📈 economics

Locational Energy Storage Bid Bounds for Facilitating Social Welfare Convergence

This paper proposes a novel, uncertainty-aware method for generating locational energy storage bid bounds that effectively align market bids with social welfare objectives, reducing system costs and increasing storage profits while mitigating market power risks.

Original authors: Ning Qi, Bolun Xu

Published 2026-05-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ning Qi, 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 electricity grid as a giant, bustling marketplace where energy is bought and sold every second. In this market, battery storage is a new kind of merchant. Unlike a coal or gas plant that sells energy based on how much fuel it costs to burn, a battery merchant is playing a complex game of "buy low, sell high" across time. They want to charge when electricity is cheap and discharge when it's expensive.

The problem is that because the future is uncertain (the sun might not shine, or a storm might hit), these battery merchants often play it too safe—or too greedy. They might hold back their energy (withholding) hoping for a massive price spike later, or they might bid wildly high prices just in case. This behavior can make the whole market inefficient and expensive for everyone.

This paper proposes a new set of "Price Caps" (or bid bounds) for these batteries. Think of these caps as a referee's rulebook that says, "You can bid as high as you want, but not higher than this specific limit based on the current weather, your battery's charge level, and how risky the market looks."

Here is a breakdown of how this works, using simple analogies:

1. The "Crystal Ball" Problem

In a perfect world, the market operator would have a crystal ball. They would know exactly how much wind and solar power will be generated tomorrow and exactly how much electricity people will use. With this perfect knowledge, they could tell the batteries exactly how much to charge and discharge to keep costs low. This is called the "Oracle" scenario.

But we don't have crystal balls. We only have guesses (uncertainty). If the operator sets rules based on a "best guess," the batteries might get tricked. If the guess is wrong, the batteries might lose money, or the market might get chaotic.

2. The "Safety Net" (Chance-Constrained Bounds)

The authors created a new mathematical tool called a "Chance-Constrained" model. Imagine you are packing for a trip.

Deterministic (Old Way): You pack for sunny weather because the forecast says "sunny." If it rains, you get soaked.
Chance-Constrained (New Way): You pack an umbrella just in case it rains, but you only pack it if there is a 5% chance of rain. You are willing to accept a tiny risk of getting wet to avoid carrying a heavy umbrella every day.

The paper uses this logic to set the price caps. It calculates a limit that is "safe" 95% of the time (or whatever confidence level the operator chooses). This limit accounts for:

How much energy is in the battery: If a battery is nearly full, its value to the system is lower (it has less room to store more), so the cap on its bid price goes down.
How uncertain the weather is: If the wind is unpredictable, the cap goes up to compensate the battery for the risk.
How risk-averse the operator is: If the operator is very nervous about blackouts, they set the caps higher to encourage batteries to participate.

3. The "Speed Limit" Analogy

Think of the battery's bid as a car speeding down a highway.

Without the rule: The battery might speed (bid high) hoping to get a huge reward, but this causes traffic jams (high system costs) for everyone else.
With the rule: The paper proposes a dynamic speed limit. If the road is clear (low uncertainty), the limit is lower. If the road is foggy (high uncertainty), the limit is higher.
The Result: The battery can still drive fast if the conditions justify it, but it can't speed recklessly to the point of crashing the market.

4. What Happened in the Test?

The authors tested this idea on a simulated version of the New England power grid (ISO-NE). They created "agents" (computer programs) that acted like greedy battery owners trying to game the system.

The Outcome: When they applied their new "Speed Limits" (bid bounds):
- System Costs Dropped: The overall cost to run the grid went down slightly (about 0.17% in their test).
- Battery Profits Went Up: Surprisingly, the batteries actually made more money (about 10% more on average).
- Why? By capping the "greedy" bids that were too high, the batteries were allowed to participate more often in the market. They stopped holding back their energy in fear, which meant they could sell more power at fair prices.

5. The "Big Battery" Effect

The paper found that the more batteries you have on the grid, the more important these rules become.

If you have a few batteries, they don't change the market much.
If you have a lot of batteries (like 35% of the grid's capacity), their collective behavior can swing prices wildly. The new rules act as a stabilizer, ensuring that a massive fleet of batteries works for the grid rather than against it.

Summary

The paper argues that we need a smarter way to regulate energy storage. Instead of guessing or using rigid rules, we should use a flexible "safety net" that changes based on how full the battery is and how risky the weather looks. This keeps the market fair, lowers costs for everyone, and actually helps the battery owners make a better profit by preventing them from playing dangerous, inefficient games.

Technical Summary: Locational Energy Storage Bid Bounds for Facilitating Social Welfare Convergence

Problem Statement
The rapid deployment of energy storage in electricity markets introduces significant challenges in regulating market power and ensuring social welfare convergence. Unlike thermal generators, whose offers are based on known fuel costs, storage participants must submit bids that account for future opportunities and uncertainties, which are difficult to quantify. Current market practices rely on strategic bidding by storage participants, often leading to economic withholding (withholding capacity to arbitrage in real-time markets or anticipating price spikes). This behavior can result in market inefficiencies, excessive price volatility, and reduced social welfare. Existing regulatory measures, such as deterministic default bids (e.g., CAISO's 4th highest day-ahead LMP), are often overly conservative, ignoring real-time uncertainties and the dependency of opportunity value on the state of charge (SoC). Furthermore, detecting market manipulation is difficult because legitimate economic withholding for capturing future opportunities is indistinguishable from manipulative withholding without a robust opportunity cost baseline.

Methodology
The paper proposes a novel framework to generate locational, unit-specific bid bounds (offer caps) for energy storage. The methodology proceeds through the following steps:

Oracle Economic Dispatch (OED): The authors first formulate a multi-period economic dispatch problem assuming perfect foresight of demand and renewable generation. This serves as a theoretical baseline to determine the optimal marginal cost of storage, which includes both physical degradation costs and opportunity costs derived from time-coupling constraints.
Chance-Constrained Economic Dispatch (CED): To address the practical inability to predict future uncertainties, the OED is extended into a chance-constrained formulation. This framework incorporates netload uncertainties (from renewables and load) and the system operator's risk preference (defined by a confidence level $1-\epsilon$ ).
Deterministic Reformulation: The chance-constrained problem is reformulated into a deterministic convex equivalent. This involves replacing probabilistic constraints with deterministic constraints using the inverse cumulative distribution function (CDF) of the netload. The authors explore various uncertainty models, including normal distributions, robust approximations (Cantelli's inequality), and versatile distributions for discrete historical data.
Derivation of Bid Bounds: The dual variables associated with the storage energy balance constraints in the CED framework represent the risk-aware marginal opportunity cost. The authors derive discharge and charge bid bounds ( $A_{s,t}$ $A_{s, t}$ and $B_{s,t}$ $B_{s, t}$ ) based on these dual variables. The bounds are defined as:
- Discharge Bound: $A_{s,t} = M_s + \max(\hat{\theta}_{s,t})/\eta_s$
- Charge Bound: $B_{s,t} = \min(\hat{\theta}_{s,t})\eta_s - M_s$
  Where $M_s$ is the physical degradation cost, $\eta_s$ is efficiency, and $\hat{\theta}_{s,t}$ is the risk-aware opportunity cost.
Agent-Based Simulation: The effectiveness of the proposed bounds is tested using a modified 8-zone ISO-NE test system with an agent-based market simulation. The simulation includes strategic storage participants employing economic withholding strategies across varying levels of renewable penetration (30%, 50%), storage capacity (20%, 35%), and system uncertainty.

Key Contributions

Chance-Constrained Bounds: The paper introduces a novel framework that provides probabilistic bid bounds for storage offers. These bounds are derived from a multi-period economic dispatch that explicitly incorporates netload uncertainty and system operator risk preferences, offering a preventive measure against market power abuse.
Theoretical Pricing Analysis: The authors provide theoretical proofs establishing key characteristics of the bounds:
- Confidence Guarantee: The bounds cap truthful marginal costs with a guaranteed confidence level ( $1-\epsilon$ ).
- SoC Monotonicity: The bid bounds decrease monotonically as the storage State of Charge (SoC) increases, reflecting diminishing marginal value.
- Uncertainty and Risk Scaling: The bounds increase monotonically with netload uncertainty and the system operator's risk aversion (lower $\epsilon$ ). This contrasts with conventional generators, whose bids are independent of such factors.
- Risk-Aware LMP: The bounds are shown to align with risk-aware Locational Marginal Prices (LMPs).
Simulation Validation: The study demonstrates through agent-based simulations that the proposed bounds outperform existing deterministic bounds. They effectively align storage bids with social welfare objectives, reducing system costs while maintaining or increasing storage profits, particularly in scenarios with high economic withholding.

Results
Numerical experiments on the ISO-NE test system yielded the following results:

System Cost Reduction: Under scenarios with 30% renewable capacity and 20% storage capacity, the proposed bid bounds reduced average system costs by 0.17% compared to unregulated strategic bidding. The reduction scaled with increased storage capacity and economic withholding.
Storage Profitability: Contrary to the intuition that capping bids reduces profits, the proposed bounds increased average storage profits by 10.16% across various scenarios. This is attributed to the bounds preventing inefficient over-withholding that leads to suboptimal clearing and missed opportunities.
Comparison with Deterministic Bounds: Deterministic bounds (based on historical LMPs) were found to underestimate opportunity costs under high uncertainty, leading to a 0.01% increase in system costs and a ~20% reduction in storage profits. The chance-constrained approach proved superior in handling uncertainty.
Sensitivity to Parameters:
- Risk Preference ( $\epsilon$ ): Lower $\epsilon$ (higher risk aversion) leads to higher bid bounds but reduced social welfare improvement. An $\epsilon$ of 5% to 10% was identified as a practical trade-off.
- Uncertainty Models: Empirical and versatile distribution models yielded better social welfare improvements than robust approximations, which produced excessively high bounds.
- Scarcity: The bounds performed particularly well in scarcity scenarios (high prices), effectively curbing excessive withholding that would otherwise drive up costs.
Computational Efficiency: While the exact reformulation scales exponentially with the number of storage units, the application of a robust relaxation (avoiding binary variables) reduced computation time linearly, allowing for the solution of systems with 10,000 units in approximately 102 seconds.

Significance and Claims
The paper claims to address the pressing need for innovative regulatory approaches to manage the surge of energy storage in electricity markets. By providing a tractable, chance-constrained framework, the authors offer system operators a tool to:

Regulate Market Power: Prevent excessive economic withholding that compromises system efficiency without stifling legitimate strategic behavior.
Facilitate Social Welfare: Align storage bidding with social welfare objectives, reducing overall system costs.
Maintain Neutrality: Allow operators to remain neutral in market clearings while upholding reliability standards, as the bounds are derived from system-wide optimization rather than specific participant data.
Adaptability: The framework is designed to be transferable to various markets (e.g., CAISO) and can be tuned based on specific uncertainty profiles and risk preferences.

The authors emphasize that their approach acknowledges the validity of economic withholding for capturing future opportunities but seeks to cap bids that exceed the truthful marginal cost derived from a risk-aware system perspective. The work is presented as a practical solution for managing storage integration as renewable and storage capacities scale up.