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Voltage Regulation Algorithms for Multiphase Power Distribution Grids

Vassilis Kekatos, Liang Zhang, Georgios B. Giannakis, Ross Baldick

arXiv:1508.06594v1eess.SYcs.SYmath.OC

TL;DR

Modern distribution grids face voltage-regulation challenges from time-varying power injections, while multiphase coupling and local control in unbalanced grids remain insufficiently characterized. The paper analyzes local reactive-power rules using approximate grid models; its single-phase accelerated scheme is faster than the compared alternatives, while the unbalanced-grid rule converges to an equilibrium that may not optimize voltage regulation.

  • Problem

    Time-varying injections challenge voltage regulation, and prior work had not characterized inter-phase voltage effects or local-control behavior in unbalanced grids.

    Source: §I · p. 1

  • Method

    The paper develops local reactive-power control rules using approximate linearized models for single- and multiphase distribution grids.

    Source: §I · pp. 1–2

  • Results

    APGD has a four-fold speedup over PGD and DPGD; under unbalanced conditions, the rule converges to an equilibrium that need not minimize voltage regulation.

    Source: §I · p. 2, A. Single-Phase Grids · p. 8

  • Takeaways & Limitations

    The schemes have convergence guarantees and numerical tests indicate resilience to topology reconfigurations.

    Source: §VI · p. 9

    Whether the unbalanced-grid equilibrium corresponds to the minimizer of a voltage-regulation problem remains an open question.

    Source: §VI · p. 9

Abstract

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Time-varying renewable energy generation can result in serious under-/over-voltage conditions in future distribution grids. Augmenting conventional utility-owned voltage regulating equipment with the reactive power capabilities of distributed generation units is a viable solution. Local control options attaining global voltage regulation optimality at fast convergence rates is the goal here. In this context, novel reactive power control rules are analyzed under a unifying linearized grid model. For single-phase grids, our proximal gradient scheme has computational complexity comparable to that of the rule suggested by the IEEE 1547.8 standard, but it enjoys well-characterized convergence guarantees. Adding memory to the scheme results in accelerated convergence. For three-phase grids, it is shown that reactive injections have a counter-intuitive effect on bus voltage magnitudes across phases. Nevertheless, when our control scheme is applied to unbalanced conditions, it is shown to reach an equilibrium point. Yet this point may not correspond to the minimizer of a voltage regulation problem. Numerical tests using the IEEE 13-bus, the IEEE 123-bus, and a Southern California Edison 47-bus feeder with increased renewable penetration verify the convergence properties of the schemes and their resiliency to grid topology reconfigurations.

I. INTRODUCTION

The paper develops local reactive-power voltage-control rules for single- and multiphase grids, addressing limited understanding of inter-phase effects and local control under unbalanced conditions. It also presents approximate grid models that relate injections to voltage magnitudes.

I. INTRODUCTION

  • Prior work had not established how inter-phase coupling affects voltage magnitudes or how local control behaves in unbalanced grids.Time-varying injections and slow utility-owned equipment make voltage regulation challenging in modern grids.Source: §I · p. 1
  • The work extends local reactive-power control to multiphase grids, revealing inter-phase voltage effects and showing that the rule may converge to an equilibrium that does not minimize voltage regulation.Numerical tests also examine convergence and resilience to topology reconfigurations.Source: §I · pp. 1–2

A. Modeling of Single-Phase Grids

  • The paper organizes the single-phase grid as a radial tree and uses an approximate linear flow model to express grid quantities in matrix-vector form.The model defines R and X from the grid’s radial structure and line resistances and reactances.Source: A. Modeling of Single-Phase Grids · p. 2
  • The approximate single-phase model relates voltage magnitudes to active and reactive injections, with each bus’s injection affecting voltages throughout the grid.The model also indicates that both active and reactive injections raise voltage magnitudes when the stated matrix properties hold.Source: A. Modeling of Single-Phase Grids · p. 2

B. Modeling of Multi-Phase Grids

The multiphase model approximates grid behavior using phase-wise quantities and a matrix-vector representation. Although power flows and injections are approximately decoupled across phases, voltage magnitudes depend on injections from all phases.

B. Modeling of Multi-Phase Grids

  • The approximation assumes all buses have three phases, small line-impedance entries, and approximately balanced phase voltages.The model drops the last summands in the voltage-drop equations under the small-impedance assumption.Source: B. Modeling of Multi-Phase Grids · p. 3
  • The model uses a matrix-vector form to relate multiphase voltages to nodal injections, with bus-wise and phase-wise vector arrangements connected by a permutation matrix.The matrix-vector formulation builds on an existing multiphase grid model.Source: B. Modeling of Multi-Phase Grids · p. 3
  • The approximate multiphase model implies that squared voltage magnitudes depend on power injections from all phases, despite approximate decoupling of injections and flows across phases.This model is used as the basis for the voltage-regulation schemes developed in the following section.Source: B. Modeling of Multi-Phase Grids · p. 3

III. SCHEMES FOR SINGLE-PHASE GRIDS

The paper relaxes voltage regulation into a localizable cost that penalizes deviations from nominal voltage. For unconstrained reactive support, a localized rule converges to the optimization minimizer for a sufficiently small step size.

III. SCHEMES FOR SINGLE-PHASE GRIDS

  • The voltage-regulation formulation penalizes squared-voltage deviations from nominal values and treats utility-owned regulating equipment as fixed because it responds more slowly than inverter controls.The formulation also allows a cost for reactive compensation and a feasible set of reactive injections.Source: §III · p. 4

A. Unconstrained Reactive Support

  • With unlimited reactive support, the unique minimizer achieves perfect voltage regulation but requires active-injection information from all buses.The minimizer can be computed locally only if each bus measures power flows on its incident lines.Source: A. Unconstrained Reactive Support · p. 4
  • A gradient-descent update uses the voltage-deviation gradient, but that update is not locally implementable; the proposed rule converges to the minimizer for a sufficiently small step size.The convergence result is stated for the rule in (18).Source: A. Unconstrained Reactive Support · p. 4

B. Constrained Reactive Support

Constrained reactive support accounts for limited, time-varying inverter capability while using a locally measurable voltage-deviation objective. The proximal-gradient rule converges linearly, although its rate depends on the grid matrix’s condition number.

B. Constrained Reactive Support

  • Reactive injections must remain within time-varying limits, and available PV capability depends on active injection and may be absent at some buses.The practical constraint is represented by a box Q; buses without reactive support can be assigned zero limits.Source: B. Constrained Reactive Support · p. 4
  • The adjacent-bus voltage-deviation cost has a locally measurable gradient but does not guarantee voltages stay within a desired range.Its gradient equals the deviation of squared voltage magnitudes from nominal, measured locally at each bus.Source: B. Constrained Reactive Support · p. 4
  • A reactive-power compensation cost penalizes unnecessary inverter use, and the resulting strongly convex objective has a unique minimizer found by proximal-gradient iterations.Positive and negative reactive injections are treated equally by the stated compensation-cost choice.Source: B. Constrained Reactive Support · p. 4
  • The control rule converges linearly with the optimal step size, but its rate is proportional to κ(X), which reaches 1.6×10^4 on the SCE 47-bus grid.The cited IEEE 34-bus benchmark has κ(X) = 5.5×10^4.Source: B. Constrained Reactive Support · p. 5

C. Accelerating Voltage Regulation Schemes

The section considers diagonal scaling and a memory-based accelerated proximal-gradient rule to improve convergence. The accelerated scheme performs better than the compared alternatives, while retaining computational requirements similar to the IEEE 1547.8 rule.

C. Accelerating Voltage Regulation Schemes

  • Diagonal scaling transforms reactive injections and yields a per-bus step size adjusted by the corresponding diagonal entry of the scaling matrix.A simple choice sets the diagonal of D^−1 to diag(X), making the diagonal entries of D^1/2XD^1/2 equal to unity.Source: C. Accelerating Voltage Regulation… · p. 5
  • The accelerated update adds memory by combining current and previous update quantities using coefficients that vary with β_t and converge to 1.The rule is described as a simple modification of the proximal-gradient control rule.Source: C. Accelerating Voltage Regulation… · p. 5
  • The memory-based accelerated rule converges linearly to an ϵ-optimal cost within a number of iterations proportional to √κ(X) log(1/ϵ).Numerical tests report superior performance over the basic and diagonally scaled rules, even without sequence resets or precise knowledge of λ_max(X).Source: C. Accelerating Voltage Regulation… · p. 5
  • The IEEE 1547.8 rule has similar computational requirements to the proposed rules, but can oscillate, whereas the proposed schemes have well-understood convergence properties.Its reactive-injection function is increasing and piecewise linear.Source: C. Accelerating Voltage Regulation… · p. 5

IV. SCHEMES FOR MULTI-PHASE GRIDS

The multi-phase section extends voltage regulation to unbalanced grids and studies the matrix properties underlying the schemes. Under stated diagonal-dominance conditions, the relevant matrices are positive definite or positive semidefinite.

IV. SCHEMES FOR MULTI-PHASE GRIDS

  • The multi-phase voltage model governs squared-voltage dependence on nodal injections, and the section studies properties of its matrices before constructing regulation schemes.The setting is motivated by unbalanced conditions associated with distributed generation and demand-response programs.Source: §IV · pp. 5–6
  • Strict diagonal dominance with positive diagonal entries in ˜X_n implies Im[˜Z_n] ≻ 0.The stated mutual- and self-reactance inequalities yield this diagonal-dominance condition and are generally satisfied by distribution-line structure.Source: §IV · p. 6
  • The decomposition separates X into a symmetric component and an anti-symmetric component, and strict diagonal dominance with positive diagonal entries in Im[Z_n] for every n implies X ⪰ 0.The decomposition links positive definiteness of X to that of the block-diagonal reactive component.Source: §IV · p. 6

A. Inter-Phase Coupling

The linearized multi-phase model reveals that reactive injections can affect voltage magnitudes across phases in different directions. These coupling effects can make local voltage correction counterproductive in another phase.

A. Inter-Phase Coupling

  • Injecting more reactive power raises voltages at all buses in the same phase.This same-phase effect is stated to match the single-phase case.Source: A. Inter-Phase Coupling · p. 6
  • Injecting more reactive power decreases voltage magnitudes across buses of the preceding phase in positive-sequence ordering.The passage identifies the effect through a negative cross-phase matrix entry.Source: A. Inter-Phase Coupling · p. 6
  • The effect on the following phase depends on differences in the phase-specific line parameters, and is less significant than the effect on the previous phase when every bus serves all phases.The comparison is conditional on every bus serving all phases.Source: A. Inter-Phase Coupling · p. 6
  • Reducing phase-b reactive injections to address over-voltage there can aggravate over-voltage on phase a, making multi-phase regulation more challenging.The passage presents this as an illustration of the inter-phase influence patterns.Source: A. Inter-Phase Coupling · p. 6

B. Unconstrained Reactive Support

With unlimited reactive support and no compensation cost, the update converges to the unique minimizer of the voltage-deviation objective. Unlike the single-phase case, that minimizer cannot be found locally in a multiphase grid.

B. Unconstrained Reactive Support

C. Constrained Reactive Support

For constrained reactive support, the paper analyzes a localized voltage-deviation cost and a proximal update under the multiphase model. The update converges under a step-size condition, but its equilibrium need not minimize a voltage-regulation problem.

C. Constrained Reactive Support

V. NUMERICAL TESTS

Numerical tests evaluate the control schemes on three benchmark feeders, including unbalanced conditions and a topology change. They report faster convergence for the accelerated scheme, close agreement between linearized and full AC voltage profiles, and an unresolved question about the multiphase equilibrium.

V. NUMERICAL TESTS

  • The tests used IEEE 13-bus, IEEE 123-bus, and SCE 47-bus feeders with solar generation data; active power was fixed and PV reactive injections initialized to zero unless stated otherwise.The IEEE 13-bus scenario used a 1.07 p.u. feeder voltage and 52% PV penetration at most buses; the SCE case included specified PV installations and load settings.Source: §V · p. 7, A. Single-Phase Grids · pp. 7–8

A. Single-Phase Grids

  • On IEEE 13-bus, APGD is four-fold faster than PGD and DPGD; on SCE 47-bus, the novel scheme converges at least six to ten times faster than alternatives.The IEEE 13-bus comparison uses µ = 0.1/λmax(X); the SCE 47-bus comparison uses the optimal µ = 1/λmax(X).Source: Figure 4 · p. 8, Figure 5 · p. 8, A. Single-Phase Grids · p. 8
  • On the SCE 47-bus feeder, linearized-model voltage magnitudes closely approximate those from the full AC model.The comparison uses voltage profiles from PGD and APGD with µ = 0.1/λmax(X).Source: Figure 6 · p. 8, A. Single-Phase Grids · p. 8
  • On IEEE 123-bus, topology reconfiguration did not significantly affect any scheme’s convergence rate, and APGD remained superior to PGD.The longer convergence period than on the IEEE 13-bus and SCE 47-bus feeders was attributed to the larger feeder size.Source: Figure 7 · p. 9, A. Single-Phase Grids · p. 9

VI. CONCLUSIONS

  • For single-phase grids, the proximal-gradient scheme has complexity comparable to the IEEE 1547.8 rule with convergence guarantees, while adding memory accelerates convergence.For unbalanced grids, the control rule converges to an equilibrium, but whether it minimizes an optimization problem remains open.Source: §VI · p. 9

B. Multi-Phase Grids

  • On the unbalanced IEEE 13-bus system, rule (38) converged within 40 iterations; smaller steps slowed convergence, while µ ≥ 3.1λmin caused divergence.The linearized and full AC voltage profiles were reported to agree closely.Source: Figure 8 · p. 9, B. Multi-Phase Grids · p. 9

APPENDIX

The appendix derives a stacked multiphase voltage relation and supports convergence claims through descent-direction and matrix-positivity arguments.

APPENDIX

  • The multiphase voltage relation is derived by stacking per-phase voltage-drop and flow equations, then substituting the resulting expressions.The derivation uses the stated inverse identities and Kronecker-product properties.Source: APPENDIX · pp. 9–10
  • For the single-phase analysis, the voltage-error vector gives a descent direction for f1(q), and the control rule converges to the minimizer of (22).The direction is not necessarily the steepest descent direction.Source: APPENDIX · p. 10
  • In the multiphase case, positive-definiteness of Xx guarantees that the voltage-error vector is a descent direction.The proof relates the direction’s inner product with the gradient to a quadratic form in Xx.Source: APPENDIX · p. 10
  • The step-size condition in Proposition 6 is equivalent to requiring Xµ = X + X⊤ − µXX⊤ to be positive definite.The appendix obtains the condition through a linear matrix inequality.Source: APPENDIX · p. 10

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