Optimal Power Flow Pursuit

Emiliano Dall'Anese; Andrea Simonetto

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Optimal Power Flow Pursuit

Emiliano Dall'Anese, Andrea Simonetto

arXiv:1601.07263v4Jan 27, 2016math.OC

TL;DR

Distribution OPF and feedback control can operate too slowly and separately for rapidly changing feeder conditions. The paper develops distributed feedback controllers based on linearized AC power flow and Lagrangian methods, and analytically establishes convergence and tracking of time-varying OPF targets. The controllers use elementary operations suitable for low-cost inverter microcontrollers.

Problem
Fast-changing distribution conditions make conventional OPF computation and setpoint dispatch potentially too slow, while existing local feedback controls do not guarantee system-level optimality.
Method
The paper combines linear approximations of AC power flow with distributed primal-dual and Lagrangian regularization methods to update inverter setpoints from voltage feedback and OPF targets.
Results
The proposed controllers analytically achieve convergence and OPF-target tracking, with Q-linear convergence to a neighborhood whose asymptotic error bounds decrease with the regularization parameter ϵ.
Takeaways & Limitations
The framework supports continuous pursuit of AC OPF solutions in distribution systems with arbitrary topologies and time-varying loads and ambient conditions.

Abstract

from arXiv · show

This paper considers distribution networks featuring inverter-interfaced distributed energy resources, and develops distributed feedback controllers that continuously drive the inverter output powers to solutions of AC optimal power flow (OPF) problems. Particularly, the controllers update the power setpoints based on voltage measurements as well as given (time-varying) OPF targets, and entail elementary operations implementable onto low-cost microcontrollers that accompany power-electronics interfaces of gateways and inverters. The design of the control framework is based on suitable linear approximations of the AC power-flow equations as well as Lagrangian regularization methods. Convergence and OPF-target tracking capabilities of the controllers are analytically established. Overall, the proposed method allows to bypass traditional hierarchical setups where feedback control and optimization operate at distinct time scales, and to enable real-time optimization of distribution systems.

I. INTRODUCTION

The paper targets fast, distributed control of inverter-interfaced resources so distribution-system operation can continuously pursue time-varying AC OPF solutions despite rapidly changing conditions.

Existing Volt/VAr, Volt/Watt, and droop-based controls regulate local voltages but do not guarantee system-level optimality.
OPF inputs and computation can be too slow for distribution dynamics, requiring inverter setpoints to be updated every second in a representative feeder.The example concerns five secondary transformers in Anatolia, California.
The proposed distributed controllers use fast inverter feedback to continuously drive outputs toward OPF-based targets while bypassing separate optimization and feedback time scales.
The framework combines linearized AC power-flow models with double-smoothing methods and supports elementary implementations on low-cost microcontrollers without requiring all feeder loads.
The approach extends prior work to AC OPF in arbitrarily topologized distribution systems with time-varying loads and ambient conditions, while establishing convergence and optimality.

A. System model

The system model represents a discretized distribution feeder with a slack transformer bus, constant-power loads, and inverter-based renewable sources subject to physical operating limits.

The feeder contains N+1 nodes, with node 0 serving as the transformer secondary and slack bus, and time is discretized into slots t=kτ.
Renewable sources are located at nodes G, and their available real generation varies with ambient conditions such as photovoltaic irradiance.
High renewable penetration can produce overvoltages when generation exceeds demand and fast output variations that cause transients and legacy-switchgear wear.
The inverter operating region captures real and reactive power limits through rated apparent power and optional minimum-power-factor constraints.
The framework can accommodate other devices, including diesel generators, fuel cells, and variable-speed drives, by representing their physical limits in the operating set.

B. Problem setup

The problem setup defines closed-loop control objectives that update renewable setpoints quickly while pursuing solutions of a time-varying AC OPF problem with voltage and operational constraints.

The controllers regulate renewable real and reactive powers in closed loop using measurements of electrical quantities generated by feeder physics and primary inverter dynamics.
Voltage magnitudes are constrained between minimum and maximum service limits at strategically selected nodes to enforce feeder-wide regulation.
The AC OPF objective combines time-varying inverter performance functions with system-level objectives such as losses or deviations from nominal voltage.
Because the OPF is nonconvex and NP-hard, centralized and distributed solvers may dispatch outdated setpoints as demand and ambient conditions change.
The control design instead aims to update inverter setpoints at a fast time scale while continuously regulating outputs to a solution of the time-indexed OPF.

A. Leveraging approximate power-flow models

The paper linearizes AC power flow around a nominal voltage profile to obtain voltage–power relations that support fast, low-complexity distributed controller design.

The approximation discards second-order voltage-deviation terms, producing a linear relationship between injected complex powers and voltages.
Net injected powers combine inverter injections with loads, while voltage magnitudes are collected into a real vector for approximate power-flow modeling.
The resulting voltage constraints become linear inequalities in real and reactive injections, with power balance intrinsically satisfied at all nodes.
Choosing the nominal voltage profile to null specific linearization terms yields a simplified linearized power-flow expression.
Matrices R and B are constructed from the real and imaginary parts of the inverse admittance matrix and the nominal voltage profile.
The approximation is justified when nominal-voltage entries dominate deviations, with analytical error bounds reported in prior work.

B. Target time-varying optimization problem

The paper formulates a convex, time-varying surrogate for AC OPF and uses regularized primal-dual optimization to pursue its solutions under changing operating conditions. The formulation supports distributed controllers without requiring strongly convex cost functions, while quantifying approximation and tracking errors.

Convex surrogate: The target OPF is approximated using convexified power-flow relations, with inverter operating regions represented by convex, closed, bounded sets and voltage regulation enforced through linear constraints.The formulation includes 2M constraints to regulate voltage magnitudes, while allowing additional OPF constraints without changing the feedback-controller design.
Convex surrogate: Under Lipschitz-gradient and Slater assumptions, the approximated problem is convex, satisfies strong duality, and has an optimizer at every time k.The optimizer is denoted {uopt,k_i}i∈G.
Regularized optimization: The regularized Lagrangian is strictly convex in primal variables and strictly concave in dual variables, enabling gradient-based saddle-point methods without strongly convex cost functions.Tikhonov regularization also avoids averaging primal and dual variables.
Approximation error: The regularized saddle-point solution can differ from the original OPF optimizer, but both decision and constraint discrepancies are bounded proportionally to √ϵ.Thus, reducing ϵ reduces the discrepancy between the regularized and original problems.
Controller iteration: The proposed primal-dual iteration uses a stepsize and projections onto inverter operating regions and nonnegative multiplier sets, but initially updates setpoints in an open-loop fashion.The paper subsequently incorporates physical feedback to adapt the updates to changing operating conditions.
Time variation: For time-varying operation, bounded changes in primal optima and constraint functions yield bounded changes in the dual variables and the combined primal-dual optimizer.Convergence of the primal-dual gradient method is investigated under Assumptions 1–4.

C. Feedback controllers pursuing OPF solutions

The proposed controllers use voltage feedback and regularized-Lagrangian updates to steer inverter setpoints toward time-varying OPF solutions. Under stated assumptions and stepsize conditions, the iterates converge linearly up to a bounded tracking error, while accommodating practical measurement and actuation effects.

Controller operation: At each update, the controller collects feeder voltage measurements and updates RES-inverter power setpoints in closed loop.The update cycle repeats by returning to the measurement step.
Information requirements: The controllers require feeder and line models but do not require load information at nodes outside the RES locations.This distinguishes the information used by the feedback scheme from traditional distributed optimization approaches described in the passage.
Controller operation: The controller’s gradient updates are ε-gradients of a regularized Lagrangian, with inverter-local updates admitting closed-form solutions for various feasible sets.Some updates can be computed at each inverter or at the utility/aggregator, depending on voltage-measurement availability.
Convergence and tracking: Under Assumptions 1–5 and suitable positive parameters, the time-varying mapping is strongly monotone and Lipschitz, supporting Q-linear convergence to the optimizer up to an asymptotic error bound.The strong-monotonicity constant is η = min{ν, ϵ}; the stepsize condition enforces a contraction factor below one.
Convergence and tracking: The tracking error reflects a trade-off in stepsize selection: smaller α reduces the gradient-error contribution but worsens convergence, whereas larger α has the opposite effect.The bound also accounts for measurement errors, power-flow approximation errors, and updates faster than inverter settling.
Scope and implementation: The framework is stated to extend from balanced networks to multi-phase unbalanced systems with arbitrary topology and inverter placement.The paper attributes this extension to the linearized model and controller embedding at any phase and node.

IV. EXAMPLE OF APPLICATION

The IEEE 37-node feeder case evaluates the proposed controllers against local Volt/VAr control under rapidly varying PV and load conditions. The controllers regulate voltages while limiting real-power curtailment and reactive-power support, and track changing voltage bounds.

Test setup: The test case uses a modified single-phase IEEE 37-node feeder with one-second Anatolia, California load and solar-irradiance data and eighteen PV systems.The PV inverters have ratings of 300 kVA, 350 kVA, or 200 kVA depending on the unit.
Test setup: PV inverters update reactive setpoints every 0.33 seconds, while the proposed controller iterations use the same interval.The comparison sets voltage limits to 1.05 pu and 0.95 pu and uses local Volt/VAr control without a deadband.
Voltage regulation: Volt/VAr control regulates voltage except from 11:30 to 13:00, when available reactive power is insufficient.The maximum voltage occurs at node 35 in the illustrated profiles.
Voltage regulation: The proposed controllers enforce voltage regulation and produce a flat voltage profile from 9:30 to 14:00 while minimizing real-power curtailment and reactive-power provision.The control objective keeps operation close to the available real-power point while satisfying voltage regulation.
Cost and comparison: The proposed controllers provide voltage regulation with minimal real-power curtailment and reactive-power support, whereas Volt/VAr cannot ensure regulation during solar-peak hours.Lower reactive-power absorption also lowers currents on distribution lines.
Time-varying targets: When V max changes from 1.05 pu to 1.035 pu and then 1.02 pu, the proposed controllers quickly regulate voltages within the desired bounds.The controller can also flatten voltage magnitudes at another value by adjusting V max.

V. CONCLUDING REMARKS

The paper develops low-complexity feedback controllers that pursue AC OPF solutions using linearized power flow and primal-dual methods. Their tracking capabilities are analytically established and numerically corroborated, including under time-varying voltage targets.

Concluding remarks: The controllers seek RES setpoints corresponding to AC OPF solutions using linearized AC power-flow equations and primal-dual methods.The resulting fast-acting controllers are designed for microcontrollers accompanying gateway and inverter interfaces.
Concluding remarks: The controllers’ tracking capabilities are analytically established and numerically corroborated.The framework is applied to time-varying voltage limits in the numerical example.

A. Proof of Theorem 1

The proof bounds the controller iterates’ distance from the time-varying optimizer using projection non-expansivity, strong monotonicity, Lipschitz continuity, and gradient-error bounds. Under a contraction condition, the iterates converge Q-linearly to a neighborhood whose asymptotic error is bounded.

The optimizer is a fixed point of the projected iterations, enabling the distance recursion used in the proof.
The analysis bounds the gradient error and combines this bound with inequalities for the iterate-distance recursion.
The mapping Φk is strongly monotone with constant η and Lipschitz continuous with constant Lν,ϵ.
If ρ(α) < 1, the resulting recursion is a contraction and can be summed using the geometric-series formula.
The controller iterates converge Q-linearly to a neighborhood of zero, with an asymptotic error bound.

B. Setpoint update

The setpoint update projects unprojected real and reactive power values onto device operating regions. The paper gives closed-form updates for several controllable-device settings, including real-power-only, reactive-power-only, and joint real/reactive-power control.

The setpoint update (18c) has a closed-form solution for a variety of renewable energy sources and other controllable devices.
Joint real and reactive power control: For joint control, the unprojected point contains real and reactive power values that are projected onto the feasible operating set.
Real power control: Real-power-curtailment-only strategies use an operating set associated with the available real power and unity-power-factor generation.
Reactive power control: Reactive-power-only control is described through a set of possible operating points for renewable energy sources with reactive-power capability.
The projection is illustrated in Fig. 7, where the red dot represents [P k−1 av,i , 0]T.
Closed-form expressions also apply when the feasible set models diesel generators and controllable loads with variable-speed drives.