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Aggregate Power Flexibility in Unbalanced Distribution Systems

Xin Chen, Emiliano Dall'Anese, Changhong Zhao, Na Li

arXiv:1812.05990v2eess.SYmath.OC

TL;DR

Massive DER deployment creates a need to aggregate distribution-level flexibility for transmission support while respecting unbalanced-network constraints. The paper develops an inner-box aggregation method with guaranteed feasible disaggregation and a distributed MPC implementation, validated on a real feeder with 126 multi-phase nodes.

  • Problem

    Distribution-level aggregation must harness heterogeneous DER flexibility despite computational burdens and the need to represent network constraints and unbalanced multi-phase systems.

  • Method

    The paper uses an inner-box approximation with two convex aggregation models and a distributed MPC framework incorporating network constraints and multi-phase unbalanced modeling.

  • Results

    The method defines an approximate net-substation-injection region, guarantees feasible disaggregation for trajectories within it, and is validated on a real feeder with 126 multi-phase nodes.

  • Takeaways & Limitations

    Aggregate distribution flexibility can be represented for transmission-distribution interaction while preserving privacy and supporting scalable implementation.

Abstract

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With a large-scale integration of distributed energy resources (DERs), distribution systems are expected to be capable of providing capacity support for the transmission grid. To effectively harness the collective flexibility from massive DER devices, this paper studies distribution-level power aggregation strategies for transmission-distribution interaction. In particular, this paper proposes a method to model and quantify the aggregate power flexibility, i.e., the net power injection achievable at the substation, in unbalanced distribution systems over time. Incorporating the network constraints and multi-phase unbalanced modeling, the proposed method obtains an effective approximate feasible region of the net power injection. For any aggregate power trajectory within this region, it is proved that there exists a feasible disaggregation solution. In addition, a distributed model predictive control (MPC) framework is developed for the practical implementation of the transmission-distribution interaction. At last, we demonstrate the performances of the proposed method via numerical tests on a real-world distribution feeder with 126 multi-phase nodes.

NOMENCLATURE

The nomenclature defines limits, capacities, states, injections, temperatures, and mathematical operators used throughout the distribution-system model.

  • Voltage and current symbols specify upper and lower limits for three-phase nodal voltages and distribution-line currents.
  • PV, energy-storage, and controllable-load notation records phase-specific active-power limits and apparent-power capacities.
  • Energy-storage symbols include apparent-power capacity and upper and lower state-of-charge limits.
  • HVAC notation covers indoor and outside temperatures, comfortable-temperature bounds, and the discretized time-slot length.
  • Power-injection vectors distinguish wye- and delta-connected devices, while p0 denotes three-phase net substation injection.
  • The notation also defines PV, storage, controlled-load, HVAC, and state-of-charge variables as phase- or device-level quantities.

I. INTRODUCTION

The introduction frames distribution systems as increasingly flexible because of DERs, but identifies computational, network-modeling, and unbalanced-system challenges. It presents an inner-box aggregation method with disaggregation guarantees and a distributed MPC framework for scalable transmission-distribution interaction.

  • I. INTRODUCTION: DER coordination can convert many small device capabilities into distribution-level flexibility that supports transmission-system operation.
  • I. INTRODUCTION: Power aggregation projects high-dimensional DER operating constraints onto the achievable net power injection at the substation.
  • I. INTRODUCTION: Existing approaches face computational burdens, while many studies omit network constraints or support only single DER types in intrinsically unbalanced networks.
  • I. INTRODUCTION: The proposed method uses an inner-box feasible region and upper and lower trajectories to quantify aggregate flexibility from heterogeneous DERs over time.
  • I. INTRODUCTION: The aggregation formulation incorporates network constraints and multi-phase unbalanced modeling, with a guaranteed feasible disaggregation solution.
  • I. INTRODUCTION: A distributed MPC framework is developed to protect DER privacy and enable scalable practical implementation of distribution-transmission interaction.
  • A. Network Model: The network model represents wye- and delta-connected devices and uses fixed-point linearization to obtain a linear multi-phase power-flow model.
  • A. Network Model: The resulting network constraints include voltage-limit and line-thermal constraints and can accommodate mixed-phase connections and meshed or radial networks.

B. Distributed Energy Resource Model

The paper models multiple DER types over a discrete-time horizon, including dispatchable PV, energy storage, directly controllable loads, and HVAC systems. These models use simplifying assumptions to balance precision with computational efficiency.

  • The DER model covers dispatchable PV units, energy storage devices, directly controllable loads, and HVAC systems over a discrete-time horizon.The time horizon is defined as T = {1, 2, · · ·, T}.
  • Energy storage output is defined as total active power across phases, with positive values for discharging and negative values for charging.The storage model also includes an efficiency factor and initial state of charge.
  • Directly controllable loads are represented at buses in Ncl with active-power limits and fixed power factors.The supplied formulation identifies Ncl as the set of buses connected with directly controllable loads and assumes constant ηd.
  • HVAC models include aggregate active load, fixed power factors, and indoor-temperature dynamics governed by building and environmental parameters.The parameter βi captures heat-capacity behavior, with its sign indicating heater or cooler operation.
  • The HVAC operation assumes that each device keeps the same heating or cooling mode throughout the considered period.This fixed-sign assumption on βi facilitates the subsequent proof of disaggregation feasibility.
  • The DER models are approximate because the paper trades model precision against computational efficiency.For storage, the simplified formulation assumes 100% charging and discharging energy conversion efficiency, while more realistic separate efficiencies would make the model nonconvex.

C. Incorporated Model

The paper models aggregate substation power flexibility using an inner-box approximation that incorporates network, DER, and multi-phase unbalanced constraints. The resulting region bounds feasible power trajectories over time and supports phase-specific flexibility quantification.

  • The method rewrites network and DER models into a compact formulation containing substation injection, time-coupled, and time-decoupled constraints.The formulation includes storage state-of-charge and HVAC temperature limits, operational constraints, and power-flow equalities represented within convex inequalities.
  • The aggregate feasible region is approximated by restricting substation net power to an interval at each time, forming an inner box over time.The actual power trajectory lies between the corresponding upper and lower trajectories.
  • The aggregate flexibility Eaf measures the potential energy flexibility level of the distribution system.Its unit is energy, reflecting the time-integrated flexibility represented by the feasible region.
  • The optimization-based method accommodates various DER models while incorporating distribution-network constraints.This distinguishes it from approaches based on Minkowski sums of polytopic DER feasible sets.
  • The inner approximation is safe: every trajectory inside the approximate region is achievable by coordinating DER devices while respecting operating constraints.The method is illustrated for phase-summed injection but can also quantify flexibility for each phase.

B. Maximal-flexibility Power Aggregation Model

The maximal-flexibility power aggregation model optimizes upper and lower operational trajectories to obtain the largest feasible inner-box region. Joint constraints ensure that any aggregate trajectory between those bounds admits a feasible disaggregation.

  • The maximal-flexibility model finds upper and lower operational trajectories that produce the largest aggregate flexibility.Its objective maximizes Eaf subject to network, DER, and joint trajectory constraints.
  • The model separates individual constraints for upper and lower trajectories from joint constraints coupling the two trajectories.The individual constraints reproduce network and DER restrictions for each trajectory.
  • Joint constraints prevent upper and lower trajectories from intersecting and preserve storage and HVAC feasibility for intermediate trajectories.They ensure the aggregate feasible region is well defined and support disaggregation feasibility.
  • The resulting maximal-flexibility model is a quadratically constrained convex programming problem.Solving it yields the largest inner-box approximation together with optimal upper and lower operational trajectories.
  • For any trajectory bounded by the feasible lower and upper aggregate trajectories, a disaggregation solution exists.The proposition applies generically to any trajectory pair satisfying the relevant constraints, independently of the objective function.

C. Economic Power Aggregation Model

The economic power aggregation model jointly schedules a base DER dispatch and flexibility reserve. It minimizes distribution-system operating cost while accounting for reserve rewards and feasibility constraints.

  • The economic model optimally schedules power dispatch and flexibility reserve using the base, upper, and lower operational trajectories.The base trajectory represents economic DER dispatch, while the upper and lower trajectories support reserve services.
  • The cost function includes storage charging and discharging effects, HVAC discomfort, PV operation and curtailment, and electricity purchases.These terms represent the principal operating costs associated with the base trajectory.
  • The model minimizes net operation cost while retaining the aggregation constraints used for network and disaggregation feasibility.Its objective combines base-trajectory costs with rewards for flexibility reserve.
  • The distribution system reports flexibility intervals that the transmission system can use for reserve coordination.According to the disaggregation property, regulation signals within those intervals can be tracked.
  • The base DER trajectories need not lie between the upper and lower trajectories to preserve disaggregation feasibility.The joint constraints and base-trajectory restrictions are sufficient for the stated guarantee.

A. Transmission-Distribution MPC Interaction Framework

The framework repeatedly coordinates transmission requests with distribution-level economic aggregation and power disaggregation. A distributed MPC solver addresses privacy and scalability by separating the system aggregator and DER computations.

  • A. Transmission-Distribution MPC Interaction Framework: The MPC framework updates aggregation after each transmission-distribution interaction because storage and HVAC operations are time-coupled and forecasts are uncertain.This rolling procedure refreshes schedules using updated DER conditions.
  • A. Transmission-Distribution MPC Interaction Framework: The transmission broadcasts prices and reserve rewards, after which each distribution system solves economic aggregation and reports its flexibility interval.The transmission then determines dispatch schemes and regulation commands for each distribution system.
  • A. Transmission-Distribution MPC Interaction Framework: Each distribution system solves a power-disaggregation problem to track transmission regulation commands and dispatch DER facilities over the next control horizon.The process advances by Td time steps and repeats with updated operating conditions.
  • B. Distributed Solution Algorithm: The distributed solver avoids the communication, computational, and privacy burdens of a centralized global optimization.It is developed for the MPA, EPA, and PD models using the predictor-corrector proximal multiplier algorithm.
  • B. Distributed Solution Algorithm: The distributed formulation separates system-aggregator and DER local variables and feasible sets.This separable structure enables local optimization problems coordinated through shared constraints.
  • B. Distributed Solution Algorithm: Each iteration uses two-way communication while the system aggregator and DER facilities solve small optimization problems in parallel.The algorithm updates virtual and dual variables before checking convergence.
  • B. Distributed Solution Algorithm: Convergence is guaranteed under strong duality with a sufficiently small positive step length, and numerical tests show fast convergence on a real-world distribution system.The MPA, EPA, and PD models are assumed strictly feasible, implying strong duality.

V. NUMERICAL TESTS

Numerical tests evaluate the proposed aggregation method on a real Southern California Edison feeder and verify aggregate flexibility and disaggregation feasibility through simulations.

  • Simulation Setup: The test feeder has 126 multi-phase buses, 366 single-phase connections, 33 PV units, 28 energy storage devices, and 5 HVAC systems.The study uses real industrial, commercial, and residential load data together with solar irradiance profiles.
  • Maximal-flexibility Power Aggregation: 37.1 MW·h of aggregate flexibility is obtained as the area between the optimal upper and lower substation-injection trajectories.The corresponding DER dispatch schemes are reported for these trajectories.
  • Maximal-flexibility Power Aggregation: The upper trajectory produces less substation power injection and larger loads than the lower trajectory.This relationship is consistent with the intended interpretation of the operational bounds.
  • Disaggregation Feasibility: Up to 5000 randomly generated regulated power trajectories are tested for disaggregation feasibility within the reported intervals.The regulated trajectories are sampled independently over time from uniform distributions.
  • Disaggregation Feasibility: The disaggregation problem is feasible for all generated power trajectories, consistent with the theoretical disaggregation-feasibility proposition.The simulations therefore support feasibility throughout the tested aggregate intervals.

C. Implementation of Transmission-Distribution Interaction

The transmission-distribution interaction framework uses receding-horizon MPC to update flexibility intervals and track transmission regulation commands with distributed optimization.

  • MPC Implementation: At each time instant, the EPA model optimizes the next 12 time steps while reporting the first-step flexibility interval before advancing the horizon.The prediction horizon is 4 hours, with Tp = 12 and Td = 1.
  • Transmission-Distribution Interaction: The implemented aggregate substation trajectory remains within the reported flexibility intervals even when individual DER trajectories do not remain between their upper and lower trajectories.The trajectories are accumulated step by step under receding-horizon operation.
  • Distributed Solution: The distributed solver converges within tens of iterations and solves the MPA model efficiently.Convergence is illustrated for the MPA model in Figure 11.
  • Computational Performance: For the 126-bus, 27-period test system, the distributed solver takes longer than the centralized solver because it uses an iterative process.The distributed implementation can solve individual DER problems successively or in parallel.
  • Computational Performance: The network optimization component handles a much larger problem than each DER, while parallel computational resources can enable a fast distributed solution process.This explains why the network-side computation dominates the distributed implementation.
  • Framework Summary: The proposed method combines unbalanced network constraints, aggregate flexibility modeling, and distributed MPC for scalable transmission-distribution interaction.The paper validates the framework through numerical tests on a real distribution feeder.

APPENDIX A LINEAR MULTI-PHASE POWER FLOW MODEL

The appendix derives a linear multi-phase power-flow model from fixed-point linearization and converts complex voltages into voltage-magnitude expressions suitable for optimization.

  • Complex-voltage formulation: The model derives a linear formulation for three-phase complex voltages from a given operational point using one fixed-point linearization iteration.The formulation uses the operational point and network admittance matrices.
  • Complex-voltage formulation: The substation three-phase complex voltage and the load-side admittance submatrix are explicit components of the linearized voltage model.The appendix defines ˜v0 and YLL as the relevant voltage and admittance quantities.
  • Network representation: A block-diagonal matrix is introduced as part of the linearized multi-phase network representation.The appendix defines this matrix after presenting the voltage formulation.
  • Voltage magnitudes: The derivation then produces a linear model for voltage magnitudes from the complex-voltage representation.The appendix applies a stated derivation rule to obtain v = |˜v|.
  • Network equations: Kirchhoff’s laws provide the matrices and vectors needed to complete the linear power-flow model.The appendix refers to a detailed prior description for the construction of B, D, b, and d.

APPENDIX B PROOF OF PROPOSITION 1

The proof establishes that every aggregate trajectory between the optimized upper and lower bounds has a feasible DER disaggregation under the model’s network and device constraints.

  • Constructing an interior trajectory: For any aggregate trajectory between the lower and upper bounds, the proof defines an interpolation coefficient λt ∈ [0, 1].The aggregate trajectory is represented as a convex combination of the two boundary trajectories.
  • Constructing an interior trajectory: The corresponding operational variables are constructed through the same interpolation and are proposed as a disaggregation solution.This construction links an interior aggregate trajectory to feasible upper and lower operational trajectories.
  • Network feasibility: Convexity of the reformulated network constraints preserves feasibility for the interpolated voltage and line-current variables.The linear power-flow model makes voltage and current affine functions of the operational variables.
  • Device feasibility: The proof verifies the interpolated solution against time-decoupled constraints and time-coupled energy-storage and HVAC-temperature limits.The SOC and HVAC constraints are handled using the corresponding upper and lower constraint relations.
  • Conclusion of proof: The resulting interpolated operational trajectory satisfies the required constraints, proving existence of a feasible disaggregation solution.This establishes the proposition for every aggregate trajectory inside the approximate feasible region.
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