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Branch Flow Model: Relaxations and Convexification (Parts I, II)

Masoud Farivar, Steven H. Low

arXiv:1204.4865v4eess.SYcs.SYmath.OC

TL;DR

Optimal power flow is generally difficult because its feasible set is nonconvex. The paper develops a branch-flow approach using angle elimination and conic relaxation, with exactness guarantees for radial networks under a load condition and a cycle test for mesh networks. It also proposes phase-shifter convexification for mesh networks.

  • Problem

    OPF is generally nonconvex and NP-hard, motivating an efficient approach that exploits branch-flow structure.

    Source: A. Motivation · p. 1

  • Method

    The approach eliminates voltage and current angles, relaxes a quadratic equality to a conic inequality, and uses angle recovery to test correspondence to the original OPF.

    Source: B. Summary · pp. 2–3

  • Results

    For radial networks, both relaxation steps are exact when there are no upper bounds on loads; for mesh networks, the conic relaxation is exact but angle recovery may fail.

    Source: §VI · pp. 11, 20

  • Takeaways & Limitations

    For mesh networks, phase shifters can make relaxed solutions optimal for a convexified network, with placement limited to lines outside a spanning tree.

    Source: §I · p. 13

    For mesh networks, an angle-relaxed solution may fail the cycle-based angle-recovery condition and then need not yield a feasible original OPF solution.

    Source: B. Summary · p. 3

Abstract

from arXiv · show

We propose a branch flow model for the anal- ysis and optimization of mesh as well as radial networks. The model leads to a new approach to solving optimal power flow (OPF) that consists of two relaxation steps. The first step eliminates the voltage and current angles and the second step approximates the resulting problem by a conic program that can be solved efficiently. For radial networks, we prove that both relaxation steps are always exact, provided there are no upper bounds on loads. For mesh networks, the conic relaxation is always exact but the angle relaxation may not be exact, and we provide a simple way to determine if a relaxed solution is globally optimal. We propose convexification of mesh networks using phase shifters so that OPF for the convexified network can always be solved efficiently for an optimal solution. We prove that convexification requires phase shifters only outside a spanning tree of the network and their placement depends only on network topology, not on power flows, generation, loads, or operating constraints. Part I introduces our branch flow model, explains the two relaxation steps, and proves the conditions for exact relaxation. Part II describes convexification of mesh networks, and presents simulation results.

I. INTRODUCTION

The paper extends branch-flow modeling and OPF relaxations from radial distribution circuits to mesh networks. It separates angle elimination from conic relaxation, characterizes when solutions recover the original OPF, and shows how phase shifters can convexify mesh networks.

C. Extensions: radial networks and equivalence

A. Motivation

  • OPF is generally nonconvex and NP-hard, while prior branch-flow work and exactness results largely focused on radial networks.The paper extends the Baran–Wu model to mesh networks and to line limits and convex objectives.Source: A. Motivation · pp. 1–2

B. Summary

  • The approach first eliminates voltage and current angles, then relaxes a quadratic equality to obtain a convex conic problem.The conic problem is an SOCP when the objective is linear.Source: B. Summary · p. 2
  • For radial networks, angle recovery holds trivially; for mesh networks, implied angle differences must sum to zero modulo 2π around every cycle.The cycle condition determines whether an angle-relaxed solution corresponds to a solution of the original OPF.Source: B. Summary · p. 3
  • Phase shifters can convexify mesh networks so relaxed solutions map to optimal solutions for the convexified network, using shifters only outside a spanning tree.The branch-flow model also provides a framework for analyzing and optimizing radial and mesh networks.Source: B. Summary · pp. 2–3

B. Optimal power flow

The OPF formulation optimizes generation and consumption subject to branch-flow equations and operating constraints. Its nonlinear feasible set is generally nonconvex, motivating an efficient solution approach that exploits the branch-flow structure.

B. Optimal power flow

  • The formulation includes generation and consumption controls, voltage-magnitude ranges, and branch-current flow limits.Real and reactive generation and consumption are represented at each node.Source: B. Optimal power flow · p. 4
  • The objective may be any convex function independent of voltage and current angles, including loss minimization, generation costs, or conservation voltage reduction.The stated assumptions also require strict increase in squared current magnitudes and nonincrease in load.Source: B. Optimal power flow · p. 5
  • The OPF feasible set is specified by nonlinear branch-flow equations, making the problem generally nonconvex and hard to solve.The stated goal is an efficient solution method that exploits the branch-flow model’s structure.Source: B. Optimal power flow · p. 5

C. Notations and assumptions

The paper defines assumptions for OPF and distinguishes the original branch-flow system from its angle-relaxed projection. In mesh networks, the relaxed equations can admit solutions that do not correspond to original power-flow solutions.

C. Notations and assumptions

  • The OPF cost is assumed convex, strictly increasing in squared branch currents, nonincreasing in load, and independent of branch powers.These are the stated A2–A3 conditions.Source: C. Notations and assumptions · p. 5

A. Relaxed branch flow model

  • The relaxed branch-flow model uses 2(m+n+1) equations in 3m+n+2 real variables, whereas a tree network has equal equation and variable counts.For practical networks, the cited discussion reports that solutions are generally unique under stated voltage and impedance conditions.Source: A. Relaxed branch flow model · p. 6
  • In a connected mesh network, the relaxed model is generally nonunique, and some relaxed solutions may be spurious.Such solutions do not correspond to solutions of the original branch-flow equations.Source: A. Relaxed branch flow model · p. 6
  • For a tree, solutions of the relaxed equations coincide with solutions of the original branch-flow equations.For a general network, equivalence requires the angle-recovery condition.Source: A. Relaxed branch flow model · p. 6

B. Two relaxations

The method first removes voltage and current angles, then relaxes a quadratic equality to form a convex conic problem. The conic relaxation is exact under the stated load condition, while recovering an original OPF solution also depends on angle consistency.

B. Two relaxations

  • Angle relaxation removes voltage and current angles, leaving OPF-ar with a quadratic equality constraint.The resulting model is still nonconvex.Source: B. Two relaxations · p. 6
  • Conic relaxation replaces the quadratic equality with an inequality, yielding a convex problem that is an SOCP for a linear objective.OPF-cr provides a lower bound to OPF-ar.Source: B. Two relaxations · p. 7

IV. EXACT CONIC RELAXATION

  • Without upper bounds on loads, every optimal OPF-cr solution is optimal for OPF-ar for radial and mesh networks.The conic relaxation is exact under this condition.Source: C. Solution strategy · p. 7, §IV · p. 7

C. Solution strategy

  • For radial networks, angle recovery always yields an optimal solution of the original OPF; for mesh networks, recovery may fail and the condition can test global optimality.The relevant condition requires implied angle differences to sum to zero modulo 2π around cycles.Source: C. Solution strategy · p. 7
  • For mesh networks, phase shifters can map relaxed solutions to optimal solutions of a convexified network, requiring placement only outside a spanning tree.The phase-shifter result is presented as a way to address cases where angle recovery fails.Source: C. Solution strategy · p. 7

IV. EXACT CONIC RELAXATION

  • The conic relaxation is convex, and its exactness proof shows that an optimum must attain equality in the relaxed quadratic constraint.The proof uses the objective assumptions to rule out a strict inequality at an optimum.Source: §IV · p. 7

V. ANGLE RELAXATION

The angle-recovery condition determines whether a relaxed solution can be mapped back to a branch flow solution, and provides a way to recover its angles. For mesh networks, that condition is nonconvex and may fail.

A. Angle recovery condition

  • A spanning-tree basis yields a direct recovery test and, when it passes, a unique set of voltage angles in (−π, π].The chosen spanning tree does not affect the theorem’s conclusion.Source: A. Angle recovery condition · pp. 9–10
  • A relaxed solution admits a unique branch flow solution with the same projection exactly when the angle-recovery condition holds.The relaxed model together with this condition is equivalent to the original branch flow model.Source: A. Angle recovery condition · p. 9
  • The angle-recovery condition holds exactly when the voltage angle differences implied by the relaxed solution sum to zero modulo 2π around every cycle.It is enough to check the basis cycles associated with links outside a spanning tree.Source: A. Angle recovery condition · pp. 9–10
  • The angle-recovery condition is nonconvex, so a relaxed solution may not correspond to any solution of the original branch flow equations.Theorem 2 characterizes precisely when the inverse projection from the relaxed solution exists.Source: A. Angle recovery condition · p. 9

B. Angle recovery algorithms

The section gives centralized and distributed procedures for recovering angles from a relaxed solution. Both can identify failure of the angle-recovery condition on a non-tree link.

B. Angle recovery algorithms

  • The centralized algorithm tests the angle-recovery condition and, if it passes, computes the unique branch flow solution projecting to the relaxed solution.It selects a basis of the reduced incidence matrix, checks the condition, then recovers the angles and inverse projection.Source: B. Angle recovery algorithms · p. 10
  • The distributed algorithm computes angles outward from a root along a spanning tree, then checks links outside the tree.A failed check identifies a non-tree link whose associated basis cycle violates the recovery condition.Source: B. Angle recovery algorithms · p. 11
  • If the distributed procedure succeeds, the relaxed solution and recovered angles form a branch flow solution.The procedure is described as more amenable to distributed implementation.Source: B. Angle recovery algorithms · p. 11

C. Radial networks

For a tree network, every relaxed solution has a unique angle recovery. Consequently, under the stated conditions, the convex problem and original OPF have corresponding optimal solutions.

C. Radial networks

  • For a tree network, every relaxed solution has a unique inverse projection that is a branch flow solution.The angle relaxation is therefore always exact for radial networks.Source: C. Radial networks · p. 11
  • For radial networks, an optimal solution of the convex problem can be mapped to a unique optimal solution of the original OPF.This follows from the equivalence of the convex problem and OPF for a tree network.Source: C. Radial networks · p. 11

VI. CONCLUSION

The conclusion presents a two-step OPF strategy: solve a conic relaxation, then recover angles if possible. It states exactness guarantees for radial networks under a load-bound condition and a topology-based phase-shifter approach for mesh networks.

VI. CONCLUSION

  • The two-step strategy guarantees a globally optimal solution for radial networks when there are no upper bounds on loads.It solves the second-order conic relaxation, then uses angle recovery to obtain an optimal solution of the original OPF.Source: §VI · p. 11
  • For mesh networks, the angle-recovery condition may fail, but it can test whether a relaxed solution is globally optimal.The conclusion distinguishes this angle-relaxation issue from the radial-network guarantee.Source: §VI · p. 11
  • The branch flow model uses variables tied directly to branch powers and currents, which the authors describe as often more intuitive than a semidefinite matrix.The conclusion also states that the number of power flow solutions depends on voltage and current magnitudes, not phase angles.Source: §VI · p. 11

Convexification (Part II)

  • For mesh networks, phase shifters outside a spanning tree convexify the network, and their placement depends only on topology.The abstract states that OPF for the convexified network can then be solved efficiently for an optimal solution.Source: Convexification (Part II) · p. 13

I. INTRODUCTION

The paper formulates OPF using branch-flow variables and applies angle and conic relaxations. It extends the model with phase shifters to convexify mesh networks and recover solutions for the convexified network.

I. INTRODUCTION

  • The conic relaxation is exact for mesh and radial networks when there are no upper bounds on real and reactive loads.It yields an SOCP when the objective is linear.Source: §I · p. 13
  • Placing phase shifters on branches outside a spanning tree can map a relaxed solution to an optimal solution for the convexified network.The convexified network’s optimal cost is no higher than the original network’s.Source: §I · p. 13
  • The minimum-phase-angle problem is NP-hard, although simulations suggest a simple heuristic works well in practice.The heuristic minimizes phase-shifter angles with phase shifters on every link.Source: §I · pp. 13–14
  • Figure 1 summarizes an algorithm for solving OPF in mesh networks with phase shifters.The paper describes it as an algorithm for the phase-shifter setting.Source: §I · p. 14, Figure 1 · p. 14
  • Phase-shifter placement outside a spanning tree depends on network topology, not on power flows, generation, loads, or operating constraints.The paper notes that this can require only a one-time deployment cost.Source: §I · p. 14

B. Model with phase shifters

  • The idealized phase shifter changes sending-end voltage and current angles, has no impedance or angle limits, and is modeled as lossless.Without phase shifters, the modified equations reduce to the original branch-flow model.Source: B. Model with phase shifters · p. 15

III. PHASE ANGLE SETTING

The phase-setting methods trade off the number of installed shifters against their angle sizes. With shifters only on non-tree branches, any relaxed solution can be recovered as a branch-flow solution of the convexified network.

A. Computing φ

  • The phase-setting problem admits different choices depending on whether the goal is to minimize the number of shifters or their angle sizes.The paper presents these as two methods for computing phase-shifter angles.Source: A. Computing φ · p. 15

1) Minimize number of phase shifters:

  • Any relaxed solution can be recovered as a branch-flow solution of the convexified network using only the m − n branches outside a spanning tree.The result holds for any spanning tree, and the associated phase settings and voltage angles are unique under the stated angle convention.Source: 1) Minimize number of phase shifters · pp. 15–16
  • The minimum-count construction places phase shifters only on links outside the selected spanning tree.For a given relaxed solution, it chooses phase angles so the inverse projection is a branch-flow solution.Source: 1) Minimize number of phase shifters · pp. 15–16

2) Minimize phase angles:

The paper contrasts minimizing phase-shifter count with minimizing angle size, then discusses recovery conditions for partial deployments and how tree-mounted shifters relate to voltage-angle changes.

2) Minimize phase angles:

  • Minimizing phase-shifter angles with shifters on every link is NP-hard, while a fixed-integer heuristic is reported to reduce angles effectively in the test cases.The heuristic fixes the integer vector so β′ = β + 2πk lies in (−π, π]^m.Source: 2) Minimize phase angles · pp. 16–17

B. Arbitrary network of phase shifters

  • With shifters on only some links, angle recovery requires implied angle differences to sum to zero modulo 2π around every loop without a shifter.Recovery is always possible when every link outside a spanning tree has a shifter.Source: B. Arbitrary network of phase shifters · p. 17

C. Other properties

  • A relaxed solution can be mapped to a branch-flow solution with phase shifters only on non-tree links.Shifters on tree links are equivalent to changes in voltage phases, while voltage and current magnitudes and branch powers remain unchanged.Source: C. Other properties · pp. 17–18

IV. OPF IN CONVEXIFIED NETWORK

The section relates optimal costs across original, relaxed, and phase-shifter OPF formulations, then identifies when angle recovery and device placement matter.

IV. OPF IN CONVEXIFIED NETWORK

  • Convexification requires a one-time deployment cost for subsequent operational simplicity.Source: §IV · p. 18
  • f* ≥ f_ar = f_ps = f_T for any spanning tree T; equality with f* holds when an OPF-ar solution satisfies angle recovery.Source: §IV · p. 18
  • When recovery holds, the original OPF solution is recoverable and adding phase shifters cannot lower cost; when it fails, no inverse projection exists from that relaxed solution.Source: §IV · p. 18
  • Allowing phase shifters can attain f_ar = f_ps, potentially strictly improving on the original network when f* > f_ar.Source: §IV · p. 19
  • The same benefit requires phase shifters only outside any chosen spanning tree, and different trees yield the same objective for a fixed relaxed solution.Source: §IV · p. 19

V. SIMULATIONS

Simulations compare OPF with and without phase shifters, while the conclusion states exactness conditions and practical implications for radial and mesh networks.

V. SIMULATIONS

  • The conic relaxation was exact with respect to OPF-ar in every test case, although constant loads violated the no-upper-bound condition; this shows that condition is sufficient, not necessary.The second-order cone constraint attained equality at optimality in all test cases.Source: §V · p. 19
  • Angle recovery failed in all test cases, so no corresponding solution was feasible for OPF without phase shifters; phase-shifter angles then produced optimal solutions for convexified networks.Source: §V · p. 19
  • Across all test cases, convexified networks achieved lower minimum loss and higher maximum loadability than original networks.The paper emphasizes design for simplicity and efficient OPF solution over the modest performance improvement.Source: §V · p. 20, Table I · p. 20, Table II · p. 20
  • Potential phase-shifter counts outside a spanning tree were 17%–37% of links, while active devices ranged from 7%–25% for loss minimization and 11%–34% for loadability maximization.Optimal phase angles were no more than 20° with the minimum number of devices and below 7° with the maximum number.Source: §V · p. 20
  • Simulation times ranged from seconds to minutes, were much faster than SDP relaxation, and appeared linear in network size.Source: §V · p. 20

VI. CONCLUSION

  • For radial networks, both relaxation steps are exact without upper bounds on loads; for meshes, angle recovery can check whether a relaxed solution is globally optimal.Source: §VI · pp. 20–21
  • Because practical networks are sparse, few phase shifters may be needed; their placement depends on topology rather than power flows, generation, loads, or operating constraints.The required deployment is a one-time cost for subsequent operational simplicity.Source: §VI · p. 21

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