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Robust stability analysis of DC microgrids with constant power loads

Jianzhe Liu, Wei Zhang, Giorgio Rizzoni

arXiv:1702.07580v1math.OC

TL;DR

Existing DC microgrid stability analyses often consider a fixed equilibrium, although uncertain CPL loading changes the equilibrium and may make stability difficult to assess. This paper formulates stability across such operating conditions as a polytopic robust-stability problem and derives structure-based sufficient conditions. In simulation, the conditions identify a stable operating range of 5–20 kW when droop gains are set to 0.2.

  • Problem

    Uncertain CPL powers change the microgrid’s equilibria, but existing stability analyses often rely on a fixed, known equilibrium that may not be available.

    Source: §I · p. 1, A. Robust Stability under Uncertain CPLs · p. 4

  • Method

    The paper formulates stability across uncertain CPL operating conditions as a polytopic robust-stability problem and derives sufficient conditions that exploit diagonal matrix uncertainty.

    Source: §I · p. 2

  • Results

    With droop gains set to 0.2, problem (10) is feasible and simulations show stability as load power varies in [5kW,20kW].

    Source: A. Simulation Validation · p. 8

  • Takeaways & Limitations

    For a given CPL power range, a voltage constraint can be computed that guarantees local exponential stability when satisfied during operation.

    Source: §I · p. 2

    The reduced-complexity condition can be conservative: some locally robustly stable microgrids may satisfy Lemma 1 but fail Lemma 2.

    Source: B. Robust Stability Analysis · p. 5

Abstract

from arXiv · show

This paper studies stability analysis of DC microgrids with uncertain constant power loads (CPLs). It is well known that CPLs have negative impedance effects, which may cause instability in a DC microgrid. Existing works often study the stability around a given equilibrium based on some nominal values of CPLs. However, in real applications, the equilibrium of a DC microgrid depends on the loading condition that often changes over time. Different from many previous results, this paper develops a framework that can analyze the DC microgrid stability for a given range of CPLs. The problem is formulated as a robust stability problem of a polytopic uncertain linear system. By exploiting the structure of the problem, we derive a set of sufficient conditions that can guarantee robust stability. The conditions can be efficiently checked by solving a convex optimization problem whose complexity does not grow with the number of buses in the microgrid. The effectiveness and non-conservativeness of the proposed framework are demonstrated using simulation examples.

I. INTRODUCTION

CPLs can destabilize DC microgrids, while existing analyses often rely on a fixed equilibrium despite loading-dependent changes. The paper develops robust-stability conditions for uncertain CPLs and models an n-bus microgrid with locally controlled sources and loads.

I. INTRODUCTION

  • The framework addresses changing CPL-dependent equilibria by formulating stability for a general microgrid as robust stability of a polytopically uncertain linear system.It derives sufficient conditions that exploit uncertainty in diagonal entries of the system matrix and can be checked by convex optimization.Source: §I · pp. 1–2

II. GENERAL DC MICROGRID MODELLING

  • An n-bus DC microgrid is represented as an undirected graph whose vertices are buses and edges are transmission lines.The modeling setup assumes one controllable voltage source and one CPL per bus, while noting it can extend to buses with multiple or no sources or loads.Source: §II · p. 2

1) Voltage Source:

  • Each bus’s controllable voltage source uses standard V-I droop control, and its output current is modeled as a state variable.The source connects to the microgrid through a resistor and inductor; internal power-electronic dynamics are ignored.Source: 1) Voltage Source · pp. 2–3

2) CPL:

The model represents each CPL as a nonlinear current sink and combines load, source, and DC-link dynamics into a system model. The resulting matrices encode system parameters, network topology, and droop control.

2) CPL:

  • A CPL is modeled as a current sink injecting p_k/v_lk, with uncertain power p_k bounded within a given interval.This term introduces the additive nonlinearity in the load dynamics.Source: 2) CPL · p. 3

3) DC Link Capacitor:

  • The DC-link capacitor voltage is a state determined by source current, load current, and currents flowing between neighboring buses.The neighbor set and transmission-line resistances encode the network connections in its dynamics.Source: 3) DC Link Capacitor · p. 3
  • Combining source, CPL, and capacitor models yields an overall closed-loop system whose matrix A depends on system parameters, topology, and droop gains.The model can be extended to other transmission-line models and control laws, with changes to parameters or state dimension as needed.Source: B. n-bus DC Microgrid · p. 3

A. Robust Stability under Uncertain CPLs

Because uncertain CPL powers change the microgrid’s equilibria, the paper studies stability across the feasible equilibria rather than at one predetermined equilibrium. It defines locally robust stability over equilibria associated with bounded CPL powers and operationally feasible voltages.

A. Robust Stability under Uncertain CPLs

  • The analysis considers CPL power vectors in a bounded set and restricts attention to equilibria whose CPL voltages lie in a known operational constraint set.Some equilibria may not be physically admissible by the circuit.Source: A. Robust Stability under Uncertain CPLs · p. 4
  • With n uncertain CPL powers, the equilibria require solving n quadratic equations, may have 2^n solutions, and cannot generally be identified as the equilibrium the system will reach.As CPL power varies, the possible equilibria cannot be characterized for a general DC microgrid.Source: A. Robust Stability under Uncertain CPLs · p. 4
  • A system is locally robustly stable when every equilibrium in the feasible set is locally exponentially stable.The feasible set comprises equilibria associated with the uncertain CPL powers and constrained CPL voltages.Source: A. Robust Stability under Uncertain CPLs · p. 4

B. Robust Stability Analysis

The analysis represents CPL-induced linearized systems as a polytopic uncertainty set and uses sufficient conditions to certify local robust stability. It reduces the computational burden of checking stability across that set, with a more tractable alternative that can be more conservative.

B. Robust Stability Analysis

  • A common Lyapunov function for the uncertainty-set vertices guarantees stability across the polytopic set, whose matrices correspond to possible equilibria and linearized systems.Lemma 1 expresses this sufficient condition as 2n LMIs, which can impose a large computational burden when n is nontrivial.Source: B. Robust Stability Analysis · pp. 4–5
  • The Lemma 2 condition replaces 2n LMIs with a convex optimization problem whose complexity does not increase with n.It checks an extreme case with each CPL at maximum power and lowest steady-state voltage, and bounds the largest difference in parametric uncertainty from other cases.Source: B. Robust Stability Analysis · p. 5
  • The Lemma 2 reduction can be conservative: some microgrids satisfying Lemma 1 may fail its condition.The text says conservativeness can be reduced by constraining the smallest eigenvalue of each Q_j, but this adds 2n constraints.Source: B. Robust Stability Analysis · p. 5
  • Proposition 1 constrains portions of each Q_j smallest eigenvalue, yielding a polynomial number of constraints and avoiding the potential conservativeness of Lemma 2.It focuses on each CPL's bound rather than only bounding the largest parameter-uncertainty difference.Source: B. Robust Stability Analysis · pp. 5–6

C. Discussions of the Proposed Framework

This subsection presents two insights into the sufficient conditions for local robust stability.

C. Discussions of the Proposed Framework

1) Critical Case Interpretation:

The framework reduces robust-stability checking to properties of a critical-case linearized matrix formed using the CPL power upper bounds and voltage lower bounds. Its guarantees depend on the chosen constraint sets, so changes to those sets may require renewed checks.

1) Critical Case Interpretation:

  • The critical case sets each CPL power at its upper bound and its steady-state voltage at its lower bound; checking its linearized matrix can guarantee stability across possible scenarios.The matrix is deterministic for given power and voltage constraint sets.Source: 1) Critical Case Interpretation · p. 6
  • Feasibility for the original constraint sets does not guarantee robust stability after those sets change, so the framework may need to be checked again.Repeated checks may be undesirable when constraint sets change constantly.Source: 1) Critical Case Interpretation · p. 6
  • Corollary 1 states that system (4) is locally robustly stable if P_l(A_z) is feasible for l=1 or 2 and δ_k ≥ δ_k for every k=1,...,n.Source: 1) Critical Case Interpretation · p. 6
  • Smaller δ_k makes the conditions easier to satisfy; increasing load capacitance, reducing the CPL power upper bound, or raising the voltage lower bound can reduce δ_k.The paper also notes that the power upper bound and voltage lower bound have less influence under the condition when the lower bound of δ_k is not constrained.Source: 1) Critical Case Interpretation · pp. 6–7

2) Application to Distributed Control:

The proposed stability framework also applies to DC microgrids with distributed voltage controllers. The paper models the controller's state together with the microgrid and applies the same polytopic robust-stability analysis.

2) Application to Distributed Control:

  • The framework can be extended to distributed controllers that replace V–I droop control.The paper introduces a distributed voltage-regulation example using communication between sources on neighboring buses.Source: 2) Application to Distributed Control · p. 7
  • Each source adjusts its output voltage using a weighted sum of neighboring buses’ voltage-tracking errors, and controller equilibria make that sum zero.The control objective is for bus voltages to track their references; stable equilibria are relevant to achieving that objective.Source: 2) Application to Distributed Control · p. 7
  • Augmenting the state with the controller voltage makes the closed-loop dynamics a polytopic uncertain linear system with uncertainty in diagonal matrix entries, to which the framework applies.The augmented state is ζ = [v^T, x^T]^T.Source: 2) Application to Distributed Control · p. 7

IV. CASE STUDY

In the nine-bus simulation, the proposed framework identifies robust stability across the tested load range at droop gain 0.2, unlike the instability observed near 16 kW at gain 0.06. The case study uses uncertain CPL powers and reports the optimization solve time.

IV. CASE STUDY

  • The nine-bus V–I droop-control case tests unknown CPL powers from 5 kW to 20 kW and feasible steady-state CPL voltages from 360 V to 440 V.Each CPL has nominal power 15 kW.Source: A. Simulation Validation · p. 7
  • At droop gain 0.06, stability at the nominal 15 kW load does not persist as load power rises to around 16 kW.Fig. 4 shows the load power and bus-1 voltage for this case.Source: A. Simulation Validation · p. 8, Figure 4 · p. 8
  • At droop gain 0.2, the microgrid remains stable as load power varies from 5 kW to 20 kW, matching the framework's robust-stability prediction.The simulation result is shown in Fig. 5.Source: A. Simulation Validation · p. 8, Figure 5 · p. 8
  • CVX solves the optimization problem in 13.48 seconds for the reported case study.The microgrid model and simulations use MATLAB/Simulink.Source: A. Simulation Validation · pp. 7–8

B. Study on Conservativeness

The section compares the conservativeness of the proposed stability conditions and connects their bounds to practical CPL power and voltage limits. Its eight-bus case study shows how changing system parameters and droop references affects the reported certificates and simulated operation.

B. Study on Conservativeness

  • The paper quantifies conservativeness using the upper bound of δₖ that keeps a condition feasible; a larger bound indicates a less conservative condition.The section compares Lemmas 1 and 2 with Proposition 1 using these upper bounds.Source: B. Study on Conservativeness · p. 8
  • When δᵤ₃ exceeds δᵤ₂, Proposition 1 can certify all microgrids certified by Lemma 2 and additional ones, so it is less conservative.Source: B. Study on Conservativeness · p. 8
  • Problem (10) is reported as a good approximation of problem (7), while problem (9) is conservative but least computationally demanding.For this case study, the replaced bound on γ can be seven times larger than the previous bound.Source: B. Study on Conservativeness · pp. 8–9
  • For transition resistance of 0.3 Ω, the approximated δᵤ values are 153 for problem (10) and 170 for problem (7).The section notes that δᵤ depends on topology and parameters including inductance, capacitance, and resistance.Source: B. Study on Conservativeness · p. 9
  • The derived inequality can set a minimum certifiable CPL voltage for a known upper power bound, or a maximum certifiable CPL power for a given voltage lower bound.Source: B. Study on Conservativeness · p. 9
  • With the CPL voltage lower bound at 360 V, Lemma 1 certifies power below 15.4 kW; supporting 20 kW requires a minimum certifiable voltage of 410 V.In simulation, load power increased from 11 kW and the system became unstable around 6 s; with droop references at 440 V, the CPL voltage stayed above 410 V and supported all five 20 kW CPLs.Source: B. Study on Conservativeness · p. 9, Figure 7 · p. 9, Figure 8 · p. 9

V. CONCLUSION

The conclusion frames the problem as stability analysis over operationally feasible equilibria when CPL uncertainty makes the actual equilibrium difficult to determine. It formulates this as robust stability of polytopic uncertain linear systems, presents sufficient conditions and convex problems, and identifies conservativeness and stabilization control synthesis as future work.

V. CONCLUSION

  • Because uncertain CPLs can place operation at different feasible equilibria, the paper studies stability across that feasible set rather than at one known equilibrium.Source: §V · pp. 9–10
  • The paper formulates the problem as robust stability analysis of linear systems with polytopic uncertainty in their system matrices and proposes sufficient conditions checked through convex problems.Source: §V · p. 10
  • Case studies demonstrate the effectiveness of the results, while further study of condition conservativeness and control synthesis for stabilization are left for future research.Source: §V · p. 10

APPENDIX A PROOF OF LEMMA 2

The proof establishes robust stability by showing that the relevant Lyapunov matrix expressions are negative definite for each vertex matrix. It bounds the perturbation terms through eigenvalue and matrix-norm arguments.

APPENDIX A PROOF OF LEMMA 2

  • Assuming problem (9) is feasible yields P, t, and γ; the proof then aims to establish negative definiteness for each vertex matrix expression.Source: Appendix A · p. 10
  • The proof substitutes the vertex-matrix expression into a bound involving Qⱼ and uses a lower bound on Qⱼ’s smallest eigenvalue to establish the required definiteness.Source: Appendix A · p. 10
  • For symmetric Qⱼ, the proof bounds its largest absolute eigenvalue using the matrix 2-norm and the uncertainty bound δmax, relating that bound to γ.Source: Appendix A · p. 10
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