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A power consensus algorithm for DC microgrids
Claudio De Persis, Erieke Weitenberg, Florian Dorfler
TL;DR
Growing deployment of DC microgrids calls for deeper system-theoretic understanding of their dynamics. This paper proposes source-power consensus control using communicated power measurements, and establishes convergence to weighted power-consensus equilibria while preserving a weighted geometric mean of source voltages.
Problem
The growing use of DC microgrids makes a deeper system-theoretic understanding of these dynamical networks important.
Source: §1 · p. 1Method
Results
Solutions from the stated initial-condition set remain bounded with positive voltages and asymptotically converge to equilibria with weighted consensus powers and preserved weighted geometric mean source voltage.
Source: §4.3 · p. 8Takeaways & Limitations
Preservation of the weighted geometric mean of source voltages allows an estimate of voltage excursion, although the scheme has no active voltage regulation.
Source: §6 · p. 12Existence of equilibria for ZIP loads is deferred to future research.
Source: §4.1 · p. 5
Abstract
from arXiv · showhide
A novel power consensus algorithm for DC microgrids is proposed and analyzed. DC microgrids are networks composed of DC sources, loads, and interconnecting lines. They are represented by differential-algebraic equations connected over an undirected weighted graph that models the electrical circuit. A second graph represents the communication network over which the source nodes exchange information about the instantaneous powers, which is used to adjust the injected current accordingly. This give rise to a nonlinear consensus-like system of differential-algebraic equations that is analyzed via Lyapunov functions inspired by the physics of the system. We establish convergence to the set of equilibria consisting of weighted consensus power vectors as well as preservation of the weighted geometric mean of the source voltages. The results apply to networks with constant impedance, constant current and constant power loads.
1 Introduction
Growing use of DC microgrids motivates stronger system-theoretic understanding, and this paper proposes a controller for power sharing among sources. Its contribution combines communicated power measurements with analysis of a nonlinear system that includes network and load constraints.
1 Introduction
- Growing renewable and storage deployment motivates DC microgrids, while HVDC networks are described as comparatively better than AC networks for low-loss long-distance transmission.The introduction also notes hybrid microgrids and mobile grids as DC applications.Source: §1 · p. 1
1.1 Literature review
- Prior work covers DC voltage and current regulation, current sharing, and HVDC control, while stability analysis remains challenging when communication and physical networks interact.The review also cites power-sharing methods without formal analysis and studies of DC power-flow feasibility.Source: §1.1 · pp. 1–2
1.2 Main contribution
- The proposed controller shares source power using communicated instantaneous measurements, with a weighted geometric average of source voltages preserved.The analysis treats ZIP loads and uses physically inspired Lyapunov arguments; the contribution extends a related AC microgrid idea to network-preserved DC models with explicit algebraic equations.Source: §1.2 · p. 2
1.3 Paper organization
- The paper introduces the microgrid model, presents the controller, analyzes general ZIP loads and then ZI loads, reports simulations, and concludes.The ZI specialization is stated to yield stronger results under weaker conditions.Source: §1.3 · p. 2
1.4 Notation
- The notation section defines vector stacking, identity and zero matrices, and element-wise logarithms.It also specifies a matrix-stacking convention for equal-column matrices.Source: §1.4 · p. 2
2 DC resistive microgrid
The DC microgrid is represented as a connected resistive network, then reduced to source-node balance equations by eliminating load voltages.
2 DC resistive microgrid
- The microgrid is modeled by a connected undirected graph whose edges are resistive lines weighted by conductance, with nodes partitioned into sources and loads.The edge weight is the reciprocal line resistance.Source: §2 · p. 2
- The resistive network’s current-potential relation is expressed using the graph incidence matrix and diagonal conductance matrix.Partitioning the nodes separates source and load currents and voltages in the balance equations.Source: §2 · pp. 2–3
- Positive-definite load and source network submatrices permit eliminating load voltages and reducing the balance equations to source nodes.The resulting conductance matrix is the Kron-reduced matrix; a separate term maps load-current injections to sources.Source: §2 · p. 3
3 Power consensus controllers
The controllers use communication among source nodes to average instantaneous injected powers and adjust source voltages. The section specifies the power-sharing behavior, load scope, and implementation interpretations of this consensus-like control.
3 Power consensus controllers
- Source nodes exchange instantaneous injected-power measurements over an undirected communication graph, and the controller uses neighboring powers to set source-terminal voltage.The communication graph’s nodes are the sources, though its edges need not match the physical network’s edges.Source: §3 · p. 3
- Positive initial source voltages remain positive because the controller renders the positive orthant positively invariant.The controller is initialized with each source voltage above zero.Source: §3 · p. 3
- The controller has a circuit interpretation; its ideal current source can also be generated by a series voltage source and resistance under the stated relation.The dynamic droop controller corresponds, in this notation, to constant capacitance and current source.Source: §3 · p. 3, Figure 1 · p. 3
- In digital implementation, neighboring sources’ injected powers are broadcast and processed with current measurements to compute terminal voltage.The closed-loop system is described by equations (9), (10), and (11).Source: §3 · pp. 3–4
- At steady state, nonzero source voltages yield proportional power sharing according to the prescribed C_i ratios.The paper states that positivity of the voltages will be established in later analysis.Source: §3 · p. 3
- A distributed averaging integral controller also yields power sharing, and numerical results report similar performance to controller (6).The two closed loops have similar dynamics near steady state when D_i = 0 and voltages are nearly unit-valued in per-unit terms.Source: §3 · p. 4
- The analysis covers parallel combinations of constant-current, constant-impedance, and constant-power loads, with stronger statements also reported for ZI loads.The load cases are denoted I, Z, and P, respectively.Source: §3 · p. 4
- The proposed power consensus differs from weighted power-mean consensus by adding communication-layer averaging and algebraic constraints, requiring physically inspired Lyapunov analysis.The comparison is made after neglecting algebraic constraints and the source-load distinction for the related algorithms.Source: §3 · p. 4
4 Power consensus algorithm with ZIP loads
This section analyzes equilibria and convergence properties of power-consensus systems with ZIP loads, including preservation of the weighted geometric mean of source voltages. It also characterizes ZI-load equilibria and convergence to a single asymptotically stable equilibrium.
4.1 Steady-state characterization
- Equilibria are characterized by load current balance and source power balance, with source injections sharing power proportionally to the coefficients Ci.The asymptotic shared power is the total current demand divided by the weighted sum of steady-state source voltages, and source and load voltages are linked through the balance equations.Source: §4.1 · p. 5
4.2 A Lyapunov function and hidden gradient form
- A Bregman function centered at an equilibrium reveals that the closed-loop system can be rewritten as a weighted gradient flow.This gradient relation underlies the Lyapunov-based convergence analysis.Source: §4.2 · p. 7
4.3 Convergence of solutions
- Under the theorem’s conditions, solutions from a compact sublevel set remain there with positive voltages and converge to equilibria preserving the weighted geometric mean of source voltages.The theorem assumes an equilibrium exists; analytical investigation of equilibrium existence is deferred.Source: §4.1 · p. 5, §4.3 · p. 8
- In the two-source example with Cs = I2C, each equilibrium source voltage equals the geometric mean of the initial source voltages, and the load voltage satisfies (19).The source-voltage product is preserved in this example.Source: §4.3 · p. 9
- With capacitors interconnecting loads, the equilibria remain unchanged and solutions still converge to EZIP ∩ ΛZIP ∩ VZIP.The convergence argument is analogous to that for the differential-algebraic model.Source: §4.3 · p. 9
4.4 The case of ZI loads
- For ZI loads, equilibria reduce to a source-voltage-only power-balance system, after which the load voltages can be calculated explicitly.The load voltages are obtained from Vl = (Yll + Y*l)^−1YlsVs.Source: §4.4 · p. 10
- For ZI loads with P_l*=0, trajectories in the stated M-sublevel sets converge to a single asymptotically stable equilibrium without requiring Yeq definiteness.The limit equilibrium also belongs to EZI ∩ ΛZI ∩ VZI.Source: §4.4 · pp. 10–11
- For ZI loads, the convergence set preserves the weighted voltage product, ∏V_i^C_i = ∏V_i^C_i(0).The limit set is EZI ∩ ΛZI ∩ VZI.Source: §4.4 · p. 10
5 Simulations
The simulations compare the proposed control strategy with a distributed averaging integral controller under gradually increasing constant-power loads. Both achieve proportional power sharing at steady state and perform similarly, with a slight overshoot for the integral controller at source 2.
5 Simulations
- Both controllers achieve proportional power sharing at steady state as the total load rises from 70 W to 245 W.Five of the seven constant-power loads are switched on gradually between 9.5 and 10.5 ms.Source: §5 · p. 11
- In simulations with seven constant-power loads, five switched on gradually from 9.5 to 10.5 ms as total load rose from 70 W to 245 W, the proposed strategy and distributed averaging integral controller performed similarly.Fig. 4 shows power measured at source nodes for both control strategies.Source: §5 · p. 11, Figure 4 · p. 11
- Fig. 3 shows voltage evolution at both the sources and the loads during the simulation.The network diagram distinguishes sources, loads, electrical interconnections, and the controllers’ communication graph.Source: §5 · p. 11, Figure 3 · p. 11, Figure 5 · p. 11
6 Conclusions
The conclusions extend the results to ZIP loads and note a passivity-based route for including terminal capacitors. The proposed scheme preserves the weighted geometric mean of source voltages but does not actively regulate voltage.
6 Conclusions
- The results apply to network-preserved DAE models of microgrids with ZIP loads.Terminal capacitors modeling Π-model lines or power-converter components can be included through passivity-based analysis.Source: §6 · pp. 11–12
- The scheme preserves the weighted geometric mean of voltages, which allows an estimate of voltage excursion, but provides no active voltage regulation.Adding voltage controllers is identified as an open problem.Source: §6 · p. 12
- The solvability of the power-flow equations induced by power consensus remains to be investigated.The conclusions also identify dynamical inductive lines and loads, and extensions to network-preserved AC microgrids, as research directions.Source: §6 · p. 12