⚡ electrical engineering

Chance-Constrained Energy Storage Pricing for Social Welfare Maximization

This paper proposes a novel two-stage, opportunity-cost-constrained framework for energy storage pricing to maximize social welfare, theoretically demonstrating that storage opportunity costs are convex and coupled with future prices, while simulation results on an ISO-NE test system show that this approach significantly reduces electricity payments and system costs compared to profit-maximizing bidding models.

Original authors: Ning Qi, Ningkun Zheng, Bolun Xu

Published 2026-05-08

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CC BY 4.0

Original authors: Ning Qi, Ningkun Zheng, Bolun Xu

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 the power grid as a vast, bustling city where electricity is the currency and "power plants" are the shops selling it. In this city, there are also massive batteries (energy storage) that can buy electricity when it is cheap and sell it when it is expensive.

The problem is that these battery owners act like shrewd traders. They try to predict the future to maximize their own profits. Sometimes their private guesses lead them to withhold electricity or bid in ways that make power more expensive for everyone else. The "city manager" (the grid operator) wants to keep prices fair and ensure the entire city runs smoothly, but it is difficult to know what the batteries should do if they were to help the city rather than just their own pockets.

This article proposes a new "rulebook" for pricing these batteries. Instead of allowing batteries to guess their own prices to maximize profit, the city manager uses a special mathematical tool to set a "standard price" that maximizes the well-being of the entire city.

Here is how the article's ideas work, broken down with simple analogies:

1. The "Weather Forecast" Problem (Uncertainty)

The power grid is unpredictable. Sometimes the sun shines brighter than expected (solar energy), and sometimes people consume more electricity than predicted.

The old way: Traditional power plants (like coal or gas) simply consider their fuel costs. They really do not care about the weather forecast when setting their prices.
The new way: The article argues that batteries are different. Since they can store energy, their value depends heavily on what might happen tomorrow. If the weather is very unpredictable, a battery becomes more valuable because it acts as a safety net. The article's new pricing model automatically adds a "safety fee" to the battery's price when the weather forecast is uncertain, thereby ensuring grid security.

2. The "Full Tank" Analogy (State of Charge)

Imagine a battery like the fuel tank of a car.

The insight: The article proves that the "fuller" the battery is, the less valuable it is to charge it again (it is difficult to fill a full tank). Conversely, the less valuable it is to sell the remaining gasoline, the emptier the tank is (you do not want to run out).
The result: The new pricing model generates a smooth curve where the price naturally drops as the battery becomes fuller. This prevents the battery from trying to charge when it is already full, which would be wasteful.

3. The "Price Cap" (Preventing Market Power)

Sometimes a large battery owner tries to act like a monopolist by withholding electricity to drive prices sky-high.

The solution: The article's model acts like a "speed limit" or a "price ceiling." It proves mathematically that the price a battery can charge is strictly tied to the current electricity price and the costs of providing backup power.
The metaphor: Imagine a taxi driver charging $1,000 for a short ride. The new rule says: "No, your price cannot be higher than the cost of gasoline plus a small fee for waiting." This prevents the battery from exploiting the system.

4. The "Group Discount" (Social Welfare)

The researchers tested this new rulebook on a simulated version of the New England power grid (a real test system).

The result: When they used this new "city-first" pricing instead of allowing batteries to chase their own profits:
- Everyone paid less: The average consumer electricity bill dropped by about 17.4%.
- The system cost less: The total costs for operating the grid decreased by 3.9%.
- The batteries still made a profit: Although the batteries earned slightly less profit than under greedy conditions, the savings for consumers were enormous. The article notes that the money saved by consumers is far greater than the small profit amount the batteries "lost."

Summary

In short, this article introduces a way for grid operators to set fair prices for batteries that account for uncertainty and prevent price gouging. It is like shifting from a "Wild West" market where the shrewdest traders win to a "team sport" where the rules are designed so that the entire team (the grid and its customers) wins, even if the star players (the batteries) have to play a bit more conservatively. The result is a cheaper, more reliable, and fairer electricity system.

Technical Summary: Chance-Constrained Energy Storage Pricing for Maximizing Social Welfare

Problem Statement
As storage capacity in power grids increases significantly (e.g., over 8.6 GW in CAISO), current market designs face a critical challenge: storage participants typically submit strategic bids based on private price forecasts to maximize their individual profits. This price-taking behavior aimed at profit maximization often conflicts with social welfare objectives and complicates the detection of market power, as it is difficult to distinguish between legitimate capacity withholding for future opportunities and anti-competitive behavior. Furthermore, existing stochastic market designs often fail to guarantee sufficient revenue adequacy for individual scenarios or provide clear standard pricing schemes. There is a lack of systematic methods for grid operators to generate standard storage bids or benchmarks that maximize social welfare while explicitly accounting for net-load uncertainty and the operator's risk preferences.

Methodology
The authors propose a novel framework based on a two-stage chance-constrained optimization model. In contrast to traditional approaches focusing on participant profit, this model is designed for the grid operator to determine storage opportunity prices prior to the main economic dispatch.

Formulation: The core model minimizes expected system operating costs (conventional generation and storage degradation) while satisfying power balance and storage dynamics. It integrates joint chance constraints to ensure that power generation, reserve allocation, and the storage state of charge (SoC) remain within physical limits with a specified confidence level ( $1-\epsilon$ ), thereby managing uncertainties in renewable generation and load demand.
Reformulation: To make the problem solvable, the authors derive deterministic reformulations:
- The expectation of the cost function is expanded using polynomial approximations (handling quadratic and super-quadratic generator cost functions).
- Joint chance constraints are decomposed using the Bonferroni approximation and further reformulated into deterministic constraints based on the inverse cumulative distribution function (CDF) of the net-load forecast error. This enables flexibility in handling various uncertainty distributions (Gaussian, versatile, or robust approximations under incomplete information).
Pricing Mechanism: The storage opportunity price is derived from the dual variables of the chance-constrained economic dispatch. This price serves as a "standard bid" for storage or as a benchmark for monitoring market power.

Main Contributions
The work provides three key contributions:

Chance-Constrained Opportunity Price Framework: A novel design deriving storage opportunity prices from a social welfare maximization objective. It systematically integrates system balance, renewable/load uncertainties, and the market operator's risk preference (via the confidence level $\epsilon$ ).
Theoretical Pricing Analysis: The authors establish rigorous theoretical properties of the proposed pricing mechanism:
- SoC Monotonicity: The storage opportunity cost function is proven to be convex, meaning the opportunity price decreases monotonically with increasing state of charge (SoC).
- Uncertainty Monotonicity: The opportunity price increases monotonically with net-load uncertainty, but only if the generator cost function is super-quadratic. For quadratic cost functions, the price is insensitive to uncertainty.
- Boundedness and Linear Coupling: The opportunity price is proven to be bounded and linearly coupled with future energy and reserve prices. This relationship holds regardless of whether the storage participates in the energy market, the reserve market, or both.
Simulation Validation: The framework is validated using an 8-zone ISO-NE test system, comparing the proposed social welfare maximization approach against a profit-maximization model for price-taking storage bids.

Results
Simulation results on the ISO-NE test system demonstrate the effectiveness of the proposed framework:

Cost Reduction: Compared to the profit-maximization baseline, the proposed mechanism reduces total system costs by an average of 3.9% and consumer electricity payments by 17.4%.
Profit versus Welfare Trade-off: While storage profit margins decrease (by an average of 25.4% in the specific comparison), this decline accounts for only a negligible portion (approximately 0.49%) of the total savings in electricity payments, indicating a net gain in social welfare.
Scalability: The benefits of the proposed mechanism scale with the capacity of both renewable energy and energy storage. Higher renewable penetration and larger storage capacities lead to greater reductions in system costs and payments.
Risk Sensitivity: The model adapts to varying levels of risk aversion; as risk aversion decreases (higher $\epsilon$ ), reserve and system costs decline, consistent with theoretical predictions.
Market Power Regulation: The derived opportunity prices are bounded by energy and reserve prices, providing grid operators with a mechanism to regulate storage market power by setting price floors and ceilings.

Significance
The work claims to fill a critical gap in market design by providing grid operators with a tool to generate standard bids or benchmarks for storage resources. By shifting the focus from individual profit maximization to social welfare maximization under uncertainty, the framework offers a robust method to:

Align storage participation with system reliability and economic efficiency.
Mitigate market power by bounding opportunity prices based on fundamental system variables (energy and reserve prices) rather than strategic speculation.
Provide a solvable, deterministic reformulation of complex stochastic problems suitable for practical implementation without disrupting the main market clearing process.

The authors conclude that given the higher integration of storage and renewable energy driven by decarbonization, such systematic, uncertainty-aware pricing mechanisms are essential for maintaining reliable and economical power system operations.

1. The "Weather Forecast" Problem (Uncertainty)

2. The "Full Tank" Analogy (State of Charge)

3. The "Price Cap" (Preventing Market Power)

4. The "Group Discount" (Social Welfare)

Summary

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