Source-linked AI summary
Data-based Distributionally Robust Stochastic Optimal Power Flow, Part I: Methodologies
Yi Guo, Kyri Baker, Emiliano Dall'Anese, Zechun Hu, Tyler H. Summers
TL;DR
Stochastic OPF must account for forecast errors, but error distributions are not generally known and finite datasets can leave decisions vulnerable to sampling error. This paper proposes a multi-stage data-based distributionally robust OPF framework using feedback policies and Wasserstein ambiguity sets centered on historical data. The framework is formulated for distribution and transmission systems and is designed to provide robustness to forecast errors and finite-data sampling errors.
Problem
Stochastic OPF methods often rely on prescribed forecast-error distributions, while decisions based on limited data can be overly dependent on that data.
Source: §I · pp. 1–2Method
The paper formulates multi-stage data-based distributionally robust OPF using feedback policies and Wasserstein ambiguity sets centered on finite training datasets.
Source: §I · p. 2, §II · p. 3, B. Data-based distributionally robust… · p. 4Results
The framework is formulated for distribution and transmission systems and is designed to provide policies robust to forecast errors and sampling errors in finite training datasets.
Source: §I · p. 2, §VI · p. 8Takeaways & Limitations
The method provides a framework for balancing operational efficiency with constraint-violation and sampling-error risks in power-system control.
Source: §I · p. 2, §VI · p. 8
Abstract
from arXiv · showhide
We propose a data-based method to solve a multi-stage stochastic optimal power flow (OPF) problem based on limited information about forecast error distributions. The framework explicitly combines multi-stage feedback policies with any forecasting method and historical forecast error data. The objective is to determine power scheduling policies for controllable devices in a power network to balance operational cost and conditional value-at-risk (CVaR) of device and network constraint violations. These decisions include both nominal power schedules and reserve policies, which specify planned reactions to forecast errors in order to accommodate fluctuating renewable energy sources. Instead of assuming the uncertainties across the networks follow prescribed probability distributions, we consider ambiguity sets of distributions centered around a finite training dataset. By utilizing the Wasserstein metric to quantify differences between the empirical data-based distribution and the real unknown data-generating distribution, we formulate a multi-stage distributionally robust OPF problem to compute optimal control policies that are robust to both forecast errors and sampling errors inherent in the dataset. Two specific data-based distributionally robust stochastic OPF problems are proposed for distribution networks and transmission systems.
I. INTRODUCTION
The paper addresses forecast-error uncertainty and finite-data sampling errors in stochastic OPF, proposing a multi-stage data-based distributionally robust framework. Its feedback-control formulation combines nominal schedules with planned reactions to forecast errors and repeatedly solves stage-wise problems.
I. INTRODUCTION
- Existing OPF methods often assume point forecasts or prescribed error distributions, while finite training data can leave decisions vulnerable to sampling error and overfitting.The paper identifies sampling errors from limited data as a gap in existing work and links overdependence on small datasets to poor out-of-sample performance.Source: §I · pp. 1–2
- The proposed framework formulates multi-stage distributionally robust OPF using limited forecast-error information from finite historical datasets.It focuses on Wasserstein balls centered on empirical distributions, with a radius that controls conservativeness.Source: §I · p. 2
II. STOCHASTIC OPF AS STOCHASTIC OPTIMAL CONTROL
- The control policy uses state and forecast-error histories to choose controllable-device inputs, including nominal plans and reactions as errors are realized.The general formulation minimizes worst-case expected operating cost and constraint-violation risk subject to system dynamics and network and device constraints.Source: §II · p. 3
- A distributionally robust model predictive control scheme repeatedly solves stage-wise problems using affine feedback policies and linear device-dynamics models.The framework permits updating forecast-error datasets and recomputing decisions with newer information.Source: §I · p. 2, §II · p. 3
A. Ambiguity sets and Wasserstein balls
The section centers uncertainty about forecast-error distributions on a finite training dataset and uses Wasserstein ambiguity sets to formulate tractable distributionally robust optimization. A model predictive control procedure applies this approach stage by stage, with online dataset updates possible.
A. Ambiguity sets and Wasserstein balls
- The Wasserstein approach seeks worst-case expectations over an ambiguity set and can be reformulated as a finite-dimensional convex problem solvable with existing convex optimization solvers.The section contrasts this with approaches that rely on specific distributional forms or other ambiguity-set constructions.Source: A. Ambiguity sets and Wasserstein balls · p. 3
- Wasserstein ambiguity sets contain distributions within a radius of the uniform empirical distribution on the training dataset, and the radius controls conservativeness and robustness to sampling errors.The radius can be chosen so the ball contains the true distribution with a prescribed confidence level and yields performance guarantees.Source: A. Ambiguity sets and Wasserstein balls · pp. 3–4
B. Data-based distributionally robust model predictive control
- The framework combines operating cost with a weighted constraint-violation risk function formed from CVaR measures for network and device constraints.The weight represents the network operator’s risk aversion, and the CVaR confidence level determines the risk level at which violation magnitude is penalized.Source: B. Data-based distributionally robust… · p. 4
- The proposed distributionally robust model predictive control algorithm optimizes over affine feedback policies using a Wasserstein ambiguity set, implementing only the immediate control decision at each stage.At the next time step it observes new forecast errors and states, resolves the problem, and repeats.Source: B. Data-based distributionally robust… · p. 4
- The forecast-error dataset defining the ambiguity-set center can be updated online or have outdated data removed to accommodate time-varying distributions.The approach permits any forecasting methodology for predicting uncertainties over the planning horizon.Source: B. Data-based distributionally robust… · p. 4
III. SYSTEM MODEL
The system model represents a steady-state power network with bus voltages, current injections, and active and reactive power injections. Because the AC power-flow equations are nonconvex, the paper uses linearization methods to support tractable stochastic OPF formulations.
A. Network model
- The network is represented by buses and connecting lines, with line admittances assembled into a matrix and currents related to voltages through the network equation.The model uses line-to-ground voltage and current-injection phasors and a standard Pi model for lines.Source: A. Network model · p. 4
- Active and reactive power injections are separated into nominal control-dependent terms and forecast-error-dependent terms.Positive injections denote generation, while negative injections denote absorption.Source: A. Network model · p. 4
- Two linearization methods are used to handle nonconvexity of the power-flow equations in transmission and distribution networks.The distribution-network approximation is described as preserving network structure for real and reactive power flows while enabling stochastic optimization.Source: A. Network model · p. 4
B. Dynamic of grid-connected devices
Grid-connected devices are modeled according to whether their power flow is controllable or fixed or uncertain. Controllable-device power flows follow discrete-time linear dynamics, which can represent internal dynamics such as storage state of charge.
B. Dynamic of grid-connected devices
- The device set includes generators, inverter-based renewable sources, loads, and storage devices, which may act as generators or loads.Fixed loads are examples of devices whose power flow is not affected by decision variables.Source: B. Dynamic of grid-connected devices · p. 5
- Controllable-device power flow is modeled as a discrete-time linear dynamical system whose state and input describe device behavior over time.The internal state can include a storage device’s state of charge, and the device input corresponds to its power injection into the network.Source: B. Dynamic of grid-connected devices · p. 5
C. Network constraints
The network constraints are represented using linear power-flow approximations because AC OPF is nonconvex and computationally difficult. The distribution model assumes a symmetric, balanced network connected to the grid at a PCC.
C. Network constraints
- AC power-flow equations make prototypical AC OPF nonconvex and NP-hard, hindering computationally affordable chance-constrained formulations using CVaR.With an accurate linear model, the framework can formulate convex constraints on line flows and voltage magnitudes.Source: C. Network constraints · p. 5
- Local device limits, including storage state of charge, power ranges, and generator ramping, can be modeled or approximated as linear inequalities.These constraints complement network constraints such as voltage limits.Source: C. Network constraints · p. 5
A. Distribution network model
- Distribution networks are modeled as symmetric and balanced, with node 0 representing the PCC and loads and generators potentially connected at each bus.Node 0 is modeled as a slack bus; the other buses are PQ buses with specified injected complex power.Source: A. Distribution network model · p. 5
- The distribution-network formulation uses a convex, computationally efficient approximation of power flow that preserves network structure for real and reactive power flows.The approximation enables stochastic optimization techniques to incorporate forecast errors.Source: A. Distribution network model · p. 5
B. Leveraging approximate power flow
The distribution-network approach linearizes the AC power-flow relationship around a nominal voltage. The resulting voltage approximations support linear voltage constraints and stochastic optimization with forecast errors.
B. Leveraging approximate power flow
- Voltage vectors are approximated as affine functions of active and reactive power injections, yielding linear voltage-magnitude expressions.The resulting approximations are used to express voltage constraints.Source: B. Leveraging approximate power flow · p. 5
- Neglecting second-order voltage-deviation terms makes the power-balance relation linear around a nominal operating point.The nominal-voltage choice sets the linearization terms Λ and Ψ to zero.Source: B. Leveraging approximate power flow · p. 6
- The voltage-magnitude approximation assumes the nominal voltage dominates its deviation, and the approach can apply to multiphase feeders with wye and delta connections.The cited approach in accounts for those connection types.Source: B. Leveraging approximate power flow · p. 6
C. Data-based stochastic OPF formulation for distribution networks
The distribution-network formulation applies data-based distributionally robust CVaR to selected affine constraints. It optimizes controllable set points using a finite training dataset while balancing operating cost, risk, and sampling error.
C. Data-based stochastic OPF formulation for distribution networks
- The operator selects a subset of affine constraints for distributionally robust treatment, with risk assessed individually rather than jointly.Joint constraint-violation risk is possible but is left for future work because it is more difficult.Source: C. Data-based stochastic OPF… · p. 6
- The decision vector includes renewable curtailment ratios and other controllable-device set points, while uncertainty includes active and reactive power forecast errors.The active-power curtailment ratio specifies the fraction of renewable injection curtailed.Source: C. Data-based stochastic OPF… · p. 6
- The optimization minimizes a weighted sum of operating cost and worst-case CVaR for affine constraints, including voltage and local device limits.The reformulated problem is a quadratic program.Source: C. Data-based stochastic OPF… · p. 7
- The quadratic program explicitly uses the training dataset; risk aversion and the Wasserstein radius balance efficiency, risk, and sampling errors.The risk-aversion parameter is ρ, and the Wasserstein radius is ετ.Source: C. Data-based stochastic OPF… · p. 7
V. DATA-BASED DISTRIBUTIONALLY ROBUST STOCHASTIC
The transmission-system formulation uses a DC power-flow approximation to relate nodal injections to line flows. It relies on lossless lines, near-unit voltage magnitudes, and neglect of reactive power.
A. Illustrative explanation for the DC approximation
- The DC approximation treats lines as lossless, assumes voltage magnitudes are constant and close to one per unit, and ignores reactive power.Under these assumptions, line flows are linearly related to nodal power injections.Source: A. Illustrative explanation for the DC… · p. 7
- A network-derived mapping converts nodal power injections or consumptions into line flows, which are constrained by line-flow limits.The mapping can be constructed from network line impedances.Source: A. Illustrative explanation for the DC… · p. 7
- The power balance constraint is expressed in the DC-model formulation using time-indexed quantities.The notation defines I_t as a t-dimensional identity matrix and ⊗ as the Kronecker product.Source: A. Illustrative explanation for the DC… · p. 7
B. Reserve policies
Reserve policies make stochastic OPF decisions responsive to forecast errors through causal feedback, while affine policies keep the optimization tractable. The transmission formulation balances operating cost against worst-case CVaR of line-flow and device-constraint violations using finite training data.
B. Reserve policies
- Affine reserve policies make device power schedules respond causally to forecast errors while avoiding the infinite-dimensional optimization over general policies.The response matrices are lower-triangular and represent planned reserves; under a convex quadratic operating cost, expected cost depends on the first and second moments of forecast errors.Source: B. Reserve policies · p. 7
C. Data-based stochastic OPF formulation for transmission systems
- The transmission OPF weighs operating cost against worst-case CVaR of individual line-flow and local device constraint violations.The constraints are expressed using decision variables for affine reserves and offsets, and the formulation is a quadratic program using the training dataset.Source: C. Data-based stochastic OPF… · p. 8
- The training dataset, risk-aversion parameter ρ, and Wasserstein radius ετ jointly set the tradeoff among efficiency, risk, and sampling error.The uncertainty set is modeled as a polytope.Source: C. Data-based stochastic OPF… · p. 8
VI. CONCLUSIONS
The framework computes multi-stage feedback policies from finite forecast-error datasets, using Wasserstein ambiguity sets and convex reformulations for tractability. Part II reports superior out-of-sample performance and a more systematic tradeoff between efficiency and risk.
VI. CONCLUSIONS
- Part II reports superior out-of-sample performance and a more principled, systematic tradeoff between efficiency and risk.Its case studies address over-voltages in a high-solar distribution network and line-flow risks in a high-wind transmission system.Source: §VI · p. 8
- Wasserstein balls centered on empirical distributions enable convex reformulations that make multi-stage feedback-policy computation efficient.The framework adapts to distribution and transmission networks using general device models and different linearizations of AC power flow.Source: §VI · p. 8
SUPPLEMENTARY MATERIALS
- Implementation codes for data-based distributionally robust stochastic OPF and DC OPF are downloadable from a link, while numerical case studies on distribution-network overvoltage and transmission-system N-1 security, with simulation discussions, appear in.Source: SUPPLEMENTARY MATERIALS · p. 9
APPENDIX
The appendix describes reformulations of probabilistic transmission line-flow constraints under affine reserve policies. It notes that analogous steps yield constraints for devices and voltage magnitude.
APPENDIX
- The appendix reformulates probabilistic line-flow constraints over the time interval [0, t] for affine reserve policies.It gives a CVaR counterpart for each individual constraint.Source: APPENDIX · p. 9
- The affine reserve-policy terms enter the line-flow constraint alongside the device input.Source: APPENDIX · p. 9
- Similar steps produce constraints for each device and for voltage magnitude.Source: APPENDIX · p. 9