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The Theory of Storage in a Power System with Stochastic Demand

Darryl Biggar, Mohammad Reza Hesamzadeh

TL;DR

This paper develops a theory of storage in a power system where periodical net demand $L$ is IID, analyzing how storage should be operated and expanded, and how to hedge storage risk without distorting decisions. It extends basic electricity-market theory by introducing a systemwide state of charge $S$ and solving a Bellman equation to obtain threshold-based storage dispatch rules, while linking investment to a stationary distribution over $S$. It shows that private, price-taking storage incentives align with the social optimum and derives a hedge construction that perfectly insulates storage from risk through a combination of caps, floors, and a novel S-shaped hedge, all interpreted via augmented price-duration curves. The worked example illustrates the qualitative and quantitative impacts of storage on price dynamics and identifies plausible storage levels (e.g., around 24% of load variation under given costs). The work provides a foundational benchmark for storage optimization, hedging, and expansion planning, while noting the IID simplifications and other idealizations as avenues for future extension.

Abstract

Electric power systems are increasingly turning to energy storage systems to balance supply and demand. But how much storage is required? What is the optimal volume of storage in a power system and on what does it depend? In addition, what form of hedge contracts do storage facilities require? We answer these questions in the special case in which the uncertainty in the power system involves successive draws of an independent, identically-distributed random variable. We characterize the conditions for the optimal operation of, and investment in, storage and show how these conditions can be understood graphically using price-duration curves. We also characterize the optimal hedge contracts for storage units.

The Theory of Storage in a Power System with Stochastic Demand

TL;DR

This paper develops a theory of storage in a power system where periodical net demand is IID, analyzing how storage should be operated and expanded, and how to hedge storage risk without distorting decisions. It extends basic electricity-market theory by introducing a systemwide state of charge and solving a Bellman equation to obtain threshold-based storage dispatch rules, while linking investment to a stationary distribution over . It shows that private, price-taking storage incentives align with the social optimum and derives a hedge construction that perfectly insulates storage from risk through a combination of caps, floors, and a novel S-shaped hedge, all interpreted via augmented price-duration curves. The worked example illustrates the qualitative and quantitative impacts of storage on price dynamics and identifies plausible storage levels (e.g., around 24% of load variation under given costs). The work provides a foundational benchmark for storage optimization, hedging, and expansion planning, while noting the IID simplifications and other idealizations as avenues for future extension.

Abstract

Electric power systems are increasingly turning to energy storage systems to balance supply and demand. But how much storage is required? What is the optimal volume of storage in a power system and on what does it depend? In addition, what form of hedge contracts do storage facilities require? We answer these questions in the special case in which the uncertainty in the power system involves successive draws of an independent, identically-distributed random variable. We characterize the conditions for the optimal operation of, and investment in, storage and show how these conditions can be understood graphically using price-duration curves. We also characterize the optimal hedge contracts for storage units.

Paper Structure

This paper contains 16 sections, 7 theorems, 76 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Literature review and key contributions
  3. Review of basic electricity market theory
  4. Social incentives for usage of, investment in, and hedging of generation
  5. Private incentives for usage of, and investment in, generation
  6. Perfect hedges for generators
  7. Optimal social use of and investment in storage in a power system
  8. Optimal use of storage in a power system
  9. Optimal investment in storage
  10. Optimal private use of and investment in storage
  11. Hedge contracts for storage
  12. Worked example
  13. Conclusion
  14. Solutions for recursive Markov chains
  15. Private incentives for use of storage
  16. ...and 1 more sections

Key Result

Theorem 1

In a power system in which consecutive draws of $L$ are independent and identically distributed, the optimal dispatch of a non-rate-limited storage system when the state of charge is $S$ and the realisation of the load is $L$, $S^*_{SL}$ is as follows: Here $P_{SL}$ is the 'spot price' (the slope of the dispatch cost) that emerges when the storage system is used optimally, and $E[P^+_{SL}]\equiv

Figures (11)

  • Figure 1: Illustration of the determination of $S^+_{SL}$ in the case where the storage capacity is 10% of the variation of the load). Load is uniformly distributed on [0,100]. Raw price is assumed to be a linear function of load $P(L)=20+1.5L$. Initial storage level in the middle of the range.
  • Figure 2: Storage size $S=10\%$ of load variation.
  • Figure 3: Storage size $S=150\%$ of load variation.
  • Figure 4: The optimal level of storage capacity is where the marginal benefit from adding storage is equal to the fixed cost of storage capacity. Here the marginal benefit is calculated holding constant the stock of generation assets.
  • Figure 5: Illustration of the effect of adding storage to the long-run price-duration curve. Load uniform on [0,100], Raw Price a linear function of load $P(L)=1.5L+20$. Storage 10%, 50% and 150% of the variation in load.
  • ...and 6 more figures

Theorems & Definitions (12)

  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • ...and 2 more