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Optimal Power Flow Pursuit via Feedback-based Safe Gradient Flow

Antonin Colot, Yiting Chen, Bertrand Cornelusse, Jorge Cortes, Emiliano Dall'Anese

Abstract

This paper considers the problem of controlling inverter-interfaced distributed energy resources (DERs) in a distribution grid to solve an AC optimal power flow (OPF) problem in real time. The AC OPF includes voltage constraints, and seeks to minimize costs associated with the economic operation, power losses, or the power curtailment from renewables. We develop an online feedback optimization method to drive the DERs' power setpoints to solutions of an AC OPF problem based only on voltage measurements (and without requiring measurements of the power consumption of non-controllable assets). The proposed method - grounded on the theory of control barrier functions - is based on a continuous approximation of the projected gradient flow, appropriately modified to accommodate measurements from the power network. We provide results in terms of local exponential stability, and assess the robustness to errors in the measurements and in the system Jacobian matrix. We show that the proposed method ensures anytime satisfaction of the voltage constraints when no model and measurement errors are present; if these errors are present and are small, the voltage violation is practically negligible. We also discuss extensions of the framework to virtual power plant setups and to cases where constraints on power flows and currents must be enforced. Numerical experiments on a 93-bus distribution system and with realistic load and production profiles show a superior performance in terms of voltage regulation relative to existing methods.

Optimal Power Flow Pursuit via Feedback-based Safe Gradient Flow

Abstract

This paper considers the problem of controlling inverter-interfaced distributed energy resources (DERs) in a distribution grid to solve an AC optimal power flow (OPF) problem in real time. The AC OPF includes voltage constraints, and seeks to minimize costs associated with the economic operation, power losses, or the power curtailment from renewables. We develop an online feedback optimization method to drive the DERs' power setpoints to solutions of an AC OPF problem based only on voltage measurements (and without requiring measurements of the power consumption of non-controllable assets). The proposed method - grounded on the theory of control barrier functions - is based on a continuous approximation of the projected gradient flow, appropriately modified to accommodate measurements from the power network. We provide results in terms of local exponential stability, and assess the robustness to errors in the measurements and in the system Jacobian matrix. We show that the proposed method ensures anytime satisfaction of the voltage constraints when no model and measurement errors are present; if these errors are present and are small, the voltage violation is practically negligible. We also discuss extensions of the framework to virtual power plant setups and to cases where constraints on power flows and currents must be enforced. Numerical experiments on a 93-bus distribution system and with realistic load and production profiles show a superior performance in terms of voltage regulation relative to existing methods.
Paper Structure (23 sections, 6 theorems, 39 equations, 6 figures, 1 algorithm)

This paper contains 23 sections, 6 theorems, 39 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Distribution system model
  4. OPF for voltage regulation in distribution systems
  5. Safe OPF pursuit
  6. Feedback-based online algorithm
  7. Remarks on the linear model
  8. Stability analysis and constraint satisfaction guarantees
  9. Extensions
  10. Numerical experiments
  11. Simulation setup
  12. Simulation parameters
  13. Volt/Var control
  14. No control
  15. Results
  16. ...and 8 more sections

Key Result

Lemma 3.4

Consider the problem eq:OPFProb2 satisfying Assumptions as:steadyStateMap-as:openloop. There exists $\boldsymbol{\mu}^*$ such that $(\boldsymbol{u}^*,\boldsymbol{\mu}^ *)$ is a KKT point for eq:OPFProb2 if and only if $\boldsymbol{u}^*$ is an equilibrium of $\dot{\boldsymbol{u}} = \eta F_{\beta}(\bo

Figures (6)

  • Figure 1: Closed-loop implementation of the proposed online feedback optimization algorithm. A central unit (blue box) receives measurements of voltages (from measurement units, green box) and of the DERs' output powers from inverters (red box); based on these measurements, it updates the DERs' setpoints based on the proposed controller $\dot{\boldsymbol{u}}(t) = \eta \hat{F}_{\beta}(\boldsymbol{u}(t), \tilde{\boldsymbol{\nu}}(t))$. Once the setpoints $\boldsymbol{u}(t)$ are computed, the central unit transmits $\boldsymbol{u}(t)$ to the DERs' inverters. Through this closed-loop scheme, the proposed controllers drive the distribution system to solutions of the AC OPF problem \ref{['eq:OPFProb2']}.
  • Figure 2: (a) Distribution network used in the simulations. (b) Aggregated load consumption ($P_L,Q_L$) and PV production profiles ($P_{max}$) used in the simulations. (c) Operational set compared to grid code requirements inspired from the IEEE Std 1547-2018, where $s_n$ is the inverter rated power.
  • Figure 3: Implementation of IEEE standard IEEE Std 1547-2018 with $Q$ the reactive power injection, $S$ the nominal apparent power of the DER, $Q/S$ represents the ratio between $Q$ and $S$, and $|v|$ is the voltage magnitude at the node.
  • Figure 4: (a) Overvoltages for NC. (b) Overvoltages for VVC. (c) Overvoltages for PDM. (d) Overvoltages for SGF. (e) Overvoltage duration times. (f) Linear map error: $||\hat{H}(\boldsymbol{u};\boldsymbol{p}_l,\boldsymbol{q}_l) - H(\boldsymbol{u};\boldsymbol{p}_l,\boldsymbol{q}_l)||$ and Jacobian error: $||J_{\hat{H}} - J_{H}(\boldsymbol{u};\boldsymbol{p}_l,\boldsymbol{q}_l)||$, where $\boldsymbol{u}$ is picked from the SGF algorithm.
  • Figure 5: Achieved values of cumulative cost \ref{['eq:costfunction']}.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Remark 2.1: Jacobian of map $H$
  • Remark 2.2: Model and notation
  • Remark 3.1: Pseudo-measurements
  • Remark 3.2: Measurement of setpoints
  • Remark 3.3: Validity of the linear approximation
  • Lemma 3.4: KKT and equilibrium
  • Lemma 3.5: Lipschitz continuity
  • Theorem 3.6: Practical local exponential stability
  • Lemma 3.7: Practical forward invariance
  • Corollary 3.8: Error-free implementation
  • ...and 1 more