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Hybrid Sequential Feedback Optimization for Wind Farm Power Maximization

Shijie Huang, Sergio Grammatico

arXiv:2609.17859v1eess.SYcs.SYmath.OC

TL;DR

Nonlinear systems can make exact steady-state sensitivity expensive to compute. The paper combines SFO’s model-based gradient with RLS- or ESC-based corrections using decaying weights. Both variants retain convergence to a neighborhood of the optimum, and the wind farm case study reports faster early power increase than SFO.

  • Problem

    Computing steady-state sensitivity can be computationally prohibitive for nonlinear systems with high-dimensional states.

    Source: §I · p. 1

  • Method

    H-SFO combines the model-based SFO gradient with a correction through a convex combination with summable, diminishing weights, using RLS sensitivity estimation or ESC gradient estimation.

    Source: §I · p. 1, §III · p. 3

  • Results

    Both H-SFO variants converge to a neighborhood of the optimal steady state, and the wind farm case study reports faster early power increase than SFO.

    Source: B. Main convergence result for H-SFO · p. 4, B. Comparison with SFO · p. 5

  • Takeaways & Limitations

    The wind farm case study suggests H-SFO can accelerate the increase in total power production while the combination weights balance the correction against convergence smoothness.

    Source: §VI · p. 6

    Validation in high-fidelity models and wind tunnel testing remains future work, as does extending the approach to dynamic models with time-varying parameters.

    Source: §VI · p. 6

Abstract

from arXiv · show

This paper considers feedback optimization for optimal steady-state operation of nonlinear discrete-time systems when the steady-state input-output map and its sensitivity are expensive to compute. We propose a hybrid extension of sequential feedback optimization (SFO) that augments the model-based SFO gradient with correction terms through a convex combination with summable diminishing weights. Two variants are studied: one based on recursive least-squares (RLS) sensitivity estimation, and another on extremum seeking control (ESC) gradient estimation. Under contractivity and smoothness assumptions, we show that both hybrid schemes preserve the convergence of SFO to a neighborhood of the optimal steady state. The proposed methods are validated through a wind farm power maximization problem using a medium-fidelity model, demonstrating improved early transient performance compared to pure SFO.

I. INTRODUCTION

Nonlinear, high-dimensional systems make exact steady-state sensitivity costly, and SFO’s measured-state linearization can be inaccurate during early transients. H-SFO combines that model-based gradient with RLS- or ESC-based corrections, aiming to improve early optimization while retaining convergence to a neighborhood of the optimum.

I. INTRODUCTION

  • RLS learns sensitivity from incremental input-output data, while ESC estimates the objective gradient using periodic perturbations and demodulation.These measurement-based estimates avoid model approximation error but typically require long learning periods for accuracy.Source: §I · p. 1
  • H-SFO combines the model-based SFO gradient with RLS- or ESC-based corrections through summable weights that decay to zero.The correction can help during early transients, while the algorithm gradually shifts toward model-based linearization.Source: §I · p. 1
  • Both hybrid variants converge to a neighborhood of the optimal steady state, and wind farm simulations show faster early power increase than SFO with comparable steady-state performance.The correction acts as a vanishing perturbation.Source: §I · p. 1

II. PROBLEM FORMULATION

  • The formulation includes wind farm power maximization, where turbine controls affect power through wake interaction dynamics.The state represents flow velocities, controls include induction factors and yaw angles, and output measures power production.Source: §II · p. 2
  • For nonlinear systems with high-dimensional states, exact sensitivity computation can be prohibitive because it requires solving for the steady state at each iteration.SFO instead linearizes the dynamics at the measured state, but that sensitivity can be inaccurate when the state differs from the steady state for the current input, particularly early on.Source: §II · p. 2

III. HYBRID SFO DESIGN

H-SFO combines a sequentially linearized model-based gradient with an adaptive correction, using a decaying convex-combination weight to transition between them. Its RLS and ESC variants construct corrections differently, and the method is distinct from prior hybrid work in combining these corrections with sequential linearization.

III. HYBRID SFO DESIGN

  • H-SFO-RLS learns an adaptive sensitivity estimate online by relating output increments to input increments through a linear regression model.The RLS estimate supplies the sensitivity-based correction gradient.Source: §III · p. 3
  • H-SFO-ESC constructs a correction using sinusoidal perturbations and filtering, without explicitly estimating the steady-state sensitivity.The method uses stable high- and low-pass filters in its filtering and demodulation structure.Source: §III · p. 3
  • Summable, time-varying weights initially emphasize the correction gradient and decay to zero, making it a vanishing perturbation as the trajectory approaches the optimal steady state.The combined gradient interpolates between the correction and SFO gradients.Source: §III · p. 3
  • Unlike the related hybrid method, H-SFO combines RLS or ESC with a sequentially linearized model-based gradient and analyzes convergence of the control sequence.The related method combines RLS sensitivity estimates with a model-free gradient estimate and focuses on dynamic regret.Source: §III · p. 3

IV. CONVERGENCE ANALYSIS

The convergence analysis relates H-SFO iterates to ideal feedback optimization. It simplifies the output map for the analysis and indicates how the general case is handled.

IV. CONVERGENCE ANALYSIS

  • The convergence analysis relates H-SFO iterates to the ideal feedback optimization scheme.The analysis sets g(x, u) = x, so the steady-state input-output map h equals ϕ.Source: §IV · p. 3
  • Setting g(x, u) = x simplifies notation for the convergence analysis.Under this choice, h = ϕ.Source: §IV · p. 3
  • The analysis states that the general output-map case follows by incorporating the Lipschitz constants of g and ∇g.Source: §IV · p. 3

A. Auxiliary lemmas

The auxiliary results establish regularity and convergence properties used in the H-SFO analysis. They bound the linearized sensitivity and correction estimates and show convergence of ideal feedback optimization under the stated conditions.

A. Auxiliary lemmas

  • The linearized sensitivity is Lipschitz continuous over the state and input domain under Assumption 1.The lemma gives a bound for its Lipschitz constant.Source: A. Auxiliary lemmas · p. 4
  • Under Assumptions 1–2 and a suitable step size, the ideal feedback optimization scheme converges to the optimal steady state.Source: A. Auxiliary lemmas · p. 4
  • The incremental regression model has bounded residuals, supporting bounded-error analysis of the RLS sensitivity estimate.The RLS estimate error is uniformly bounded under the stated excitation and tuning conditions.Source: A. Auxiliary lemmas · p. 4
  • The ESC correction is uniformly bounded when the filters are stable and the stated boundedness assumptions hold.The result follows from bounded states, cost gradients, sinusoidal signals, and stable filters.Source: A. Auxiliary lemmas · p. 4

B. Main convergence result for H-SFO

Under the theorem’s stated assumptions, both H-SFO variants converge to a neighborhood of the optimal steady state, with the correction acting as a summable perturbation. The numerical section evaluates the variants on a wind-farm power-maximization problem and describes its simulation setup.

B. Main convergence result for H-SFO

  • Both H-SFO variants converge to a neighborhood of the optimal steady state; the summable correction does not add a term to the asymptotic bound.The bound may be conservative, especially when ρf or ρ(M) is close to one; persistent probing contributes terms for H-SFO-RLS, while those terms vanish for H-SFO-ESC.Source: B. Main convergence result for H-SFO · p. 4

V. NUMERICAL RESULTS

  • The numerical section evaluates H-SFO-RLS and H-SFO-ESC against SFO, examines combination-weight decay, and compares the methods with an alternative hybrid approach.Source: §V · p. 4

A. Simulation setup

  • The simulations use WFSim for a 9-turbine NREL 5MW farm arranged in a 3 × 3 grid, with a medium-fidelity flow model on a 50 × 25 grid.The computational domain is 2518.8 m × 1558.4 m, with steady uniform flow at ub = 8 m/s and vb = 0 m/s.Source: A. Simulation setup · pp. 4–5
  • Each turbine’s controls are thrust coefficient and yaw angle, and its steady-state power is represented through WFSim’s nonlinear input-output mapping.Controllers start from the greedy steady state; SFO linearizes at each iteration, RLS estimates sensitivities online, and ESC uses dither signals.Source: A. Simulation setup · p. 5

B. Comparison with SFO

H-SFO improves early power increase over SFO, while weight schedules shape the transient trade-offs and an alternative hybrid method struggles with stability or speed. The paper concludes that further validation in high-fidelity models and wind-tunnel testing remains future work.

B. Comparison with SFO

  • Both H-SFO variants achieve moderately higher steady-state power than SFO and improve transient performance, while SFO rises slowly early and overshoots before settling.The comparison covers total farm power against SFO and the greedy controller.Source: Figure 1 · p. 5, B. Comparison with SFO · p. 5

C. Influence of weight decay rate

  • Both weight schedules improve early performance over SFO, but their decay choices trade initial gain, convergence speed, and smoothness differently for RLS and ESC.For RLS, slower asymptotic decay brings faster initial gain but more oscillations, while p = 2 balances speed and smoothness; T = 3000 gives the fastest, smooth finite-time transient. ESC remains smooth; faster decay or shorter switching time speeds convergence.Source: C. Influence of weight decay rate · p. 5, Figure 2 · p. 6

D. Comparison with alternative hybrid FO method

  • For, fixed step sizes destabilize upstream controls, while diminishing steps remain near greedy inputs after 5x10^6 iterations, making convergence impractically slow.The comparison presents control trajectories because fixed-step power fluctuations make power-based visualization uninformative.Source: D. Comparison with alternative hybrid… · p. 6, Figure 3 · p. 6

VI. CONCLUSION

  • Validation in high-fidelity models such as SOWFA and wind-tunnel testing remains future work, as does extension to dynamic models with time-varying parameters.Source: §VI · p. 6

APPENDIX A PRELIMINARY BOUNDS

The appendix establishes preliminary bounds for the proof, using contractivity, smoothness, compactness, and a lemma for handling vanishing weights.

APPENDIX A PRELIMINARY BOUNDS

  • The appendix specializes to g(x,u)=x, so estimated output equals state, and uses Euclidean, induced 2-norm, and Frobenius norm conventions.Source: Appendix A · p. 6
  • Uniform contractivity bounds the inverse (I − ∇_x f)^−1 by 1/(1−ρ_f), supporting finite sensitivity-related estimates.Source: Appendix A · p. 6
  • Lipschitz continuity of h and compactness of U bound the output gradient and control increments; the appendix also establishes bounded states and state increments.Source: Appendix A · pp. 6–7
  • Lemma 6 supplies a bound for handling vanishing weights by controlling geometrically weighted sequences and their limiting upper bounds.Source: Appendix A · p. 7

APPENDIX B PROOF OF LEMMA 3

The proof uses an integral mean-value argument along an interpolated state-control path, then bounds the residual terms independently of the iteration.

APPENDIX B PROOF OF LEMMA 3

  • An integral mean-value formula along an interpolated state-control path relates the output and input increments to local linearized terms.Source: Appendix B · p. 7
  • The increment identities yield the stated output-increment relation and the evolution equation for h_k.Source: Appendix B · p. 7
  • Both residual bounds are finite and independent of k, using bounded sensitivity, control increments, state increments, and matrix norms.Source: Appendix B · p. 7

APPENDIX C PROOF OF LEMMA 4

The proof establishes bounded correction estimates for both hybrid variants. For RLS, persistent excitation and the estimator conditions bound sensitivity error; for ESC, stable filters and bounded signals bound the correction without using extremum tracking.

APPENDIX C PROOF OF LEMMA 4

  • Persistent excitation and the estimator conditions yield a uniformly bounded RLS sensitivity-estimation error.The recursion has exponentially stable homogeneous error dynamics, and the bounded residuals imply a uniformly bounded estimation error.Source: Appendix C · pp. 7–8
  • Stable ESC filters and bounded signals yield a bounded correction term.The proof uses bounded impulse-response norms and bounded cost gradients; it does not rely on extremum tracking, which requires additional dither-amplitude and time-scale-separation conditions.Source: Appendix C · p. 8

APPENDIX D PROOF OF THEOREM 1

The proof compares each hybrid trajectory with the ideal feedback-optimization trajectory and bounds their mismatch. Summable diminishing weights and bounded correction terms yield convergence to a neighborhood of the optimal steady state.

APPENDIX D PROOF OF THEOREM 1

  • The correction bound is variant-specific: RLS uses its sensitivity-estimation error bound, while ESC uses a bound on its gradient correction.These bounds feed into the hybrid trajectory mismatch argument.Source: Appendix D · pp. 8–9
  • A comparison system bounds the hybrid-versus-ideal input and output errors using a Perron–Frobenius weighted error measure.The proof applies the limit bounds to the comparison recursion; the summable weights make one error contribution vanish.Source: Appendix D · p. 9
  • The hybrid scheme converges to a neighborhood of the optimal steady state by comparison with the convergent ideal FO scheme.The proof bounds the limiting trajectory mismatch and combines it with convergence of ideal FO.Source: Appendix D · p. 9
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