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Decentralized Stochastic Optimal Power Flow in Radial Networks with Distributed Generation

Mohammadhafez Bazrafshan, Nikolaos Gatsis

arXiv:1601.04023v2eess.SYcs.SY

TL;DR

Uncertain solar availability can create inadequate or excess generation, threatening user satisfaction and voltage regulation if unaccounted for. The paper formulates a scenario-based stochastic power-management program and solves it with decentralized ADMM; reported case results relate solar conditions to voltage drops and show relaxation exactness approaching zero across tested stepsizes.

  • Problem

    Uncertain solar availability can cause inadequate or excess generation, threatening user satisfaction, voltage regulation, and equipment when unaccounted for.

    Source: §I · p. 1

  • Method

    The paper formulates a scenario-based convex stochastic program with an SOCP power-flow relaxation and solves it by ADMM using closed-form node- and scenario-specific updates.

    Source: §I · p. 1, §VII · p. 11

  • Results

    Cloudy-day generation produces a higher voltage drop than improved solar conditions, and SOCP relaxation exactness eventually approaches zero for all tested ρ values.

    Source: C. Case study for different day types · p. 8, F. Effect of stepsize ρ in convergence… · p. 10

  • Takeaways & Limitations

    The solver targets scalability through node- and scenario-specific closed-form updates and communication only between neighboring nodes.

    Source: B. Contributions and outline · p. 2

Abstract

from arXiv · show

This paper develops a power management scheme that jointly optimizes the real power consumption of programmable loads and reactive power outputs of photovoltaic (PV) inverters in distribution networks. The premise is to determine the optimal demand response schedule that accounts for the stochastic availability of solar power, as well as to control the reactive power generation or consumption of PV inverters adaptively to the real power injections of all PV units. These uncertain real power injections by PV units are modeled as random variables taking values from a finite number of possible scenarios. Through the use of second order cone relaxation of the power flow equations, a convex stochastic program is formulated. The objectives are to minimize the negative user utility, cost of power provision, and thermal losses, while constraining voltages to remain within specified levels. To find the global optimum point, a decentralized algorithm is developed via the alternating direction method of multipliers that results in closed-form updates per node and per scenario, rendering it suitable to implement in distribution networks with large number of scenarios. Numerical tests and comparisons with an alternative deterministic approach are provided for typical residential distribution networks that confirm the efficiency of the algorithm.

I. INTRODUCTION

Uncertain solar generation can undermine user satisfaction and voltage regulation, motivating joint control of programmable loads and PV inverter reactive power. The paper develops a stochastic, decentralized framework for this control and reports benefits in user satisfaction, thermal losses, and voltage regulation.

I. INTRODUCTION

  • Solar uncertainty can cause insufficient or excess generation, with potential reductions in user satisfaction, poor voltage regulation, and equipment failure.Programmable loads can reduce peak demand during inadequate generation, while inverter reactive power can be used to improve voltage regulation.Source: §I · p. 1

A. Prior Art

  • Prior deterministic reactive-power methods assume known user consumption and PV generation; some local policies are computationally attractive but lack optimality guarantees.The cited stochastic approaches differ in control scope or solution architecture: [32] omits controllable loads and uses centralized optimization, while uses static rather than adaptive PV reactive-power control.Source: A. Prior Art · p. 1, B. Contributions and outline · p. 2

B. Contributions and outline

  • The framework jointly optimizes controllable-load real-power consumption and PV-inverter reactive power while modeling PV injections across a finite set of scenarios.The stochastic formulation distinguishes it from earlier approaches that assume known, deterministic PV injections.Source: B. Contributions and outline · p. 2
  • The ADMM solver decomposes updates by node and scenario, uses closed-form updates, and requires communication only between neighboring nodes.The paper extends its decentralization approach to stochastic programming and incorporates SOCP power-flow relaxation for tree networks.Source: B. Contributions and outline · p. 2
  • Numerical tests with realistic PV models and few representative scenarios report user-satisfaction and thermal-loss benefits, improved voltage regulation, and reduced thermal losses versus.The comparison also includes networks with shunt capacitors.Source: B. Contributions and outline · p. 2

II. NETWORK MODEL AND DECISION VARIABLES

  • The network model represents a radial distribution system as a tree, with node 0 as the substation and the remaining nodes as users.Each non-root node has a unique ancestor; terminal nodes have no children.Source: §II · p. 3

A. User load model

  • Users have non-elastic real and reactive loads, while programmable demand-response consumption allows real power use to vary within a range.Elastic reactive consumption is linearly related to elastic real consumption, with power factor specified in the supplied model passage.Source: A. User load model · p. 3

B. PV generation model

  • PV real-power injections are modeled as random variables over finitely many scenarios with probabilities, and PV inverters can generate or consume reactive power.The supplied model also identifies inverter apparent-power capacity as a constraint on reactive output.Source: B. PV generation model · p. 3

C. Power flow equations

The formulation models scenario-dependent power flows and voltage limits with an SOCP relaxation, then combines demand response, power-provision cost, and thermal losses in a convex stochastic program. Load consumption is chosen before uncertainty is known, while inverter reactive power and network states adapt by scenario.

C. Power flow equations

  • Scenario-dependent real and reactive PV injections make the network power flows scenario-dependent, and the SOCP equations represent those flows.The formulation also constrains voltage levels and line-current magnitudes for each scenario.Source: C. Power flow equations · pp. 3–4

III. OPTIMAL POWER FLOW FORMULATION

  • The objective combines user utility, expected power-provision cost, and expected line losses, with a selectable weight on loss minimization.The loss weight KLoss reflects the system operator’s relative priority for minimizing losses.Source: §III · p. 4
  • The power-provision cost can be convex and piecewise linear to distinguish buying from selling, with auxiliary variables enabling a smooth objective.The paper gives an equivalent formulation for the piecewise-linear example.Source: §III · p. 4
  • The two-stage stochastic program jointly optimizes elastic load consumption and scenario-adaptive inverter reactive power and network flows.Load decisions are made independently of uncertainty; inverter reactive power, voltages, and currents adapt to each scenario.Source: §III · p. 4
  • The SOCP relaxation is exact under sufficient conditions for the stochastic program, rather than unconditionally for every instance.The paper derives sufficient conditions for the inequality in (7) to hold as equality.Source: §III · p. 4

IV. SOLUTION ALGORITHM

The solution algorithm reformulates the stochastic program for ADMM by introducing auxiliary variables that separate node-level decisions. This construction targets decomposition across nodes and scenarios.

IV. SOLUTION ALGORITHM

  • An equivalent formulation introduces auxiliary variables to make the stochastic program amenable to ADMM decomposition by node and scenario.The paper groups the resulting variables into x and z vectors for the ADMM algorithm.Source: §IV · p. 4, Table II · p. 5

A. Equivalent problem

  • Power-flow equations are the remaining obstacle to fully separating nodes because they couple neighboring-node variables.The reformulation introduces copies of variables to decouple each node from its children and ancestors.Source: A. Equivalent problem · p. 5
  • The equivalent problem separates coupling constraints from individual equality constraints, enabling the ADMM formulation.Source: A. Equivalent problem · p. 5

B. Review of ADMM

The paper casts the reformulated problem in ADMM form and derives per-node updates. Auxiliary variables separate inequality handling from the node-level quadratic update, yielding closed-form steps.

B. Review of ADMM

  • ADMM alternates minimizing the augmented Lagrangian over x and z, then updates multipliers for the linear equality constraints.The general form uses convex functions and sets that represent the individual equalities and inequalities.Source: B. Review of ADMM · p. 5, C. Updates · p. 6
  • Tilde variables let the z-update handle individual inequality constraints separately, while the x-update is an equality-constrained quadratic program.This variable separation is essential to obtaining closed-form updates.Source: B. Review of ADMM · p. 5

C. Updates

  • The x-update minimizes the augmented Lagrangian separately for each node and scenario, subject to that node’s equality constraints.The paper gives a distinct expression for the root node.Source: C. Updates · p. 6
  • The non-scenario-dependent load variable has a scalar box-constrained convex update, which is closed form for a quadratic utility function.The remaining listed z-variables are scenario-dependent and have closed-form updates.Source: 2) zi-update · p. 6

V. DECENTRALIZED IMPLEMENTATION

Each node maintains its own variables and multipliers, exchanging information with neighboring nodes during ADMM iterations. The algorithm uses closed-form updates and aims to preserve user privacy while achieving global optimality.

V. DECENTRALIZED IMPLEMENTATION

  • Each node maintains and updates its variables and multipliers, with communication restricted to neighboring nodes.Nodes exchange required variables before the x- and z-updates and update multipliers after the z-step.Source: §V · p. 6
  • The algorithm achieves global optimality without transmitting user-consumption bounds, power factor, or user utility to a central agent.Source: §V · p. 7
  • The algorithm uses closed-form updates per node and scenario, and per-node computational effort does not increase with network size.Source: §V · p. 7

VI. NUMERICAL TESTS

The numerical tests use a radial network with specified voltage limits and programmable-load bounds. Solar uncertainty is represented by a small reduced set of scenarios, and the SOCP relaxation is numerically exact in the tested cases.

A. Network setup

  • The test network uses constant line impedance per distance, 0.2 km node spacing, and a 7.2 kV fixed substation voltage.Nodal voltages may vary within 5% of nominal voltage; non-elastic demand is 0.1 MW and elastic demand is bounded from 0 to 0.05 MW per user.Source: A. Network setup · p. 7

B. Methodology for generating scenarios

  • Seven representative scenarios are selected from 1,000 equiprobable draws using fast forward reduction.The reduction aggregates excluded-scenario probabilities onto the closest retained scenario; M = 7 was chosen so the smallest retained probability is at least 0.01.Source: B. Methodology for generating scenarios · pp. 7–8
  • The SOCP inequality holds as equality for every scenario in all tested problems.The authors report this as a numerical verification for the problems considered.Source: B. Methodology for generating scenarios · p. 8

C. Case study for different day types

Across day types, higher PV availability accompanies greater scheduled elastic demand and lower voltage drops. In a separate comparison, the stochastic approach keeps voltage deviations below the required threshold in the tested scenarios, unlike the local policy in some cases.

C. Case study for different day types

  • As conditions shift from cloudy to sunny, satisfied elastic demand increases and expected thermal losses decrease.On cloudy days the schedule meets non-elastic demand; with more PV, larger portions of elastic demand are guaranteed.Source: C. Case study for different day types · p. 8, Figure 4 · p. 8
  • Cloudy-day voltage drops are higher than those under improved solar conditions across the worst-case scenario profiles.The voltage rise at node 30 corresponds to the larger terminal-node generation, while the drop at node 20 corresponds to a network branch.Source: C. Case study for different day types · p. 8, Figure 5 · p. 8

D. Comparison with distributed local control of [16]

E. Case study on a feeder with shunt capacitors

The feeder tests show that reactive-power compensation can prevent voltage drops that otherwise make scenarios infeasible. Additional experiments examine ADMM stepsizes and scenario-count scalability, and the conclusion describes the decentralized stochastic formulation and solver.

COMPENSATION OF PV INVERTERS IS NOT ALLOWED

F. Effect of stepsize ρ in convergence of ADMM

  • ρ = 2 and adaptive ρ have similar residual performance, while ρ = 20 and ρ = 100 perform poorly on dual-residual reduction.For ρ = 100, an accurate objective value is not found within 3,000 iterations; exactness eventually approaches zero for every tested stepsize, but does so slowly after 1,000 iterations for ρ = 2 and adaptive ρ.Source: F. Effect of stepsize ρ in convergence… · p. 10, Figure 8 · p. 10, Figure 9 · p. 10

G. Effect of number of scenarios on convergence

VII. CONCLUSION

  • The framework jointly optimizes programmable-load real consumption and PV-inverter reactive power under finite-scenario PV uncertainty.Its convex stochastic objective includes negative utility, expected power-provision cost, and expected thermal losses; ADMM updates are closed form per node and scenario.Source: §VII · p. 11

APPENDIX A PROOF THAT AT MOST ONE OF THE TWO VARIABLES P m

The proof establishes that the two variables cannot both be positive at the solution by deriving a feasible objective-improving perturbation under that assumption.

APPENDIX A PROOF THAT AT MOST ONE OF THE TWO VARIABLES P m

  • Assume both variables in the proposed solution are positive.The proof compares that solution with one in which the objective term is replaced by a linear expression.Source: Appendix A · p. 11
  • Reducing each positive variable by a sufficiently small ε preserves feasibility.Source: Appendix A · p. 11
  • Because a > b, the perturbation strictly lowers the objective, contradicting optimality.Source: Appendix A · p. 11

APPENDIX B ON THE EXACTNESS OF THE SOCP RELAXATION

The appendix gives sufficient conditions for exactness of the SOCP relaxation in a modified problem and sketches a contradiction proof. It also gives a more stringent condition independent of the scenario and reports that it holds for the tested network.

APPENDIX B ON THE EXACTNESS OF THE SOCP RELAXATION

  • The modifications assume strictly increasing root power cost, remove current-magnitude upper bounds, model shunt capacitors as fixed injections, and set KLoss = 0.The listed modifications also include an additional voltage constraint, but its full specification is not given here.Source: Appendix B · pp. 11–12
  • The modified SOCP is exact when sufficient condition (37) holds for every scenario; the condition can be checked a priori.The result applies to a modified problem close to problem (12).Source: Appendix B · pp. 12–13
  • If the relaxation constraint is slack at a node in a scenario, the proof constructs a feasible solution with lower objective, contradicting optimality.The contradiction uses strict increase of C(P0).Source: Appendix B · pp. 12–13
  • A more stringent sufficient condition independent of the scenario accounts for the lowest consumptions and was numerically verified for the tested network.It is formed using the maximum scenario values of the relevant quantities.Source: Appendix B · p. 13

APPENDIX C CLOSED-FORM UPDATES FOR ˜P m

This appendix derives closed-form solutions for the z-update optimization, extending a prior approach to include constraint (61e). It uses KKT conditions and case enumeration, with a bound check determining whether a candidate solution is optimal.

APPENDIX C CLOSED-FORM UPDATES FOR ˜P m

  • The z-update for the variables is reduced to solving an optimization problem, with cases covering an interior, lower-bound, or upper-bound value of z3 when µ > 0.Source: Appendix C · p. 13, B. Case 2: γ > 0 · pp. 13–14
  • The closed-form method extends [22, Appendix I] to problem (61) with constraint (61e), using KKT conditions and enumerating cases for γ.Problem (61) without constraint (61e) already has a closed-form solution in [22, Appendix I].Source: Appendix C · p. 13
  • When a case yields a unique closed-form solution, it is optimal for (61) if z4 ≤ zmax4; otherwise, the next case is examined.Source: A. Case 1: γ = 0 · p. 13

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