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Optimal Distributed Control of Reactive Power via the Alternating Direction Method of Multipliers

Petr Šulc, Scott Backhaus, Michael Chertkov

arXiv:1310.5748v2math.OCeess.SY

TL;DR

Reactive-power control in photovoltaic distribution circuits must balance inverter capacity and voltage limits, while local policies can be suboptimal. The paper develops distributed dual-ascent and ADMM protocols using local computations and neighbor communication, finding that ADMM converges significantly faster and achieves optimal control.

  • Problem

    Local policy-based reactive-power control can become suboptimal when inverter saturation prevents desired injections, motivating distributed control that retains optimality without full centralization.

  • Method

    The paper exploits separable convex optimization to design distributed dual-ascent and consensus-ADMM protocols using local measurements, computations, and nearest-neighbor communication.

  • Results

    ADMM converges within tens of iterations for cases 1–5 and dramatically outperforms dual ascent when initialization is not near the optimum, while constrained cases require about 10^3 iterations.

  • Takeaways & Limitations

    The distributed scheme achieves globally optimal reactive-power flows using local measurements, local computations, and nearest-neighbor communication.

Abstract

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We formulate the control of reactive power generation by photovoltaic inverters in a power distribution circuit as a constrained optimization that aims to minimize reactive power losses subject to finite inverter capacity and upper and lower voltage limits at all nodes in the circuit. When voltage variations along the circuit are small and losses of both real and reactive powers are small compared to the respective flows, the resulting optimization problem is convex. Moreover, the cost function is separable enabling a distributed, on-line implementation with node-local computations using only local measurements augmented with limited information from the neighboring nodes communicated over cyber channels. Such an approach lies between the fully centralized and local policy approaches previously considered. We explore protocols based on the dual ascent method and on the Alternating Direction Method of Multipliers (ADMM) and find that the ADMM protocol performs significantly better.

I. INTRODUCTION

The paper develops a decentralized, communication-assisted approach to reactive-power control that seeks optimality using local measurements and nearest-neighbor information. It exploits separability in a convex formulation and evaluates dual ascent and ADMM, with faster ADMM convergence reported across diverse circuit configurations.

  • Motivation: Existing reactive-power control approaches include centralized optimization and local policy-based methods, representing contrasting levels of coordination and information.Centralized optimization assumes a single authority with full observability, while the cited introduction contrasts this with local policy approaches.
  • Motivation: Inverter saturation can make local policy actions suboptimal when real-power injection approaches apparent-power capacity, leaving little reactive-power capability.The desired local control action may exceed inverter capability when combined with local power-flow conditions.
  • Contribution: The proposed scheme restores optimality and adaptability through iterative message passing while using local computations, local measurements, and current data from nearest-neighbor nodes.The approach uses limited communications but remains decentralized and relies on node-local computations.
  • Method and evaluation: Separability of the convex formulation motivates distributed dual ascent and ADMM algorithms, with faster ADMM convergence confirmed experimentally across seven distribution-circuit configurations.The experiments vary node counts, photovoltaic penetration, and load profiles while minimizing real-power loss subject to nominal voltage bounds.

II. DISTRIBUTED FLOW FORMULATION

The section models radial distribution-circuit power flows with DistFlow equations and adopts the LinDistFlow approximation under small voltage variations and small real and reactive losses. Node power combines local consumption and generation, with photovoltaic-inverter reactive generation adjustable within limits.

  • DistFlow equations restate Kirchhoff’s laws in terms of power flows for radial or tree-like distribution circuits with discrete loads.For the radial case, the equations describe flows between successive nodes.
  • The manuscript adopts the LinDistFlow approximation, assuming voltage variations are much smaller than upstream voltage magnitude and real and reactive losses are much smaller than their respective flows.
  • Node complex power extracted from the circuit comprises local consumption minus local generation, represented by active and reactive components.
  • Active power contributions are uncontrolled, whereas photovoltaic-inverter reactive generation can be adjusted within limits.

III. CONTROL OF INVERTERS AS A GLOBAL OPTIMIZATION

The section formulates inverter control as a global optimization minimizing real-power losses under voltage and inverter-capacity constraints. Under LinDistFlow, the problem is convex and motivates decentralized algorithms based on nearest-neighbor message passing, including ADMM and dual ascent.

  • Global optimization: The optimization minimizes total real-power loss while enforcing nominal voltage limits and each inverter’s apparent-power capacity.The voltage tolerance is typically ǫ = 0.05 under ANSI C84.1-2006.
  • LinDistFlow formulation: Under normal operating conditions, small voltage changes and relatively small power losses justify the LinDistFlow approximation.The approximation applies when voltage variations are small relative to voltages and real- and reactive-power losses are small relative to flows.
  • LinDistFlow formulation: The LinDistFlow formulation is convex because it has a quadratic objective and linear constraints, enabling efficient optimization.A centralized solution would require communication with an authority that computes and distributes optimal inverter values.
  • Distributed algorithms: The work develops a decentralized algorithm that solves the LinDistFlow problem by passing messages between nearest neighbors on the network.The section presents an ADMM-based algorithm and compares it with dual ascent for a simplified problem without voltage constraints.

A. ADMM consensus distributed algorithm

The paper adapts consensus ADMM so neighboring nodes independently optimize local variable copies while enforcing agreement with shared global variables. Iterative minimization, averaging, and multiplier updates yield a feasible global solution and faster convergence than dual ascent.

  • Algorithm formulation: Consensus ADMM assigns each node a local objective and constraints over a shared global variable, with local copies constrained to converge to the global optimum.The consensus formulation makes the distributed problem equivalent to the original optimization when local and global copies agree.
  • Iteration steps: Each ADMM iteration performs independent local convex minimization, nearest-neighbor averaging of shared variables, and local Lagrange-multiplier updates.The minimization has a quadratic objective and linear constraints, while averaging requires neighboring nodes to exchange local variables.
  • Communication and consensus: Nearest-neighbor communication exchanges local reactive-power and voltage beliefs, whose consensus convergence identifies the global optimal solution.The exchanged local variables represent each node’s beliefs about reactive-power flows and voltage magnitudes.
  • Feasible solution: Upon convergence, local variables form an optimized feasible solution, and each node can calculate its inverter reactive-power injection from those variables.The inverter output remains within its allowed bounds because the local variables satisfy the formulation’s feasibility conditions.
  • Performance comparison: ADMM converges faster than the dual ascent algorithm used as the comparison method for a simplified LinDistFlow problem without voltage constraints.The comparison is motivated by faster convergence being a general main advantage of ADMM.

B. Dual ascent algorithm for distributed control of the inverters with no voltage constraints

The dual ascent algorithm solves the simplified inverter-control problem without voltage constraints through parallel node updates and neighbor communication. Although local computations are simple, convergence can require many iterations and is sensitive to the stepsize parameter α.

  • Formulation: Dual ascent targets the simplified optimization problem after excluding the voltage constraints.Its Lagrangian is formed without constraints (3d, 3e).
  • Algorithm: Each iteration minimizes the Lagrangian over inverter reactive powers and then updates the dual variables.The primal update uses fixed dual variables, followed by dual-variable updates.
  • Distributed implementation: The method supports parallel implementation in which each node exchanges local dual variables and reactive-power values with neighboring nodes.Node j receives ζ+ values from its right neighbor, sends ζ+ values leftward, computes Q_j, and communicates the result to adjacent nodes.
  • Distributed implementation: Its node computations require only trivial algebraic operations and synchronous communication of Q, ζ+, and ζ− with neighbors.These operations are simpler than solving Eq. (6).
  • Convergence: Dual ascent may converge slowly because α has a limited feasible range: excessively large values prevent convergence, whereas excessively small values slow it.The method is expected to require a large number of iterations to converge.
  • Comparison: The paper compares dual ascent with ADMM and uses ADMM-noV as the corresponding ADMM variant without voltage constraints.ADMM-noV is obtained by excluding voltage variables U_j, U_j+ and removing constraints (4e, 4f, 4g).

IV. EXPERIMENTS WITH DISTRIBUTED GLOBAL OPTIMIZATION ON DIFFERENT DISTRIBUTION CIRCUIT CASES

The experiments use simulations across multiple distribution-circuit cases to compare distributed optimization algorithms with a sub-optimal local scheme. They also assess the validity of the LinDistFlow approximation against DistFlow calculations.

  • Circuit cases and algorithm comparisons: Simulations evaluate distributed optimization algorithms across a range of distribution-circuit cases.The cases are introduced as feeder-line configurations.
  • Circuit cases and algorithm comparisons: The experiments compare dual ascent, ADMM, and ADMM-noV performance on the considered circuit cases.These are the distributed algorithms examined in the simulations.
  • Circuit cases and algorithm comparisons: Global optimization results are compared with the sub-optimal local optimization scheme proposed in.The local scheme uses only each node’s local information about q(c)_j.
  • Approximation validation: LinDistFlow and DistFlow losses and voltages are compared for the same injected reactive power q(g) to check the underlying approximation.The comparison uses the same set of injected reactive-power values.

A. Distribution circuit test cases

The simulations use feeder configurations characterized by PV penetration, inverter capacity, node loads, and fixed line parameters. Cases include uncontrolled minimum- and maximum-voltage violations under high-load and high-generation conditions.

  • Circuit configurations: PV-pen denotes the percentage of nodes with PV generation, whose inverters inject the same p(g), while smax defines inverter apparent-power capacity.Nodes with PV generation use sj = smax; all other nodes use sj = 0.
  • Circuit configurations: Real consumption p(c)j is uniformly distributed from 0 to p(c)max, while reactive consumption is generally q(c)j = 0.25p(c)j with exceptions in cases 5 and 7.Case 5 uses fj uniformly between 0.01 and 1.0, whereas case 7 uses fj uniformly between 0 and 1.0.
  • Circuit configurations: All circuit lines use resistance 0.33 Ω/km, reactance 0.38 Ω/km, neighbor spacing 0.25 km, and starting voltage V0 = 7.2 kV.These parameters are fixed across the considered cases.
  • Circuit configurations: The test circuits are based on feeder-line configurations analyzed in previous work.The earlier work examined suboptimal, policy-based control schemes.
  • Circuit configurations: Case 6 models higher loads with no generation and 100% penetration, while case 7 models high generation and low consumption causing uncontrolled voltage-limit violations.Case 6 violates the minimum-voltage constraint without q(g) control; case 7 violates the maximum-voltage constraint.

B. Global vs local optimization

The globally optimal reactive-power control reduces real-power losses and generally outperforms or matches the local policy scheme. However, under severe voltage violations, only global optimization fully restores voltage constraints, at the cost of higher losses.

  • Global vs local optimization: The local policy is approximately optimal until voltage limits are approached, but it cannot fully respond once those limits become active.The passage attributes the local policy’s limitation to its inability to fully correct severe voltage conditions.
  • Global vs local optimization: Both global and local control schemes lower total real-power losses relative to the no-optimization scheme.The no-optimization scheme sets all reactive powers to zero, corresponding to PV inverters operating at nominal power factor 1.0.
  • Global vs local optimization: In cases 1–5, the local policy performs nearly as well as the globally optimal solution except in cases 3 and 5.This similarity is expected when loading or over-generation does not push voltages beyond normal operational limits.
  • Global vs local optimization: For cases 6 and 7, the no-optimization scheme violates nominal voltage limits, while local control fails to fully correct the situation and worsens case 7’s voltage.The voltages are evaluated using the exact DistFlow equations with reactive powers from the global, local, and no-optimization schemes.
  • Global vs local optimization: Global optimization corrects the voltage to respect the constraints, which naturally leads to higher losses.This tradeoff is observed for the heavily loaded or over-generated cases where voltage limits are violated.

C. Performance of distributed optimization algorithms

ADMM-noV and ADMM achieve the same minimum real-power losses as dual ascent when voltage constraints are already satisfied, but converge much faster. ADMM’s performance depends on voltage-constraint violations, initialization, and parameter choices.

  • Convergence performance: For cases 1–5, dual ascent, ADMM-noV, and ADMM yield identical minimum real-power losses when voltage constraints are satisfied without explicit optimization constraints.Their performance differs in convergence rather than the resulting optimum.
  • Parameter selection: Empirical experiments identify α = 0.05/V_0^2 and ρ = 1/V_0^2 as choices giving good or optimal convergence, while values that are too large or small can slow or destabilize convergence.The convergence rate is sensitive to ρ, and α choices caused numerical instability or slower convergence.
  • Convergence performance: ADMM-noV and ADMM converge to optimal q(g) within tens of iterations for cases 1–5, substantially outperforming dual ascent when the initial guess is not near optimal.Dual ascent requires the order of 104 iterations in cases 1, 2, and 5, but only a few tens in cases 3 and 4.
  • Convergence performance: The ADMM algorithm requires order of 103 iterations in cases 6 and 7 because violated voltage constraints must propagate throughout the feeder.Bare ADMM-noV converged in tens of iterations for both tested initializations.
  • Initialization: ADMM convergence can be improved by running ADMM-noV first and then initializing ADMM voltages from measured node values.This addresses inaccurate voltage guesses that can otherwise cause slow convergence.

D. Validity of LinDistFlow approximation

The LinDistFlow approximation closely matches exact DistFlow results, with relative losses agreeing within a few percent and voltage differences of order 10^-3. However, in cases 6 and 7, small voltage estimation errors affect constraint satisfaction and relative-loss comparisons.

  • Comparison with nonlinear optimization: The decentralized LinDistFlow solution produces relative losses close to those from centralized global DistFlow optimization.The comparison uses a centralized CVX solution of the nonlinear DistFlow problem.
  • Accuracy of the approximation: Relative losses from DistFlow agree with LinDistFlow predictions within a few percent across the considered cases.DistFlow relative losses are slightly lower than LinDistFlow losses in most cases.
  • Accuracy of the approximation: The maximum difference between normalized voltages computed by LinDistFlow and DistFlow is of order 10^-3.This precision is considered satisfactory for voltage constraints between 0.95 and 1.05.
  • Voltage-regulation cases: For cases 6 and 7, LinDistFlow slightly overestimates the exact voltage magnitude, making actual voltages lower by O(10^-3).These cases require global reactive-power control to maintain voltage regulation within the allowed bounds.
  • Voltage-regulation cases: In case 6, LinDistFlow optimization reaches a minimum voltage of 0.947, below the lower bound 0.95.In case 7, its maximum normalized voltage is 1.045, whereas global DistFlow optimization attains the upper bound 1.05.

V. DISCUSSION

The work presents three exact distributed algorithms for globally optimal reactive power flows, using separable costs and constraints with local measurements, computations, and nearest-neighbor communication. The ADMM approach supports several extensions, including locally adapted objectives, asynchronous communication, more complex topologies, nonlinearities, and binary inverter selection.

  • Contributions: Three exact distributed algorithms achieve globally optimal reactive power flows using separable costs and constraints, local measurements, local computations, and nearest-neighbor communication.The algorithms were tested on multiple circuits with high PV penetration.
  • Generalizations: The ADMM algorithm can generalize to different objective functions, including node-specific voltage penalties such as (V − V0)^2.These objective rules can change locally and on-the-go without other nodes being aware, potentially improving voltage quality at that node.
  • Generalizations: Node communications need not be synchronized and may tolerate some delay, but robustness to delays, errors, corruptions, and targeted attacks remains important to analyze.The distributed nature is expected to provide greater security against localized, non-correlated attacks than centralized algorithms.
  • Generalizations: The algorithm extends naturally to more complicated tree-like topologies and some separable nonlinearities in power flows.These nonlinearities are of the type previously discussed for the dual ascent algorithm.
  • Generalizations: The framework also supports binary PV-inverter selection.This extension is discussed in prior work cited by the paper.
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