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Day-Ahead Offering for Virtual Power Plants: A Stochastic Linear Programming Reformulation and Projected Subgradient Method

Weiqi Meng, Hongyi Li, Bai Cui

Abstract

Virtual power plants (VPPs) are an emerging paradigm that aggregates distributed energy resources (DERs) for coordinated participation in power systems, including bidding as a single dispatchable entity in the wholesale market. In this paper, we address a critical operational challenge for VPPs: the day-ahead offering problem under highly intermittent and uncertain DER outputs and market prices. The day-ahead offering problem determines the price-quantity pairs submitted by VPPs while balancing profit opportunities against operational uncertainties. First, we formulate the problem as a scenario-based two-stage stochastic adaptive robust optimization problem, where the uncertainty of the locational marginal prices follows a Markov process and DER uncertainty is characterized by static uncertainty sets. Then, motivated by the outer approximation principle of the column-and-constraint generation (CC&G) algorithm, we propose a novel inner approximation-based projected subgradient method. By exploiting the problem structure, we propose two novel approaches to improve computational tractability. First, we show that under mild modeling assumptions, the robust second-stage problem can be equivalently reformulated as a linear program (LP) with a nested resource allocation structure that is amenable to an efficient greedy algorithm. Furthermore, motivated by the computational efficiency of solving the reformulated primal second-stage problem and the isotonic structure of the first-stage feasible region, we propose an efficient projected subgradient algorithm to solve the overall stochastic LP problem. Extensive computational experiments using real-world data demonstrate that the overall projected subgradient descent method achieves about two orders of magnitude speedup over CC&G while maintaining solution quality.

Day-Ahead Offering for Virtual Power Plants: A Stochastic Linear Programming Reformulation and Projected Subgradient Method

Abstract

Virtual power plants (VPPs) are an emerging paradigm that aggregates distributed energy resources (DERs) for coordinated participation in power systems, including bidding as a single dispatchable entity in the wholesale market. In this paper, we address a critical operational challenge for VPPs: the day-ahead offering problem under highly intermittent and uncertain DER outputs and market prices. The day-ahead offering problem determines the price-quantity pairs submitted by VPPs while balancing profit opportunities against operational uncertainties. First, we formulate the problem as a scenario-based two-stage stochastic adaptive robust optimization problem, where the uncertainty of the locational marginal prices follows a Markov process and DER uncertainty is characterized by static uncertainty sets. Then, motivated by the outer approximation principle of the column-and-constraint generation (CC&G) algorithm, we propose a novel inner approximation-based projected subgradient method. By exploiting the problem structure, we propose two novel approaches to improve computational tractability. First, we show that under mild modeling assumptions, the robust second-stage problem can be equivalently reformulated as a linear program (LP) with a nested resource allocation structure that is amenable to an efficient greedy algorithm. Furthermore, motivated by the computational efficiency of solving the reformulated primal second-stage problem and the isotonic structure of the first-stage feasible region, we propose an efficient projected subgradient algorithm to solve the overall stochastic LP problem. Extensive computational experiments using real-world data demonstrate that the overall projected subgradient descent method achieves about two orders of magnitude speedup over CC&G while maintaining solution quality.

Paper Structure

This paper contains 40 sections, 4 theorems, 57 equations, 8 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Literature Review
  4. Contribution
  5. Paper Organization
  6. Problem Statement and Optimization Formulation
  7. Problem Setup and Market Setting
  8. Device Modeling
  9. Aggregated Load and PV Unit Model
  10. Energy Storage Model
  11. Power Balance and Imbalance Power
  12. Uncertainty Modeling
  13. PV Availability Uncertainty via a Static Budgeted Set
  14. Day-Ahead Price Scenarios via a Markov Process
  15. Optimization Formulation
  16. ...and 25 more sections

Key Result

Proposition 1

Given a price scenario $\omega\in\Omega$ and a PV realization $\hat{p}^{\mathrm{PV}}\in\mathcal{U}$. As long as i) the imbalance settlement rates in eq:imb_rates satisfy $\lambda^{\mathrm{imp}}_{t,\omega}, \lambda^{\mathrm{exp}}_{t,\omega} > 0$ for all $t\in\mathcal{T}$ (that is, $\lambda^{\mathrm{D for the optimal solution to the second-stage problem eq:cost-to-go function without constraint eq:E

Figures (8)

  • Figure 1: Two ways of approximation of the value function.
  • Figure 2: Pre-determined uncertainty parameters.
  • Figure 3: Performance comparison of the proposed greedy algorithm and the benchmark commercial LP solver under different numbers of price trajectories. The top row illustrates the iteration process and optimality, while the bottom row reports the computational time per iteration (maximum iterations = 50, $\Gamma = 6$).
  • Figure 4: Performance comparison of the iteration process, optimality and computational time for the PSM-PAVA and traditional CC&G algorithm for the 2S-ARO problem (Maximum iteration =600; $\Gamma=6$).
  • Figure 5: Offering curves in the day-ahead market for time slots 1, 11, 12, and 22.
  • ...and 3 more figures

Theorems & Definitions (9)

  • Remark 1
  • Proposition 1
  • Proposition 2: Decoupling of the second-stage problem
  • Proposition 3
  • Proposition 4
  • proof
  • proof
  • proof
  • proof