⚡ electrical engineering

Peak-Load Pricing and Investment Cost Recovery with Duration-Limited Storage

This paper extends the classic peak-load pricing model to incorporate duration-limited energy storage, demonstrating that under solar-dominated conditions, scarcity premiums primarily reflect fixed storage investment costs recovered per peak event rather than variable efficiency losses, with binding duration constraints fundamentally altering optimal pricing and capacity investment strategies.

Original authors: Daniel Shen, Marija Ilic, John Parsons

Published 2026-03-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Daniel Shen, Marija Ilic, John Parsons

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

The Big Idea: The "Time-Traveling" Battery

Imagine you have a magical bucket that can hold water. You can fill it up when the water is cheap and plentiful (like at night), and then pour it out when water is scarce and expensive (like during a drought).

In the world of electricity, this "bucket" is energy storage (like giant batteries). The paper asks a very specific question: How do we set the price of electricity so that building these batteries makes financial sense for investors?

If the prices aren't right, nobody will build the batteries, and we won't have enough power when the sun goes down or when everyone turns on their AC at once.

The Problem: The "Short-Lived" Battery

Most power plants (like coal or gas) are like a faucet. As long as you pay for the fuel, the water keeps flowing. They can run for days or weeks.

Batteries are different. They are duration-limited. Think of a battery like a soda can.

You can pour it out fast (high power).
But once the soda is gone, it's gone. You can't keep pouring for 24 hours straight; you only have enough for a few hours.

The paper argues that because batteries run out of "soda" quickly, the rules for how we pay for them must be different from how we pay for regular power plants.

The Two Main Costs of a Battery

To build a battery, you pay two types of costs:

The "Pump" Cost (Power): How fast can it pour? (Measured in Megawatts).
The "Bucket" Cost (Energy): How much can it hold? (Measured in Megawatt-hours).

The paper discovers something surprising about how these costs are recovered through electricity prices:

1. The "Leak" isn't the main problem (Variable Costs)

When you charge and discharge a battery, some energy is lost as heat (like a leaky bucket). This is called round-trip efficiency.

The Old Thinking: People thought the high price of electricity during peak times was mostly to pay for this "leak."
The Paper's Finding: Actually, the leak is small. The main reason electricity prices spike is to pay for the bucket itself (the fixed investment cost).

2. The "Event" vs. The "Marathon" (Fixed Costs)

This is the most important insight.

Regular Power Plants: If a gas plant runs for 100 hours a year, you spread its cost over those 100 hours. It's like paying for a marathon runner by the mile.
Batteries: Because a battery runs out of energy quickly, it can't run for 100 hours straight. It runs for one "event" (e.g., a 4-hour evening peak) and then needs to recharge.
- The paper says: You don't pay for the battery by the hour. You pay for it per event.
- Analogy: Imagine you hire a lifeguard. A gas plant is like a lifeguard who works the whole summer. A battery is like a lifeguard who only works the 2 hours when the kids are swimming. You pay the battery a "surge fee" for showing up to save the day, regardless of whether the emergency lasts 1 hour or 4 hours.

The "Scarcity Premium"

In a normal market, electricity prices are low at night and high during the day.

Without Storage: Prices go up just enough to pay for the expensive gas plants that turn on when demand is high.
With Storage: Prices need to go up even higher. Why? Because the battery needs to make enough money in those few short hours to pay for the entire cost of the battery (the "bucket" and the "pump").

The paper calculates that for modern lithium-ion batteries, about 80% of the extra high price you see during peak times is actually to pay for the battery's construction, not to pay for the energy lost during charging.

What Happens if We Get the Prices Wrong?

The authors warn regulators: Don't cap the prices too low.

If the government says, "Electricity can never cost more than $100," the battery investors will lose money.

They won't build the batteries.
We will be left with only gas plants.
The system becomes less reliable and more expensive in the long run.

The Takeaway in Plain English

Batteries are special: They are limited by how long they can last, not just how fast they can work.
Prices must spike: To make batteries profitable, electricity prices during peak times must be high enough to cover the cost of the battery itself, not just the cost of the electricity inside it.
Pay per event, not per hour: We need to think of battery value as a "surge fee" for a specific emergency, rather than a steady wage for hours worked.
Efficiency is secondary: While losing energy during charging is annoying, the cost of building the battery is the real driver of high prices, not the efficiency loss.

In short: If we want a future with lots of solar and wind power (which are only available part of the time), we need batteries. And for those batteries to exist, we must allow electricity prices to spike high enough to pay for the "buckets" that hold the energy.

1. Problem Statement

The rapid integration of grid-scale energy storage (projected to be 29% of new US generation capacity by 2025) necessitates a re-evaluation of electricity market pricing and investment mechanisms. Unlike conventional generators, storage is duration-limited: it cannot supply power indefinitely once its stored energy is depleted.

Existing literature often models storage as either:

Duration-unlimited: Assuming infinite energy capacity relative to power (ignoring the depletion constraint).
Power-unlimited: Assuming infinite power relative to energy.

The authors identify a gap in understanding how simultaneous constraints on power capacity ( $K_s$ ) and stored energy capacity ( $E$ ) affect equilibrium prices and the recovery of fixed investment costs. Specifically, they ask: What market prices incentivize efficient storage investment, and how do these prices relate to storage technology characteristics (efficiency, duration, fixed costs)?

2. Methodology

The authors extend the classic Steiner/Boiteux two-period peak-load pricing model to incorporate duration-limited storage.

Model Formulation:
- A social planner maximizes total surplus (consumer utility minus generation and storage costs) over $n$ annual cycles.
- The system includes two periods: Off-peak (low demand) and On-peak (high demand).
- Decision variables include investments in baseload ( $K_B$ ), peaker ( $K_P$ ), and storage power ( $K_S$ ) and energy ( $E$ ) capacities, alongside operational dispatch ( $q$ ).
- Key Constraints:
  - Power balance (supply = demand).
  - Power capacity limits for all units.
  - Energy capacity limit ( $E$ ): The total energy discharged cannot exceed stored energy.
  - Round-trip efficiency ( $\eta$ ): Energy out = $\eta \times$ Energy in.
  - Aggregate energy balance: Charging in off-peak must equal discharging in on-peak (adjusted for efficiency).
Simplifying Assumptions:
To derive closed-form analytical solutions, the authors assume a solar-dominated system profile:
1. Price Ordering: The price difference between on-peak and off-peak is sufficient to induce storage investment.
2. Sufficient Off-Peak Duration: The off-peak period is long enough that the storage power constraint does not bind during charging (only binds during on-peak discharge).
3. No Inter-Period Carryover: Stored energy is fully depleted by the end of the on-peak period (zero continuation value).
Analytical Approach:
The authors utilize the Karush-Kuhn-Tucker (KKT) conditions of the Lagrangian to derive the dual variables (shadow prices) associated with the constraints. These duals represent the marginal value of energy and capacity, allowing for the derivation of equilibrium price equations.

3. Key Contributions

The paper provides three primary theoretical insights:

Decomposition of Scarcity Prices: The on-peak price is shown to consist of a variable-cost component (driven by round-trip efficiency losses) and a fixed-cost component (driven by investment costs). Crucially, the fixed-cost component dominates the price premium for realistic technologies (e.g., Lithium-ion).
Per-Event Cost Recovery: Unlike conventional peaker plants, where fixed costs are amortized over the total annual duration of peak hours, duration-limited storage recovers its energy capacity costs on a per-peak-event basis. The energy capacity premium is independent of the total annual peak duration ( $n \times T_{on}$ ) and depends only on the number of peak events ( $n$ ).
Investment Cost Recovery Guarantee: The authors prove that if storage is optimally sized and operated, the standard dual-based energy prices (derived from the power balance constraint) are sufficient to guarantee full recovery of both power and energy investment costs.

4. Key Results and Findings

A. The On-Peak Price Equation

The derived equilibrium on-peak price ( $\lambda_{onp}$ ) is:
$\lambda_{onp} = \frac{\lambda_{offp}}{\eta} + \frac{1}{n} \left( I_{s,E} + \frac{I_{s,q}}{T_{onp}} \right)$
Where:

$\lambda_{offp}/\eta$ : Variable cost of purchasing and storing energy.
$I_{s,E}$ : Fixed cost of energy capacity ($/MWh-year).
$I_{s,q}$ : Fixed cost of power capacity ($/MW-year).
$T_{onp}$ : Duration of the peak period.
$n$ : Number of peak cycles per year.

Critical Insight: The term $\frac{I_{s,E}}{n}$ represents the energy capacity cost. Because it is divided by $n$ (number of events) rather than $n \times T_{onp}$ (total hours), the scarcity premium per hour is significantly higher than in conventional generation models.

B. Numerical Example (Lithium-Ion Storage)

Using parameters from the NREL Annual Technology Baseline:

Scenario: 4-hour on-peak, 20-hour off-peak, 365 cycles/year.
Result: The fixed-cost component contributes $109.6/MWh to the on-peak price, while the efficiency-loss component contributes only $23.5/MWh.
Implication: Fixed costs account for nearly 5x the price impact of efficiency losses. Equilibrium prices must rise well above variable operating costs to incentivize storage.

C. System Impact (With vs. Without Storage)

Price Spreads: Storage narrows the price spread between on-peak and off-peak periods compared to a system without storage.
Capacity Mix: Storage replaces peaker generation ( $K_P$ drops from 3.8 GW to 0) and synergizes with baseload generation ( $K_B$ increases slightly).
Investment Recovery: The numerical solution confirms that the revenue from the price spread exactly covers the annualized investment costs for both power and energy capacity.

5. Significance and Regulatory Implications

Market Design: The paper argues that standard capacity markets or administrative price caps may fail to incentivize storage if they do not account for the unique "per-event" cost recovery mechanism. If scarcity prices are capped without compensation for the energy capacity constraint, storage will be under-invested.
Valuation of Scarcity: The results emphasize that scarcity pricing is the primary mechanism for storage cost recovery, not just energy arbitrage. The "scarcity premium" is largely a reflection of the fixed cost of the energy capacity constraint.
Policy: Regulators must recognize that storage investment costs cannot be amortized over total peak hours like conventional generators. Pricing mechanisms must allow for high scarcity prices during peak events to ensure the "per-event" recovery of energy capacity costs.

In conclusion, the paper establishes that for duration-limited storage, scarcity prices are driven primarily by fixed investment costs rather than operational inefficiencies, and these costs must be recovered on a per-peak-event basis to ensure economic viability.