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Sparsity-Promoting Optimal Wide-Area Control of Power Networks

Florian Dörfler, Mihailo R. Jovanovic, Michael Chertkov, Francesco Bullo

arXiv:1307.4342v2math.OC

TL;DR

Inter-area oscillations are difficult to control with local decentralized methods, motivating wide-area control with remote signals. This paper jointly identifies a sparse control structure and optimizes performance, achieving nearly centralized performance with one communication link on the IEEE 39 New England model.

  • Problem

    Inter-area oscillations can remain poorly controlled or difficult to stabilize with decentralized power-system stabilizers, while conventional wide-area control typically fixes its architecture in advance.

  • Method

    The approach uses sparsity-promoting optimal control with static state feedback, combining an l1 penalty on the feedback matrix with slow-coherency-inspired quadratic performance objectives.

  • Results

    Within 1.5882% of optimal centralized performance, the sparsity-promoting controller requires only a single wide-area communication link on the IEEE 39 New England model and retains favorable robustness properties.

  • Takeaways & Limitations

    The method can simultaneously identify a sparse wide-area control architecture and optimize closed-loop performance, while also supporting robustness assessment and alternative control design.

  • Takeaways & Limitations

    Further evaluation on other power-system models is needed because the identified controller structure and achieved performance depend on the system model and the selected state and control weights.

Abstract

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Inter-area oscillations in bulk power systems are typically poorly controllable by means of local decentralized control. Recent research efforts have been aimed at developing wide- area control strategies that involve communication of remote signals. In conventional wide-area control, the control structure is fixed a priori typically based on modal criteria. In contrast, here we employ the recently-introduced paradigm of sparsity- promoting optimal control to simultaneously identify the optimal control structure and optimize the closed-loop performance. To induce a sparse control architecture, we regularize the standard quadratic performance index with an l1-penalty on the feedback matrix. The quadratic objective functions are inspired by the classic slow coherency theory and are aimed at imitating homogeneous networks without inter-area oscillations. We use the New England power grid model to demonstrate that the proposed combination of the sparsity-promoting control design with the slow coherency objectives performs almost as well as the optimal centralized control while only making use of a single wide-area communication link. In addition to this nominal performance, we also demonstrate that our control strategy yields favorable robustness margins and that it can be used to identify a sparse control architecture for control design via alternative means.

I. INTRODUCTION

Inter-area oscillations become increasingly lightly damped as power transfers grow and transmission capacity is strained, while local decentralized control may fail to stabilize or optimally control them. The paper develops sparsity-promoting wide-area control and demonstrates near-centralized performance with minimal communication on the IEEE 39 New England model.

  • Growing demand, remote renewables, and deregulation increase long-distance transfers, shrinking stability margins and producing more lightly damped inter-area modes.
  • Decentralized PSS actions can interact adversely, fail to stabilize inter-area modes, or require many carefully tuned controllers.
  • Wide-area control uses remote measurements and control signals, enabled by PMUs, fast communication networks, and FACTS devices.
  • Conventional approaches typically fix sensor, actuator, and communication structures a priori, whereas structural optimization can be difficult and nonconvex.
  • The proposed method jointly optimizes quadratic performance and sparse linear state feedback using slow-coherency-inspired objectives, emphasizing delay, gain, and operating-condition robustness.
  • 1.5882% is the performance gap between the sparse controller and optimal centralized control on the IEEE 39 New England model using one WAC link.

B. Review of slow coherency theory

Slow coherency theory represents inter-area behavior through aggregate variables for coherent groups of generators. The resulting reduced dynamics connect inertia, damping, and network coupling through aggregated matrices.

  • The generator state is partitioned into rotor angles, frequencies, and remaining states that typically represent fast electrical dynamics.
  • The linearized swing dynamics describe generator interactions through a Laplacian-weighted graph when transfer conductances are absent.
  • Inter-area oscillations can arise from non-uniform inertia and damping, coherent machine clusters, sparse interconnections, and large power transfers.
  • Each coherent area is represented by an aggregate variable δα describing its center-of-mass motion.
  • The slow inter-area dynamics are captured by ˜M δ¨ + ˜D δ˙ + ˜Lδ = 0.
  • The aggregated matrices ˜M, ˜D, and ˜L represent area-level inertia, dissipation, and Laplacian coupling.

C. Local and wide-area control design

The control design combines local stabilization with sparse wide-area feedback for global inter-area damping. An l1-regularized H2 objective identifies essential measurement-to-input links while trading performance against communication sparsity.

  • The two-level strategy uses u(t) = uloc(t) + uwac(t), combining local and wide-area control laws.
  • Local control stabilizes isolated components using local measurements, while wide-area control enhances global behavior and suppresses inter-area oscillations.
  • Wide-area feedback is static state feedback uwac(t) = −Kx(t), with communication architecture determined by the sparsity pattern of K.
  • The weighted l1 norm approximates the number of nonzero gain entries and identifies essential pairs of control inputs and measured outputs.
  • ADMM provides an iterative solution strategy for the sparsity-promoting optimal-control formulation.
  • The objective combines closed-loop H2 performance with an l1 penalty weighted by γ, where γ = 0 recovers standard state-feedback H2 control.

E. Choice of optimization objectives

The objectives combine slow-coherency-inspired state costs with control-effort and sparsity penalties to shape inter-area dynamics while selecting a sparse feedback architecture.

  • State cost: The state cost models the kinetic and potential energy of a homogeneous network, penalizing frequency violations and angular differences.A small regularization term ensures numerical stability and detectability.
  • Mode-specific objective: For a specific inter-area mode involving groups Vα and Vβ, an alternative cost penalizes their aggregate-variable difference.The gains ℓ and m are positive, while ε is a small nonnegative regularization parameter.
  • State cost: The parameters ℓ and m primarily tune damping of machine difference angles and frequency deviations, respectively.These costs reflect slow coherency theory and can promote readily available control variables.
  • Control cost: The control effort is penalized as uT Ru, with larger diagonal entries of positive definite R producing smaller control effort.
  • Sparsity promotion: An l1-regularized optimization produces a γ-parameterized family of feedback gains whose off-diagonal sparsity pattern identifies communication structure.The gains and control effort depend on the model and design choices Q, R, B1, and γ; B1 can represent noise or load and generation uncertainties.

F. Robustness, time delays, and gain uncertainties

Wide-area control must account for communication delays, asynchronous signals, and uncertain channel dynamics. The proposed controller family is evaluated for phase and gain robustness, with margins degrading gradually as sparsity promotion increases.

  • Uncertainty sources: Wide-area signals may experience communication delays, asynchronous measurements, multiple data rates, and differing time stamps from local signals.
  • Uncertainty model: Gain uncertainties ∆g and multiplicative uncertainties ∆m model uncertain or unmodeled wide-area channel dynamics, including time delays.The uncertainty blocks are unknown, stable, proper systems satisfying norm bounds.
  • Reference margins: The centralized optimum guarantees ±60° phase margins, a 0.5 lower gain margin, and infinite upper gain margins for diagonal gain uncertainty.
  • Sparse-controller robustness: For sparsity-promoting controllers, phase and gain margins decay gracefully as γ increases.Known delays can alternatively be incorporated into the plant through Padé approximations.

III. COORDINATED SUPPLEMENTARY PSSS DESIGN

The proposed wide-area control strategy is validated on the IEEE 39 New England power grid model.

  • The IEEE 39 New England model contains 39 buses and 10 two-axis generator models, with generator 10 represented by an equivalent aggregated model.

A. Local control design and inter-area dynamics

The local PSS design stabilizes the otherwise unstable open-loop model and provides damping for local modes, while the system retains five dominant inter-area modes for analysis.

  • Local control design: The model uses nonlinear differential-algebraic and linear state-space representations, with local PSS excitation controllers based on washout and lead/lag elements.
  • Local control design: The local PSS transfer function applies gain, washout, and two lead/lag elements to each generator’s speed deviation.
  • Local control design: The selected PSS parameters provide good damping for local modes and stabilize the otherwise unstable open-loop system without severely distorting inter-area mode shapes.The latter condition may depend on the operating point and careful tuning.
  • Inter-area dynamics: Closed-loop modal analysis identifies five dominant inter-area modes, whose coherent machine groups and eigenvector frequency components are illustrated in Fig. 3.
  • Inter-area dynamics: Table I reports the inter-area modes of the New England power grid with PSSs.

B. WAC design and nominal performance

The proposed WAC design trades feedback sparsity against closed-loop performance, producing a single-link architecture that nearly matches centralized control while improving inter-area damping and retaining favorable robustness.

  • Sparse architecture: The sparsity-promoting controller uses γ to regulate the number of nonzero feedback entries, with larger γ producing only slight increases in local-feedback entries and promoting sparse architectures.The resulting sparsity patterns identify which measurements and control channels require communication.
  • Nominal performance: The identified 9 → 1 communication channel appears necessary to suppress inter-area mode 2, which is mainly dominated by generators 1 and 9.The New England model contains five dominant inter-area modes associated with coherent generator groups.
  • Closed-loop modes: Compared with open loop, WAC improves weakly damped modes mainly by increasing damping ratios and distorting eigenvectors rather than substantially shifting their real parts.One reported pair is at −0.6044 ± i 2.5203 with damping ratio 0.239, while other complex pairs lie left of Real(s) = −12.74.
  • Closed-loop modes: The controller suppresses poorly damped power-flow oscillations between areas Vα = {1, . . . , 9} and Vβ = {10}, while significantly improving damping ratios of other weakly damped modes.The closed-loop least-damped mode no longer corresponds to generators oscillating against each other.
  • Robustness and implementation: The multivariable phase margin drops by about 4° across γ ∈ [10^-4, 100], while gain margins deteriorate gracefully and PSS gains may vary within approximately [0.2, 5] even for γ = 1.The WAC signal is designed to have approximately the same magnitude as the local control signal, avoiding input saturation.
  • Robustness and implementation: A 58.303° phase margin corresponds to a tolerable remote-feedback delay of 3.964 s, and a 750 ms delay does not significantly affect closed-loop performance.The delayed case produces only slight distortion in generator 1 trajectories.
  • Robustness and implementation: The identified sparsity pattern remains identical under PSS-gain changes within [0.5, 4] and load-demand changes within ±25% of nominal demand.These perturbations generate different operating points and linearization matrices.

D. Sparsity identification and alternative control schemes

The sparsity-promoting framework identifies a single crucial wide-area control channel and supports a proportional implementation using rotor-angle differences. Nonlinear simulations show improved performance, including under substantial communication delay, although delay slightly degrades nonlinear performance.

  • Sparsity identification: The framework identified a single crucial WAC channel from rotor angle θ9 at generator 9 to the AVR control at generator 1.This channel establishes the communication direction 9 → 1 for the considered model.
  • Alternative control schemes: The resulting proportional WAC signal uses θ9(t) − θ1(t), so absolute angle measurements are unnecessary.The signal can be implemented by integrating the frequency difference ˙θ9(t) − ˙θ1(t).
  • Alternative control schemes: The proportional WAC signal, together with appropriately retuned local PSSs, yields the implemented WAC control signal uwac(t).The study applies this signal to the full nonlinear differential-algebraic power-network model equipped with PSSs.
  • Nonlinear evaluation: A three-phase fault at line {3, 4}, cleared after 0.1 s, was used to evaluate nonlinear responses with and without WAC and with or without a 750 ms delay.The simulations compare generator-frequency and power-output responses under local control, WAC, and delayed WAC conditions.
  • Nonlinear evaluation: The WAC signal reduces stabilizing-control effort, while the nonlinear performance slightly degrades under delays compared with the corresponding linear case.Generator 10 still slightly swings against the remaining generators in WAC closed loop, but the conclusions persist with a severe delay.
  • Alternative control schemes: The identified channel 9 → 1 can guide more sophisticated single-input-single-output controllers beyond the illustrative proportional feedback.The proportional strategy is presented primarily as an example of architecture identification.

IV. CONCLUSIONS

The paper proposes sparsity-promoting wide-area control for inter-area oscillations, combining slow-coherency-inspired objectives with sparse feedback design. On the IEEE 39 New England model, the approach achieves nearly optimal performance with low communication requirements and robustness, while its broader applicability remains to be tested.

  • Contributions: The proposed approach applies sparsity-promoting optimal control to wide-area control of inter-area oscillations.Its performance objectives are inspired by slow coherency theory.
  • Validation: The conclusion section reports simulations comparing local PSS control, local PSS control with WAC, and local PSS control with delayed WAC.The nonlinear model simulation uses a three-phase fault at line {3, 4}, cleared at 0.2 s after occurring at 0.1 s.
  • Validation: The IEEE 39 New England model demonstrated nearly optimal performance, low communication requirements, and robustness of the sparsity-promoting controller.The conclusions summarize these properties at the model level.
  • Limitations and future work: Further scrutiny on other power-system models is needed because controller structure and performance depend on the underlying model and selected state and control weights.The authors identify testing other systems as future work.
  • Limitations and future work: The paper also identifies future extensions to decentralized control with supplementary WAC and to sparse differential-algebraic and structure-preserving models.It further raises questions about state-space representations amenable to sparsity-promoting design and fundamental performance limitations.

APPENDIX

The appendix summarizes a homotopy-and-ADMM procedure for sparsity-promoting control, followed by weight updates and structured polishing. Under a local convexity assumption, the approach is convergent and guarantees closed-loop stability.

  • Algorithmic approach: The optimization algorithm traces a homotopy path from the optimal centralized controller at γ = 0 to a desired γ value.This warm-start strategy continuously increases γ during optimization.
  • Algorithmic approach: For each γ along the path, the optimization problem is solved iteratively using the alternating direction method of multipliers.ADMM is applied for γ ∈ [0, γdes].
  • Algorithmic approach: ADMM weights are updated as wij = 1/(|Kij| + ε), with ε = 10^-3, for five update steps.The reweighted procedure is part of the sparsity-identification stage.
  • Algorithmic approach: After the sparsity pattern K is identified, the method solves a structured optimal-control problem to polish the controller.The appendix gives the associated Lyapunov equation for the closed-loop matrix A − B2K.
  • Guarantees and implementation: The iterative approach is convergent under a local convexity assumption, and A − B2K is guaranteed to be stable.The algorithms and numerical power-network data were implemented and made available in MATLAB.
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