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Geometry of Power Flows and Optimization in Distribution Networks

Javad Lavaei, David Tse, Baosen Zhang

arXiv:1204.4419v3math.OCcs.ITeess.SY

TL;DR

The paper asks when nonconvex injection regions in tree-network OPF can be safely convexified. It decomposes the region into two-bus flow regions and shows that realistic angle limits preserve Pareto-optimality, support efficient convex relaxations, and also ensure uniqueness and nonnegative locational marginal prices.

  • Problem

    The paper addresses how to convexify and efficiently solve nonconvex OPF problems in tree distribution networks while preserving their optimal solutions.

  • Method

    The paper analyzes the injection region geometrically as a linear transformation of a product of two-bus flow regions and studies its Pareto-front.

  • Results

    Under practical angle assumptions, the Pareto-front remains unchanged by convexification, including the relaxed semidefinite-programming formulation.

  • Takeaways & Limitations

    The convexified OPF can attain a global solution of the original problem using simpler linear and norm constraints, while the angle assumption also gives uniqueness and nonnegative locational marginal prices.

Abstract

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We investigate the geometry of injection regions and its relationship to optimization of power flows in tree networks. The injection region is the set of all vectors of bus power injections that satisfy the network and operation constraints. The geometrical object of interest is the set of Pareto-optimal points of the injection region. If the voltage magnitudes are fixed, the injection region of a tree network can be written as a linear transformation of the product of two-bus injection regions, one for each line in the network. Using this decomposition, we show that under the practical condition that the angle difference across each line is not too large, the set of Pareto-optimal points of the injection region remains unchanged by taking the convex hull. Moreover, the resulting convexified optimal power flow problem can be efficiently solved via }{ semi-definite programming or second order cone relaxations. These results improve upon earlier works by removing the assumptions on active power lower bounds. It is also shown that our practical angle assumption guarantees two other properties: (i) the uniqueness of the solution of the power flow problem, and (ii) the non-negativity of the locational marginal prices. Partial results are presented for the case when the voltage magnitudes are not fixed but can lie within certain bounds.

I. INTRODUCTION

The paper studies how the geometry of injection regions in tree distribution networks can make nonconvex optimal power flow efficiently solvable. Under realistic angle-difference limits, local geometric structure preserves the Pareto-front under convexification and strengthens prior results.

  • Motivation: Tree-network OPF is motivated by growing interest in optimizing distribution flows with demand response and distributed generation.
  • Problem formulation: The injection region contains feasible bus real-power injections, while its Pareto-front contains boundary points where no component can decrease without increasing another.
  • Geometric decomposition: For tree networks, the power-flow region decomposes into a product of two-bus line-flow regions intersected with linear bus-power constraints.
  • Main result: If every two-bus region has a convex Pareto-front, then the overall injection region does too; this holds when each line’s angle difference is below about 45°.
  • Computational implication: The resulting convexification preserves the Pareto-front even under a more relaxed semidefinite-programming relaxation, enabling efficient solution of the convexified OPF problem.
  • Relation to prior work: Compared with earlier tree-network results, the approach uses elementary geometry and replaces active-power lower-bound assumptions with adjacent-bus voltage-angle constraints.
  • Additional consequences: The same angle assumption guarantees a unique power-flow solution and nonnegative locational marginal prices when all bid functions are positive.

II. MODEL

The model represents a connected AC network through bus voltages, currents, admittance relationships, and real-power injections. The real-power injection vector is derived from complex voltage and current variables.

  • The network is modeled as a connected graph with buses as vertices and transmission lines as edges.
  • The admittance matrix includes shunt admittances and is symmetric.
  • Bus voltages and currents are complex vectors, with each current representing the total flow out of a bus into the network.
  • Ohm’s law and Kirchhoff’s current law relate current and voltage through i = Yv.
  • The real-power injection vector p = (P1, P2, . . . , Pn) contains each bus’s net active power.
  • Real-power injections are computed as p = Re(v ⊙ conj(i)) = Re(v ⊙ (Yv)).

III. FIXED VOLTAGE PARETO OPTIMAL POINTS

With fixed voltage magnitudes, a two-bus injection region is generated by varying the voltage-angle difference and has an elliptical geometric structure. Under angle constraints, its Pareto front is preserved by convexification, making optimization over the original and convexified regions equivalent.

  • Two-Bus Network: The two-bus injection region is parameterized by the angle difference θ = θ1 − θ2 while voltage magnitudes remain fixed.
  • Two-Bus Network: Because (cos(θ), sin(θ)) parameterizes a circle, the injection region is an affine transformation of a circle and therefore forms an ellipse.
  • Two-Bus Network: For lossy lines, the injection region is a hollow ellipse, whereas for lossless lines it degenerates into a line through the origin.
  • Angle, Thermal and Flow Constraints: With fixed voltage magnitudes, thermal, line-flow, and loss constraints can be expressed as angle constraints.
  • Pareto Optimality: The Pareto front of the nonconvex injection region is the same as the Pareto front of its convex hull.
  • Pareto Optimality: For strictly increasing objectives, optimization over the injection region and its convex hull has the same solution.

B. General Network With Local Constraints

For tree networks with fixed voltage magnitudes and local angle constraints, the feasible injection region is a linear image of decoupled two-bus flow regions. Under these conditions, its Pareto front is unchanged by convexification.

  • The injection vector is related linearly to feasible flows through p = Af, implying conv(P) = A conv(F).
  • The feasible flow set is a product of one two-bus flow region for each network line.Flows on different lines are decoupled under the tree structure.
  • Fixed voltage magnitudes and local angle constraints yield O(P) = O(conv(P)) for tree-network injection regions.This extends the two-bus Pareto-front result to arbitrary tree networks.
  • The result follows by applying the two-bus Pareto property independently to each factor of the product flow region.

C. Two-Bus Network with Bus Constraints

The two-bus analysis combines angle-constrained flows with bus power constraints. Under a practical angle condition, the Pareto front survives convexification, and the tree-network extension also gives uniqueness of flows.

  • Two-Bus Network with Bus Constraints: Bus power constraints intersect the angle-constrained injection region, producing several possible region shapes depending on upper and lower bounds.The paper distinguishes cases with upper bounds only from cases that include lower bounds.
  • Two-Bus Network with Bus Constraints: Active power lower bounds can destroy O(P) = O(conv(P)) when angle differences range over the full interval from −π to π.The failure is illustrated by the case where both buses have lower bounds.
  • Two-Bus Network with Bus Constraints: Thermal and flow constraints often restrict line angle differences to small values, such as less than 10° or 7° in typical examples.
  • Two-Bus Network with Bus Constraints: If the angle-constrained region satisfies Pθ = O(conv(Pθ)), then O(P) = O(conv(P)), including the case where P is empty.
  • Two-Bus Network with Bus Constraints: For a non-empty constrained tree network under the angle assumption, every injection has a unique feasible flow and O(P) = O(conv(P)).The theorem also states P = O(P).

E. Numerical Algorithms for Convexification

The paper replaces the difficult convex hull of the full injection region with tractable convexified flow constraints. The resulting OPF can be solved efficiently while preserving the original optimum when feasible.

  • Numerical Algorithms for Convexification: Replacing each two-bus flow constraint by its convex hull produces a convexified OPF with a simple algebraic representation.The angle-constrained injection region is a linear mapping of the product flow region.
  • Numerical Algorithms for Convexification: The convexified OPF can be solved efficiently as a second-order cone program or interpreted as a semidefinite program.
  • Numerical Algorithms for Convexification: The convexified OPF uses conv(Pθ) ∩ PP rather than the generally unavailable representation conv(Pθ ∩ PP).Convexification and intersection do not generally commute.
  • Numerical Algorithms for Convexification: The sets conv(P) and conv(Pθ) ∩ PP have identical Pareto fronts, despite being different sets.
  • Numerical Algorithms for Convexification: If the convexified solution is feasible for the original OPF, it is globally optimal; if infeasible, the original OPF is infeasible.

F. Nonnegative Locational Marginal Prices

Under the practical angle assumption, locational marginal prices are well-defined through the convexified OPF and are nonnegative. The same assumption also supports the paper’s power-flow uniqueness result.

  • Nonnegative Locational Marginal Prices: The paper defines LMPs through the change in optimal objective value caused by perturbing the load vector.Differentiability at zero load perturbation is required for the definition.
  • Nonnegative Locational Marginal Prices: Theorem 5 identifies each LMP with the Lagrange multiplier of the corresponding power-balance equation in the convexified OPF.
  • Nonnegative Locational Marginal Prices: The practical angle assumption guarantees that every LMP λi is nonnegative.
  • Nonnegative Locational Marginal Prices: The nonnegativity result establishes that the load over-satisfaction assumption used in earlier convexification work holds under the practical angle assumption.

IV. VARIABLE VOLTAGE PARETO OPTIMAL POINTS

The variable-voltage analysis studies how angle-constrained injection regions and their convexifications relate when voltage magnitudes vary within bounds. It addresses the noncommutativity of convex hulls and unions by exploiting flow decomposition and convex matrix representations.

  • Region construction: Variable-voltage injection regions are represented as unions of angle-constrained regions indexed by feasible voltage-magnitude vectors.The overall region is expressed through the regions Pθ(ṽ) and voltage-magnitude constraints.
  • Convexification challenge: The convex hull of a union cannot generally be obtained by independently convexifying each constituent region.This is the central obstacle in extending fixed-voltage results to variable voltage magnitudes.
  • Flow decomposition: Flow decomposition addresses this obstacle by associating each network edge with a 2 × 2 positive semidefinite Hermitian edge submatrix.The edge submatrices are formed from the corresponding entries of a global Hermitian matrix.
  • Flow-region relaxation: Dropping the rank constraint convexifies each flow region, although the resulting set may differ from its exact convex hull.The distinction is explicitly retained in the notation for the convexified flow region.
  • Convexity relation: Without the angle constraint, the relevant convexification and union operations commute, yielding the same convexified injection region.The argument uses convexity of the Hermitian-matrix set together with linear transformations.
  • Main result: The section establishes the main variable-voltage result by combining the convexified region construction with the Pareto-optimality relations.The theorem is introduced after defining the convex set used for the variable-voltage case.

A. Convexification via SOCP and SDP Relaxations

The paper convexifies OPF through semidefinite and second-order cone relaxations after showing that the relevant injection-region geometry supports optimization over a convexified set. For tree networks, the SDP and SOCP relaxations have the same projected injection set, while SOCP is easier to solve computationally.

  • Convexification strategy: The geometric relation is used to replace a difficult optimization over the injection region with a convexified OPF formulation.The paper notes that the convexity relation alone is not directly sufficient for solving the hard optimization problem.
  • Variable-voltage formulation: Variable voltage magnitudes are treated as optimization variables, and each nonlinear edge-flow constraint is replaced by a convexified flow-region constraint.This gives the operational formulation used to obtain the convex relaxation.
  • Exactness: The angle condition guarantees that OPF and its convexified formulation have the same solution.The resulting convexified OPF is identified as an SOCP problem.
  • SDP–SOCP relation: For tree networks, SDP and SOCP relaxations project onto the same reduced feasible set in bus-injection space.This equality is stated for the feasible sets after projection onto bus injections.
  • Computational comparison: SOCP is computationally easier to solve than SDP despite producing the same injection-region convexification.The stated reason is the total number of variables involved in the optimization.

B. Inclusion of Lower Bound on Bus Injection

This section extends the results to finite lower bounds on bus injections. The extension reduces the variable-voltage case to fixed-voltage results while retaining angle and non-emptiness assumptions.

  • Lower-bound extension: The extension removes the earlier assumption that every active-power lower bound equals negative infinity.Finite lower bounds are allowed instead.
  • Proof strategy: The variable-voltage problem is first reduced to a fixed-voltage-magnitude problem and then treated using the earlier Pareto-front theorem.This is the stated proof strategy for the generalization.
  • Assumptions: The general result requires non-emptiness of the injection region and an angle condition on every line.These assumptions are inherited from the fixed-voltage theorem and generalized for the variable-voltage setting.
  • Pareto-front relation: Under the angle condition, the two-bus flow region and its convexified counterpart share the same Pareto front.This relation is used in the variable-voltage extension.

V. CASE STUDY

The case study evaluates rank-relaxation tightness on 34-bus and 123-bus IEEE test feeders under randomized power-bound settings. The paper concludes that convexification preserves relevant global solutions under the practical angle condition, with variable-voltage magnitudes treated separately.

  • Experimental setup: The experiments use 34-bus and 123-bus IEEE test feeders with active and reactive power bounds determined by two methods.The network data and bound-generation procedures are described for the simulation study.
  • Randomized cases: 1000 runs are performed for each case, with randomly selected variable power bounds on 20% of buses in one setting.The randomized bounds are used across both feeder sizes.
  • Simulation result: Rank relaxation is tight in all test runs reported in Table I.The table caption states this outcome for both cases A and B.
  • Paper objective: The paper frames the study as an investigation of geometric properties of injection regions in tree-shaped power networks.These regions are linked to the nonconvexity of optimal power flow.
  • Fixed-voltage conclusion: For fixed voltage magnitudes, the Pareto fronts of the injection region and its convex hull are identical under a practical angle condition.This supports convexifying the OPF without changing its global solution.
  • Optimization implication: Replacing nonlinear constraints with linear and norm constraints can still attain a global solution of the original OPF.The paper also studies injection regions with variable voltage magnitudes, but reports that case separately.

APPENDIX

The appendix converts current limits into thermal-loss and angle constraints, then proves that convexification preserves Pareto-optimal injection points under the stated tree-network conditions.

  • Thermal Constraints of Distribution Networks: Current limits are converted into |I|^2r thermal-loss constraints and then into line-angle constraints.The appendix illustrates these limits using a 13-bus test feeder operating at 2.4 kV line to neutral.
  • Thermal Constraints of Distribution Networks: Each line is assigned an operating angle α and a thermally derived limit β.α denotes the typical angle between related buses, while β denotes the thermal angle limit.
  • Proof of Pareto-Set Preservation: The proof establishes O(S) ⊆ O(P) by showing that Pareto points of the convexified set correspond to feasible line-flow pairs in the original flow regions.The argument propagates flow inequalities through a maximal connected subtree and ultimately proves equality of paired line flows.
  • Proof of Pareto-Set Preservation: The subtree argument uses boundary nodes, leaf-node inequalities, and induction toward the root to derive equality across the subtree.Figure 8 supplies the geometric flow-region argument for leaf edges and neighboring nodes.
  • Proof of Pareto-Set Preservation: The reverse inclusion O(P) ⊆ O(S) follows by contradiction from P ⊆ S and the already established inclusion O(S) ⊆ O(P).A strictly dominating point in O(S) would also belong to P, contradicting Pareto optimality in P.
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