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Decentralized Charging Control of Electric Vehicles in Residential Distribution Networks

Mingxi Liu, Phillippe K. Phanivong, Yang Shi, Duncan S. Callaway

arXiv:1710.05533v1math.OC

TL;DR

Uncontrolled or network-blind EV charging can harm distribution networks, while decentralized charging must also meet individual requirements and network constraints. The paper proposes an SPDS-based control framework; in simulation, the load profile reached an acceptable form by iteration 15, with a 0.05% normed error relative to iteration 25.

  • Problem

    EV charging can elevate demand peaks, cause voltage drops and transformer overloading, and harm distribution networks when network physics and constraints are neglected.

    Source: §I · p. 1

  • Method

    The paper proposes decentralized EV charging control using SPDS to achieve valley-filling while meeting heterogeneous charging requirements and network constraints.

    Source: §I · p. 2

  • Results

    At iteration 15, the total load had converged to an acceptable profile, with a 0.05% normed error between iterations 15 and 25.

    Source: B. Decentralized control · p. 8

  • Takeaways & Limitations

    The framework is designed to meet heterogeneous charging needs while regulating nodal voltages within the service range.

    Source: §I · p. 2

    Transformer-overloading constraints are not included in the approach and are identified as future work.

    Source: §V · p. 11

Abstract

from arXiv · show

Electric vehicle (EV) charging can negatively impact electric distribution networks by exceeding equipment thermal ratings and causing voltages to drop below standard ranges. In this paper, we develop a decentralized EV charging control scheme to achieve "valley-filling" (i.e., flattening demand profile during overnight charging), meanwhile meeting heterogeneous individual charging requirements and satisfying distribution network constraints. The formulated problem is an optimization problem with a non-separable objective function and strongly coupled inequality constraints. We propose a novel shrunken primal-dual subgradient (SPDS) algorithm to support the decentralized control scheme, derive conditions guaranteeing its convergence, and verify its efficacy and convergence with a representative distribution network model.

I. INTRODUCTION

The paper develops decentralized EV charging control for valley-filling while meeting heterogeneous charging needs and distribution-network voltage constraints. Its SPDS-based approach is designed for local control and assumes individual charging windows cover a common valley-filling window.

I. INTRODUCTION

  • The problem's coupled, non-separable objective and strongly coupled voltage constraints are not handled by the general decentralized algorithms discussed.The proposed SPDS approach is presented as avoiding a regularized Lagrangian while reducing iterations and communication loads.Source: §I · p. 2
  • The network-aware framework targets valley-filling while meeting heterogeneous charging requirements and keeping nodal voltages within the service range.The paper states that the framework can be extended to other grid services and additional network constraints.Source: §I · p. 2
  • The control scheme requires only a single-layer communication network, adds no bus computing hardware, and keeps customers' SOC information off the network.Chargers solve their own control problems, allowing device-level charging constraints to be met with little information sent to a central operator.Source: §I · p. 2
  • SPDS can run at charging points or support parallel computing by a central operator, using simple Euclidean projections.The approach does not require a regularized Lagrangian.Source: §I · p. 2

A. EV charging model

  • Each EV's charging state tracks energy still needed to reach its individual requirement, and its control signal is a charging rate between 0 and 1.The state evolves according to a discrete-time charging model, with local constraints intended to ensure drivers' requirements are met.Source: A. EV charging model · pp. 2–3
  • The EV model assumes heterogeneous initial SOCs, charging efficiencies, battery capacities, and maximum powers, with each designated window covering the common charging window.Under this assumption, the algorithm delivers each vehicle's required energy within the common deadline, without charging outside that window.Source: A. EV charging model · p. 3

B. Distribution network model

The network model uses a radial feeder and linearized DistFlow relations to connect EV charging power with nodal voltages. It represents voltage effects relative to baseline load and excludes several other network resources and controls.

B. Distribution network model

  • A linearized DistFlow model relates bus power, including EV charging power, to nodal voltages in the radial distribution network.The model is fitted into the valley-filling problem.Source: B. Distribution network model · pp. 3–4
  • The simulation network is a modified single-phase IEEE-13 Node Test Feeder, with the Node 4–5 transformer and Node 6–7 switch removed and no capacitor banks.The feeder is used throughout the paper.Source: B. Distribution network model · p. 3, Figure 1 · p. 3
  • The modeled feeder excludes distributed energy resources, reactive power supplies, and voltage regulation, and EVs are assumed to consume only real power.Only EVs are treated as adjustable load in the model.Source: B. Distribution network model · pp. 3–4
  • EV charging enters the voltage model as a linear term relative to the baseline-load voltage drop.The paper defines a matrix mapping charging rates to voltage changes and rewrites the model using that mapping.Source: B. Distribution network model · p. 4

III. CONTROLLER DESIGN

The controller targets valley-filling while meeting EV charging requirements and keeping nodal voltages within service range, using a centralized optimization that minimizes total-load variance.

A. Centralized controller

  • The baseline load Pb aggregates distributed uncontrollable loads across nodes over the control period.It is used as the baseline load in the centralized formulation.Source: A. Centralized controller · p. 5
  • Valley-filling is formulated as minimizing the variance of aggregated feeder load to flatten the total-load profile.The controller seeks this objective while satisfying individual EV and network constraints.Source: §III · p. 4, A. Centralized controller · p. 5
  • Each EV's charging rate must satisfy local constraints that allow it to reach its desired state of charge.The feasible set permits charging power from zero to the EV's maximum rate.Source: A. Centralized controller · p. 5
  • Network constraints keep nodal voltage magnitudes within the service range, with only a lower voltage bound considered.The model excludes distributed energy resources, reactive power supplies, and voltage regulation.Source: A. Centralized controller · p. 5
  • The lower-bound-only formulation assumes baseline voltages are within range and EV charging does not raise voltages above baseline.The non-adjustable loads are assumed to be well estimated; designs with estimation errors are left for future work.Source: A. Centralized controller · p. 5

B. Decentralized control scheme

Decentralizing valley-filling is challenging because EV decisions are coupled in both the objective and network constraints. The proposed SPDS approach avoids Lagrangian regularization while enabling parallel charger computation and limited information exchange.

B. Decentralized control scheme

  • The valley-filling objective couples individual EV decisions, while linearly coupled inequality constraints jointly restrict charging through nodal voltages.These couplings make distributing computation to individual chargers challenging.Source: B. Decentralized control scheme · p. 5
  • The decentralized optimality formulation assumes Slater's condition, under which strong duality holds.The paper uses variational inequalities and parameterized natural maps to express the primal and dual optimality conditions.Source: B. Decentralized control scheme · p. 6
  • SPDS circumvents Lagrangian regularization while solving the coupled charging problem in a decentralized manner.The cited RPDS approach uses regularization for convergence, but its terms introduce errors relative to the original optimum.Source: B. Decentralized control scheme · p. 6
  • SPDS shrinks primal updates, moves them along the Lagrangian descent direction, then expands and projects them again to support convergence without regularization.Its two-tier projection applies shrinkage and expansion to primal and dual variables.Source: B. Decentralized control scheme · p. 6
  • Chargers can compute in parallel without communicating with one another, transmitting their control sequences to the operator for dual-variable updates.The operator computes and broadcasts the Lagrangian gradient and dual variable.Source: B. Decentralized control scheme · p. 6
  • The communication arrangement protects drivers' private information, especially state of charge, by limiting what chargers transmit to the operator.Chargers send their calculated control sequences for the operator's dual update.Source: B. Decentralized control scheme · p. 7

C. SPDS Convergency analysis

The convergence analysis gives sufficient conditions for SPDS convergence with constant step sizes. It also explains how shrinking parameters affect convergence conditions and speed.

C. SPDS Convergency analysis

  • The analysis considers constant primal and dual step sizes; adaptive step sizes are deferred to future work.The updates use α and β rather than iteration-varying step sizes.Source: C. SPDS Convergency analysis · p. 7
  • When the step-size tuple satisfies condition (41), the SPDS sequence converges as the iteration count tends to infinity.The condition involves the strongly monotone constant of the mapping Φ(ζ).Source: C. SPDS Convergency analysis · p. 7
  • Shrinking parameters must lie in (0, 1) and should be close to, but not too close to, 1 to balance convergence conditions and speed.The paper empirically tunes the step sizes and shrinking parameters and leaves analytical selection of the latter for future work.Source: C. SPDS Convergency analysis · p. 7

IV. SIMULATION RESULTS

The simulations use a residential distribution network with heterogeneous EV and charging parameters. The supplied passages describe the setup but do not report simulation outcomes.

IV. SIMULATION RESULTS

  • Each eligible node connects 70 houses with level-2 chargers rated up to 6.6 kW, and battery capacities are distributed uniformly from 18 to 20 kWh.Initial state of charge ranges uniformly from 0.3 to 0.5, and designated state of charge from 0.7 to 0.9.Source: §IV · p. 7
  • The simulation uses empirically tuned parameters, at most 25 iterations, and a tolerance of 1×10^-4; parameters should be retuned for a different network model.The reported shrinking parameters are τU = τλ = 0.974.Source: §IV · p. 7
  • The valley-filling period runs from 19:00 to 8:00, with the voltage lower bound set to 0.954 to accommodate line losses omitted from LinDistFlow.Baseline load data comes from Southern California Edison and is scaled for the simulations.Source: §IV · p. 7

A. Centralized control

Centralized control starts EV charging at 22:00 to fill the overnight valley, with total load smoothing to a flat profile by 22:30. Network and local charging constraints shape that transition and the individual EV charging profiles.

A. Centralized control

  • The aggregated total load starts filling the valley at 22:00 and decays smoothly to a flat profile by 22:30 under centralized control.The transition is not strictly flat, but is described as the optimal performance given nodal voltage and local charging constraints.Source: A. Centralized control · p. 7, Figure 2 · p. 8
  • Node 11 voltage stays in line with 0.954 p.u. during 23:15–5:15 in the controlled case.Voltages are computed using LinDistFlow equations.Source: A. Centralized control · p. 8, Figure 3 · p. 8
  • Nodes 8 and 10 remain slightly above 0.954 p.u. during 1:15–4:00 under control.Source: A. Centralized control · p. 8, Figure 3 · p. 8
  • Without appropriate control, nodal voltages fall far below the lower bound; after 23:15, total load cannot increase without violating voltage limits.The optimal load profile therefore transitions from high to flat as early and smoothly as possible.Source: A. Centralized control · p. 8, Figure 3 · p. 8
  • EV charging profiles differ by location and downstream impedance: feeder-proximate EVs can charge more at peak need, while farthest Group 4 EVs stay low.Groups 2 and 3 have intermediate, parabola-like or trapezoid-like profiles with lower peaks; the profiles are shown by EV index in Fig. 4.Source: A. Centralized control · p. 8, Figure 4 · p. 8
  • All EVs reach their designated SOCs under centralized control.SOC evolution figures are omitted.Source: A. Centralized control · p. 8

B. Decentralized control

The decentralized controller approaches a valley-filling load profile while meeting voltage and charging requirements, and its final EV profiles closely match centralized control. Simulations also compare SPDS with RPDS and assess voltage-model discrepancies and computational time.

B. Decentralized control

  • The total load reaches an acceptable profile by iteration 15; its normed error from iteration 25 is 0.05%.The figure compares baseline load with total loads across 25 decentralized-control iterations.Source: B. Decentralized control · p. 8, Figure 5 · p. 9
  • Voltage magnitudes initially fall below 0.954 p.u., then converge through a damped-oscillation-like process to profiles satisfying the constraints.The figure distinguishes baseline voltages from those at iterations 1 and 25, with each line representing one node.Source: B. Decentralized control · pp. 8–9, Figure 6 · p. 9
  • At iteration 25, EV charging profiles essentially match centralized control, with errors up to 0.05 mainly among Group 4 EVs that determine voltage constraints.All EVs reach their designated SOCs under decentralized control.Source: B. Decentralized control · pp. 9–10, Figure 7 · p. 9, Figure 8 · p. 9
  • SPDS converges faster than RPDS and has no objective gap once converged.The comparison uses the same primal and dual step sizes; RPDS uses a dual-regularization coefficient of 0.1.Source: B. Decentralized control · p. 9, Figure 9 · p. 9
  • At the final stage, positive dual variables correspond to Node 11, whose voltage constraint is the only binding one.Node 11 has the largest downstream impedance, which makes its voltage more sensitive to charging power.Source: B. Decentralized control · p. 10, Figure 10 · p. 10

C. Network model discrepancies

  • During charging peaks, DistFlow voltages are about 0.004 p.u. lower than LinDistFlow voltages; the lowest DistFlow magnitude is about 0.95 p.u., satisfying ANSI C84.1.Before charging begins, the discrepancy is about 0.001 p.u.; the authors recommend keeping voltage near the ideal value or reserving a buffer for line losses.Source: C. Network model discrepancies · p. 10, Figure 11 · p. 10
  • The decentralized control sequences take 6.5 seconds for 25 iterations, versus 334 seconds for the centralized algorithm.The cited decentralized projection complexities are independent of EV population size, although the projection onto an ℓ2-norm ball may become an issue for large distribution networks.Source: C. Network model discrepancies · p. 10

V. CONCLUDING REMARKS

  • The framework uses SPDS for decentralized valley-filling with local and strongly coupled network constraints, and simulations verify the algorithm and control scheme.Chargers can update in parallel without a communication network among them; future work includes adding transformer-overloading constraints and addressing imprecise network information.Source: §V · pp. 10–11

APPENDIX PROOF OF THEOREM 2

The proof establishes regularity properties of the mappings in the analysis and uses them to choose step-size parameters that ensure convergence of the iterate sequence.

APPENDIX PROOF OF THEOREM 2

  • The proof bounds terms in the convergence inequality using convexity, Cauchy–Schwarz, and the vector-valued mean-value theorem.It applies these bounds to establish the condition needed for the final convergence argument.Source: APPENDIX PROOF OF THEOREM 2 · pp. 12, 14
  • The proof establishes strong monotonicity of the mapping Φ and Lipschitz continuity of ∇_U G and d(U).The stated constants are given as the strong-monotonicity constant, L∇G, and Ld, respectively.Source: APPENDIX PROOF OF THEOREM 2 · pp. 12–13
  • A step-size pair (α, β) can be chosen so that 0 < ρ < 1, which makes the sequence {ζ(ℓ)} converge.Source: APPENDIX PROOF OF THEOREM 2 · p. 14
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