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A Robust Consensus Algorithm for Current Sharing and Voltage Regulation in DC Microgrids

Michele Cucuzzella, Sebastian Trip, Claudio De Persis, Xiaodong Cheng, Antonella Ferrara, Arjan van der Schaft

arXiv:1708.04608v2math.OCeess.SY

TL;DR

DC microgrids require simultaneous proportional current sharing and voltage regulation despite unknown loads and modelling uncertainties. The paper proposes a communication-based consensus-like controller with a manifold constraint and sliding-mode strategies, formally showing current-sharing convergence and weighted-average voltage regulation.

  • Problem

    The paper addresses simultaneous proportional current sharing and voltage regulation in DC microgrids with unknown load demand, modelling uncertainties, and arbitrary network topology.

  • Method

    The controller exchanges generated-current information over a communication network, constrains the system to a suitable manifold, and uses two sliding-mode strategies to reach it in finite time.

  • Results

    The analysis proves proportional current sharing and convergence of the microgrid’s weighted-average voltage to the weighted average of the voltage references.

  • Takeaways & Limitations

    The proposed distributed scheme achieves the two control objectives using local voltage measurements and neighboring current information without exact knowledge of network parameters or current demand.

Abstract

from arXiv · show

In this paper a novel distributed control algorithm for current sharing and voltage regulation in Direct Current (DC) microgrids is proposed. The DC microgrid is composed of several Distributed Generation units (DGUs), including Buck converters and current loads. The considered model permits an arbitrary network topology and is affected by unknown load demand and modelling uncertainties. The proposed control strategy exploits a communication network to achieve proportional current sharing using a consensus-like algorithm. Voltage regulation is achieved by constraining the system to a suitable manifold. Two robust control strategies of Sliding Mode (SM) type are developed to reach the desired manifold in a finite time. The proposed control scheme is formally analyzed, proving the achievement of proportional current sharing, while guaranteeing that the weighted average voltage of the microgrid is identical to the weighted average of the voltage references.

I. INTRODUCTION

DC microgrids integrate diverse sources and loads while addressing voltage regulation and proportional current sharing under uncertain network conditions. The paper develops a distributed, communication-based robust control approach using sliding-mode strategies and a microgrid model with locally available measurements and bounded unknown parameters.

  • I. INTRODUCTION: DC microgrids can connect sources and loads directly through DC-DC converters, avoiding several AC conversion and synchronization issues.The cited passage lists reduced DC-AC and AC-DC conversion, absence of reactive power and harmonics, no frequency synchronization, and absent skin effect.
  • I. INTRODUCTION: Voltage regulation supports proper load operation, while current sharing prevents source overstressing and can allocate demand according to generation capacity.Conventional hierarchical schemes address both objectives, but current sharing generally prevents every node voltage from matching its individual reference.
  • I. INTRODUCTION: The proposed algorithm combines proportional current sharing with voltage regulation through a communication-based consensus-like controller and a suitable manifold.Its voltage objective is weighted-average regulation rather than necessarily matching every node reference.
  • I. INTRODUCTION: Second- and third-order sliding-mode controllers drive the system to the designed manifold in finite time while addressing modelling uncertainties.The third-order design additionally produces a continuous control signal suitable for the converter duty cycle and avoids the variable switching-frequency issue identified for SOSM.
  • I. INTRODUCTION: The model represents Buck converter-based DGUs and resistive-inductive interconnecting lines using network equations derived from Kirchhoff’s laws.The network topology is represented by an incidence matrix, while converter output voltage is the control input and may be expressed as duty cycle times source voltage.
  • I. INTRODUCTION: Controller design assumes local state measurements, constant unknown network parameters and current demand with known bounds, while seeking independence from the unknown reduced network topology.The paper states that the proposed strategy remains applicable even if the constant-parameter assumption is removed.

III. CURRENT SHARING AND VOLTAGE BALANCING

The control objectives are proportional allocation of generated current and weighted-average voltage balancing. Because the required feedforward input depends on unavailable network and load information, the paper motivates distributed controllers using local voltage measurements and communicated generated currents.

  • III. CURRENT SHARING AND VOLTAGE BALANCING: Proportional current sharing requires each DGU’s generated current to scale inversely with its capacity weight, so w_iI_ti = w_jI_tj.The weights w_i represent the generation capacities of the corresponding converters.
  • III. CURRENT SHARING AND VOLTAGE BALANCING: The current-sharing condition fixes voltage differences across the network but leaves a common voltage shift undetermined.The freedom follows because equal shifts do not change the voltage differences represented by B^T V.
  • III. CURRENT SHARING AND VOLTAGE BALANCING: Voltage balancing keeps the weighted average of PCC voltages equal to the weighted average of their references when individual voltage matching conflicts with current sharing.Weights are selected as 1/w_i so higher-capacity sources experience relatively smaller voltage deviations.
  • III. CURRENT SHARING AND VOLTAGE BALANCING: Equal weights reduce the objectives to equal current sharing and equality between the arithmetic averages of microgrid voltage and voltage references.This is presented as a special case of the weighted objectives.
  • III. CURRENT SHARING AND VOLTAGE BALANCING: Achieving both objectives determines an optimal steady-state Buck-converter input, but calculating it requires nearly all network parameters and the current demand.The proposed distributed controllers avoid that unavailable information by using local voltages and exchanging generated currents among neighboring DGUs.

A. Steady state voltages

Under the two control objectives, changing the voltage references shifts every steady-state node voltage by the same quantity. This voltage-shifting property can help tune references to avoid undesired node-voltage levels.

  • A. Steady state voltages: The steady-state node voltages depend on both the load currents and the voltage references.The references are therefore available as design variables for influencing the resulting voltage levels.
  • A. Steady state voltages: Changing the reference vector while preserving the objectives produces a uniform shift in all steady-state node voltages.The result follows because current sharing preserves voltage differences, while voltage balancing determines the weighted-average shift.
  • A. Steady state voltages: Any node’s steady-state voltage can be lowered or increased by adjusting its own voltage reference.The paper presents this property as support for designing references that avoid excessively low or high voltages.

IV. A MANIFOLD-BASED CONSENSUS ALGORITHM

The proposed solution combines distributed integrators, a communication-based consensus protocol, and a designed manifold to achieve current sharing and voltage regulation. Under an undirected connected communication graph and suitable initialization, the manifold preserves the weighted reference-voltage average; changing network topology remains outside the main analysis.

  • Algorithm structure: The algorithm augments the microgrid with distributed integrators and designs a manifold for simultaneous current sharing and voltage regulation.The manifold is W^-1(V − V⋆) − θ = 0.
  • Consensus protocol: Communication dynamics use generation-capacity weights and a weighted Laplacian, allowing a steady state with proportional generated currents.The communication graph may differ from the reduced microgrid topology.
  • Assumptions: The controller assumes an undirected, connected communication graph and integrator initialization satisfying 1_n^Tθ(0) = 0.Initializing all integrator states to zero is the straightforward choice that satisfies this condition.
  • Voltage regulation: Preserving the average integrator state enables the weighted average voltage to match the weighted average voltage reference on the desired manifold.The preserved quantity follows from the communication Laplacian property 1_n^TL_c = 0.
  • Scope: The main results assume a constant network topology; analysis of plugging converters in or out as a switched or hybrid system is outside scope.The paper describes integrator initialization extensions for topology changes without analyzing the resulting switched or hybrid dynamics.

V. SLIDING MODE CONTROLLERS

The paper uses distributed second- and third-order sliding mode controllers to drive the augmented microgrid to the desired manifold in finite time. The controller choice depends on implementation requirements.

  • Controller design: Distributed SOSM and 3SM laws steer the augmented system to the desired manifold in finite time.The two laws are presented as alternatives whose selection depends on the desired implementation.
  • Sliding-mode concepts: Sliding mode control is introduced through sliding functions, r-sliding manifolds, and controllers that enforce the corresponding manifold after finite-time reaching.The controller order matches the order of the sliding mode it enforces.
  • Sliding-mode concepts: An r-order sliding mode requires the state to reach the r-sliding manifold and remain there for all subsequent time.The manifold imposes zero conditions on the sliding function and its derivatives through order r−1.
  • Controller design: The sliding function is constructed from the desired manifold and serves as the output driven to zero by the sliding-mode controllers.The paper introduces this sliding function after defining the desired manifold.

A. Second order SM control: variable switching frequency

The second-order sliding mode design exploits a relative degree of two to reach the desired manifold in finite time despite bounded uncertainties. Its discontinuous control signal can produce variable switching frequency and increased losses.

  • SOSM design: The sliding function has relative degree two, so an SOSM controller drives σ and σ̇ to zero in finite time.The associated auxiliary variables are ξ1 = σ and ξ2 = σ̇.
  • Robustness assumptions: The auxiliary-system formulation uses bounded model terms and disturbance mappings to support robust SOSM control.The entries of b and Gd are assumed to have known bounds.
  • Implementation: Only ξ1i = wi(Vi − V⋆i) − θi is required to generate the control signal ui.The paper notes that this quantity can be detected using a peak detector.
  • Implementation limitation: The discontinuous SOSM signal can directly switch the Buck converter, but its switching frequency cannot be fixed a priori and power losses could be high.A continuous duty-cycle signal requires a different implementation such as PWM.

B. Third Order SM control: duty cycle

The third-order sliding mode design integrates the discontinuous signal so the converter input becomes continuous and can serve as a duty cycle. This raises the relative degree to three and requires finite-time control of σ, σ̇, and σ̈.

  • Duty-cycle implementation: Integrating the discontinuous sliding-mode signal produces a continuous converter input that can be used as the Buck converter duty cycle.The new input is the derivative of the continuous control signal.
  • 3SM design: The new control input gives the system relative degree three, requiring a 3SM controller to reach σ = σ̇ = σ̈ = 0 in finite time.The auxiliary variables are ξ1 = σ, ξ2 = σ̇, and ξ3 = σ̈.
  • Robustness assumptions: The 3SM design assumes bounded derivatives of the uncertain model term and bounded disturbance mappings.The bound on ḃ is represented by the known positive constant βmaxi.
  • Implementation: The controller can replace unmeasurable derivatives with finite-time estimates from a second-order differentiator.The measured current and voltage are used to retrieve σ̇ and σ̈, and the estimated variables replace the originals in the control law.
  • Distributed implementation: The distributed design requires only current information from communicating DGUs, and its local synthesis complexity does not depend on microgrid size.The result is independent of the particular SOSM or 3SM controller choice.

VI. STABILITY ANALYSIS

The stability analysis shows finite-time convergence to sliding manifolds that enforce voltage balancing, followed by exponential convergence to a constant state. Both SOSM and 3SM controllers establish the required manifold conditions under the stated assumption.

  • Post-manifold convergence: After the sliding manifold is attained, the system solutions converge exponentially to a constant point, additionally achieving Objective 1.The equivalent reduced-order system describes the controlled dynamics for t ≥ Tr together with algebraic state relations.
  • Finite-time manifold convergence: The SOSM controller drives the system in finite time to the sliding manifold defined by σ = ˙σ = 0.This result holds under Assumption 1 and uses the SSOSM control law.
  • Finite-time manifold convergence: The 3SM controller drives the system in finite time to the higher-order manifold defined by σ = ˙σ = ¨σ = 0.The associated differentiator estimates the required quantities in finite time before the controller enforces these conditions.
  • Voltage balancing: Reaching σ = 0 is sufficient to achieve Objective 2, identified as voltage balancing.The analysis uses sliding-mode order reduction to study the dynamics after the manifold is reached.

B. Exponential convergence and objectives attainment

The equivalent reduced-order system is shown to converge: line currents and voltages approach constant values, enabling the proposed controllers to attain proportional current sharing and weighted-average voltage balancing after finite-time manifold reaching.

  • Convergence analysis: The equivalent reduced-order system is analyzed using semistability to establish convergence of its current and voltage states.The analysis first proves convergence of line currents, then uses it to establish convergence of node voltages.
  • Current convergence: A positive definite diagonal line-resistance matrix satisfies the required condition, making the equivalent system semistable and ensuring that line currents converge.The convergence holds for all initial conditions after the finite reaching time T_r.
  • Current-sharing objective: The proposed distributed SSOSM and 3SM controllers make generated currents converge exponentially after finite time to the proportional-sharing subspace W^-1 1_n 1_n^T.This establishes proportional current sharing under Assumptions 1–3.
  • Voltage-balancing objective: After finite-time reaching, the weighted average voltage satisfies 1_n^T W^-1 V(t) = 1_n^T W^-1 V⋆ for all t ≥ T_r.The result follows from the manifold relation and the initialization condition on the controller integrators.
  • Communication robustness: If communication fails, each DGU can still converge in finite time to its own voltage reference, although the communication layer is required for current sharing and voltage balancing.The integrator states used for sharing and balancing are not needed for local voltage regulation.
  • Scope and assumptions: Violating the integrator initialization assumption shifts the weighted-average voltage, while stability and proportional current sharing remain guaranteed.The shift is determined by the initial controller-state aggregate 1_n^T W^-1 θ(0).

VII. SIMULATION RESULTS

The simulation evaluates the proposed third-order sliding-mode controller on a four-DGU microgrid with specified communication weights and parameter settings.

  • VII. SIMULATION RESULTS: The simulations use four interconnected DGUs with a communication network and current-demand parameters defined for each unit.Edge weights are γ12 = γ23 = γ34 = 10, and each controller uses αi = 2.4.

A. Scenario 1: proportional current sharing

After a current-demand variation, the controller preserves weighted-average voltage regulation and achieves proportional current sharing while using local measurements and neighboring current information.

  • A. Scenario 1: proportional current sharing: The weighted average PCC voltage remains equal to the weighted average of the voltage references after the demand variation.The result corresponds to the voltage-regulation objective.
  • A. Scenario 1: proportional current sharing: Each DGU’s generated current converges to its desired value, achieving proportional current sharing.The generated currents and their desired values are shown in Figure 4.
  • A. Scenario 1: proportional current sharing: The 3SM controllers use local PCC-voltage measurements and neighboring generated-current information received through the communication network.The resulting current-sharing signals and control inputs are reported in Figure 5.

B. Scenario 2: opening of a distribution line

The simulations examine line opening, plug-and-play DGU changes, and communication failure, while reporting voltage regulation, current sharing, and the associated control signals.

  • B. Scenario 2: opening of a distribution line: Opening the distribution line between DGUs 1 and 4 is used to assess controller performance under an electric-fault condition.A current-demand variation is then applied as in Scenario 1.
  • Plug-and-play operation: Disconnecting DGU 4 leaves equal current sharing among DGUs 1, 2, and 3, while reconnection restores sharing among all DGUs.During isolation, DGU 4 supplies its local load.
  • Scenario configurations: The considered configurations represent four microgrid scenarios, with resistive-inductive line impedances and failed or removed components marked in red.The configurations cover the scenarios used for the simulation studies.
  • Simulation outputs: The simulations compare line currents and integrated control inputs with optimal feedforward inputs across the tested scenarios.The figures report these quantities for the relevant scenario responses.
  • Communication-failure robustness: Communication failure is tested by interrupting the link between DGUs 3 and 4, while voltage deviations remain within 0.3% of the 380 V nominal value in the reported scenarios.The practical reference cited in the passage is a 5% voltage-deviation range.
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