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Optimal Power Flow in Stand-alone DC Microgrids

Jia Li, Feng Liu, Zhaojian Wang, Steven H Low, Shengwei Mei

arXiv:1708.05140v1math.OC

TL;DR

Stand-alone DC-MG optimal power flow is non-convex, and prior DC-network assumptions about an unconstrained-power, fixed-voltage substation do not apply. The paper develops an SOCP relaxation and proves exactness under mild assumptions. The exactness result does not depend on topology or operating mode, and the relaxed problem has at most one optimum.

  • Problem

    Stand-alone DC-MG OPF is non-convex, and the prior DC-network approach assumes an unconstrained-power, fixed-voltage substation that stand-alone systems lack.

    Source: §I · p. 1

  • Method

    The paper transforms the OPF into an equivalent rank-constrained formulation and removes the rank constraints to form an SOCP relaxation.

    Source: §I · p. 1

  • Results

    Under uniform nodal voltage upper bounds and positive network loss, the SOCP relaxation is exact regardless of topology or operating mode, and its optimal solution is unique if it exists.

    Source: §VI · pp. 7–8

  • Takeaways & Limitations

    The topology- and mode-independent exactness result supports optimal operation across grid-connected and stand-alone modes and target setting in optimal control.

    Source: §VI · pp. 7–8

    When line constraints are considered, SOCP-relaxation exactness may not hold, and the paper proposes heuristic methods for approximate feasible solutions.

    Source: §I · p. 2

Abstract

from arXiv · show

Direct-current microgrids (DC-MGs) can operate in either grid-connected or stand-alone mode. In particular, stand-alone DC-MG has many distinct applications. However, the optimal power flow problem of a stand-alone DC-MG is inherently non-convex. In this paper, the optimal power flow (OPF) problem of DC-MG is investigated considering convex relaxation based on second-order cone programming (SOCP). Mild assumptions are proposed to guarantee the exactness of relaxation, which only require uniform nodal voltage upper bounds and positive network loss. Furthermore, it is revealed that the exactness of SOCP relaxation of DC-MGs does not rely on either topology or operating mode of DC-MGs, and an optimal solution must be unique if it exists. If line constraints are considered, the exactness of SOCP relaxation may not hold. In this regard, two heuristic methods are proposed to give approximate solutions. Simulations are conducted to confirm the theoretic results.

I. INTRODUCTION

Stand-alone DC microgrids lack the unconstrained-power, fixed-voltage substation assumed in prior DC-network OPF work, motivating an SOCP approach for their non-convex OPF. The paper formulates the problem and transforms it into a rank-constrained form before relaxation.

I. INTRODUCTION

  • Stand-alone DC-MG OPF is non-convex, and unlike the prior DC-network setting, stand-alone systems lack an unconstrained-power, fixed-voltage substation.Stand-alone DC-MGs are used in shipboard, aircraft, automotive, and remote-rural applications.Source: §I · p. 1
  • The solution approach removes the rank constraint for SOCP relaxation, then converts the relaxed problem to a branch-flow model for numerical stability.The branch-flow conversion is equivalent to the relaxed formulation.Source: §I · p. 1
  • The proposed SOCP relaxation is exact under mild assumptions, and exactness and optimal-solution uniqueness do not depend on network topology or operating mode.When line constraints are considered, exactness may fail, so the paper also proposes heuristic feasible-solution methods.Source: §I · pp. 1–2

A. Basic Formulation

  • The stand-alone OPF model bounds nodal voltages and power injections, with each bus’s objective term strictly increasing in its power injection.The model assumes each bus has nonpositive power injection; buses may be generation-only, load-only, or mixed load and generation.Source: A. Basic Formulation · p. 2

B. Equivalent Transformation

  • Introducing slack variables converts the nonlinear power-injection equation into a rank constraint, yielding an equivalent OPF formulation.The equivalence holds without relying on a particular network topology.Source: B. Equivalent Transformation · pp. 2–3

III. EXACTNESS OF CONIC RELAXATION

The paper relaxes the rank constraint in the transformed DC-MG OPF to obtain an SOCP problem. With uniform nodal voltage upper bounds and positive network loss, this relaxation is exact.

A. SOCP Relaxation of OPF in DC-MGs

  • RL1 is formed by removing the rank constraint from the equivalent non-convex OPF formulation, and exactness requires every optimal solution to satisfy that constraint.RL1 and OPF2 differ only by the removed rank constraint.Source: A. SOCP Relaxation of OPF in DC-MGs · p. 3

B. Assumptions

  • The assumptions require equal upper bounds on all nodal voltages and positive total network loss.These assumptions accommodate stand-alone DC-MGs without requiring an unconstrained power injection.Source: B. Assumptions · p. 3

C. Exactness of Conic Relaxtion

  • The SOCP relaxation RL1 is exact when all nodal voltage upper bounds are uniform and total network loss is positive.The voltage-bound assumption is described as reasonable for DC-MGs, and positive total loss is described as trivial.Source: C. Exactness of Conic Relaxtion · p. 3, B. Assumptions · p. 3
  • The exactness proof rules out an optimal rank-violating solution by improving the objective when endpoint injections are not binding and propagating rank violations across neighboring lines when they are binding.The propagation preserves power injections and uses connectivity of the network.Source: C. Exactness of Conic Relaxtion · p. 3

D. Topological Independence

Under the proposed assumptions, SOCP-relaxation exactness does not depend on DC-MG network topology. Combined with the grid-connected result, this applies across grid-connected and stand-alone operation.

D. Topological Independence

  • Under the assumptions, exactness is independent of network topology in both stand-alone and grid-connected DC-MGs.The paper combines its topology-independence theorem with the prior grid-connected result.Source: D. Topological Independence · p. 4

E. Uniqueness of the Optimal Solution

Under the proposed assumptions, the SOCP relaxation has a unique optimum and its exactness is independent of DC-MG topology and operating mode. A branch-flow reformulation is equivalent and improves numerical stability.

F. Branch Flow Model

  • The branch-flow conversion avoids subtracting numerically close voltage variables, improving numerical stability.The paper identifies RL1 as ill-conditioned because equation (4a) subtracts v_i and W_ij; converting RL1 to a branch-flow model avoids these subtractions.Source: F. Branch Flow Model · p. 4
  • RL1 and its branch-flow reformulation RLS1 are equivalent, so RLS1 is a convex relaxation of the corresponding non-convex OPF3.The feasible-set mapping preserves the optimal solution when the relaxation is exact.Source: F. Branch Flow Model · p. 4

E. Uniqueness of the Optimal Solution

F. Branch Flow Model

  • The OPF1–OPF3 formulations omit line-flow constraints, and the authors cannot rule out their violation.They propose heuristic approximate solutions for such circumstances.Source: F. Branch Flow Model · p. 5

IV. CASE STUDIES

Case studies report exact SOCP relaxation across tested network topologies and operating modes, with computation-time results suggesting possible use in larger systems. The studies also compare network loss across modes and topologies, while line-constrained exactness is not theoretically guaranteed.

A. 16-bus System

  • The 16-bus study evaluates four combinations of grid-connected or stand-alone operation with tree or mesh topology.The modified three-feeder system includes six distributed generators, each with 5 MW capacity.Source: A. 16-bus System · p. 5, Figure 4 · p. 5

B. Exactness and Comparison

A. 16-bus System

  • Mesh topology reduces network loss in both operating modes, and grid-connected operation has less loss than stand-alone operation at the same topology.The authors attribute these comparisons to higher nodal voltages supported by mesh connections and greater power injection from feeders in grid-connected mode.Source: A. 16-bus System · p. 5, Table I · p. 5

B. Exactness and Comparison

A. OPF with Line Constraints

B. Exactness Conditions When Considering Line Constraints

With line constraints, the SOCP relaxation is exact under specified conditions; when it is not, the paper proposes heuristic constructions of nearly optimal solutions. The direct construction method preserves power balance, voltage, and line constraints, while power-injection bounds may be violated within limits.

B. Exactness Conditions When Considering Line Constraints

  • RL2 is exact if every line constraint is nonbinding, or if each binding line has at least one endpoint whose power-injection lower bound is nonbinding.These are the two sufficient conditions in Theorem 6.Source: B. Exactness Conditions When… · p. 6
  • If power-injection adjustments at both endpoints are allowed, a rank-satisfying RL2 solution can be constructed with a lower objective value.The paper notes that demand response or energy storage may provide these adjustments.Source: B. Exactness Conditions When… · p. 6

C. Constructing Approximate Optimal Solutions

  • When RL2 is not exact, two heuristic methods construct nearly optimal solutions; an RL2 solution satisfying the rank constraint is globally optimal for OPF1.The rank constraint is checked after solving RL2.Source: C. Constructing Approximate Optimal… · p. 6
  • If a binding line violates the rank constraint at an RL2 optimum, at least one endpoint power injection must be at its lower bound.Otherwise, a feasible point with a smaller objective value can be found.Source: C. Constructing Approximate Optimal… · p. 6

1) Direct construction method:

  • The direct construction method uses the relaxed solution’s v* to build an approximate (p, V) for the original problem.It derives each power injection by substituting the constructed voltage into the power-balance equation.Source: 1) Direct construction method · pp. 6–7
  • The constructed solution satisfies power balance, voltage, and line constraints, but may violate power-injection bounds by limited amounts.The passage does not quantify those bounds.Source: 1) Direct construction method · p. 7

2) Slack variable method:

The slack-variable method addresses rank-constraint violations that coincide with binding power-injection lower bounds by adding penalized slack to the power-balance constraint. The conclusions state that, under the proposed assumptions, SOCP exactness and uniqueness are independent of network topology and operating mode.

2) Slack variable method:

  • At buses with both a rank-constraint violation and a binding power-injection lower bound, the slack-variable method adds ε to the power-balance constraint and minimizes ε in the objective.This raises the affected injection above its lower bound in the next RL2 iteration.Source: 2) Slack variable method · p. 7

VI. CONCLUSIONS

  • Under uniform voltage upper bounds and positive network loss, the proposed assumptions guarantee exactness of the SOCP relaxation for stand-alone DC-MG OPF.The conclusion describes these as mild assumptions.Source: §VI · p. 7
  • Under the assumptions, exactness does not depend on grid operating mode or network topology.The conclusion notes that topology changes from line switching can still permit an equivalent convex SOCP formulation.Source: §VI · p. 7
  • The global optimum of the SOCP problem is unique, independently of DC-MG topology and operating mode.The conclusion says this uniqueness can help set a target in optimal control.Source: §VI · p. 8

APPENDIX

The appendix proves the slack-variable formulation’s equivalence, establishes uniqueness in the voltage mapping, and uses feasible-point constructions to support the rank and objective arguments.

A. Proof of Lemma 1

  • For positive v_i, Lemma 1 maps each v to the unique positive voltage V_i = √v_i satisfying (2).Existence follows from the square-root construction; uniqueness follows because the negative root violates positivity.Source: A. Proof of Lemma 1 · p. 9

B. Proof of Theorem 1

  • Theorem 1’s equivalence proof shows a one-to-one map between the feasible sets of OPF1 and OPF2, with p uniquely determined by either representation.The proof establishes that the map is both into and onto.Source: B. Proof of Theorem 1 · p. 9

C. Proof of Lemma 2

  • The Lemma 2 proof divides the endpoint injection cases and uses Assumption 1 to relate neighboring voltages when the rank constraint is violated.It considers whether the injection is negative at its lower bound or zero at its lower bound.Source: C. Proof of Lemma 2 · p. 9

D. Proof of Lemma 3

  • For Lemma 3, a small ε perturbation constructs a feasible point with lower objective while retaining the constraints used in the proof.The construction changes W and adjusts p at the endpoints of the rank-violating line.Source: D. Proof of Lemma 3 · p. 9

E. Proof of Lemma 4

The Lemma 4 proof constructs a perturbed feasible point with a lower objective for a case where one endpoint injection is at its lower bound and the other is above it.

E. Proof of Lemma 4

  • With one endpoint injection at its lower bound and the other above it, a small ε perturbation constructs a feasible point with a lower objective.The proof constructs W′ and adjusts p and v while checking the constraints.Source: E. Proof of Lemma 4 · p. 10

F. Proof of Lemma 5

The proof constructs a rank-constraint-violating feasible point for Lemma 5, then uses casewise contradictions to establish that every RL1 optimum satisfies the rank constraint.

F. Proof of Lemma 5

  • Lemma 5 constructs a feasible point that violates (4g) on the selected line and all its neighboring lines.A small perturbation preserves the stated constraints, while connectedness supplies a neighboring line for the construction.Source: F. Proof of Lemma 5 · p. 10

G. Proof of Theorem 2

  • For the other considered injection cases, feasible points with smaller objective values rule out the assumed violating solution as optimal.The proof invokes Lemma 3 for one case and Lemma 4 for another.Source: G. Proof of Theorem 2 · p. 11
  • When the relevant injections are equal, propagation across the connected network produces a rank-constraint violation on every line, contradicting Assumption 2.The resulting point cannot be an RL1 optimum.Source: G. Proof of Theorem 2 · p. 11
  • RL1 is exact because every optimal solution must satisfy the rank constraint (4g).The proof rules out optimal solutions violating that constraint across the considered injection cases.Source: G. Proof of Theorem 2 · p. 11

H. Proof of Theorem 4

These proofs establish uniqueness for RL1, a one-to-one correspondence between RL1 and RLS1 feasible sets, and exactness under the stated line-constraint conditions.

H. Proof of Theorem 4

  • RL1 has at most one optimal solution.The proof uses strict increase of f_i(p_i) to conclude that two optimal solutions must coincide.Source: H. Proof of Theorem 4 · p. 11

I. Proof of Theorem 5

  • The map g is one-to-one between the feasible sets of RL1 and RLS1, with each feasible point in either set corresponding to one in the other.The proof establishes the correspondence in both directions.Source: §I · p. 11

J. Proof of Theorem 6

  • When line constraint (7) is binding and the bounds on p_s and p_t are not binding, an optimal solution to RL2 must satisfy rank constraint (4g).The proof constructs a feasible point with a smaller objective if the assumed RL2 optimum violates (4g).Source: J. Proof of Theorem 6 · p. 11

K. Proof of Theorem 7

The proof perturbs a point violating the rank constraint, restores equality on the selected line, and obtains a feasible point with a lower objective.

K. Proof of Theorem 7

  • A point violating (4g) on s→t has W_st < √(v_s v_t), enabling a perturbation that sets W′_st = √(v_s v_t).The construction also sets W′_ij = √(v_i v_j) for the other considered line pairs.Source: K. Proof of Theorem 7 · pp. 11–12
  • The constructed point satisfies the listed constraints, including line constraint (7).The proof verifies (4a), (4c)–(4f), and (7).Source: K. Proof of Theorem 7 · p. 11
  • The construction lowers p′, giving a lower objective because the objective function is strictly increasing in p_i.This completes the proof's comparison with the original point.Source: K. Proof of Theorem 7 · p. 12
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