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Exact Convex Relaxation of Optimal Power Flow in Radial Networks

Lingwen Gan, Na Li, Ufuk Topcu, Steven H. Low

arXiv:1311.7170v1math.OC

TL;DR

The paper addresses the nonconvex OPF problem in distribution networks. It proves that, for radial networks, a modified OPF admits an exact SOCP relaxation under a condition that can be checked a priori. The condition is supported by empirical studies on four IEEE networks and two real-world networks, while the modification excludes points near voltage upper bounds.

  • Problem

    OPF is nonconvex because of the physical laws governing power flow in distribution networks.

  • Method

    The paper imposes an additional constraint restricting power injections to Svolt, making one exactness condition automatic and leaving exactness to condition C1.

  • Results

    The modified OPF has an exact SOCP relaxation when C1 holds, and C1 holds with large margin in the studied networks.

  • Takeaways & Limitations

    For radial distribution networks, global OPF optima can be recovered through SOCP after a slight feasible-set shrinkage under an a priori-checkable condition.

  • Takeaways & Limitations

    The exactness guarantee also depends on C2, which cannot be checked a priori and is ensured here by restricting injections to Svolt.

Abstract

from arXiv · show

The optimal power flow (OPF) problem determines power generation/demand that minimize a certain objective such as generation cost or power loss. It is nonconvex. We prove that, for radial networks, after shrinking its feasible set slightly, the global optimum of OPF can be recovered via a second-order cone programming (SOCP) relaxation under a condition that can be checked a priori. The condition holds for the IEEE 13-, 34-, 37-, 123-bus networks and two real-world networks, and has a physical interpretation.

I. INTRODUCTION

OPF is a nonconvex optimization problem whose common approximations and local methods have important limitations for distribution networks. This paper develops an SOCP-based approach for radial networks that is exact after a slight feasible-set modification under an a priori-checkable condition.

  • OPF chooses power generations and demands to minimize objectives such as generation cost or power loss, but its governing power-flow laws are nonconvex.
  • DC power flow linearizes the laws under small resistance, near-nominal voltage, and small adjacent-bus angle differences, reducing OPF to a linear program.
  • DC power flow is effective for transmission networks but does not apply to distribution networks or problems requiring explicit reactive-power or voltage-deviation optimization.
  • Local-optimum algorithms can be empirically successful, but generally lack guarantees of convergence or near-optimality.
  • The modified OPF imposes additional power-injection constraints, eliminating only feasible points close to voltage upper bounds while ensuring exactness under the stated condition.

II. THE OPTIMAL POWER FLOW PROBLEM

The paper formulates distribution-network OPF using a branch-flow model with controllable power injections, device-specific feasible sets, voltage bounds, and an SOCP relaxation of a nonconvex constraint.

  • Power flow model: The branch-flow variables include complex bus injections, sending-end line flows, squared voltages, squared currents, and substation injection.For each line, impedance is zij = rij + ixij, sending-end power is Sij = Pij + iQij, and ℓij = |Iij|2.
  • Controllable devices: The feasible injection set can represent generators, inverters, controllable loads, and shunt capacitors, including nonconvex or disconnected device sets.Shunt capacitors provide a two-point injection set, while solar panels and controllable loads impose device-specific bounds.
  • OPF formulation: OPF controls bus power injections to minimize generation cost or total network loss subject to power-flow, injection, and voltage constraints.The objective sums bus generation costs fi; choosing fi(x) = x gives total power loss.
  • Power flow model: Distribution networks are modeled as rooted trees with a fixed substation voltage and directed lines toward the root.The root bus 0 connects to the transmission network, while the remaining buses are indexed 1 through n.
  • SOCP relaxation: Relaxing the nonconvex equality constraint produces an SOCP relaxation, and any SOCP solution feasible for OPF is a global OPF optimum.The relaxation is termed exact when every SOCP solution satisfies the original equality constraint.

III. A SUFFICIENT CONDITION

The paper gives a sufficient exactness condition for SOCP based on a priori network and device parameters plus a voltage-region condition that may require solving SOCP. The condition motivates a modified OPF whose corresponding SOCP is exact under the a priori part alone.

  • A sufficient condition: The sufficient condition has two parts, C1 and C2, with C1 depending only on SOCP parameters and C2 depending on SOCP solutions.C1 reflects the physical intuition that reducing line losses increases upstream reverse power flows.
  • Upper-bound construction: Linear DistFlow quantities provide affine upper bounds on branch flows and squared voltages for feasible branch-flow points with nonnegative currents.The bounds are S ≤ ˆS(s) and v ≤ ˆv(s); equality holds exactly when every line loss zijℓij is zero.
  • Voltage region: The region Svolt contains injections whose Linear DistFlow voltage upper bounds satisfy the voltage upper limits.Because v ≤ ˆv(s), requiring s ∈ Svolt ensures the relevant upper-bound condition.
  • Theorem 1: Theorem 1 states that SOCP is exact when C1 holds, assuming strictly increasing root-bus cost and upper bounds on real and reactive injections.C1 is expressed through products of the matrices A_l along every leaf path, while C2 requires every SOCP solution to lie in Svolt.
  • Modified OPF: C2 cannot be checked a priori, so the authors modify OPF so that the corresponding SOCP is exact under C1.This modification addresses the theorem’s solution-dependent condition while retaining the parameter-checkable component.

B. Interpretation of C1

C1 links reductions in line-current losses to positive increases in upstream reverse power flows along every relevant radial-network segment. It depends only on SOCP parameters, so it can be checked efficiently before solving the problem.

  • Linear-network interpretation: C1 requires the product of A matrices over any highlighted segment, multiplied by u_t, to be componentwise strictly positive.For the linear network, the only leaf is n and its path is P_n = {n → n −1 → · · · →1 →0}.
  • Checkability: C1 can be checked a priori in O(n) time because A and u are simple functions of (r, x, p, q, v), with at most n(n + 1)/2 inequalities.The condition therefore does not require an SOCP solution.
  • Sufficient cases: If every bus consumes real and reactive power, then C1 holds because the relevant transformed power flows are nonpositive and A_i = I.This is stated by Proposition 2 for (p, q) ≤ 0.
  • Physical interpretation: For practical parameter ranges, A_i is close to I because line parameters are small, flows are O(1), and voltage lower bounds are near 1 p.u.; C1 therefore often holds.Numerical studies report C1 holding for several test networks, including networks with high distributed-generation penetration.
  • Physical interpretation: Physically, reducing a line's power loss is associated with increased upstream reverse power flows along the path toward the root, which C1 formalizes through Jacobian products.The proof uses forward and backward sweeps to construct a feasible SOCP point with lower objective value, contradicting optimality when the relaxation is not exact.
  • Proof idea: Under C1 and C2, the constructed point w′ is feasible for SOCP and has a smaller objective value, yielding a contradiction and proving SOCP exactness.The argument establishes the needed voltage and flow inequalities through Claims 1 and 2.

IV. A MODIFIED OPF PROBLEM

The modified OPF problem restricts injections so voltage upper bounds do not bind, making the otherwise solution-dependent C2 condition automatic. Under C1, the corresponding SOCP relaxation is exact, while the feasible sets remain similar in practice.

  • Motivation and construction: C2 depends on SOCP solutions and cannot be checked a priori, motivating a modified OPF problem with an additional injection constraint.The modification is designed to make C2 hold automatically.
  • Motivation and construction: The added constraint is equivalent to affine inequalities ˆv_i(s) ≤ v_i, ensuring voltage upper bounds do not bind because v_i ≤ ˆv_i(s).This removes the need to verify C2 from an SOCP solution.
  • Scope and practical relation: The modified and original OPF feasible sets are similar in practice because ˆv_i(s) is close to v_i, although the modification is necessary for the exactness guarantee.SOCP-m itself is not necessarily convex when the cost or injection sets are nonconvex.
  • Exactness result: Under the stated bounded-injection assumption, SOCP-m is exact whenever C1 holds.The theorem assumes f_0 is strictly increasing and each S_i lies within specified real and imaginary upper bounds.
  • Exactness result: Restricting injections to S_volt, where voltage upper bounds do not bind, yields exactness under the mild, a priori-checkable condition C1.This is the main implication of Theorem 2.
  • Uniqueness: If the modified relaxation is convex and exact, it has at most one solution.Theorem 3 requires convex generation-cost functions and convex injection sets.

V. CONNECTION WITH PRIOR RESULTS

The paper positions its sufficient condition relative to prior exactness results by trading restrictions on power injections against restrictions on voltage magnitudes. It also shows that C1 unifies and generalizes earlier conditions, including several physically interpretable network patterns.

  • Prior condition categories: Existing sufficient conditions mainly involve patterns in power injections, bounds on phase-angle differences, or relaxed voltage upper bounds.These form three broad categories identified in the comparison with prior work.
  • Comparison with prior work: The condition in relaxes lower bounds on power injections but permits arbitrary voltage-magnitude constraints, whereas Theorem 1 relaxes voltage upper bounds but permits arbitrary injection constraints.The comparison highlights a different allocation of assumptions between the two results.
  • Comparison with prior work: Theorem 1 requires the objective to be strictly increasing in s_0, unlike the condition in, which requires strict increase in each ℓ_ij and nondecrease in each s_i.The two conditions therefore impose different objective-function requirements.
  • Generalization: Theorem 4 assumes each injection set is bounded above in real and imaginary parts before applying its sufficient cases for C1.The assumption is stated as the existence of p_i and q_i bounding S_i.
  • Generalization: When voltage upper bounds are absent, prior results become exact under any of conditions (i)–(v), and Theorem 4 implies C1 under those same cases.In this setting, C2 holds automatically.

VI. CASE STUDIES

The case studies evaluate SOCP and SDP computational efficiency, Condition C1, and the similarity between the original and modified OPF feasible sets on IEEE and real-world radial networks.

  • B. SOCP is more efficient to compute than SDP: SOCP is much more efficient to compute than SDP, with efficiency gains increasing as network size grows.SOCP and SDP have similar computation times for small networks, while SOCP is expected to be much more efficient for medium networks.
  • A. Test networks: The test networks comprise modified IEEE 13-, 34-, 37-, and 123-bus networks plus SCE 47-bus and 56-bus real-world networks.The SCE networks have 56.6% and 130.4% DG penetration, respectively.
  • A. Test networks: The IEEE networks are adapted by splitting loads across three phases, decoupling phases, and modeling switches, regulators, transformers, and distributed loads.The resulting networks are treated as three identical single-phase networks.

C. C1 holds with a large margin

The experiments show that Condition C1 is satisfied with substantial margin across the test networks, while shrinking the voltage-feasible region changes the feasible set only slightly.

  • C. C1 holds with a large margin: The C1 margin η∗ measures how much distributed generation and shunt capacitors can be scaled before C1 fails.C1 holds when η∗ > 1, and larger η∗ indicates that the condition holds more safely.
  • C. C1 holds with a large margin: The minimum C1 margin is 1.30, meaning distributed generation and shunt capacitors can be scaled up by 1.39 before C1 breaks down.The IEEE 37-bus network has infinite C1 margin because it has no distributed generation or shunt capacitors.
  • C. C1 holds with a large margin: C1 margins exceed 10 for all IEEE networks but are much smaller for SCE networks with high distributed-generation penetration.The SCE 56-bus network has over 130% DG penetration and can still scale up DG by a factor of 1.30 before C1 breaks down.
  • D. Modification gap: OPF-m removes feasible OPF points near voltage upper bounds, which may be undesirable for robust operation.The modification gap ε quantifies the difference between the feasible sets of OPF and OPF-m.
  • D. Modification gap: Monte Carlo simulations estimate a small modification gap for every test network.The estimate uses 1000 samples, power-flow solutions, and the maximum sampled voltage deviation.
  • D. Modification gap: For the IEEE 13-bus network, εset = 0.0362 changes the voltage constraints from 0.81 ≤ vi ≤ 1.21 to 0.81 ≤ vi ≤ 1.1738 for OPF-m.The stricter voltage bound defines OPF-ε, whose feasible set is contained in that of OPF-m.

APPENDIX A

The appendix proves exactness by constructing a feasible SOCP point from any solution violating the relaxation’s key equality, while obtaining a strictly smaller objective value.

  • Proof of Theorem 1: The appendix uses induction from leaf lines to establish inequalities relating original and reconstructed line flows.These inequalities imply S_ij ≤ Ŝ_ij(s) and v_i ≤ v̂_i(s) for the relevant network elements.
  • Construction of w′: If an SOCP solution violates equality (5d), the proof constructs a new point w′ through initialization, forward-sweep, and backward-sweep steps.The construction modifies variables along a path from a violating leaf bus and preserves the required constraints.
  • Feasibility and Superiority of w′: The new point w′ satisfies the SOCP constraints and has a smaller objective value than the original solution w.This contradicts the optimality of w, proving that SOCP is exact under C1 and C2.
  • Proof of Claim 3: Condition C1 ensures upstream reverse power flows increase along the affected path during the construction.The proof establishes ΔS_k,k−1 > 0 for each path segment using matrix inequalities and Lemma 3.
  • Feasibility and Superiority of w′: The constructed voltages satisfy v′ ≥ v from the flow changes and v′ ≤ v from Condition C2.Together these inequalities yield v ≤ v ≤ v′ ≤ v, ensuring the voltage bounds remain satisfied.

APPENDIX C

Appendix C proves the needed matrix monotonicity by comparing matrices associated with two power-injection vectors and applying Lemma 3.

  • APPENDIX C: When (p, q) ≤ (p′, q′), the matrices associated with (p, q) and (p′, q′) satisfy the componentwise comparison needed for the proof.The argument applies to every leaf path and uses the corresponding matrices A and A′.
  • APPENDIX C: Lemma 3 transfers positivity through products of the compared matrices along each leaf path.The resulting inequalities establish the proposition for every leaf and all relevant indices.

APPENDIX D

Under the stated convexity and exactness assumptions, the proof shows that SOCP-m has a unique solution. It does so by proving that any two solutions must coincide.

  • APPENDIX D: Any two arbitrary SOCP-m solutions are shown to be equal, which is sufficient for uniqueness.The proof begins by selecting two solutions and reducing the goal to proving their equality.
  • APPENDIX D: Averaging two SOCP-m solutions yields another SOCP-m solution because the relaxation is convex.Exactness then preserves the quadratic equality for the averaged solution.
  • APPENDIX D: Equality in the averaged quadratic relation forces equal normalized magnitudes and angles of the corresponding power-flow variables.The proof concludes that ˜Sij/˜vi = ˆSij/ˆvi on every edge.
  • APPENDIX D: Network connectivity propagates the equality from the reference bus to all buses, implying that the two complete solutions are identical.The scaling factors satisfy η0 = 1 and connectivity gives ηi = 1 for every i.

APPENDIX E

Appendix E proves several sufficient conditions for C1 through Claims 5–9 and supports Claim 9 with an induction-based lemma. The claims use line-parameter ratios, flow-sign conditions, and positivity arguments along radial paths.

  • APPENDIX E: Theorem 4 follows from Claims 5–9.This identifies the claims as the components supporting the theorem.
  • Claim 6: The proof of Claim 6 uses backward induction to show that recursively defined αs and βs remain positive along each path.The base case uses line resistance and reactance, while the induction preserves the ratio αs/βs = η.
  • Claim 9: Claim 9 establishes C1 under a condition involving adjusted active and reactive power terms, with a special case yielding Alk = I.The proof analyzes products of matrices along paths from the reference bus to leaves.
  • Lemma 4: Lemma 4 is proved by induction on i, first applying the induction hypothesis to truncated vectors and then combining the first two components.The construction uses c1c2, d1 + d2, e1 + e2, and f1f2 to establish the next induction step.
  • Claim 9: Applying Lemma 4 in Claim 9 proves that As ··· At−1ut > 0 for every path position, completing the claim.The proof first establishes positivity of the relevant factors and then invokes the lemma before concluding the result.
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