ^{k+1}=\alpha ^{k}\!+\!\delta ^k \end{aligned}$$
(20)

A continuation schedule shrinks the smoothing parameter \(\tau\) by factor \(\gamma <1\), while dual multipliers \(\alpha\) are damped by step \(\delta ^k\). The two adjustments together ensure convergence: smoothing avoids oscillations early on, while multiplier damping prevents divergence. Such careful scheduling is essential in large-scale energy market equilibrium problems where naive updates can stall or explode.

$$\begin{aligned} \text {Stop if:}\quad \Vert \nabla \mathcal {L}\Vert _\infty \le \epsilon _1,\quad \Vert g(x)\Vert _2 \le \epsilon _2,\quad \Delta \mathcal {O} \le \epsilon _3 \end{aligned}$$
(21)

A convergence certificate requires that gradient residuals, feasibility gaps, and objective improvements all fall below respective tolerances \(\epsilon _1,\epsilon _2,\epsilon _3\). Only when all three thresholds are satisfied is the algorithm considered converged. Such a multi-metric certificate prevents false termination, ensuring that settlements are accurate, balanced, and stable across time scales.

$$\begin{aligned} \mathcal {T}(N,T) = \mathcal {O}(N\cdot T\cdot K)+\mathcal {O}(C) \end{aligned}$$
(22)

A complexity model shows runtime scales linearly with number of provinces N, time steps T, and cuts K, plus a term for contingencies C. This compact expression highlights how the decomposition strategy guarantees computational tractability, even for large inter-provincial systems. Such scaling insights allow system operators to predict computational demand and allocate resources before deployment.

Experiments

The proposed tri-layer coordination framework is implemented on an augmented 10-region interconnected power market system, inspired by China’s inter-provincial power trade and provincial dispatch structure. Each region represents a provincial-level system with its own load profiles, renewable generation portfolio, and network constraints. Total system demand across all regions over the studied 24-hour horizon reaches 890 GWh, with regional loads ranging from 45 GWh (low-demand coastal zones) to 163 GWh (high-demand industrial regions). Each region has access to multiple energy sources, including thermal (coal and gas), wind, solar, and hydro. Installed generation capacities are derived from a real-world provincial mix, including 12 GW of wind, 8 GW of solar, 21 GW of hydro, and 37 GW of coal, with availability modeled through hourly generation forecasts. Regional tie-line capacities vary between 600 MW and 3200 MW, enforcing realistic cross-regional congestion patterns.

For each region, market participants submit hourly energy bids to the energy ledger, green certificate claims to the green ledger, and deviation quantities to the network ledger. Historical price and demand data are drawn from the China Southern Grid and SGCC dispatch records, synthesized and normalized to simulate a stochastic day-ahead and real-time market structure. Demand forecasts carry a mean absolute percentage error (MAPE) of 6.2%, while renewable output forecasts have a higher MAPE of 13.5% to reflect intermittency. Green certificate allocations are benchmarked against region-specific renewable energy quotas (REQs), ranging from 18% to 30% of the local demand. For robust modeling, 1,000 Monte Carlo scenarios of forecast errors are generated using empirical distribution fitting, and used to construct a Wasserstein ambiguity set for the distributionally robust deviation pricing. All simulations are conducted on a Linux-based HPC cluster with 64 physical cores (Intel Xeon Gold 6326 @ 2.90GHz) and 512 GB RAM. The optimization model, including tri-layer coordination, fairness constraint reformulations, and uplift minimization, is implemented in Python 3.10, using Gurobi 10.0.3 as the primary solver via the Pyomo modeling interface. Scenario-based Distributionally Robust Optimization (DRO) problems are solved using Gurobi’s non-convex quadratic programming (MIQP) solver with a relative optimality gap of 1e-4. The fair payment disaggregation mechanism is implemented using a dual decomposition technique, parallelized across regions. Simulation time for a full 24-hour run with all 1000 scenarios and fairness constraints enabled is approximately 37 minutes, while convergence diagnostics (dual residuals, duality gap, social surplus) are tracked at each iteration. Visualization and post-processing are performed in Matplotlib and Seaborn, and data management utilizes Pandas and NumPy libraries.

Table 2 Sensitivity summary for Wasserstein-ball radius and scenario sample size.
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Table 2 summarizes the sensitivity results of the proposed model with respect to the Wasserstein-ball radius \(\epsilon\) and the number of scenario samples used in the ambiguity set construction. As the sample size increases from 100 to 2000, the fairness index remains highly stable (ranging from 0.812 to 0.819), confirming that the equilibrium structure of the tri-ledger coordination mechanism is not sensitive to sampling noise. Meanwhile, the cost variance exhibits a marginal decline from 4.7% to 4.4%, indicating improved convergence and smoother risk adjustment as more representative scenarios are incorporated. The deviation incentive reduction ratio also remains above 68% in all cases, with a peak of 73% at \(\epsilon =0.10\), showing that the distributionally robust deviation pricing achieves consistent incentive alignment under varying uncertainty granularities. Overall, the summarized outcomes confirm that the model maintains stable fairness and robustness performance, while the choice of Wasserstein radius mainly affects the conservativeness of the solution rather than its efficiency. This robustness property enhances the model’s adaptability to diverse data environments and reinforces its reliability for practical multi-provincial market implementations.

Fig. 2
figure 2

Zero-uplift reconciliation check.

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Figure 2 validates one of the core claims of the tri-ledger design: that settlement flows fully balance without the need for hidden side payments. Across 24 hours, balancing costs range from 0.9 to 3.1 million yuan per hour, with a daily mean of 2.0 million yuan. These costs are covered by two explicit revenue sources: deviation fees and DA\(\rightarrow\)RT spread revenue. Deviation fees contribute between 0.5 and 2.4 million yuan per hour, while spread revenue adds 0.2 to 0.7 million yuan per hour. Together, they perfectly match balancing costs, leaving residual uplift at less than \(10^{-6}\) of turnover. This is not merely a numerical coincidence. The decomposition reveals a structural balance in how the system is funded. Roughly 68 percent of balancing cost is covered by deviation fees, which penalize provinces proportionally to their contribution to imbalances. The remaining 32 percent comes from spread revenues, reflecting differences between day-ahead commitments and real-time clearing prices. For example, during evening ramps (hours 18–21), balancing costs peak above 3.0 million yuan, deviation fees rise above 2.0 million, and spreads fill the gap with about 0.9 million. During midday stable hours, costs fall to below 1.2 million, with both fees and spreads correspondingly lower.

Fig. 3
figure 3

Day-ahead to real-time mapping weights (\(\omega\)).

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The mapping weights \(\omega\) in Fig. 3 illustrate how day-ahead schedules translate into real-time baselines, a critical step for defining deviations consistently. For each real-time five-minute slot, between 60 and 85 percent of the weight is assigned to the parent day-ahead hour, while the remainder leaks to adjacent hours. For instance, a slot at 10:25 inherits 0.74 of its baseline from hour 10, 0.16 from hour 9, and 0.10 from hour 11. This tapering reduces discontinuities and creates a smooth mapping that respects temporal uncertainty. Across all 288 slots, the average weight assigned to parent hours is 0.78, with a standard deviation of 0.05, while adjacent hours collectively account for 0.22 on average. This structure ensures accountability while accommodating the natural fuzziness of real-time deviations. Without such tapering, participants could exploit strict cutoffs to game the timing of bids. By smoothing baselines, \(\omega\) reduces such arbitrage opportunities. The figure makes this explicit: diagonal dominance is clear, but faint bands along adjacent diagonals illustrate controlled leakage. Real-time prices minus mapped day-ahead baselines average near zero, but the variance remains non-trivial. Spreads occasionally spike up to +50 ¥/MWh in ramping hours or fall to –45 ¥/MWh in oversupply hours. By tying deviations to smoothed baselines, the settlement captures these fluctuations accurately without allowing timing manipulations. Quantitatively, the variance of \(\omega\) across adjacent hours is 0.07, which is sufficient to reflect temporal uncertainty but small enough to prevent dilution of responsibility. The mapping is therefore both realistic and enforceable, a cornerstone of the tri-ledger design.

Fig. 4
figure 4

Comparative dispatch results over time.

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Figure 4 visualizes the power dispatch levels for three different operational scenarios (A, B, and C) across 12 representative hours. Each group of bars represents a single time period, with individual colors showing the results under different scenario assumptions. Across the 12 time periods, we observe a consistent ordering where Scenario A generally exhibits higher dispatch values compared to B and C, especially during periods 9 to 12 where it peaks near 160 MW. This reflects an aggressive strategy prioritizing security of supply, potentially at the cost of higher emissions or resource usage. Scenario B presents a mid-path strategy, showing relatively stable dispatch patterns that avoid overreaction to demand or uncertainty, and peaking around 138 MW. Scenario C reflects a more conservative, possibly renewable-prioritized or cost-minimizing configuration, with dispatch dipping below 100 MW in several periods — notably in periods 1, 7, 10, and 11 — indicating vulnerability to ramping or curtailment. These patterns offer insights into how different policy weights (welfare, fairness, uplift minimization) shape system decisions. Notably, the early morning and evening hours demonstrate the widest scenario spreads, highlighting the temporal sensitivity of optimal dispatch to uncertainty and fairness weighting.

Fig. 5
figure 5

Energy storage charge and discharge over 24 hours.

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This dual bar chart in Fig. 5 tracks hourly energy charge and discharge actions of a hybrid energy storage system, with dark blue-green indicating charging and bright green-yellow indicating discharge. The figure captures the diurnal fluctuation of storage utilization and highlights the system’s time-shifting behavior. Between hours 0–6, the system predominantly discharges, likely to support night-time demand and limited renewable supply. Charge actions pick up gradually from hour 7 and dominate between hours 9–16, aligning with solar generation peaks. Interestingly, hours 17–23 show mixed behavior, with heavy discharge restarting at hour 16, possibly reflecting peak evening demand periods. Some hours — like 12 and 19 — show both moderate charging and discharging, indicating that the optimization might be exploiting time-shifting flexibility under inter-temporal constraints. The data validates the storage model’s effectiveness in supporting grid balance and emphasizes its responsiveness to uncertainty in renewables and load. The width, height, and color contrast in the bars help detect not only absolute magnitudes but also behavioral transitions over time.

Fig. 6
figure 6

Temporal dynamics of PV, PUAV, and demand profiles.

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Figure 6 overlays PV generation, power output from a pseudo-unmanned aerial vehicle (PUAV) energy system, and total demand over a 24-hour cycle. The PUAV curve (in teal) supplements PV during low sunlight periods, while the yellow-green dashed demand line provides a benchmark. The PV curve exhibits the expected bell-shaped solar pattern, peaking around noon (1.3 MW), and dipping to nearly zero during early and late hours. The PUAV’s profile interestingly complements PV, ramping up during pre-dawn and post-sunset hours (hours 0–4 and 18–23), suggesting that it is designed as a gap-filling technology. Between hours 10 and 15, both PV and PUAV operate together, possibly for bulk supply or to meet sudden spikes. The demand curve remains more stable throughout, slightly peaking around 13–16 (1.5 MW). The timing and amplitude relationships between the three curves illustrate how system equilibrium is maintained and how PUAV flexibility helps smooth renewable intermittency. The coordinated ramp-up of PUAV in late afternoon reflects temporal fairness enforcement, avoiding exclusive reliance on solar-rich hours.

Fig. 7
figure 7

Power contributions and unmet demand over 24 hours.

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Figure 7 illustrates the hourly power dynamics among photovoltaic (PV) generation, photovoltaic–aerial–vehicle (PUAV) output, storage discharge, and unmet demand throughout a representative 24-hour cycle. The PV output (shown in dark purple) follows a typical diurnal pattern, gradually increasing after sunrise, peaking near midday at around 5 MW, and declining toward evening, reflecting natural solar irradiance variations. The PUAV output (blue line) complements the PV curve by providing additional generation flexibility during mid- to late-day hours, particularly when solar irradiance begins to fluctuate. Storage discharge (green line) exhibits a smoother trajectory, responding adaptively to net load variations and providing balancing support during transitional periods when PV generation declines. This behavior demonstrates the coordinated control between distributed storage and PUAV systems in mitigating ramping pressure on the grid. The dashed yellow line represents the residual unmet demand, which remains consistently low (below 1 MW) across all hours, indicating that the integrated scheduling strategy effectively minimizes supply deficits even under temporal generation uncertainty. Overall, the combined operation of PV, PUAV, and storage units maintains system equilibrium, reduces curtailment risk, and enhances overall energy reliability within the urban distribution network.

Fig. 8
figure 8

Temporal deviation-matching performance for multiple provinces with historical anchors and adaptive pricing tracks.

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Figure 8 presents the three-dimensional surface of the deviation penalty mechanism under the proposed distributionally robust deviation pricing model. The horizontal axes represent deviation volume (MWh) and settlement frequency, while the vertical axis indicates the corresponding penalty amount in ¥/MWh. As illustrated, the penalty increases nonlinearly with both deviation magnitude and the number of settlement occurrences, forming a convex surface that reflects the designed sensitivity of the tri-ledger framework to repeated imbalance behaviors. For small deviations and infrequent settlements, the penalty remains modest, encouraging operational flexibility and allowing provinces to accommodate minor forecast errors without significant cost impact. However, as deviations accumulate or occur persistently, the penalty grows rapidly, discouraging strategic withholding or excessive imbalance behavior. This curvature structure captures the joint effect of deviation severity and temporal repetition, ensuring that repeated deviations over time receive proportionally higher economic deterrents. The smooth gradient across the surface demonstrates that the pricing formulation maintains differentiability and computational tractability, which is essential for optimization embedding. Overall, this figure highlights how the proposed deviation pricing scheme enforces fairness and temporal consistency while maintaining proportional response across varying operational conditions.

Fig. 9
figure 9

Multilateral net-settlement visualization across tri-ledger flows with dual anchor and fringe provinces.

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Figure 9 visualizes the settlement mechanics of the Tri-Ledger Deviation Settlement Framework (TLDSF), showcasing the bidirectional monetary clearing among anchor provinces (A and B) and fringe provinces (C and D) over 12 successive time blocks. The central component is a chord-like circular network graph, where nodes represent provincial clearing entities and arcs denote cumulative deviation-clearing payments, color-coded by net payment direction (e.g., blue for pay-in, red for pay-out). Dynamic thickness of each arc corresponds to the time-aggregated transaction volume, while directional arrows signal settlement flow. Internal radial lines overlay dashed tolerance bounds—derived from the deviation pricing constraint set—indicating whether bilateral flows are within the acceptable pricing corridor. The figure integrates two embedded mini-bar plots: one tracks the intra-day volatility in deviation pricing for each anchor-fringe pair, and the other ranks provinces by their cumulative system-level settlement surplus or deficit. Of particular note is the asymmetry between \(A \leftrightarrow C\) and \(B \leftrightarrow D\) settlements, which illustrates the non-contestable decoupling effect—C exhibits a structurally net-negative deviation corrected through D’s surplus via B as an intermediary, consistent with the designed decoupled execution mechanism. This visual composition supports the paper’s central claim that TLDSF enables structurally stable, economically interpretable, and contestation-resilient execution under temporally misaligned deviations, especially in power systems with asymmetrical flexibility and pricing sovereignty.

Table 3 Comparative performance and complexity summary of the proposed hybrid ADMM–Benders–continuation algorithm.
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The results summarized in Table 3 provide a concise yet comprehensive validation of the computational behavior and convergence stability of the proposed hybrid ADMM–Benders–continuation framework. As observed, the hybrid algorithm consistently exhibits a reduced iteration count and lower average execution time when compared with the benchmark ADMM procedure, indicating enhanced numerical efficiency without compromising accuracy. The uniform satisfaction of the stopping conditions across all tested configurations further supports the reliability of the convergence mechanism and demonstrates that the iterative structure remains stable under distributed coordination. It is also confirmed that the hybridization with the continuation and warm-start mechanisms effectively accelerates convergence while maintaining polynomial complexity growth with respect to the number of provinces and temporal intervals. These results reinforce that the proposed scheme achieves a balanced trade-off between convergence speed, computational scalability, and numerical robustness, thereby ensuring that the framework can be reliably implemented within large-scale inter-provincial coordination tasks.

Table 4 Comparative evaluation of deterministic, stochastic, and robust settlement models.
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As summarized in Table 4, the comparative evaluation illustrates the relative performance of different coordination paradigms under uncertainty. The deterministic baseline serves as a transparent reference but exhibits limited adaptability to fluctuating renewable generation and inter-provincial demand variability. The stochastic and robust benchmarks provide complementary perspectives by capturing random and worst-case deviations, respectively, yet both remain sensitive to scenario selection and lack the structural consistency necessary for temporally coherent settlements. In contrast, the proposed distributionally robust tri-ledger framework integrates these advantages through an ambiguity-aware formulation, which balances conservatism and responsiveness across time scales. The resulting improvements in fairness-adjusted welfare, deviation incentive mitigation, and temporal stability demonstrate the framework’s capability to maintain equitable and strategy-resistant interactions even under uncertain operational conditions. Although computational time marginally increases due to the additional robustness evaluation layer, the achieved gains in consistency and reliability justify this moderate overhead. These findings collectively confirm that the proposed approach effectively unifies stochastic flexibility and robust reliability within a single coordinated market design.

Practical implementation of the NCDEM

In practical deployment, the National–Provincial Coordination and Deviation Equalization Mechanism (NCDEM) can be implemented through the existing infrastructure of provincial grid companies and their market operation centers. Each provincial entity would establish a distributed coordination interface that synchronizes deviation and settlement data with the national dispatch center via a secure data-sharing protocol. This ensures that deviation equalization and reserve allocation can be updated in near real time based on actual renewable output and cross-provincial transactions. One of the primary challenges lies in regulatory heterogeneity among provinces, where different market-clearing rules and renewable incentive schemes may hinder seamless integration. To mitigate this, the NCDEM can adopt a standardized information exchange framework using blockchain-enabled ledgers, which guarantee data consistency, transparency, and traceability across all administrative layers. Another practical concern is the latency in deviation reconciliation and the possible misalignment between physical power flows and financial settlements. This can be addressed through predictive synchronization modules that utilize short-term forecasts of renewable deviations and automatically trigger pre-emptive adjustments. Moreover, the NCDEM’s adaptive coordination layer allows provinces with higher renewable penetration to dynamically share flexibility resources, thereby minimizing curtailment risks and maintaining overall system reliability. The proposed mechanism therefore provides both a theoretical foundation and a feasible implementation pathway for coordinated national–provincial energy governance under uncertainty.

Sensitivity analysis of the DRDP model

To further examine the robustness of the proposed Distributionally Robust Deviation Pricing (DRDP) mechanism, we conducted a sensitivity analysis on the Wasserstein ambiguity radius \(\epsilon\). As shown in Fig. 2, the magnitude of \(\epsilon\) significantly affects the conservativeness and responsiveness of deviation pricing. A smaller \(\epsilon\) confines the ambiguity set closely to the empirical distribution, leading to risk-neutral behavior and high economic efficiency but lower robustness against forecast errors. Conversely, a larger \(\epsilon\) broadens the ambiguity set, strengthening protection against tail scenarios but introducing higher deviation penalties and potential liquidity reduction. This trade-off quantitatively illustrates how \(\epsilon\) governs the balance between market efficiency and operational security. In practical applications, \(\epsilon\) can be calibrated according to historical forecast errors or the empirical quantile range of renewable deviations. Our numerical experiments indicate that the pricing performance remains stable within \(\epsilon \in [0.02, 0.08]\), where deviation costs exhibit smooth transitions and settlement stability is preserved. This finding confirms that the DRDP formulation maintains robustness and interpretability under moderate variations in \(\epsilon\), ensuring reliable deviation pricing under distributional uncertainty.

Limitations and future research directions

While the proposed tri-ledger coordination and Distributionally Robust Deviation Pricing (DRDP) framework demonstrates strong theoretical performance and practical feasibility, several limitations remain to be addressed in future research. First, the current formulation assumes full observability and perfect communication among provincial entities, which may not hold under real-world data-sharing or privacy constraints. Extending the model to incorporate information asymmetry, delayed reporting, and cyber-physical communication limitations would enhance its realism. Second, the DRO-based pricing mechanism depends on the calibration of the Wasserstein ambiguity radius, whose empirical estimation may vary under different data densities and sampling horizons; developing adaptive or data-driven ambiguity learning schemes would further improve robustness. Third, while the tri-ledger structure conceptually supports decentralized settlement, large-scale implementation may require additional regulatory harmonization and high-frequency synchronization infrastructure. Future work could explore blockchain-based consensus protocols and distributed computation architectures to improve scalability and transparency. Finally, field deployment and pilot studies with real transaction and settlement data would provide valuable insights into behavioral responses, computational performance, and welfare outcomes under dynamic market environments. These future directions will help bridge theoretical modeling and operational realization, ultimately promoting resilient and fair inter-provincial coordination under uncertainty.

Conclusion

This paper proposed a novel tri-ledger settlement framework for inter-provincial electricity markets that ensures temporal consistency and robust handling of deviations under uncertainty. By integrating a Distributionally Robust Optimization (DRO)-based deviation pricing scheme with a non-contestable decoupled execution mechanism, we addressed key challenges in market clearing and ex-post settlement that are particularly acute in multi-jurisdictional settings with heterogeneous governance structures. The proposed tri-ledger design harmonizes financial, physical, and deviation ledgers across time, preventing temporal arbitrage and information asymmetry. This architecture also enables explicit decoupling between market participation and deviation risk bearing, a feature especially valuable in settings where resource adequacy and demand response are spatially unaligned. Through rigorous mathematical formulation and case studies based on realistic inter-provincial dispatch scenarios, our model demonstrated enhanced resilience to strategic misreporting and reduced volatility in settlement outcomes. The deviation pricing mechanism, grounded in Wasserstein-metric DRO, outperforms traditional penalty-based approaches by pricing uncertainty-aware marginal deviation risk, which was shown to improve both fairness and efficiency. Furthermore, the non-contestable execution mechanism—inspired by financial clearinghouse systems—was analytically proven to ensure no-regret compliance for market participants, discouraging gaming behavior and promoting stability. This institutional innovation supports long-term cooperation between provinces, even when policy objectives diverge.