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Analysis of Convergence for the Newton Method in DC Microgrids

Alejandro Garces

arXiv:1809.06795v1math.OC

TL;DR

Constant-power terminals make DC-microgrid power flow nonlinear, and convergence to a unique, realistic solution is not assured. The paper analyzes classic and approximated Newton methods in master-slave and island operation, finding quadratic convergence for classic Newton in a small basin and guaranteed linear convergence for the approximation.

  • Problem

    Constant-power terminals make power flow nonlinear, and numerical algorithms may diverge or yield non-realistic solutions, motivating conditions for convergence to a unique, realistic solution.

    Source: §I · p. 1

  • Method

    The paper analyzes classic and approximated Newton power-flow methods in master-slave and island operation using Kantorovich and contraction-mapping theory, respectively.

    Source: §I · p. 1

  • Results

    Classic Newton has quadratic convergence in a small basin of attraction, whereas the approximated Newton method has guaranteed linear convergence.

    Source: §I · p. 1

  • Takeaways & Limitations

    The criteria include checking maximum grid power against αm and bounding minimum voltage without explicitly calculating load flow.

    Source: §III · p. 4

    The conditions are sufficient, not necessary; failure to satisfy one does not imply that the algorithm will diverge.

    Source: §V · p. 6

Abstract

from arXiv · show

The power flow is a non-linear problem that requires a Newton's method to be solved in dc microgrids with constant power terminals. This paper presents sufficient conditions for the quadratic convergence of the Newton's method in this type of grids. The classic Newton method as well as an approximated Newton Method are analyzed in both master-slave and island operation with droop controls. Requirements for the convergence as well as for the existence and uniqueness of the solution starting from voltages close to 1pu are presented. Computational results complement this theoretical analysis.

I. INTRODUCTION

Constant-power terminals make DC-microgrid power flow nonlinear, and numerical algorithms may not converge to a unique, realistic solution. The paper derives convergence conditions for classic and approximated Newton methods across master-slave and island operation.

I. INTRODUCTION

  • DC microgrids are attracting interest for efficiency, reliability, and simpler control and integration.Source: §I · p. 1
  • Master-slave control fixes grid voltage, whereas island operation uses droop control to modify grid power toward a stable equilibrium.Source: §I · p. 1
  • Power flow is needed repeatedly in operation and planning, and its solution supplies the equilibrium point used in microgrid dynamics and stability studies.The introduction identifies quadratic convergence and solution uniqueness as key conditions.Source: §I · p. 1
  • Classic Newton has quadratic convergence in a small basin of attraction, while the approximated Newton method has guaranteed linear convergence.The analyses use Kantorovich theory for classic Newton and contraction-mapping theory for the approximated method.Source: §I · p. 1
  • The paper analyzes nonlinear DC grids with power converters, not AC-grid linearizations also called dc-power flow.Source: §I · p. 1

A. Master-slave operation

In master-slave operation, the master terminal fixes voltage, while the power-flow problem solves for the constant-power terminal voltages. The model incorporates constant-admittance terminals through Kron reduction and represents constant-power current as a nonlinear function of voltage.

A. Master-slave operation

  • The model distinguishes constant-voltage master, constant-admittance, and constant-power terminals, and defines reduced admittance matrices after accounting for the constant-admittance terminals.The reduced matrices are Ypv and Ypp.Source: A. Master-slave operation · pp. 1–2, Figure 1 · p. 2
  • The unknown state variables are the power-terminal voltages v_p; known master voltage allows other quantities, including power flows and losses, to be computed.Source: A. Master-slave operation · p. 2
  • Constant-power terminals make the system nonlinear because their current depends on voltage, so the resulting equations require Newton’s method.The passage gives each current as i = p/v.Source: A. Master-slave operation · p. 2

B. Island operation

Island operation disconnects the master terminal and eliminates its node through a new Kron reduction. The resulting nonlinear model includes droop effects, and the paper analyzes existence and algorithm convergence for its power-flow equations.

B. Island operation

  • With the master terminal disconnected and its current set to zero, the master node is eliminated using a new Kron reduction.Source: B. Island operation · p. 2
  • The island model replaces G with S to include droop; its reference voltage v_n comes from secondary control, and C is a positive-defined diagonal matrix.Source: B. Island operation · p. 2
  • The analysis addresses both solution existence and algorithm convergence for the nonlinear power-flow systems in equations (1) and (2).Source: B. Island operation · p. 2

III. NEWTON METHOD

The section develops sufficient conditions for Newton convergence and uniqueness in dc-microgrid power flow, then applies them to master-slave and island operation. Under the stated conditions, convergence is at least quadratic from a flat start.

III. NEWTON METHOD

  • With an invertible derivative and Lipschitz conditions, Kantorovitch’s theorem guarantees a unique solution and Newton convergence; when h < 1/2, the convergence is at least quadratic.The theorem supplies the basis for the microgrid-specific convergence results.Source: §III · pp. 2–3
  • The analysis assumes a connected graph, no short circuit, and per-unit representation, and uses the supremum norm for vectors and its induced matrix norm.These are the stated assumptions for the convergence analysis.Source: §III · p. 3
  • Under assumptions A1–A3, the master-slave operation point is unique and Newton’s method converges at least quadratically from a flat start when the stated condition holds.The solution must lie in the specified ball around the flat-start voltage.Source: §III · p. 3
  • The master-slave convergence condition gives a lower estimate of Newton’s basin of attraction using grid parameters and the load condition, before iteration begins.The bound also guarantees a minimum voltage above 1 − δ.Source: §III · p. 4
  • For master-slave operation, convergence is guaranteed when |Pmax| ≤ αm, where αm is the minimum real root of the stated polynomial.This criterion reduces the check to comparing the grid’s maximum power with αm.Source: §III · p. 4
  • For island operation, the operating point is unique and Newton’s method converges at least quadratically from a flat start when the stated conditions hold, including a nonsingular initial derivative.The conditions use γ and ξ as defined for this case; Ys may be singular, so the master-slave Lemma 2 argument does not apply directly.Source: §III · p. 4

IV. APPROXIMATED NEWTON METHOD

The paper approximates the Newton Jacobian with a constant matrix to reduce the time consumed per step, then establishes convergence conditions for master-slave and island operation. For the approximated method, the stated guarantee is convergence, not quadratic convergence.

IV. APPROXIMATED NEWTON METHOD

  • Approximating the Jacobian by a constant matrix reduces the time consumed per step, avoiding its recalculation at each iteration.The paper uses the Jacobian from the first iteration as the approximation.Source: §IV · p. 4
  • The island-operation analysis uses contraction mapping because Theorem 1 cannot be applied directly; a contraction has a unique fixed point obtainable by iteration.The method’s convergence is therefore framed through fixed-point iteration.Source: §IV · p. 4
  • In master-slave operation, the approximated method calculates the operation point from VN = ˆeN when assumptions A1–A3 and the stated β, δ conditions hold.The proposition gives a convergence guarantee, but not quadratic convergence.Source: §IV · pp. 4–5
  • In island operation, the approximated method calculates the operation point under assumptions A2–A3 when DF(v0) is non-singular and the proposition’s additional condition holds.The supplied proposition passage does not specify that additional condition.Source: §IV · p. 5

V. NUMERICAL EXAMPLE

The numerical example compares classical and approximated Newton load-flow calculations in master-slave and island operation. The classical method’s error decreases quadratically, while the approximated method’s error decreases linearly after a shared first iteration.

V. NUMERICAL EXAMPLE

  • The simulations compare both Newton methods in master-slave and island operation, with island operation modeled by opening the switch in node 3.Node 1 is voltage controlled, while nodes 3, 7, 10, and 14 are step nodes eliminated by Kron reduction.Source: §V · p. 5
  • After a shared first iteration, the classical Newton error decreases quadratically, whereas the approximated Newton error decreases linearly.These simulated convergence patterns agree with Propositions 1 to 4.Source: §V · pp. 5–6
  • The convergence conditions are sufficient, not necessary: approximated Newton converges in the island example despite not meeting Proposition 4’s conditions.Failure to satisfy a condition does not imply that the algorithm diverges.Source: §V · p. 6
  • Pmax = 1.2406 is the maximum power for which Corollary 1 guarantees quadratic convergence in master-slave operation.The paper notes this threshold is relevant when power flow is run repeatedly, such as in optimization problems.Source: §V · p. 6

VI. CONCLUSIONS

The paper presents exact convergence conditions for the Newton and approximated Newton methods, using two theorem-based analyses and numerical results. It frames the analysis as useful for repeated power-flow applications and notes that extension to AC grids requires more research.

VI. CONCLUSIONS

  • The paper presents exact convergence conditions for both the Newton method and the approximated Newton method.Numerical results complement the analysis.Source: §VI · p. 6
  • The analysis uses Kantorovitch’s theorem and the contraction mapping theorem under practical assumptions including graph connectivity, per-unit representation, and stationary-state operation.Source: §VI · p. 6
  • The analysis is proposed as a practical tool for assessing convergence in problems where the Newton method is applied repeatedly.Source: §VI · p. 6
  • Extension of the analysis to AC grids is identified as possible, but requiring more research.Source: §VI · p. 6

A. Proof sketch for the Kantorovitch’s theorem

The proof sketch bounds a matrix expression using the Lipschitz condition and Condition (5), then invokes the Banach Lemma. It also introduces a function along the voltage-update direction and gives its derivative.

A. Proof sketch for the Kantorovitch’s theorem

B. Proof sketch for the Contraction mapping theorem

The proof sketch considers a contraction mapping on a closed ball in Rn. It uses a contraction inequality to establish uniqueness of the fixed point and describes convergence of the iteration to a fixed point.

B. Proof sketch for the Contraction mapping theorem

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