⚡ electrical engineering

Multi-Region Optimal Energy Storage Arbitrage

This paper proposes a computationally efficient mixed-integer linear programming model for optimizing grid-scale battery arbitrage across two interconnected electricity markets, demonstrating that cross-border participation can increase revenue by over 40% compared to single-market operation while accounting for interconnector losses, congestion, and battery cycle constraints.

Original authors: Md Umar Hashmi, Harsha Nagarajan, Dirk Van Hertem1

Published 2026-04-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Md Umar Hashmi, Harsha Nagarajan, Dirk Van Hertem1

Original paper licensed under CC BY 4.0 (http://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 you own a giant, high-tech water tank (a battery) sitting right on the border between two countries: Belgium and the UK. These two countries have their own separate electricity markets, meaning the price of electricity changes differently in each place, hour by hour.

Usually, people use these batteries to play a simple game: Buy low, sell high. You charge the battery when electricity is cheap in your local country and sell it when it's expensive.

But this paper asks a bigger question: What if you could play this game in both countries at the same time?

Here is the simple breakdown of what the researchers did, using some everyday analogies.

1. The "Two-Headed" Strategy

Imagine you are a trader with a magic wallet. In the past, you could only buy apples in Market A and sell them in Market A. But now, you have a bridge (an interconnector) connecting Market A and Market B.

The researchers realized that if you only look at Market A, you might miss a huge opportunity. Maybe electricity is cheap in Belgium but super cheap in the UK, and expensive in both places at different times. By connecting your battery to both markets, you can:

Charge your battery when it's cheap in Belgium.
Charge it when it's cheap in the UK.
Sell it when prices are high in either place.

The Result: The paper found that by playing in both markets, the battery made 40% more money than if it just stayed home and played in one market.

2. The "One-Way Street" Problem (The Tricky Part)

Here is the catch: Your battery is physically located on one side of the bridge. It can't be in two places at once.

If you try to charge from the UK side while simultaneously discharging to the Belgium side, you are essentially creating a loop where electricity flows in a circle, losing energy and money. You can't be "charging" and "discharging" at the exact same moment.

Mathematically, this is a nightmare to solve. It's like trying to tell a computer, "You can either go Left OR go Right, but never both." Most old math models would get confused or make mistakes here, especially when electricity prices go negative (which happens sometimes when there is too much wind or solar power).

The Solution: The authors created a new "traffic light" system for the math. They used a clever trick (called disjunctive linearization) to force the computer to make a clear choice: "At this specific hour, we are either charging from the left OR charging from the right, but never doing both." This turned a messy, unsolvable puzzle into a clean, fast calculation.

3. The "Toll Booth" and the "Traffic Jam"

The bridge between the two countries isn't infinite. It has a size limit (capacity).

The Toll: Using the bridge costs money (rent).
The Traffic Jam: Sometimes, the bridge is already full of other electricity (like wind power) trying to cross.

The researchers' model is smart enough to check the traffic. If the bridge is jammed, the battery knows it can't move as much power. If the "toll" is too high, it might decide the trip isn't worth it. They found that if the bridge is congested, the extra profit can drop by about 36%. It's a reminder that having a bridge is great, but you need to know if it's open for business.

4. The "Smart Filter" (Saving the Battery)

Batteries are expensive, and they wear out. Every time you charge and discharge them, they get a little older.

The researchers noticed that sometimes the battery would make a tiny profit (like 10 cents) by doing a quick charge/discharge cycle. But that tiny profit wasn't worth the "wear and tear" on the battery.

They introduced a "Pseudo-Efficiency" filter. Think of this as a strict manager who says, "We are only going to do the big, juicy trades. If the profit is too small, we skip it."

What happened? The total money made went down a bit, but the number of times the battery had to work dropped by 74%.
The Win: The battery lasted much longer, and the money made per cycle went up significantly. It's like a taxi driver who refuses short, low-paying trips so they can save gas and focus on the long, high-paying ones.

5. The Bottom Line

This paper is essentially a blueprint for how to make grid-scale batteries much more profitable.

Old Way: Play in one market, get stuck with local prices, and wear out the battery doing small, unprofitable trades.
New Way: Play in two markets simultaneously, use a smart math model to avoid mistakes, check the "bridge traffic," and skip the small trades to save the battery's health.

The Takeaway: By connecting batteries to multiple markets and using this new "smart" math, energy companies can squeeze significantly more value out of their expensive equipment, helping to make the green energy transition more financially viable.

1. Problem Statement

The increasing interconnection of power systems via AC and DC links allows energy storage units to access multiple electricity markets. However, most existing arbitrage models are limited to single-market participation, failing to capture the economic value and operational constraints of cross-border storage operations.

Key challenges addressed include:

Multi-Market Participation: How to optimize a single battery's operation across two distinct day-ahead markets (e.g., Belgium and UK) simultaneously.
Physical Constraints: Accounting for interconnector capacity limits, transmission losses, and converter inefficiencies.
Non-Convexity: Handling the mathematical challenge of preventing simultaneous charging and discharging (a non-convex constraint) while dealing with negative electricity prices, which render traditional linear programming (LP) formulations suboptimal or inapplicable.
Battery Degradation: Balancing revenue maximization with battery cycle life, particularly by avoiding low-value charging/discharging cycles.

2. Methodology

The authors propose a Mixed-Integer Linear Programming (MILP) formulation for Multi-Region Energy Arbitrage (MREA).

A. System Modeling

Configuration: A grid-scale battery is located at one end of an interconnector (Region A) but trades against prices in both Region A and Region B. A "pseudo-agent" concept is used to model trading at the remote end.
Cost Functions: The model incorporates:
- Buying ( $P^B$ ) and selling ( $P^S$ ) prices for both regions.
- Interconnector rent ( $\zeta$ ) and line/converter losses ( $\eta_{line}, \eta_{conv}$ ).
- Effective prices for Region B transactions are adjusted for losses and rent (e.g., $eP^{B,B}_i = (P^{B,B}_i + \zeta_i) / \eta_{line}$ ).
Battery Dynamics: Includes state-of-charge (SoC) limits, ramping constraints, and efficiency factors ( $\eta^*_{ch}, \eta^*_{dis}$ ).

B. Mathematical Reformulation (The Core Contribution)

The primary innovation is the exact reformulation of the non-convex constraint that prevents simultaneous charging and discharging across regions.

The Constraint: The battery must either charge or discharge in both markets simultaneously ( $x^A_i \cdot x^B_i \geq 0$ ). This bilinear term is non-convex.
Disjunctive Linearization: Instead of using weak relaxations (like McCormick), the authors introduce binary variables ( $z^{ch}_i, z^{dis}_i$ $z_{i}^{c h}, z_{i}^{d i s}$ ) to enforce a shared operating mode.
- If $z^{ch}_i=1$ , the battery charges in both regions.
- If $z^{dis}_i=1$ , the battery discharges in both regions.
- This creates a piecewise linear feasible region that is exactly representable in MILP.
Epigraph Transformation: The objective function (minimizing cost/maximizing profit) is transformed using epigraph variables to handle the piecewise linear cost functions, ensuring the problem remains solvable as an MILP.

C. Handling Constraints and Uncertainty

Interconnector Congestion: The model adjusts ramping limits based on the actual flow of the interconnector ( $L^{AB}_i$ ). If the interconnector is congested by other flows, the available capacity for the battery is reduced dynamically.
Pseudo-Efficiency ( $\eta_{pseudo}$ ): A tunable parameter is introduced to artificially increase the cost of low-margin cycles. This allows operators to discard unprofitable cycles, thereby reducing battery wear and increasing revenue per cycle.
Uncertainty: While the base model is deterministic (using known day-ahead prices), the authors outline extensions for Stochastic Programming (Sample Average Approximation) and Model Predictive Control (MPC) for real-time implementation.

3. Key Contributions

Optimal MREA Formulation: A novel MILP model enabling simultaneous participation in multiple energy markets with exact handling of the "no simultaneous charge/discharge" constraint via disjunctive binary modeling.
Techno-Economic Framework: A comprehensive evaluation framework that accounts for interconnector rent, losses, battery degradation (cycle life), and negative price scenarios.
Pseudo-Efficiency Mechanism: Introduction of a tunable efficiency term to optimize the trade-off between total revenue and battery cycle life, improving revenue per cycle.
Benchmarking: Rigorous comparison against classical LP, MILP, and heuristic "no-discharge" (noDis) models, specifically under negative price conditions where traditional models often fail.

4. Results and Case Studies

The model was validated using 8 years of historical day-ahead price data (2017–2024) for the NEMO Link interconnector between Belgium and the UK, simulating a 1 MW / 1 MWh battery.

Revenue Increase: Multi-region participation increased arbitrage revenue by >40% compared to single-market operation.
- Example (2022): MREA profit was €95,545 vs. ~€53,000 for single-market MILP.
Impact of Congestion: Interconnector congestion significantly impacts gains. When flow direction opposed the optimal arbitrage strategy, revenue decreased by up to 36%.
Battery Location: In the specific case study, locating the battery at the UK end yielded slightly higher profits than the Belgian end, though total 8-year profits were comparable.
Cycle Efficiency:
- Standard MREA increased total revenue but did not significantly improve revenue per cycle compared to single-market models.
- Applying the pseudo-efficiency ( $\eta_{pseudo} = 0.7$ ) reduced total cycles by 74% (from 6,281 to 1,593) while reducing total revenue by only 45%. This resulted in a significant increase in revenue per cycle, extending the asset's economic life.
Computational Performance: The MILP model is highly efficient. Annual simulations (8 years of data) completed in <30 seconds on a standard laptop, making it suitable for real-time or near-real-time decision-making.
Negative Prices: The proposed MILP formulation successfully handled negative electricity prices, outperforming the "noDis" heuristic which arbitrarily forbids discharging during negative price periods.

5. Significance and Conclusion

This paper provides a computationally efficient and mathematically rigorous framework for operating grid-scale storage in interconnected power systems.

Economic Viability: It demonstrates that cross-border market access is a critical value driver for storage assets, potentially doubling revenue potential in specific scenarios.
Operational Realism: By explicitly modeling interconnector congestion and losses, the study offers a realistic assessment of arbitrage potential, warning against overestimating gains when interconnectors are heavily utilized for other grid services.
Asset Management: The introduction of the pseudo-efficiency term offers a practical tool for asset owners to prioritize high-value cycles, thereby maximizing the return on investment per unit of battery degradation.

The authors conclude that while the current model focuses on two markets, the formulation is scalable to $N$ regions. Future work will explore stochastic extensions to handle price uncertainty and co-optimization with ancillary services (e.g., frequency regulation).