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Decentralized Data-Enabled Predictive Control for Power System Oscillation Damping

Linbin Huang, Jeremy Coulson, John Lygeros, Florian Dörfler

arXiv:1911.12151v4eess.SYcs.SY

TL;DR

Conventional model-based wide-area damping requires accurate system models that are difficult to obtain in practice. This paper uses DeePC with measured input/output data and develops a decentralized Min-Max variant for multiple VSC-HVDC stations. The authors report attenuation of inter-area oscillations under noisy measurements, nonlinear loads, load fluctuations, and communication delays, with improved performance from Page matrices and SVD noise filtering.

  • Problem

    Conventional optimal wide-area control relies on accurate parametric models, which are difficult to obtain amid uncertain device parameters, changing grid models, and commissioning demands.

    Source: §I · p. 1

  • Method

    The paper uses input/output-data-based DeePC for multiple VSC-HVDC stations and develops a decentralized Min-Max formulation, while exploring Page-matrix predictors and SVD filtering.

    Source: §I · p. 2, C. DeePC with Page Matrix · p. 3

  • Results

    DeePC attenuates inter-area oscillations despite noisy measurements, nonlinear loads, load fluctuations, and communication delays, while Page matrices with SVD filtering improve performance.

    Source: §I · p. 2, §VI · p. 11

  • Takeaways & Limitations

    The paper concludes that data-driven DeePC can provide decentralized, robust, optimal wide-area control for oscillation damping in its simulated power-system setting.

    Source: §IV · p. 8, §VI · p. 11

Abstract

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We employ a novel data-enabled predictive control (DeePC) algorithm in voltage source converter (VSC) based high-voltage DC (HVDC) stations to perform safe and optimal wide-area control for power system oscillation damping. Conventional optimal wide-area control is model-based. However, in practice detailed and accurate parametric power system models are rarely available. In contrast, the DeePC algorithm uses only input/output data measured from the unknown system to predict the future trajectories and calculate the optimal control policy. We showcase that the DeePC algorithm can effectively attenuate inter-area oscillations even in the presence of measurement noise, communication delays, nonlinear loads and uncertain load fluctuations. We investigate the performance under different matrix structures as data-driven predictors. Furthermore, we derive a novel Min-Max DeePC algorithm to be applied independently in multiple VSC-HVDC stations to mitigate inter-area oscillations, which enables decentralized and robust optimal wide-area control. Further, we discuss how to relieve the computational burden of the Min-Max DeePC by reducing the dimension of prediction uncertainty and how to leverage disturbance feedback to reduce the conservativeness of robustification. We illustrate our results with high-fidelity, nonlinear, and noisy simulations of a four-area test system.

I. INTRODUCTION

Model-based wide-area damping depends on accurate system models that are often difficult to obtain, motivating control from measured input/output data. The paper applies DeePC to multiple VSC-HVDC stations and develops a decentralized Min-Max variant; its preliminaries frame prediction using past and future data trajectories.

I. INTRODUCTION

  • Accurate, detailed models are needed for conventional predictive control, but device parameters, grid models, and commissioning effort constrain their practical availability.Model mismatch or uncertainty may also impair model-based control performance.Source: §I · p. 1
  • DeePC predicts system behavior and chooses control using input/output measurements from an unknown system rather than a parametric model.Earlier work applied it to local control at a VSC-HVDC station in a two-area system.Source: §I · pp. 1–2
  • The paper applies DeePC across multiple VSC-HVDC stations for optimal wide-area oscillation damping and develops a decentralized, robust Min-Max approach.The authors also investigate predictor matrix structures and denoising.Source: §I · p. 2

A. Preliminaries and Notation

  • Offline input/output data are arranged in Hankel matrices and partitioned into past and future blocks; past data implicitly estimate the initial condition, while future data predict trajectories.The collected input can be persistently exciting of order T_ini + N + n.Source: A. Preliminaries and Notation · p. 2
  • The Fundamental Lemma characterizes system trajectories through the span of the data matrix, with T_ini setting the initial trajectory and N the forward prediction length.When T_ini is at least the system lag, the future output is uniquely determined for a given input trajectory.Source: A. Preliminaries and Notation · pp. 2–3

B. Review of DeePC

DeePC uses measured input/output trajectories to optimize future control without constructing a parametric system model. Its formulation regularizes noisy-data predictions and operates in a receding horizon.

B. Review of DeePC

  • DeePC uses input/output data from the unknown system to predict future behavior and optimize safe control without a parametric system representation.It constructs Hankel matrices from measured trajectories before solving for future control inputs.Source: B. Review of DeePC · p. 3
  • A quadratic penalty on the data coefficient g helps avoid overfitting with noisy samples and corresponds to worst-case performance optimization for bounded data disturbances.The formulation also uses a slack variable for feasibility of the initial-condition equality constraint.Source: B. Review of DeePC · p. 3
  • DeePC applies an optimized input sequence for k ≤ N steps, updates its recent input/output history, and resolves the optimization.Terminal constraints are omitted because observed performance was similar with and without them, and solving is faster without them.Source: B. Review of DeePC · p. 3
  • For σ_y = 0, DeePC corresponds to a least-norm solution satisfying the trajectory constraints and offers more flexibility than a particular identified ARX model.The paper relates that identified-model approach to certainty-equivalence MPC.Source: B. Review of DeePC · p. 3

C. DeePC with Page Matrix

The paper replaces the usual Hankel arrangement with Page matrices as data-driven predictors and explores SVD-based noise filtering. The structure avoids repeated entries, allowing noise filtering without breaking the predictor structure, though constructing Page matrices requires more data.

C. DeePC with Page Matrix

  • DeePC can use block Page matrices as an alternative arrangement of input/output data for predicting future system behavior.The method partitions the Page matrices into past and future blocks and uses them in place of Hankel matrices.Source: C. DeePC with Page Matrix · p. 3
  • Page matrices have no repeated entries, so output measurement noise can be filtered using SVD and small-singular-value truncation without breaking the data-matrix structure.The stated filtering assumption is that noise is uncorrelated across measurements.Source: C. DeePC with Page Matrix · p. 4
  • The paper reports that Page structure and singular-value noise filtering can significantly improve DeePC performance; denoising Hankel matrices instead destroys their predictor structure and tends to perform poorly.Page matrices require more data to construct than Hankel matrices.Source: C. DeePC with Page Matrix · p. 4

III. CENTRALIZED WIDE-AREA CONTROL

The section applies DeePC as a centralized wide-area controller for VSC-HVDC stations. Its setup considers the dynamic interaction between both stations of an HVDC link.

III. CENTRALIZED WIDE-AREA CONTROL

  • DeePC is applied to VSC-HVDC stations for centralized optimal wide-area control aimed at mitigating low-frequency oscillations.Source: §III · p. 4
  • The study considers a VSC-HVDC link and the dynamic interaction between its two stations, rather than a single station.The authors describe the setup as more realistic, large-scale, and challenging than prior work.Source: §III · p. 4
  • The controller is evaluated in a large-scale system setup.The passage characterizes this system as more realistic and challenging than the earlier setup.Source: §III · p. 4

A. Descriptions of a Four-area Test System

The four-area, 208-state test system extends a two-area benchmark with a VSC-HVDC link and sustains low-frequency oscillations without auxiliary damping control; the paper applies centralized DeePC using wide-area power-flow measurements to command both stations.

A. Descriptions of a Four-area Test System

  • Station 1 regulates active power and station 2 DC voltage; their conventional control structures need auxiliary control for oscillation damping.The algorithms may also be applied to synchronous-generator excitation systems using different control inputs.Source: A. Descriptions of a Four-area Test… · p. 4
  • Without auxiliary damping, the four-area system sustains low-frequency oscillations, and its dominant poles lie close to the imaginary axis.The 208-state system extends a two-area benchmark by replicating its model twice to integrate a VSC-HVDC link; the paper omits modal analysis.Source: A. Descriptions of a Four-area Test… · p. 4

B. Centralized Wide-area Control Using DeePC

  • The centralized DeePC controller collects three interface power-flow measurements and distributes four transient-control inputs to the two VSC-HVDC stations.The inputs affect transient performance but not steady state, which is determined by the outer-loop PI regulators.Source: B. Centralized Wide-area Control Using… · p. 4
  • Unknown constant communication and measurement delays can be incorporated into the data matrices; significantly varying delays would require adaptive DeePC.The authors suggest updating Hankel-matrix data when time-stamped measurements reveal significant delay changes.Source: B. Centralized Wide-area Control Using… · p. 5

Configuration of the DeePC Algorithm

The DeePC setup uses persistently exciting input data and a 120-step prediction horizon, then evaluates oscillation damping against PEM-MPC in four-area simulations. The results include repeated cost comparisons and tests with missing measurements.

Configuration of the DeePC Algorithm

  • The DeePC sampling time is 0.02 s, with Tini = 60 and prediction horizon N = 120.The cost parameters are R = I, Q = 400 × I, λg = 20, and λy = 2000; persistently exciting white-noise inputs are injected for 30 s to construct the Hankel matrix with T = 1500.Source: Configuration of the DeePC Algorithm · p. 5
  • DeePC attenuates inter-area oscillations after activation at 10 s; smaller k improves damping but requires more frequent optimization.The study uses a nonlinear four-area system with constant-power loads and noise-free measurements in this base case.Source: Configuration of the DeePC Algorithm · p. 5, Figure 3 · p. 6
  • VSC-HVDC active-power fluctuations during transients are described as acceptable and can be limited through input constraints.Penalizing input-rate changes is also suggested to smooth station transients.Source: Configuration of the DeePC Algorithm · p. 6
  • Across 100 runs, DeePC consistently achieves superior closed-loop performance to certainty-equivalence PEM-MPC.The cost comparison covers 10–30 s with k = 60.Source: Figure 4 · p. 6, Configuration of the DeePC Algorithm · p. 6
  • DeePC stabilizes the system with P2 or both P2 and P3 unavailable, with a modest reduction in performance.The lost measurement rows are removed from the Hankel matrices and initial trajectory.Source: Figure 5 · p. 6, Configuration of the DeePC Algorithm · p. 6

C. Nonlinear, Delayed and Noisy Implementation

With nonlinear loads, noisy measurements and load fluctuations, and 100 ms communication delays, DeePC mitigated low-frequency oscillations and outperformed PEM-MPC on average. Parameter sweeps examined how prediction horizon, initialization history, data length, and regularization affect closed-loop cost.

C. Nonlinear, Delayed and Noisy Implementation

  • Across 100 simulations, DeePC performed better than PEM-MPC on average, while PEM-MPC sometimes produced instability and closed-loop performance above 8000.The simulations used different data sets and random seeds for measurement and load noise.Source: C. Nonlinear, Delayed and Noisy… · p. 7

D. DeePC Hyperparameter Tuning

  • Closed-loop cost drops as prediction horizon N increases, then remains within an acceptable range.Fig. 8 plots the cost for different DeePC parameters.Source: D. DeePC Hyperparameter Tuning · p. 7, Figure 8 · p. 7
  • Closed-loop cost drops as Tini increases from 5 to 40, then remains nearly unchanged once Tini ≥ ℓ in the deterministic case.At Tini ≥ ℓ, the system state is uniquely determined.Source: D. DeePC Hyperparameter Tuning · p. 7, Figure 8 · p. 7
  • Increasing T from 800 to 1000 significantly lowers closed-loop cost, which then remains nearly unchanged; a minimum also appears near T = 1439 for a square Hankel matrix.The results indicate that adding more data does not necessarily improve performance.Source: D. DeePC Hyperparameter Tuning · p. 7, Figure 8 · p. 7
  • DeePC performs satisfactorily across a wide range of λg, but values that are too small weaken robustness and values that are too large can make control conservative.Proper regularization, together with sufficiently large N, Tini, and T, supports superior damping performance.Source: D. DeePC Hyperparameter Tuning · pp. 7–8

E. Comparison of Hankel Matrix and Page Matrix

Page matrices achieve better DeePC performance than Hankel matrices in the reported comparison, with SVD filtering further improving Page-matrix performance. The paper then formulates Min-Max DeePC for decentralized control under bounded future disturbances and derives a tractable robust counterpart using a relaxation.

E. Comparison of Hankel Matrix and Page Matrix

  • Page matrices outperform Hankel matrices even without SVD filtering; SVD filtering further improves Page-matrix performance, especially with larger control horizon k, but degrades Hankel-matrix performance.The authors attribute the Page-matrix advantage to its use of more data and the Hankel degradation to loss of structure after truncating small singular values.Source: E. Comparison of Hankel Matrix and… · p. 8

IV. MIN-MAX DEEPC

  • Min-Max DeePC models neighboring-subsystem inputs as disturbances, enabling decentralized wide-area damping control and robustness to measured disturbances.The centralized DeePC formulation is not resilient to communication failures, especially as more VSC-HVDC stations are considered.Source: §IV · p. 8

A. Basic Formulation

  • The formulation uses measured past disturbances and bounds unknown future disturbances, then optimizes control inputs against the worst case within those bounds.It is implemented in a receding-horizon fashion.Source: A. Basic Formulation · p. 8
  • Eliminating g and y and deriving a robust counterpart yields a conic program, but assumes adjustable g and y are linear in the disturbance and relaxes the original equality constraints.The authors note that equality constraints in a min-max problem can be conservative or infeasible.Source: A. Basic Formulation · p. 9

B. Disturbance-Feedback (DF) Min-Max DeePC

Open-loop Min-Max DeePC can be conservative because it ignores feedback available as the receding-horizon problem is repeatedly solved. The disturbance-feedback variant parameterizes that feedback, while using semidefinite relaxation to address computational difficulty.

B. Disturbance-Feedback (DF) Min-Max DeePC

  • The open-loop control sequence from Min-Max DeePC ignores receding-horizon feedback from measured actions and disturbances, potentially making the sequence conservative.The control horizon is typically shorter than the prediction horizon, so feedback updates occur when the problem is re-solved.Source: B. Disturbance-Feedback (DF) Min-Max… · p. 9
  • The disturbance-feedback policy uses an affine control response, with a strictly lower block triangular Toeplitz matrix L to enforce causality and reduce its independent entries to mq(N − 1).Source: B. Disturbance-Feedback (DF) Min-Max… · p. 9
  • Optimizing v and L lets the disturbance-feedback term Lw emulate receding-horizon feedback; the first k elements of v are applied after solving.Source: B. Disturbance-Feedback (DF) Min-Max… · p. 9
  • A semidefinite relaxation handles the bilinear Lw term, and setting λg = 0 reduces the resulting matrix inequality's dimension to lower computational burden.Source: B. Disturbance-Feedback (DF) Min-Max… · p. 9

C. Downsampling of Future Disturbance Trajectory

A long prediction horizon can make future-disturbance uncertainty high-dimensional and increase the cost of solving the robust optimization problem. The proposed downsampling and interpolation reduce that dimension, but restrict the disturbance trajectories represented by the uncertainty set.

C. Downsampling of Future Disturbance Trajectory

  • Long prediction horizons can make future-disturbance parameterization high-dimensional and increase the computational burden of robust optimization.Source: C. Downsampling of Future Disturbance… · pp. 9–10
  • Selecting every Mth disturbance sample creates a lower-dimensional representation ˜w, which is linearly interpolated into an extended trajectory ˆw at the sampling rate of u and y.The downsampled trajectory is less accurate than the original future-disturbance trajectory.Source: C. Downsampling of Future Disturbance… · p. 10
  • Because ˆw spans only a subspace of w, the reduced uncertainty may omit the worst case unless the disturbance is smooth and satisfies the interpolation condition.The authors report satisfactory performance with this condition for low-frequency oscillations.Source: C. Downsampling of Future Disturbance… · p. 10

V. DECENTRALIZED WIDE-AREA CONTROL

The section applies decentralized Min-Max DeePC in a four-area system and compares its damping performance and real-time feasibility with disturbance-feedback Min-Max DeePC. The conclusion highlights the decentralized approach’s oscillation damping and identifies adaptive DeePC for time-varying dynamics as future work.

V. DECENTRALIZED WIDE-AREA CONTROL

  • The four-area system is partitioned into two two-area subsystems, each controlling its VSC-HVDC station against worst-case future trajectories of P3 and Pdc.Each controller uses one local measurement and two wide-area measurements.Source: §V · p. 10
  • Downsampling represents only the first and last disturbance-trajectory points as uncertain, with intermediate values linearly interpolated; forcing x to zero can yield a suboptimal solution if the system is not LTI or noise-free.The prediction horizon is N = 40, and the reduction factor M is 40.Source: §V · p. 10
  • DF Min-Max DeePC sustains almost unchanged damping at k = 24, where plain Min-Max DeePC cannot eliminate oscillations; decentralized control effectively mitigates inter-area oscillations without a centralized controller.The comparison uses k = 16 and k = 24 for DF Min-Max DeePC, and reports that plain Min-Max DeePC has significantly lower damping at k = 16 than at k = 8.Source: Figure 13 · p. 11, §V · p. 11, §VI · p. 11
  • With an i5 7200U CPU and 8GB RAM, setting k ≥22 enables real-time implementation of the DF Min-Max DeePC algorithm.Source: §V · p. 11

VI. CONCLUSIONS

  • The authors identify adaptive DeePC for time-varying dynamics, particularly time-varying delays, as future work.Source: §VI · p. 11
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