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Volatility of Power Grids under Real-Time Pricing

Mardavij Roozbehani, Munther A Dahleh, Sanjoy K Mitter

arXiv:1106.1401v1eess.SYmath.DSmath.OCq-fin.PR

TL;DR

Real-time retail pricing couples consumers to wholesale markets, creating feedback that may amplify power-grid volatility. The paper models this dynamic and shows volatility increases with relative price-elasticity, potentially causing instability.

  • Problem

    Directly coupling retail consumers to wholesale prices creates operational challenges under information asymmetry, motivating analysis of resulting market dynamics and stability.

  • Method

    The paper develops dynamical models of wholesale supply and retail demand under information asymmetry and analyzes their price dynamics.

  • Results

    RVR = 51.12, with prices extremely volatile and the system practically unstable under real-time pricing in the reported simulation.

  • Takeaways & Limitations

    Volatility and robustness can be characterized by relative price-elasticity, with instability arising when the consumer-to-producer elasticity ratio exceeds one.

Abstract

from arXiv · show

The paper proposes a framework for modeling and analysis of the dynamics of supply, demand, and clearing prices in power system with real-time retail pricing and information asymmetry. Real-time retail pricing is characterized by passing on the real-time wholesale electricity prices to the end consumers, and is shown to create a closed-loop feedback system between the physical layer and the market layer of the power system. In the absence of a carefully designed control law, such direct feedback between the two layers could increase volatility and lower the system's robustness to uncertainty in demand and generation. A new notion of generalized price-elasticity is introduced, and it is shown that price volatility can be characterized in terms of the system's maximal relative price elasticity, defined as the maximal ratio of the generalized price-elasticity of consumers to that of the producers. As this ratio increases, the system becomes more volatile, and eventually, unstable. As new demand response technologies and distributed storage increase the price-elasticity of demand, the architecture under examination is likely to lead to increased volatility and possibly instability. This highlights the need for assessing architecture systematically and in advance, in order to optimally strike the trade-offs between volatility, economic efficiency, and system reliability.

I. INTRODUCTION … 2) Stability:

The paper motivates real-time pricing as a way to improve power-grid efficiency while warning that feedback between markets and physical operations can increase volatility and threaten stability. It formalizes volatility and stability using incremental signal measures and Lyapunov-based asymptotic-stability concepts.

  • I. INTRODUCTION: Real-time pricing and demand response may improve efficiency and integrate renewable generation, but power systems’ uncertainty, tight economic-physical coupling, and reliability constraints create distinctive challenges.Consumers are encouraged to respond to price signals, while supply must match demand continuously.
  • I. INTRODUCTION: Pricing mechanisms must be designed with stability in mind because volatility, reliability, and economic or environmental efficiency involve competing trade-offs.The paper emphasizes analyzing consumer responses to price signals before implementing real-time pricing.
  • I. INTRODUCTION: The paper’s contribution is to characterize volatility through maximal relative elasticity and uncertainty in consumer behavior using global properties of a full nonlinear model.The model differs from prior work on dynamic power-market stability by analyzing the full nonlinear system.
  • A. Notation: The notation establishes conventions for positive and nonnegative numbers, differentiable functions, vector components and norms, sequence spaces, Jacobians, derivatives, and Lebesgue measure.These conventions support the paper’s mathematical definitions and subsequent system analysis.
  • 1) Volatility:: Scaled incremental mean volatility measures fast-timescale deviations of a signal from its moving average, emphasizing high-frequency behavior rather than deviations from its overall average.The paper contrasts incremental volatility with sample variance and coefficient of variation.
  • 1) Volatility:: Scaled incremental aggregate volatility complements mean volatility by quantifying fast-decaying signals with zero or small scaled incremental mean volatility.Log-scaled incremental volatility is used for price, supply, and demand in electricity markets.
  • 2) Stability:: The paper uses standard asymptotic stability for discrete-time systems, beginning with Lyapunov stability of an equilibrium under sufficiently small initial perturbations.Lyapunov stability requires trajectories starting sufficiently close to remain arbitrarily close.
  • 2) Stability:: An equilibrium is globally asymptotically stable when it is Lyapunov stable and every trajectory converges to it as time tends to infinity.This definition extends local stability to all initial states in the domain.

C. Market Structure · 1) The Consumers and the Producers:

The market model comprises price-taking consumers and suppliers, with an independent ISO clearing supply and demand under network constraints. Consumer heterogeneity and uncertainty make demand time-varying, while aggregation separates exogenous inelastic demand from price-responsive demand.

  • C. Market Structure: The market has price-taking, profit-maximizing suppliers and consumers, while an independent profit-neutral ISO clears supply and demand to maximize social welfare.The ISO performs market clearing subject to network constraints.
  • 1) The Consumers and the Producers:: Producer cost functions are strictly increasing and convex, whereas consumer value functions are strictly increasing and concave.These functions define production costs and consumers’ monetary value from electricity consumption.
  • 1) The Consumers and the Producers:: Consumers’ uncertain value functions are modeled through multiplicative or additive perturbations, allowing heterogeneous, time-varying behavior.Under these models, the same price may induce different consumption at different times depending on load type and composition.
  • 1) The Consumers and the Producers:: Several consumers or producers can be represented by a representative agent, although its explicit utility function may be difficult to derive.For identical consumers, aggregate demand is equivalent to representative-consumer demand.
  • 1) The Consumers and the Producers:: Aggregate demand combines an exogenous inelastic component with an elastic component responding to the real-time price.The inelastic term can represent non-participating consumers, while the elastic term represents consumers subject to real-time pricing.
  • 1) The Consumers and the Producers:: The ISO matches supply and demand subject to Kirchhoff’s laws, transmission limits, generator capacities, and reserve requirements.Additional constraints may be specific to the ISO, and real-time operation linearizes constraints near the steady-state operating point.

2) The Independent System Operator (ISO): · III. DYNAMIC MODELS OF SUPPLY-DEMAND UNDER INFORMATION ASYMMETRY

The ISO model assumes an unconstrained network and sufficient reserves, enabling optimal supply–demand matching and LMP-based competitive clearing. The dynamic framework then studies discrete-interval real-time markets under asymmetric information, including ex-ante pricing based on predicted demand.

  • 2) The Independent System Operator (ISO):: Under negligible losses, uncongested lines, unconstrained generators, and sufficient reserves, network and reserve parameters are omitted from the ISO matching problem.The marginal cost of reserves is assumed equal to the marginal generation cost.
  • 2) The Independent System Operator (ISO):: Capacity-market effects and transmission or generator capacity constraints are outside scope; related dynamic-pricing analyses are cited as,.
  • 2) The Independent System Operator (ISO):: The ISO optimization problem is formulated under these assumptions, with the balance constraint central to market clearing.
  • 2) The Independent System Operator (ISO):: A price λ* exists such that optimal demand and supply solve the corresponding allocation problems, with λ* equal to the balance-constraint Lagrangian multiplier.The lemma is adopted from.
  • 2) The Independent System Operator (ISO):: Defining market prices as balance-constraint Lagrange multipliers creates a competitive environment whose collective self-interested behavior achieves a system-wide optimum.
  • 2) The Independent System Operator (ISO):: When retail consumers do not bid value functions, the operator uses predicted demand, sets price at minimum-cost marginal production, and balances forecast error through reserves.Consumers may still adjust consumption in response to the wholesale clearing price.
  • III. DYNAMIC MODELS OF SUPPLY-DEMAND UNDER INFORMATION ASYMMETRY: The dynamic models address wholesale-supply and retail-demand interaction when the operator knows supply costs but lacks demand-side resource valuations.
  • III. DYNAMIC MODELS OF SUPPLY-DEMAND UNDER INFORMATION ASYMMETRY: Real-time markets clear at discrete intervals, with ex-ante prices based on predicted demand and ex-post prices based on consumption realized at interval end.

A. Price Dynamics under Exant´e Pricing · B. Price Dynamics under Ex-post Pricing

The paper models ex-ante pricing as a forecast-driven feedback system and ex-post pricing as a price-prediction system whose dynamics depend on consumer behavior. Under persistence forecasting, ex-post dynamics match ex-ante dynamics, but consumers bear the associated price uncertainty and risk.

  • A. Price Dynamics under Exant´e Pricing: Under ex-ante pricing, the ISO sets λ(t) as the marginal cost of predicted supply for predicted demand over [t, t+1].Because each consumer’s valuation is privately known, the ISO relies on an estimate of demand.
  • A. Price Dynamics under Exant´e Pricing: The ISO forecasts next-period demand from actual demand in previous intervals using a function φ(d(t), …, d(t−T)).The prediction can be implemented with linear autoregressive models.
  • A. Price Dynamics under Exant´e Pricing: The persistence model is a special ex-ante case in which predicted next-period demand equals demand at the previous time step.The paper identifies persistence as a popular forecasting approach used by system operators.
  • A. Price Dynamics under Exant´e Pricing: Aggregating producers and consumers into representative agents reduces the ex-ante dynamics to relationships governed by a convex production cost and concave consumption value.The representative-agent construction is referenced to.
  • B. Price Dynamics under Ex-post Pricing: Under ex-post pricing, the consumption price is declared after consumption materializes, so consumers must predict it using previous ex-post prices.Consumer j’s prediction is represented as φ_j(λ(t), …, λ(t−T)).
  • B. Price Dynamics under Ex-post Pricing: With persistence price forecasting, ex-post dynamics become identical to ex-ante dynamics, while consumers bear the price uncertainty and associated risks.More generally, ex-post dynamics depend on how each individual consumer predicts future prices.
  • B. Price Dynamics under Ex-post Pricing: The ex-post formulation assumes optimally dispatched generators and does not model discrepancies between forecast-based ex-ante dispatch and ideal ex-post dispatch.Settlement of these discrepancies is acknowledged as practically important but excluded from the model.

C. Demand Dynamics under Exant´e or Ex-post Pricing · IV. THEORETICAL FRAMEWORK · A. Stability Analysis

The paper formulates demand dynamics separately for exanté and ex-post pricing and develops a Lyapunov-based framework for analyzing stability, robustness, and volatility in real-time electricity markets. Generalized elasticity and the market’s maximal relative price-elasticity provide stability criteria, including MRPE < 1 for suitable exponents.

  • C. Demand Dynamics under Exanté or Ex-post Pricing: Exanté pricing makes demand depend on contemporaneous predicted clearing-price changes, whereas ex-post pricing allows dependence on a history of price changes.Under representative-agent models, these become respectively demand dynamics driven by current demand history or by lagged price responses.
  • IV. THEORETICAL FRAMEWORK: The theoretical framework targets implicit dynamical systems arising when real-time optimization is embedded in feedback loops, supporting analysis of market dynamics, disturbances, price stability, and volatility.The subsequent stability analysis applies this framework to clearing-price dynamics formulated earlier.
  • A. Stability Analysis: Theorem 1 establishes stability using continuously differentiable functions f and g whose monotonicity and asymptotic conditions make a Lyapunov function decrease along system trajectories.The proof shows the Lyapunov sequence is strictly decreasing and bounded, while trajectories are bounded and converge; standard arguments then establish Lyapunov stability.
  • A. Stability Analysis: For price dynamics, the framework decomposes the system through inverse cost and value functions, yielding sufficient stability criteria that can be difficult to satisfy globally for typical supply and demand functions.A monotone transformation ρ can relax the direct comparison conditions, while identity transformations recover simpler criteria.
  • A. Stability Analysis: Generalized price-elasticities extend standard elasticity through an exponent l, and MRPE is defined as the market’s maximal ratio of generalized demand elasticity to generalized supply elasticity.For l = 1, generalized elasticity reduces to the standard notion.
  • A. Stability Analysis: The persistence-model systems are stable when MRPE is less than one for some l ≥ 0, while the analogous cost-value system is stable when MRRA is less than one for some k ≥ 0.These criteria are obtained from the transformed stability conditions using logarithmic or power transformations.
  • A. Stability Analysis: In the power-function example, the system diverges for β > 2 − α^-1, while with α = β = 2 and u > 0, simulations find asymptotic instability for u < 1/4.At u = 1/4 the system is at least marginally stable, and generalized elasticity can prove stability even when traditional price elasticity exceeds one.

B. Invariance Analysis

Theorem 2 establishes an invariant set for autoregressive state-space dynamics under continuously differentiable dynamics and a monotonic state transformation. A local corollary shows that these conditions suffice near equilibrium and that strict inequalities bound the transformed invariant average variation, supporting analysis of market and demand dynamics.

  • Theorem 2: Theorem 2 establishes that Ω0 is invariant: once n+1 consecutive states lie in Ω0, all subsequent states remain there.The set Ω0 is defined by |g(x) − f(x,z)| ≤ γ0, with γ0 depending only on the first n+1 initial states.
  • Theorem 2: When condition (61) holds strictly, the g-scaled IAV of x is bounded from above.The bound follows from the nonnegativity of the Lyapunov-like function V along the trajectory.
  • Corollary 4: Near equilibrium, invariance requires conditions (60)–(61) only locally over a properly defined subset of R^(n+1).Corollary 4 states that the local set eΩ0 remains invariant under (59), and strict inequality yields the bound in (63) when the initialization vector lies in eΩ0.
  • Applications: Theorem 2 and Corollary 4 apply to generic autoregressive models for price dynamics under ex-ante or ex-post pricing and to aggregate demand dynamics.Their invariant sets characterize persistence of bounded differences between predicted demand and actual supply, possibly after scaling.

1) Analysis of Market Dynamics under Generic Autoregressive Prediction Models:

This section analyzes market stability under linear autoregressive price-prediction models using sufficient conditions derived from Theorem 2 and examines their implications near equilibrium. The analysis suggests that linear autoregressive prediction may be unable to globally stabilize systems that are unstable under persistent prediction.

  • 1) Analysis of Market Dynamics under Generic Autoregressive Prediction Models:: The sufficient conditions are complicated and generally demand numerical computation for verification.
  • 1) Analysis of Market Dynamics under Generic Autoregressive Prediction Models:: At equilibrium, the local condition reduces to a generalized producer elasticity satisfying ϵ_p ≤ 1, independently of the prediction-model order l.This condition follows by evaluating the sufficient conditions at an equilibrium price.
  • 1) Analysis of Market Dynamics under Generic Autoregressive Prediction Models:: If the closed-loop market is unstable under persistent prediction, these sufficient conditions cannot verify global stability for any linear autoregressive model of the stated form.The analysis therefore suggests that globally stabilizing such systems through linear autoregressive prediction may be difficult, although it is based on sufficient criteria.

2) Analysis of Dynamics of Markets with Exogenous Inputs:

The analysis establishes invariant-set and volatility bounds for state-space market dynamics with exogenous inputs, then applies them to multiplicative and additive perturbations. Stability depends on relative price-elasticity, perturbation size, and the extent of real-time-pricing participation.

  • a) Multiplicative Perturbation: As the multiplicative model’s MRPE θ∗ approaches 1, small perturbations can cause extremely large supply fluctuations, with theoretical upper bound 1/(1−θ∗).The invariant-set size is bounded by ζκ(θ∗), and volatility can also be characterized by e^θ∗.
  • b) Additive Perturbation: For additive perturbations, the log-scaled supply variation is upper bounded by u0/(1−eθ∗), while the invariant-set size is bounded by ζκ(eθ∗).A nonzero minimum demand allows the stability condition to be checked only over λ ≥ c˙(u0/2).
  • b) Additive Perturbation: Larger inelastic demand makes the additive model’s stability condition easier to satisfy, whereas exposing a large population share to real-time pricing raises stability concerns.The analysis therefore indicates that limited participation in real-time pricing has no severe destabilizing effect under the stated assumptions.
  • b) Additive Perturbation: When the additive model’s input is periodic and the contraction condition holds, every solution converges to a periodic trajectory with the same period.This result applies when u(t) has period T and condition (76) is satisfied.

C. Volatility

The section relates market relative price elasticity and relative adjustment measures to volatility. Under parameters below one, log-scaled supply and price variation measures are bounded by inverse gaps to one, with generalized and local extensions.

  • C. Volatility: When θ∗ < 1 and η∗ < 1, log-scaled IMV of supply and price are upperbounded by C/(1 − θ∗) and C/(1 − η∗), respectively.Here, θ∗ and η∗ are the market model’s MRPE and MRRA, and C depends only on disturbance size.
  • C. Volatility: With linear autoregressive prediction, log-scaled IAV of supply and price are upperbounded by C/(1 − θ∗) and C/(1 − η∗), respectively, when θ∗ < 1 and η∗ < 1.The bounds apply to the market model (59).
  • C. Volatility: Generalized bounds use θ∗(l) and η∗(l): ρ_l-scaled IMV of supply and price are bounded by C/(1 − θ∗(l)) and C/(1 − η∗(l)).When prices remain bounded in an invariant set, local relative measures can replace the global parameters under the stated corollary conditions.

D. Robustness and Incremental L2-Gain

This section defines the ρ-scaled incremental L2-gain as a robustness metric for discrete-time dynamical systems. Larger gains indicate greater sensitivity of outputs to input deviations and external disturbances, including in the market model with multiplicative uncertainty.

  • The ρ-scaled incremental L2-gain is the minimal γ ≥ 0 satisfying the specified inequality for a discrete-time system with input u and output h.
  • The gain condition must hold for all input/output pairs (u, h), making the metric applicable to arbitrary system trajectories.
  • Larger gains generally mean that small deviations from nominal inputs produce larger output deviations, so the gain measures robustness and sensitivity to external disturbances.
  • The market model with multiplicative uncertainty admits a log-scaled incremental L2-gain from the perturbation δ(·) to the specified output.

V. DISCUSSION

The discussion relates the framework to prior full-information market-volatility work, highlights how demand-side ramp limits or inelastic demand can stabilize responses, and motivates quantifying information’s value in closed-loop electricity markets.

  • Relation to prior work: Cho and Meyn study power-market volatility in a full-information dynamic general-equilibrium model with instantaneous clearing and producer ramp constraints.The system operator knows producers’ and consumers’ cost and value functions, while supply and demand are matched without time lag.
  • Stabilizing mechanisms: Consumer ramp constraints would stabilize the system by limiting responsiveness to price signals and reducing consumer elasticity.The framework captures this effect qualitatively through an inelastic demand component that limits demand’s rate of change and has a stabilizing effect.
  • Open question: The discussion motivates quantifying the value of information in closed-loop electricity markets, although real-time learning of heterogeneous, time-varying consumer values and responses appears difficult.Consumers could provide real-time estimates of their inelastic and elastic consumption, but the passage does not specify the resulting information-value method.

VI. NUMERICAL SIMULATIONS · VII. CONCLUSIONS AND FUTURE WORK

Numerical simulations show that real-time pricing can substantially increase market volatility, especially as demand price-elasticity rises, while the conclusions emphasize the need for better demand models and market-design analysis before large-scale implementation.

  • VI. NUMERICAL SIMULATIONS: Demand is modeled as a combination of exogenous inelastic demand and a real-time-price-responsive term, with random disturbances included in the inelastic-demand process.The exogenous component is d1(t) = a0 + a1 sin(t) + a2 sin(2t) + δ1(t).
  • VI. NUMERICAL SIMULATIONS: The simulations normalize µ1 and µ2 so average real-time-pricing demand remains nearly equal to open-loop demand, isolating differences in price volatility.This removes the effect of higher or lower average demand from the comparison.
  • VI. NUMERICAL SIMULATIONS: The simulations compare closed-loop and open-loop markets using the Relative Volatility Ratio, defined as the ratio of their log-scaled IAV values.Average demand was approximately 4 GW per five minutes in both markets, with prices updated every 5 minutes over 24 hours.
  • VI. NUMERICAL SIMULATIONS: RVR=2.33 in the second simulation, where real-time pricing increases demand volatility despite the market being less volatile than in the first simulation.Because the cost is quadratic, the unshown price follows a similar pattern to demand.
  • VI. NUMERICAL SIMULATIONS: Volatility increases as a decreases or µ1 decreases, because both changes increase the price-elasticity of demand.This trend was estimated across 50 randomized simulations for four value functions, v(x) = x1/a with a = 4, 4.5, 5, 5.5.
  • VII. CONCLUSIONS AND FUTURE WORK: Real-time retail pricing creates a closed-loop feedback system that can be very volatile or unstable, with stable-system volatility and disturbance robustness characterized by relative price-elasticity.The conclusions summarize the paper’s investigation of stability and volatility in electricity markets under real-time pricing.
  • VII. CONCLUSIONS AND FUTURE WORK: Large-scale implementation requires more sophisticated demand models, deeper understanding of consumer responses to real-time prices, and analysis of alternative market mechanisms and system architectures.The paper identifies these topics as necessary before real-time pricing can be implemented at scale.
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