Source-linked AI summary
Dynamic Pricing and Distributed Energy Management for Demand Response
Liyan Jia, Lang Tong
TL;DR
The paper studies how a retailer can use day-ahead hourly pricing to manage demand while balancing consumer surplus and retail profit. It models retailer–consumer interaction as a Stackelberg game and derives an affine demand response for thermostatic loads. The resulting Pareto trade-offs are characterized alongside renewable and consumer-side storage effects.
Problem
The paper addresses how dynamic retail pricing can balance consumer surplus and retail profit while supporting distributed demand response.
Method
A Stackelberg game models retailer price-setting and consumer real-time response, with thermostatic demand represented through an affine price-response model.
Results
The paper fully characterizes the consumer-surplus–retail-profit Pareto front; renewable benefits initially accrue to the retailer and increasingly benefit consumers as capacity grows.
Takeaways & Limitations
Optimized day-ahead hourly pricing provides a framework for selecting retail prices while accounting for renewable access and consumer-side storage.
Takeaways & Limitations
The affine aggregated-load mapping assumes known system parameters, although obtaining them in practice requires machine-learning techniques; wholesale-market feedback is also excluded.
Abstract
from arXiv · showhide
The problem of dynamic pricing of electricity in a retail market is considered. A Stackelberg game is used to model interactions between a retailer and its customers; the retailer sets the day-ahead hourly price of electricity and consumers adjust real-time consumptions to maximize individual consumer surplus. For thermostatic demands, the optimal aggregated demand is shown to be an affine function of the day-ahead hourly price. A complete characterization of the trade-offs between consumer surplus and retail profit is obtained. The Pareto front of achievable trade-offs is shown to be concave, and each point on the Pareto front is achieved by an optimal day-ahead hourly price. Effects of integrating renewables and local storage are analyzed. It is shown that benefits of renewable integration all go to the retailer when the capacity of renewable is relatively small. As the capacity increases beyond a certain threshold, the benefit from renewable that goes to consumers increases.
I. INTRODUCTION
The paper studies day-ahead hourly pricing as a consumer-responsive alternative to direct demand control, characterizing trade-offs between consumer surplus and retail profit while examining renewable and storage integration.
- I. INTRODUCTION: Day-ahead hourly pricing lets consumers adjust consumption while giving retailers day-ahead price certainty and flexibility to respond to wholesale-market conditions.The scheme sets hourly retail prices one day ahead and can be adjusted daily.
- A. Summary of Main Results: The paper models retailer–consumer interactions as a Stackelberg game and derives affine optimal aggregated demand for thermostatically controlled loads.The retailer sets DAHP, consumers respond, and optimal pricing can be obtained through convex optimization.
- A. Summary of Main Results: The achievable consumer-surplus–retail-profit trade-off is characterized by a concave, monotonically decreasing Pareto front, with each point associated with an optimized DAHP.Social-welfare pricing yields zero retail profit, while constant, time-of-use, and proportional-markup pricing are generally inside the Pareto front.
- A. Summary of Main Results: Renewable integration benefits the retailer entirely at small capacity, while increasing capacity increases the benefit accruing to consumers.The paper also formulates consumer-side storage within the same Stackelberg framework.
- I. INTRODUCTION: Simulations with realistic electricity prices and home energy-management models illustrate DAHP-based demand response, benchmark comparisons, and renewable effects on the trade-off curve.The paper positions its contribution as a full characterization of achievable retail-profit and consumer-surplus trade-offs.
II. A STACKELBERG GAME MODEL
The model represents retailer–consumer interaction as a Stackelberg game under day-ahead hourly pricing. Consumers optimize real-time consumption and surplus, yielding an affine demand response for thermostatically controlled loads.
- A. DAHP and a Stackelberg game model: The retailer sets a 24-dimensional day-ahead hourly price vector, while consumers respond in real time by adjusting consumption and possibly storage.The retailer meets aggregated demand through wholesale purchases and possibly its own renewables.
- B. Consumer action: optimal demand response: Consumers maximize surplus, defined as utility from consumption minus electricity payment, using real-time measurements and current energy state.The resulting optimal control policy represents rational demand response to the announced DAHP.
- B. Consumer action: optimal demand response: The thermostatic-load model describes indoor-temperature dynamics using HVAC power, outdoor conditions, process noise, and measurement noise.The analysis focuses on cooling, with results also applying to heating under the stated sign convention.
- B. Consumer action: optimal demand response: Under mild price-variation and utility-weight conditions, backward induction produces an affine relationship between optimal aggregated demand and DAHP.The demand sensitivity matrix is −G and is unaffected by the realization of randomness.
- B. Consumer action: optimal demand response: The affine demand result means price changes have deterministic effects on expected demand, without requiring the retailer to estimate every consumer parameter.The model applies to price-elastic demands resulting from optimal control of a linear system.
C. Retailer’s action: optimal dynamic pricing
The retailer chooses day-ahead hourly prices in a Stackelberg game, using expected retail costs and a payoff based on consumer surplus and retail profit. Under social-welfare maximization, the optimal price equals expected real-time retail cost and yields zero retailer profit.
- The retailer is modeled as a price taker in wholesale real-time markets, with expected hourly marginal costs covering wholesale, distribution, and service costs.
- Social welfare is defined as the sum of consumer surplus and retail profit.
- The social-welfare-maximizing day-ahead hourly price equals the expected real-time retail cost.
- Social-welfare maximization gives the retailer zero profit because the day-ahead price matches expected retail cost.
III. PARETO FRONT OF TRADEOFFS
The paper characterizes consumer-surplus and retail-profit trade-offs through weighted retailer optimization and shows that optimal day-ahead prices generate a concave Pareto front. The attainable region is bounded by this front, while common fixed or markup-based prices are generally suboptimal.
- The retailer weights profit against consumer satisfaction through parameter η, with η = 1 representing social welfare and η = 0 representing profit maximization.
- Varying the weighting parameter produces the Pareto front of consumer-surplus and retail-profit trade-offs through optimized day-ahead prices.
- The Pareto front is concave and decreasing, with points on or below it achievable under day-ahead pricing while points above it are infeasible.
- The social-welfare price yields zero retail profit, while regulated-monopoly pricing selects a Pareto-front point with a specified profit level.
- Constant prices, time-of-use prices, and markups on day-ahead wholesale prices generally lie inside the Pareto front and are therefore suboptimal.
- A profit-maximization constraint formulation is equivalent to the weighted formulation for obtaining the same Pareto front.
IV. EFFECTS OF RENEWABLE INTEGRATION
Renewable integration expands the achievable consumer-surplus and retail-profit trade-off region, but its benefits are initially captured by the retailer. As renewable capacity grows, consumers receive an increasing share of the integration benefit.
- The retailer uses owned or contracted renewable generation to compensate real-time loads, while excess renewable power is spilled.
- Renewable integration changes retail profit through the renewable power actually used and the positive-part treatment of residual demand.
- Expected retail profit remains concave in the day-ahead hourly price, so profit maximization can be solved by convex programming.
- Renewables enlarge the achievable consumer-surplus and retail-profit trade-off region and determine how integration benefits are distributed.
- Small renewable capacity sends all integration benefit to the retailer because low-cost renewable energy reduces fulfillment costs without requiring higher consumption.
- As renewable capacity increases, additional renewable power is spilled unless consumption rises, encouraging lower electricity prices and increasing consumer benefits.
- Under social-welfare optimization, consumer benefit approaches the full renewable-integration benefit while retail profit approaches zero as capacity tends to infinity.
V. EFFECTS OF STORAGE AT THE CONSUMER SIDE
Consumer-side storage adds energy-arbitrage flexibility under net metering while preserving the original linear relationship between HVAC consumption and retail price. The resulting storage problem supports an induced consumer-surplus–retail-profit tradeoff curve.
- Storage optimization: Under net metering, storage separates into HVAC control and energy-arbitrage subproblems.The HVAC component remains the previous optimal stochastic control problem, while the second component performs energy arbitrage.
- Storage optimization: Storage does not change the original linear relationship between actual HVAC consumption and retail price.Its consumer benefit arises through arbitrage options rather than a changed HVAC price response.
- Storage optimization: The optimal charging vector r∗(π) is obtained from a deterministic linear program for a given day-ahead price.The formulation incorporates battery dynamics, charging and discharging limits, and storage efficiency factors.
- Tradeoff analysis: Solving the weighted payoff problem produces the induced tradeoff curve between consumer surplus and retail profit as η varies.The storage formulation replaces the no-storage retailer payoff with the storage-adjusted payoff.
VI. NUMERICAL SIMULATIONS
The simulations use real temperature and wholesale-price records from Hartford, Connecticut, together with a specified HVAC thermal-dynamic model. These inputs support numerical evaluation of optimized day-ahead hourly pricing and demand response.
- Simulation setup: The simulations use actual Hartford temperature records from July 1–30, 2012.The study also uses day-ahead wholesale prices for the same location and period.
- Simulation setup: The HVAC thermal-dynamic model uses α = 0.5, β = 0.1, and µ = 0.5.These parameters define the model used in the numerical simulations.
A. Benchmark comparisons
Optimized DAHP forms the benchmark comparison’s Pareto frontier, outperforming constant, time-of-use, and proportional-markup schemes in the reported tradeoff analysis. At fixed regulated retail profit, DAHP provides quantified gains over ToU and CP.
- Evaluation basis: The reported comparisons use tradeoff curves and consumer-surplus tables to evaluate pricing schemes at regulated retail-profit levels.The supplied table passages identify consumer-surplus comparisons but do not provide additional readable values.
- Benchmark definitions: Constant pricing uses one price for all hours, ToU uses distinct peak and normal prices, and PMP indexes retail prices to wholesale prices.These schemes trace their respective CS–RP performance curves by varying their pricing parameters.
- Benchmark comparisons: The trade-off curves between consumer surplus and retail profit were downward and concave.The social-welfare operating point for DAHP resulted in zero retail profit, while zero-profit points for other schemes were close to it.
- Benchmark comparisons: DAHP achieved 10.7% gain over ToU and 17.1% gain over CP when regulated RP was fixed at 85.PMP performed close to DAHP, while ToU and CP performed worse because they lacked flexibility.
B. Characteristics of DAHP
DAHP balances retail profit and consumer surplus through hour-specific prices, with profit-maximizing prices tracking demand-supply dynamics and social-welfare prices remaining more consistent. Renewable integration enlarges the trade-off set, while increasing renewable capacity shifts more benefits toward consumers.
- DAHP trade-offs: Profit-maximizing DAHP prices were higher during peak hours, matching demand-supply dynamics.Consumers used less energy in response to higher prices, and consumer surplus decreased along the Pareto front.
- DAHP trade-offs: Social-welfare-optimal prices were more consistent across hours than profit-maximizing prices.
- Renewable integration: Renewable integration enlarged the consumer-surplus versus retail-profit trade-off curve and made social-welfare pricing yield positive retail profit.The social-welfare-optimal prices became economically viable after renewable integration.
- Renewable integration: At K = 20, the renewable-integration trade-off curve moved upward, indicating benefits to the retailer.
- Renewable integration: As K increased, the fraction of renewable-integration benefits going to consumer surplus increased, with substantial consumer gains as η increased.The fraction approaches 3−2η as K goes to infinity.
- Contribution: DAHP provides a full characterization of consumer-surplus and retail-profit trade-offs while allowing pricing to account for renewable access.
APPENDIX
The appendix derives consumer control and aggregated-demand properties under the model’s one-hour control-period assumption, then connects the resulting optimization to the concave trade-off curve. Its derivation relies on deterministic positive-definite system matrices and first-order optimality conditions.
- Assumptions: The appendix assumes a one-hour control period, while other lengths require transformations of the proof.The paper refers to for general control-period cases.
- Consumer control: Backward induction obtains the optimal control for each consumer.
- Consumer control: The consumer’s total demand is expanded from the optimal-control solution using temperature estimates and ancillary values.
- Matrix properties: G(j) is deterministic, diagonally dominant, and positive definite, supporting the demand characterization.
- Trade-off characterization: The optimal retail profit for a required consumer-surplus level lies on a concave trade-off curve.The derivative of consumer surplus decreases as consumer surplus increases, and no feasible CS-RP pair lies above the curve.
Proof of Theorem 4
The proof compares optimal demand before and after renewable integration using first-order conditions and the renewable-power distribution. It shows that sufficiently small renewable capacity leaves demand unchanged, whereas larger capacity adds a nonnegative adjustment.
- Proof setup: The proof begins from the retailer’s optimized price solution and its associated consumer surplus.
- Demand comparison: Before renewable integration, first-order conditions define the optimal demand d(η); after integration, they define d^w(η).
- Renewable model: The renewable-power distribution enters through the cdf F and pointwise multiplication represented by the Hadamard product.
- Demand comparison: When K < min_i d_i(η), renewable integration leaves optimal demand unchanged; otherwise, demand receives a nonnegative adjustment.For the larger-capacity case, d^w(η) = d(η) + Gδ/(2 − η), with δ ≥ 0.
- Demand comparison: As K approaches infinity, the renewable-integrated optimal demand remains bounded.