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An Optimal and Distributed Method for Voltage Regulation in Power Distribution Systems

Baosen Zhang, Albert Y. S. Lam, Alejandro Dominguez-Garcia, David Tse

arXiv:1204.5226v4math.OCcs.ITeess.SY

TL;DR

Deep DER penetration creates voltage-regulation challenges from fast changes in renewable generation and storage-capable loads. The paper formulates loss-minimizing voltage control, proves conditions for exact convex relaxation, and develops a distributed algorithm whose case studies demonstrate convergence and voltage regulation. Its scope assumes separated slow and fast control time scales.

  • Problem

    Deep penetration of DERs can cause voltage-regulation problems in distribution networks because active generation and consumption can change rapidly.

  • Method

    The paper formulates voltage regulation as constrained loss minimization, establishes exactness conditions for an SDP relaxation, and develops a distributed algorithm for tree networks.

  • Results

    The distributed algorithm converges to the optimum in all tested 34-bus cases and most tested 123-bus cases, while the voltage-regulation method keeps displayed voltages at their reference value.

  • Takeaways & Limitations

    Reactive-power-capable DERs and some active-power control from storage-capable DERs and DRRs can supplement conventional devices for faster voltage variations.

  • Takeaways & Limitations

    The analysis assumes separated slow and fast control time scales and leaves their coupling and associated trade-offs for future study.

Abstract

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This paper addresses the problem of voltage regulation in power distribution networks with deep-penetration of distributed energy resources, e.g., renewable-based generation, and storage-capable loads such as plug-in hybrid electric vehicles. We cast the problem as an optimization program, where the objective is to minimize the losses in the network subject to constraints on bus voltage magnitudes, limits on active and reactive power injections, transmission line thermal limits and losses. We provide sufficient conditions under which the optimization problem can be solved via its convex relaxation. Using data from existing networks, we show that these sufficient conditions are expected to be satisfied by most networks. We also provide an efficient distributed algorithm to solve the problem. The algorithm adheres to a communication topology described by a graph that is the same as the graph that describes the electrical network topology. We illustrate the operation of the algorithm, including its robustness against communication link failures, through several case studies involving 5-, 34-, and 123-bus power distribution systems.

I. INTRODUCTION

Deep DER penetration creates fast voltage-variation challenges that conventional devices are poorly suited to handle alone. The paper formulates voltage regulation as an optimization problem and develops conditions for exact convexification plus a distributed solution method.

  • Motivation: Deep penetration of renewable generation and storage-capable loads can create voltage-regulation problems in distribution networks.DERs include renewable-based generation and resources such as plug-in hybrid electric vehicles.
  • Motivation: Conventional mechanical voltage-regulation devices manage slow variations but may face increased operations from faster DER-driven changes.The paper distinguishes hour-scale device adjustments from minute-scale active-power variations.
  • Proposed direction: Power-electronic DER interfaces can provide reactive power, while storage-capable DERs and DRRs can shape active-power injections for voltage control.These mechanisms use existing communication capabilities and supplement conventional devices.
  • Optimization formulation: The voltage-regulation problem minimizes network losses subject to voltage, active/reactive injection, line-flow, and line-loss constraints.Bus voltages are decision variables, while reactive and partly active power injections provide the control mechanism.
  • Contributions: The paper establishes sufficient conditions for solving the nonconvex problem through an equivalent SDP and develops a distributed algorithm for tree networks.The distributed approach addresses scalability and communication limitations of centralized SDP solvers.
  • Contributions: The method extends prior voltage-regulation results by incorporating reactive power injections and tight voltage-magnitude constraints.Earlier results variously ignored reactive power, active lower bounds, or voltage upper bounds.

B. Voltage Control in Networks with Deep DER Penetration

The paper envisions hierarchical voltage control: conventional devices set slower reference conditions, while DERs and DRRs respond to faster injection changes. The fast problem minimizes network losses while enforcing voltage, injection, flow, and line-loss limits.

  • Voltage-control architecture: A hierarchical architecture separates slow conventional-device settings from fast voltage control through active and reactive power injection shaping.The paper describes conventional settings as being optimized hourly and fast regulation as operating within each hour.
  • Voltage-control architecture: The fast optimization supplies bus-level active and reactive power references so DERs and DRRs can track voltage references.The optimization is envisioned to run at regular intervals such as every minute.
  • Operating scope: The method focuses on correcting voltage deviations caused by inter-hour variations around the injection profile used to configure conventional devices.The paper assumes the slower device-setting problem has already established the reference values.
  • Problem formulation: The optimization minimizes network losses while enforcing voltage-reference, active/reactive injection, line-flow, and individual-line loss constraints.Power-control bounds represent the limited local ability to produce or consume active and reactive power.
  • Computational challenge: The problem is difficult because voltage-power relations are quadratic and network size can produce many variables and constraints.The paper addresses nonconvexity through convexification and scale through a distributed algorithm.

III. CONVEX RELAXATION

The paper rewrites voltage regulation in matrix form and relaxes its rank constraint to obtain a convex SDP. Under conditions on adjacent-bus angle differences and reactive-power lower bounds, this relaxation is exact.

  • Main result: The main theoretical result gives conditions under which the nonconvex voltage-regulation problem is solved exactly by its convex SDP relaxation.The conditions concern angle differences between adjacent buses and lower bounds on reactive power injections.
  • Matrix formulation: Bus active and reactive power injections and line flows are represented as trace expressions involving the voltage outer product vv^H.Matrices A_i, B_i, and A_ik encode bus injections and line-flow quantities.
  • Matrix formulation: The outer product vv^H is positive semidefinite and rank 1, allowing the voltage problem to be expressed as a matrix optimization problem.The rank-1 condition preserves correspondence between the matrix variable and a voltage vector.
  • Convexification: Dropping the rank-1 constraint produces the convex relaxation, whereas retaining it makes the matrix problem nonconvex.Exactness depends on whether the relaxed solution satisfies the required rank condition.

B. Convexification

The paper convexifies voltage regulation through an SDP relaxation and identifies conditions under which the relaxation exactly solves the original problem. Its geometric formulation represents line-flow constraints as angle constraints and characterizes feasible regions for active and reactive flows.

  • Exactness conditions: For tree networks satisfying the theorem’s angle and reactive-injection conditions, the relaxed problem determines feasibility and solves the original voltage-regulation problem exactly.If the relaxed optimum has rank 1, it yields an optimal voltage solution; higher rank implies infeasibility of the original problem.
  • Geometric formulation: With unit voltage magnitudes, active and reactive line-flow regions are ellipses related by an invertible linear transformation.The active ellipse is centered at [gik, gik]T and the reactive ellipse at [bik, bik]T.
  • Geometric formulation: Thermal-loss and line-flow limits become linear constraints on these regions, while equivalent angle constraints restrict the feasible line-flow portions.The paper denotes the resulting angle-constrained active and reactive regions by Fθ,ik and Gθ,ik.

B. Feasible Region of a Two-Bus Network

The two-bus geometry explains when the convex relaxation is tight and how angle and reactive-power bounds affect that property. The optimal loss-minimizing point lies in the lower-left portion of the relaxed feasible region.

  • Two-bus geometry: The relaxation fills the feasible line-flow ellipses, and loss minimization drives the relaxed optimum toward their lower-left region.For a two-bus system, P1=P12 and P2=P21, with analogous equalities for reactive power.
  • Tightness: The relaxation is tight when the relaxed optimum lies on an ellipse boundary, allowing recovery of a rank-1 solution.An interior optimum instead produces a higher-rank solution.
  • Angle constraints: An angle constraint on the maximum difference across a line can restrict feasibility to the lower-left half of the line-flow ellipse.The paper uses this geometry to explain the theorem’s tightness cases.
  • Tightness: With upper bounds on both bus powers, the lower-left optimum lies on the ellipse boundary and the relaxation is tight; with lower bounds, the optimum can be interior and the original problem infeasible.These correspond respectively to the theorem’s rank-1 and infeasibility cases.
  • Conditions and failure cases: Thermal data suggest the angle condition is usually satisfied in practical networks, while tight reactive lower bounds can still yield rank-2 relaxed optima despite original feasibility.The reactive lower-bound condition prevents this phenomenon by keeping those bounds from being tight.

V. A DISTRIBUTED ALGORITHM FOR SOLVING THE CONVEXIFIED PROBLEM

The paper decomposes the convexified SDP across buses and coordinates neighboring local solutions through consensus. This exploits the tree topology to require only local communication while preserving convergence to the global optimum.

  • Motivation: General-purpose SDP solvers are poorly scalable and centralized, motivating a distributed method for fast voltage regulation in large networks.The proposed method communicates only between neighboring buses, matching the physical network structure.
  • Distributed algorithm: Each iteration has local optimization and consensus stages: buses solve local problems, exchange neighboring multipliers, and update them to align shared line-flow variables.The consensus update operates independently for each edge using the two endpoint solutions.
  • Algorithm derivation: The SDP is rewritten using bus-centered submatrices, with local positive-semidefinite constraints equivalent to the global constraint on a tree network.The adjacent-bus sets contain all maximal cliques, and consistency constraints assemble the local matrices into the global matrix.
  • Algorithm derivation: Relaxing local matrix-consistency constraints dualizes the coupling and divides the problem into n separable bus subproblems.Each subproblem is defined over a local feasible region Ci and uses Lagrangian multipliers associated with incident lines.
  • Convergence: Because the convexified problem has zero duality gap at optimal multipliers, the distributed subgradient algorithm converges to its optimal solution.The paper allows constant or non-summable diminishing step-size rules such as α[t]=a/t.

B. Feasibility

The distributed setting requires buses to detect infeasibility locally because independently chosen power limits can create an empty feasible region. An infeasible local subproblem or persistent edge inconsistency provides a basis for declaring global infeasibility.

  • B. Feasibility: Independent bus limits can make the overall optimization problem infeasible, leaving an empty feasible region.Without a central authority holding all bus power information, buses must declare infeasibility themselves.
  • B. Feasibility: An infeasible subproblem at any bus is sufficient to conclude that the whole problem is infeasible.The affected bus should adjust its active and reactive power limits to make its subproblem feasible.
  • B. Feasibility: Persistent failure to match W^(i)_ik across an edge identifies either endpoint as contributing to infeasibility.This condition is presented as necessary and sufficient for infeasibility when Algorithm 1 evolves.

C. Numerical Performance Enhancements

The numerical enhancements address slow convergence of the distributed algorithm by tightening local feasible regions and exploiting directional power-flow assumptions. These changes improve convergence in the five-bus example.

  • C. Numerical Performance Enhancements: At iteration 20, the unenhanced five-bus algorithm remained around 20% away from the optimal objective value.The example shows that even a small network may require substantial time to converge globally.
  • C. Numerical Performance Enhancements: Assuming nonfeeder buses are net consumers and active power flows forward simplifies the algorithm but is not required for the theoretical results.The authors note that practical protection issues currently restrict DERs from causing reverse current flow.
  • C. Numerical Performance Enhancements: Replacing edge-flow constraints with bus-specific forms constructs smaller feasible regions that reduce discrepancies between neighboring local variables.The modification is applied from both endpoints of each edge and extended to all edges incident to a bus.
  • C. Numerical Performance Enhancements: The modified algorithm converges faster than the original algorithm in the five-bus example.The comparison is shown in Fig. 6(b).

2) Feasible Solution Generation:

The feasible-solution procedure coordinates local copies of shared line variables through dual multipliers and messages. Its convergence follows a leaf-to-feeder pattern, while monitoring power variations supports repeated updates.

  • 2) Feasible Solution Generation:: Algorithm 1 manipulates line-associated λ_ik values so each bus finds its optimal active and reactive power pair.Each bus solves a local subproblem, and neighboring copies of shared variables are reconciled through multiplier updates.
  • 2) Feasible Solution Generation:: Leaf buses converge first, followed successively by buses connected to them, until convergence propagates toward the feeder.A leaf has only one edge, so its local power pair is fixed before more highly connected buses.
  • 2) Feasible Solution Generation:: A converged leaf bus passes its fixed shared line variable to its neighboring bus as a local constraint.The receiving bus incorporates the message through an equality involving the real part of the transmitted variable.
  • 2) Feasible Solution Generation:: The global solution W* can be constructed from the edge variables obtained after the network is progressively reduced by removing fixed buses.The reduction removes buses whose incident variables have been fixed and continues until all edge variables are determined.
  • 2) Feasible Solution Generation:: Power injections are monitored over time because fixing a nonoptimal bus pair can induce incorrect values in subsequently processed neighboring buses.The method tracks recent active-power values and applies a similar condition to reactive power.

3) Hot Start:

The hot-started distributed algorithm is evaluated on time-varying 34- and 123-bus systems with local PV and storage resources. It generally reaches the centralized optimum, maintains voltage references, and remains convergent under packet drops, although the comparison centralized solver has convergence issues.

  • 3) Hot Start:: Hot starting initializes λ_ik[0] with the previous optimal λ*_ik because optimal line angle differences usually vary little between problem instances.This exploits the small variation expected between successive updates of active and reactive limits.
  • 3) Hot Start:: The simulations use IEEE 34- and 123-bus systems, with Algorithm 1 run every minute over irradiance-driven operating conditions.Each simulation was terminated after 300 iterations, and successive instances use the previous iteration’s multipliers as starting points.
  • 3) Hot Start:: The distributed algorithm reaches the optimum in every 34-bus case and most 123-bus cases, while failed 123-bus runs exhaust the 300-iteration limit.In nonconvergent cases, the previous solution is used; the centralized solver failed on both systems because of convergence issues.
  • 3) Hot Start:: Voltages remain at their reference values during a one-hour period with highly variable PV injections in both test systems.The result illustrates mitigation of fast-varying power injections arising from PV systems.
  • 3) Hot Start:: Computational-time comparisons count SDP-solver CPU time while neglecting communication overhead.The distributed curve sums the longest subproblem CPU time per iteration under the sequential implementation.
  • 3) Hot Start:: Convergence is always achieved for the 34-bus network when packet-drop probabilities are p = 0, p = 0.1, and p = 0.3.The failures are modeled as independent losses of transmitted Lagrangian multipliers on individual edges.

VII. CONCLUDING REMARKS

The paper presents a convex-optimization method for voltage regulation and a scalable distributed algorithm, while identifying unresolved coupling between fast and slow control timescales.

  • The voltage-regulation problem is formulated as loss minimization and can be solved through convex relaxation under broad, practically likely conditions.The relaxation provides the basis for the proposed convex optimization method.
  • A distributed algorithm is proposed for networks with many buses and demonstrated effective and robust in two case studies.
  • The method assumes separate slow timescales for conventional-device settings and fast timescales for the proposed voltage-regulation method.
  • Future work should study coupling between timescales and trade-offs involving conventional voltage-regulation devices, DERs, and reactive-power-capable resources.

APPENDIX

The appendix develops the convex-relaxation argument by characterizing feasible flow regions, their convexity, and the Pareto-front relationship between the original and relaxed problems.

  • The proof reduces the key theorem to showing that any higher-rank relaxed solution cannot yield a feasible rank-1 solution.
  • The appendix assumes reactive-power lower bounds are never tight, requiring a condition such as Qi < βi for network nodes.
  • The active flow region is represented using angle constraints and a generalized edge-to-bus incidence matrix.
  • Reactive-power flow regions are transformed globally through a block-diagonal matrix, yielding the angle-constrained reactive injection region.
  • The real- and reactive-power flow regions are convex because they are defined by linear inequalities, whereas the angle-flow region is not.
  • The relaxed feasible region is convex, contains the original feasible region, and is expressed using the convex hull of the angle-flow region.
  • Because strictly increasing objectives attain optima on Pareto fronts, the proof uses the lemma P = O(S) when the original feasible region is nonempty.
  • A locally improved feasible flow would contradict optimality, completing the relevant contradiction argument.
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