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Breaking the Hierarchy: Distributed Control & Economic Optimality in Microgrids

Florian Dörfler, John Simpson-Porco, Francesco Bullo

arXiv:1401.1767v2math.OC

TL;DR

Microgrid control must meet diverse primary, secondary, and tertiary objectives while remaining flexible and plug-and-play without hierarchical timing or centralized decisions. The paper analyzes nonlinear droop control, distributed secondary regulation, and constrained AC economic dispatch. It finds that averaging-based distributed control supports these objectives, while droop-control steady states correspond exactly to economic-dispatch minimizers.

  • Problem

    Microgrid control must coordinate varied objectives under plug-and-play requirements without hierarchical decision making or time-scale separation.

  • Method

    The paper combines first-principles droop-control analysis with centralized, decentralized, and distributed secondary-control strategies and constrained AC economic-dispatch analysis.

  • Results

    Droop-control steady states are in one-to-one correspondence with minimizers of the AC economic-dispatch problem.

  • Takeaways & Limitations

    Distributed averaging-based PI control can address primary, secondary, and tertiary objectives using local measurements and nearest-neighbor communication without time-scale separation.

Abstract

from arXiv · show

Modeled after the hierarchical control architecture of power transmission systems, a layering of primary, secondary, and tertiary control has become the standard operation paradigm for islanded microgrids. Despite this superficial similarity, the control objectives in microgrids across these three layers are varied and ambitious, and they must be achieved while allowing for robust plug-and-play operation and maximal flexibility, without hierarchical decision making and time-scale separations. In this work, we explore control strategies for these three layers and illuminate some possibly-unexpected connections and dependencies among them. Building from a first-principle analysis of decentralized primary droop control, we study centralized, decentralized, and distributed architectures for secondary frequency regulation. We find that averaging-based distributed controllers using communication among the generation units offer the best combination of flexibility and performance. We further leverage these results to study constrained AC economic dispatch in a tertiary control layer. Surprisingly, we show that the minimizers of the economic dispatch problem are in one-to-one correspondence with the set of steady-states reachable by droop control. In other words, the adoption of droop control is necessary and sufficient to achieve economic optimization. This equivalence results in simple guidelines to select the droop coefficients, which include the known criteria for power sharing. We illustrate the performance and robustness of our designs through simulations.

I. INTRODUCTION

Microgrid control must coordinate diverse objectives under variable conditions while supporting plug-and-play operation without centralized decision making or enforced time-scale separation. The paper develops primary, secondary, and tertiary strategies connecting droop control, distributed averaging, and economic dispatch.

  • Motivation: Primary and secondary loops can interact adversely unless time-scale separation, careful gain tuning, or load estimates are available.These restrictions motivate control strategies with less reliance on hierarchical timing and detailed system information.
  • Motivation: Low inertia and distributed, variable operation require fast, reliable control that adapts to unknown loads and network conditions.The architecture must support near plug-and-play operation while maintaining voltage, frequency, and power-flow tolerances.
  • Approach: The framework combines nonlinear differential-algebraic modeling, decentralized primary droop control, and networks with nonzero resistance-to-reactance ratios.This extends conventional lossless-line models to heterogeneous microgrid components and control objectives.
  • Secondary control: Distributed averaging-based secondary controllers regulate frequency while preserving primary injections and stability, without time-scale separation, including when only some generators participate.The strategies use communication among generating units and have been validated experimentally.
  • Tertiary control: The AC economic-dispatch minimizers correspond one-to-one with steady states reachable through decentralized droop control.The resulting droop design links optimal coefficients to marginal generation costs and includes proportional power sharing as a special case.
  • Overall design: Simple distributed averaging-based PI controllers address primary, secondary, and tertiary objectives using local measurements and nearest-neighbor communication.The claimed design is model-free with respect to network topology, parameters, loading profile, and source count.

B. Primary Droop Control

Primary droop control links inverter frequency errors to measured power injections in islanded microgrids. It balances power and synchronizes frequency but induces a steady-state frequency error that secondary inputs must remove.

  • Droop model: Frequency droop control makes each inverter’s frequency error proportional to its measured power injection.D_i is the inverse droop coefficient, while P_i* is the nominal injection setpoint.
  • Droop model: The droop-controlled microgrid is represented by nonlinear differential-algebraic equations.The model describes the coupled inverter and network dynamics used for primary-control analysis.
  • Steady-state behavior: Droop control produces a static steady-state frequency error determined by the scaled power imbalance.The synchronous frequency is zero exactly when nominal injections balance the relevant demand.
  • Steady-state behavior: Unknown, variable loads prevent reliable selection of nominal injections that balance demand, while arbitrarily large droop coefficients can slow or destabilize primary control.Thus, reducing frequency error through droop coefficients alone creates a speed and stability constraint.
  • Secondary correction: Secondary control inputs augment primary droop control to eliminate the frequency error.The secondary-control equations introduce u_i(t) as additional inputs to the primary dynamics.

D. Tertiary Operational Control

Tertiary control formulates economic dispatch as a constrained optimization over steady-state angles and secondary inputs. The paper relates this nonlinear AC problem to tractable DC formulations and shows model extensions preserve relevant control properties.

  • Economic dispatch: Tertiary operation minimizes an economic-dispatch objective representing the cost of accumulated generation.The cost coefficient α_i is associated with source i.
  • Economic dispatch: The dispatch decision variables are voltage angles and secondary inputs subject to nonlinear steady-state, branch-security, and generation constraints.The AC security constraint limits power flow on each branch, while generation constraints limit source operation.
  • Model extensions: Synchronous machines, inverter measurement delays, and frequency-dependent loads can be incorporated while preserving the stated equilibrium and local-stability results.The paper therefore focuses on the simpler microgrid model while asserting applicability of the control strategies to these extensions.

III. DECENTRALIZED PRIMARY CONTROL STRATEGIES

The primary droop-controlled microgrid has rotational symmetry, so synchronized trajectories can be analyzed as equilibria in a rotating frame. Under flow-feasibility conditions, these equilibria are unique modulo rotation and locally exponentially stable.

  • Symmetry and synchronization: A synchronized solution is a one-dimensional manifold because the microgrid equations are invariant under rigid rotation of all voltage angles.Stability and uniqueness are therefore understood modulo rotational symmetry.
  • Symmetry and synchronization: Transforming to a frame rotating at the synchronous frequency converts a synchronized trajectory into an equilibrium of the shifted control system.The shifted secondary input is u_i = −D_iω_sync, and the resulting injections remain balanced.
  • Equivalences: Primary, constant-input secondary, and shifted systems have equivalent stable synchronization or equilibrium manifolds and identical power injections.The correspondence preserves the synchronization manifold while changing coordinates or representing the frequency correction as a constant input.
  • Droop scaling: Scaling all droop coefficients by a positive common factor changes synchronous frequency inversely but leaves equilibria and stability properties unchanged.The invariance follows because D_iω_sync remains constant and time can be rescaled.
  • Existence and stability: A stable unique equilibrium manifold exists exactly when the shifted power flow is feasible under the stated angle constraints.The theorem characterizes feasibility through the unique KCL-consistent branch-flow vector and the condition involving A^-1ξ.

C. Power Flow Constraints and Proportional Power Sharing

The section characterizes feasible inverter power injections and shows when decentralized droop control can realize them. Proportional droop coefficients yield rating-based load sharing, while reachability is equivalent to power-flow feasibility under stated stability and constraint conditions.

  • Proportional Power Sharing: Proportional droop coefficients produce fair load sharing among inverters according to their ratings, including in lossy and meshed circuits.
  • Proportional Power Sharing: Theorem 3.3 links proportional droop selection with power-flow constraints and proportional power sharing under equilibrium conditions.
  • A feasible power injection setpoint satisfies power balance, load invariance, and branch-flow feasibility within the angle bound γ.
  • Power Flow Shaping: Coefficient selection is equivalent to feasibility: droop coefficients can realize exactly those steady-state injections that are γ-feasible.
  • Power Flow Shaping: The reachable equilibrium is locally exponentially stable if and only if β(P_i*) is nonnegative for every inverter.
  • Generation Constraints: A γ-feasible injection setpoint does not generally guarantee that each inverter respects its generation constraint P_set,i ∈ [0, P_i].

IV. CENTRALIZED, DECENTRALIZED, AND DISTRIBUTED SECONDARY CONTROL STRATEGIES

Decentralized secondary integral control faces fundamental trade-offs: it may fail exact frequency regulation, disrupt power sharing, or leave steady-state injections underdetermined. These limitations motivate distributed strategies that recover primary injections while regulating frequency.

  • Decentralized secondary control uses local frequency-error integration to eliminate the static frequency error caused by primary droop control.
  • Decentralized Control: With one secondary-integrating inverter, frequency regulation is achieved but power sharing fails, concentrating the total power imbalance on one generator.
  • Decentralized Control: With multiple decentralized integrators, equilibrium source injections are determined only up to a subspace whose dimension equals the number of integrators.
  • Decentralized Control: Because steady-state injections depend on initial values, loads, and disturbances, decentralized integral control cannot recover primary objectives such as proportional sharing or power-flow shaping.
  • Decentralized Control: For small ϵ and large k, decentralized integral control achieves practical stabilization but not exact frequency regulation.

B. Centralized Averaging PI (CAPI) Control

CAPI uses averaged frequency feedback to restore frequency while preserving primary injections, but its implementation requires centralized communication and carefully tuned gains. Its stability is tied to the stability condition of primary droop control.

  • CAPI Limitations: CAPI does not generally preserve power sharing unless the droop coefficients and integral gains are carefully tuned.
  • CAPI Implementation: The CAPI secondary variable can be implemented centrally by collecting inverter frequency measurements and broadcasting a common control signal.
  • Stability: Modulo rotational symmetry, the relevant generalized eigenvalues are real and negative for the perturbed stability analysis.
  • Stability: Stability persists as the perturbation vanishes because the only zero mode corresponds to rotational symmetry.
  • CAPI preserves the primary power injections while restoring frequency, and its stable unique equilibrium exists under the primary droop stability condition.
  • CAPI Limitations: The CAPI design requires all-to-all communication among inverters and a restrictive gain selection.

C. Distributed Averaging PI (DAPI) Control

DAPI replaces centralized averaging with communication over a connected sparse graph. It regulates frequency, preserves primary power injections, and allows its droop and integral gains to be selected independently.

  • DAPI uses averaging proportional-integral control over a weighted, connected, undirected communication graph among inverters.
  • Stability: DAPI stability is equivalent to satisfaction of the primary droop stability condition and yields a locally exponentially stable unique equilibrium manifold.
  • Performance: DAPI regulates network frequency and preserves the power injections established by primary control while requiring only sparse communication.
  • Design: The DAPI gains D_i > 0 and k_i > 0 can be chosen independently.

D. Partial Secondary Control

Partial secondary control lets only a connected subset of inverters regulate frequency while preserving stability, proportional sharing, and lower communication complexity. The participating inverters can share load according to their power ratings.

  • The scheme partitions inverters into primary-only units and units performing secondary CAPI or DAPI control over a complete or connected communication graph.
  • Partial secondary control stabilizes the microgrid and regulates frequency when primary droop control satisfies its stability condition.
  • The secondary-regulating inverters share load proportionally according to their power ratings.
  • Only a subset of inverters needs to participate in secondary control, reducing communication complexity and increasing microgrid adaptivity and modularity.

A. Convex Reformulation of the AC Economic Dispatch

The paper reformulates constrained AC economic dispatch through a DC problem for acyclic networks and links its optima to appropriately designed droop-controlled steady states. This establishes conditions under which decentralized droop control achieves economic dispatch.

  • The nonlinear, non-convex AC economic dispatch is approximated by a corresponding convex DC economic dispatch.
  • In acyclic networks, AC and DC economic dispatch are equivalent when their respective security-constrained problems are feasible.
  • The equivalence follows from a bijective change of variables between AC and DC branch flows under appropriate security constraints.
  • The AC–DC equivalence relies on acyclic networks; for cyclic networks, the two dispatch problems are generally not equivalent.
  • Any AC economic-dispatch minimizer can be achieved by appropriately designed droop control, and every droop-controlled steady state is optimal for suitably chosen dispatch parameters.
  • Droop coefficients should be larger for desirable sources with smaller cost coefficients, while proportional selection also supports proportional power sharing.

VI. SIMULATION CASE STUDY

Simulations on an islanded IEEE 37 distribution grid compare primary droop, decentralized integral, and distributed averaging-based control after a load step. Distributed control regulates frequency while preserving proportional sharing and economic optimality.

  • Distributed and averaging-based PI controllers address primary, secondary, and tertiary objectives without hierarchical implementation, time-scale separation, or detailed system knowledge.
  • The controllers are evaluated on an islanded IEEE 37 distribution grid with 16 sources and a communication network among distributed generators.
  • After a load step, primary droop produces frequency deviation, whereas decentralized integral control and DAPI quickly regulate frequency.
  • DAPI preserves proportional power sharing and economic optimality, unlike decentralized integral control, while voltage dynamics remain similar across controllers.

VII. CONCLUSIONS

The paper studies decentralized and distributed control across primary, secondary, and tertiary microgrid layers, relaxing conventional information-structure and time-scale restrictions. The analysis remains limited by local and network-model assumptions, with several dynamic and communication factors left for future work.

  • Contributions: Decentralized and distributed primary, secondary, and tertiary strategies connect microgrid control layers while reducing reliance on conventional hierarchy and time-scale separation.These designs target greater applicability to microgrids and distribution-level systems.
  • Limitations: The analysis is local and formally restricted to acyclic networks with constant resistance-to-reactance ratios.These restrictions define an important boundary on the theoretical results.
  • Limitations and future work: More detailed models incorporating reactive power flows, voltage dynamics, and ramping constraints on inverter injections remain for future work.The paper also identifies sampled or event-triggered communication schemes with delays as a future direction.
  • Simulation illustrations: Simulations illustrate controller behavior on the IEEE 37 microgrid, including frequency and voltage regulation, power-injection dynamics, and marginal-cost dynamics.The figures compare control responses and show dynamics following a step change in load.
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