⚡ electrical engineering

Multi-Microgrid Electricity–Heat Energy Sharing Based on Spatio-Temporal Distributionally Robust Optimization and Benefit Allocation

This paper proposes a distributed electricity–heat energy sharing framework for multi-microgrids that integrates spatio-temporally correlated distributionally robust optimization to handle source–load uncertainties and an asymmetric Nash bargaining mechanism to ensure fair benefit allocation, ultimately achieving low-cost, low-carbon, and robust coordinated operation.

Original authors: Yuntao Yue, CuiPing Yang, Dong Liu

Published 2026-06-25

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CC BY 4.0

Original authors: Yuntao Yue, CuiPing Yang, Dong Liu

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a neighborhood where every house has its own solar panels, wind turbines, and batteries. Sometimes, House A has too much wind power at night, while House B is struggling to keep the lights on. Other times, House C has extra heat from its furnace that it doesn't need.

In the past, these houses might have just bought expensive power from the main grid or wasted their extra energy. This paper proposes a smarter way for these "microgrids" (the houses) to work together as a team. It's like organizing a neighborhood potluck where everyone brings what they have, shares it, and splits the savings fairly.

Here is how the paper breaks down this solution, using simple analogies:

1. The Problem: The "Weather Guessing Game"

The main challenge is that the sun doesn't always shine, and the wind doesn't always blow. Predicting exactly how much energy a house will generate is like trying to guess the exact number of raindrops in a storm.

Old ways: Some methods were too pessimistic (buying too much backup power just in case), while others were too optimistic (getting caught with no power when the wind stopped).
The Paper's Solution: The authors created a new "weather guessing" tool called WDRCC. Think of this as a super-smart map that doesn't just look at the weather in one spot, but understands how the wind in one neighborhood affects the wind in the next, and how the weather today relates to tomorrow. It uses a mathematical "rubber band" (Wasserstein distance) to stretch and shrink its predictions based on how much data it has, ensuring the neighborhood is safe from surprises without wasting money.

2. The Low-Carbon Upgrade: The "Recycling Factory"

One of the houses (Microgrid 2) gets a special makeover to become "green."

The Setup: It installs a system that captures the "smoke" (carbon dioxide) from its power generator. Instead of letting it pollute the air, it turns that smoke into fuel.
The Magic Trick: It uses electricity to split water into hydrogen, then mixes that hydrogen with the captured smoke to create methane (natural gas).
The Result: This house turns its own waste into fuel, which it can use or sell to neighbors. It's like a house that eats its own trash and turns it into a pizza.

3. The Teamwork: The "Potluck Bargain"

Now that the houses can share electricity and heat, they need to decide who gets what and who pays whom.

The Negotiation: They use a method called Asymmetric Nash Bargaining. Imagine a group of friends splitting a bill. Instead of just splitting it 50/50, they look at who brought the most food, who drove the car, and who set up the chairs.
The Fairness Factor: The paper introduces a "Contribution Score." If House A brings a lot of extra electricity, House B provides heat, and House C helps stabilize the grid when things get shaky, they all get a share of the savings proportional to their help. This ensures no one feels like they are being taken advantage of.

4. The Privacy Guard: The "Blindfolded Dance"

To make this work, the houses need to talk to each other to coordinate, but they don't want to reveal their secret recipes or private financial data.

The Solution: They use an algorithm called ADMM. Imagine a group of people dancing in a circle, blindfolded. They can feel the person next to them and adjust their steps to stay in sync, but they never need to see the other person's face or know their personal secrets. This allows the whole neighborhood to optimize its energy use without anyone spying on the others' private data.

5. The Results: A Win-Win-Win

When the authors tested this system with three different "houses" (one with wind, one with solar, and one with the recycling factory):

Money Saved: The total cost for the neighborhood went down. By sharing energy and using the "recycling factory," they bought less expensive power from the main grid.
Less Pollution: Carbon emissions dropped significantly because the recycling factory turned waste into fuel.
Fairness: The house that contributed the most (the one with the recycling factory and the most flexible resources) got the biggest share of the savings, while everyone still made more money than if they had acted alone.

In a Nutshell

This paper presents a blueprint for a neighborhood of energy producers to act like a single, efficient team. It uses a smart "weather map" to handle uncertainty, a "recycling factory" to turn pollution into fuel, and a "fairness calculator" to ensure everyone gets paid for their help, all while keeping everyone's private secrets safe. The result is a cleaner, cheaper, and fairer way to power our future.

Technical Summary: Multi-Microgrid Electricity–Heat Energy Sharing Based on Spatio-Temporal Distributionally Robust Optimization and Benefit Allocation

Problem Statement
The coordinated dispatch of multi-microgrid (MMG) systems faces significant challenges arising from the coupling of low-carbon operational requirements and the uncertainty of renewable source–load predictions. Existing approaches often fail to simultaneously address three critical gaps:

Uncertainty Modeling: Traditional methods (interval, stochastic, or robust optimization) either rely on manually defined intervals, depend heavily on specific distributional assumptions, or yield overly conservative worst-case results. Furthermore, they frequently neglect the spatio-temporal correlations of prediction errors across multiple microgrids and time periods, potentially underestimating system risks.
Low-Carbon Integration: While low-carbon technologies like Carbon Capture and Storage (CCS) and Power-to-Gas (P2G) exist, there is a lack of unified frameworks that integrate these technologies into a coordinated multi-energy (electricity–heat–gas) dispatch model that leverages inter-microgrid cooperation for emission reduction.
Benefit Allocation: Existing profit-sharing mechanisms, such as the Shapley value or symmetric Nash bargaining, often struggle to balance computational efficiency with fairness, particularly when contributions are multi-dimensional (e.g., energy sharing, carbon reduction, and flexibility) and asymmetric.

Methodology
The paper proposes a comprehensive framework integrating spatio-temporally correlated distributionally robust optimization with coalition game theory. The methodology is structured into three main components:

Low-Carbon Multi-Energy Microgrid Model:
The authors construct a model for individual microgrids incorporating CCS, P2G, hydrogen storage, and methanation. This setup allows for the capture of $CO_2$ from Combined Heat and Power (CHP) units and its subsequent conversion into methane via P2G, creating an "electricity–gas–carbon" coupling pathway. The model minimizes operating costs, including fuel, maintenance, and carbon trading costs.
Spatio-Temporal Correlated Wasserstein Distributionally Robust Chance-Constrained (WDRCC) Model:
To address source–load uncertainty, the paper develops a WDRCC model. Unlike standard models, this approach:
- Constructs an ambiguity set centered on the empirical distribution of historical forecast errors using a Wasserstein ball.
- Introduces a covariance-weighted Wasserstein distance metric. This metric incorporates spatio-temporal statistical characteristics (covariance matrices) into the distance calculation, allowing the model to account for the anisotropy of uncertainty (i.e., stronger correlations in specific directions).
- Formulates a two-stage optimization problem: a day-ahead deterministic stage and an intraday rescheduling stage. The objective minimizes the sum of day-ahead costs and the worst-case expected intraday adjustment costs within the ambiguity set.
- Linearizes the problem using duality theory and auxiliary variables to ensure tractability.
Asymmetric Nash Bargaining for Benefit Allocation:
The coordinated operation is modeled as a coalition game.
- Cost Optimization: The Alternating Direction Method of Multipliers (ADMM) is employed to solve the alliance-wide cost minimization problem in a distributed manner, preserving the privacy of individual microgrids by exchanging only boundary variables (energy exchange quantities and dual variables).
- Revenue Allocation: An asymmetric Nash bargaining model is formulated. Unlike symmetric models, this approach weights the bargaining power of each microgrid based on a multi-dimensional contribution evaluation system. This system quantifies contributions across four dimensions: energy sharing, carbon emission reduction, regulation capability (flexibility), and power balance support (reliability). The final profit allocation is proportional to these comprehensive contribution indices.

Key Contributions

Covariance-Weighted WDRCC: The development of a distributionally robust model that explicitly integrates spatio-temporal statistical dependencies into the Wasserstein distance metric. This improves the representation of uncertainty anisotropy and enables a tunable trade-off between economic performance and robustness without assuming a specific probability distribution.
Integrated Low-Carbon MMG Model: The formulation of a multi-microgrid model that extends low-carbon retrofitting (CCS and P2G) from a single-unit perspective to a coordinated multi-microgrid level, facilitating $CO_2$ capture and resource utilization across the alliance.
Contribution-Based Asymmetric Bargaining: The establishment of an energy-sharing architecture that decomposes the problem into cost optimization and revenue allocation. The proposed asymmetric Nash bargaining mechanism ensures that profit distribution aligns with multi-dimensional contributions, addressing limitations of single-dimension allocation methods.

Results
Case studies involving three microgrids (two wind-based, one PV-based, with one undergoing low-carbon retrofit) demonstrate the following:

Low-Carbon Retrofit Effectiveness: The retrofit of Microgrid 2 reduced actual carbon emissions by approximately 28% and total operating costs by 2.63% compared to pre-retrofit scenarios, validating the economic viability of CCS-P2G integration.
Algorithm Convergence: The ADMM algorithm converged efficiently (within 73 iterations for the main problem), confirming the computational stability of the distributed solution.
Operational Synergy: The results show complementary energy exchange patterns. For instance, wind-rich microgrids supply power during low-demand periods, while PV-rich microgrids support others during peak daylight hours. Thermal energy sharing further enhances flexibility.
Fair Benefit Distribution: The proposed allocation mechanism increased the total alliance revenue by over 127,000 CNY. Microgrids with higher comprehensive contributions (e.g., MG2 with high carbon reduction and flexibility) received larger shares of the redistributed benefits, demonstrating the mechanism's ability to balance fairness and efficiency.
Model Comparison: Compared to Deterministic (DDRCC), Stochastic (SDRCC), and Moment-based (MDRCC) models, the proposed WDRCC model achieved the lowest total operating cost. It successfully balanced the high conservatism of MDRCC (which incurred high day-ahead costs) and the high rescheduling/penalty costs of DDRCC.
Sensitivity Analysis: The study confirmed that the WDRCC model's performance is sensitive to the confidence coefficient and sample size, allowing operators to adjust the robustness-cost trade-off. Furthermore, neglecting spatio-temporal correlations leads to underestimation of uncertainty risks and higher operational costs.

Significance and Claims
The paper claims that its proposed framework provides a unified solution for the coordinated, low-carbon, and fair operation of multi-microgrid systems. By integrating spatio-temporal uncertainty modeling with a contribution-based bargaining mechanism, the approach addresses the dual challenges of source–load uncertainty and equitable benefit distribution. The authors assert that the method reduces operating costs and carbon emissions while maintaining system robustness and fairness.

The paper modestly notes that while the proposed method effectively incorporates spatio-temporal dependence through covariance-weighted metrics, it does not explicitly construct a joint spatio-temporal ambiguity set. The authors suggest that future work could explore more rigorous modeling techniques, such as vine Copula-based dependence modeling or conditional Wasserstein ambiguity sets, to further refine the handling of coupled spatio-temporal uncertainty.