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Approximating Energy-Regulation Feasible Region of Virtual Power Plants: A Data-driven Inverse Optimization Approach
Ruike Lyu, Hongye Guo, Qixin Chen
TL;DR
VPPs need aggregate energy-regulation feasible regions for market bidding, but analytical aggregation methods have limited generality and adaptability. The paper fits a virtual-battery aggregate model by inverse optimization on multi-scenario dispatch data, with numerical tests showing improved accuracy under coupled DER operation.
Problem
Existing analytical feasible-region aggregation methods have limited generality and adaptability, and few studies address coupled energy-regulation feasible regions for VPPs.
Method
The method fits virtual-battery aggregate feasible-region parameters through inverse optimization using optimal dispatch data generated by original DER operating models.
Results
With coupled DER feasible regions, the proposed method reduced errors by over 40% compared with outer approximation, while outer-approximation errors exceeded 10%.
Takeaways & Limitations
The method can directly determine VPP feasible-region parameters for ISO bidding without relying on the analytical form of DER operating models.
Abstract
from arXiv · showhide
System operators will probably allow virtual power plants (VPPs) to submit their feasible region (FR) for market clearing and dispatch. A VPP needs to determine its FR to submit as a whole based on the individual operation model of its internal distributed energy resources (DERs), which is an FR aggregation problem. Existing FR aggregation approaches rely on analytical methods, which have issues with generality and adaptability. In this paper, we propose a data-driven approach to approximate the energy-regulation FR of VPPs. It adopts the virtual battery model to approximate the aggregate FR of a VPP and determines the model parameters through inverse optimization based on generated multi-scenario operation data using the original operation model. Numerical tests verified the accuracy of the proposed method. We believe that our work helps to better leverage the flexibility of DERs.
I. INTRODUCTION
VPPs aggregate DERs to submit feasible regions for energy and regulation markets, but existing analytical aggregation methods have limited generality and adaptability. The paper proposes fitting a virtual-battery aggregate model from multi-scenario operation data using inverse optimization.
- Motivation: VPPs aggregate small-capacity DERs so they can submit a collective feasible region for market participation.Aggregation is required for VPPs to reach market participation thresholds.
- Aggregation challenge: Aggregate feasible regions combine multiple resource sets through a Minkowski sum rather than simple parameter addition, lacking an effective general solution.Approximation methods such as inner or outer approximations are therefore commonly used.
- Limitations of existing methods: Analytical aggregation methods assume resource relationships and feasible-region forms that may not match coupled energy-regulation operation or ISO bidding models.Examples include independence assumptions and box or ellipse representations.
- Research gap: Few studies address aggregation of VPP energy-regulation feasible regions, while data-driven aggregation research remains at an early stage.The paper focuses on this underexplored combination.
- Proposed approach: The proposed method fits virtual-battery feasible-region parameters through inverse optimization using optimal dispatch data generated by original DER models.It directly determines bidding parameters without relying on specific DER operating-model forms.
II. PROBLEM DESCRIPTION
The VPP feasible region is the projection of detailed DER operating constraints onto energy and regulation variables. Because exact projection is often infeasible for thousands of DERs, the paper replaces the original model with a lower-dimensional approximation whose parameters must be fitted carefully.
- Feasible-region formulation: The VPP feasible region contains baseline energy output and regulation capacity trajectories satisfying the original DER operating constraints.The region is represented over T scheduling intervals and I DERs.
- Computational challenge: Exact feasible-region computation projects out internal DER variables from the original constraints, an NP-hard task that is often infeasible for thousands of DERs.The original constraints include power and energy limits across the VPP’s resources.
- Approximation strategy: The proposed formulation approximates the original constraints with a much smaller model and constructs an approximate feasible region from those constraints.This reduces the dimensionality of the representation.
- Approximation strategy: If the approximate model has far fewer DERs than the original VPP, deriving or submitting its feasible region is less computationally intensive.The paper gives I = 1000 electric vehicles and an approximate model with Ia = 2 as an example.
- Parameter fitting: The central fitting problem is to choose approximate-model parameters so its feasible region remains sufficiently close to the original region.A loss function J measures the distance between the two regions.
III. METHODOLOGY
The methodology uses virtual-battery constraints to represent coupled energy-regulation operation and aggregates DER outputs into VPP bids. It does not require formal consistency between the approximate and original model formulations.
- Virtual-battery model: The virtual-battery model represents DER operation through time-coupled power and energy constraints for storage, electric vehicles, and thermostatically controlled loads.The model captures common flexible-resource operating behavior.
- Virtual-battery model: The model enforces power limits, energy limits, energy transitions, and initial-energy conditions for each resource.These constraints define each resource’s hourly output behavior.
- Regulation constraints: Regulation provision adds nonnegative-capacity, power-capacity, and maintenance-time requirements.The required maintenance duration is represented by Δt_req.
- Aggregation: VPP bids aggregate the DER baseline-power and regulation-capacity outputs across resources.The internal variables include each DER’s power, regulation capacity, and energy.
- Model generality: The proposed method does not depend on formal consistency between approximate and original models, and tests include differing formulations.This supports testing across different individual-resource operating models.
B. Determining the parameters of the approximate model
The approximate model parameters are identified by comparing dispatch outcomes from the original VPP model across energy-regulation price scenarios with those produced by the approximate feasible region.
- The approximate feasible region is considered sufficiently close when it reproduces dispatching results close to those obtained using the original feasible region across scenarios.This comparison provides the basis for determining the approximate model parameters.
- The energy-regulation dispatch problem includes energy price, regulation capacity price, performance, mileage price, expected mileage, and resource degradation cost.The degradation cost is assumed proportional to the VPP’s total discharge power.
- Price-regulation scenarios are used to generate corresponding VPP scheduling results from the original operational model.The VPP is treated as a price taker under perfect competition, so scenarios can be represented by energy-regulation prices.
- The generated price-dispatch pairs form a dataset indexed by scenario, containing energy-regulation prices and scheduling results.The dataset includes the scenario index k and total number of scenarios K.
- The original VPP constraints can replace the projected feasible-region constraint without changing the optimization problem’s optimal solution.This follows from the invariance of the optimum before and after projection.
2) Parameter fitting:
Parameter fitting formulates generated scheduling outcomes as optimal solutions of the approximate model and enforces their feasibility and optimality conditions through DDIO.
- 2) Parameter fitting:: The DDIO problem fits approximate-model parameters so generated scheduling results are treated as optimal solutions of the approximate dispatch problem.The data may include noise, while the fitted parameters are constrained by the approximate model.
- 2) Parameter fitting:: Primal feasibility requires each fitted scheduling outcome to satisfy the approximate model constraints.This condition is represented by ha(pa(k)).
- 2) Parameter fitting:: Strong duality, primal feasibility, dual feasibility, and stationarity conditions jointly guarantee optimality of the fitted scheduling outcomes.The Lagrangian dual function and corresponding constraints are provided in the appendix.
C. An iterative algorithm for solving the DDIO problem
The DDIO parameters are solved iteratively, updating the model with one scenario at a time and averaging recent parameter estimates until convergence or a preset iteration limit.
- C. An iterative algorithm for solving the DDIO problem: The DDIO parameter-fitting problem is a high-dimensional nonlinear optimization that becomes especially difficult when the number of scenarios K is large.A scenario-by-scenario update strategy is adopted instead of relying on Newton’s method for a specific optimization form.
- C. An iterative algorithm for solving the DDIO problem: Algorithm 1 initializes the approximate-model parameters and repeatedly solves a scenario-specific update problem.The algorithm cycles through scenarios using k = n mod K + 1.
- C. An iterative algorithm for solving the DDIO problem: The parameters are updated by averaging the latest min{K, n} scenario estimates and stopping when the parameters converge or the maximum iteration count is reached.The algorithm outputs an approximate model parameterized by θa.
- C. An iterative algorithm for solving the DDIO problem: The scenario setting combines typical PJM energy-regulation prices with EV arrival and departure times.These are the two dimensions shown in Fig. 2.
- C. An iterative algorithm for solving the DDIO problem: A regularization term supports convergence, its coefficient α increases adaptively, and feasible solutions at a computation-time limit can still update the parameters.The subproblem can be submitted directly to commercial solvers.
IV. NUMERICAL RESULTS
The numerical optimization problems were solved using Gurobi and YALMIP on a specified workstation.
- IV. NUMERICAL RESULTS: Gurobi V10.0.0 was used to solve the optimization problems.
- IV. NUMERICAL RESULTS: YALMIP provided the modeling environment for the optimization problems.
- IV. NUMERICAL RESULTS: The computations ran on an Intel Core i9-10900X CPU at 3.7 GHz with 128 GB RAM.
A. Scenario setting and data generation
The study uses 4000-EV energy-regulation scenarios and tests the proposed aggregation method with both decoupled and coupled DER operation models.
- Scenario setting: 4000 EVs were modeled for joint energy and regulation services using bidirectional 7.68kW chargers and an 80% departure energy requirement.Batteries started at 0kWh, had 50kWh maximum capacity, and discharging compensation was set at $0.1/kWh.
- Scenario setting: Historical PJM energy and regulation market prices from July 2022 represented different dispatching scenarios.
- Data generation: Data from July 1–20 generated fitting samples, while July 21–30 tested errors against the original 4000-EV model.Each fitting scenario contained 2×T matrices with T = 16.
- Model comparison: The experiments evaluated both decoupled and coupled DER regulation models to test whether the method applies to different individual FR forms.The outer approximation method from Ref. was used for comparison.
B. Results and comparison
The proposed and outer-approximation aggregate FR parameters are close in a typical case, while dispatch accuracy differs substantially when DER regulation constraints are coupled.
- Aggregate FR parameters: The proposed method and outer approximation produced close typical aggregate FR parameters.These parameters are shown for the decoupled operation model in Fig. 3.
- Dispatch comparison: With decoupled DER operation, both aggregate FR approximation methods produced dispatch errors below 2%.Errors were measured as MAE divided by the maximum true value from the original model.
- Dispatch comparison: With coupled DER feasible regions, outer-approximation errors exceeded 10%, while the proposed method reduced errors by over 40% relative to that method.
- Overall result: The numerical results indicate that the proposed method was at least not inferior in accuracy and performed better in some tested scenarios because of its broader generality.The conclusion states that coupled-constraint cases reduced aggregate FR error by over 40%.
APPENDIX A DUAL FUNCTION AND CONSTRAINTS OF PROBLEM (7)
The appendix presents the Lagrangian dual function and dual-feasibility constraints associated with problem (7).
- Dual function: The appendix introduces the Lagrangian dual function for problem (7).
- Notation: The formulation omits the superscript k in the displayed appendix notation.The omission is stated as applying for all k.
- Dual constraints: It specifies dual-feasibility conditions for every indexed entity i.The passage explicitly marks the condition as applying ∀i.