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Real-Time Feedback-Based Optimization of Distribution Grids: A Unified Approach

Andrey Bernstein, Emiliano Dall'Anese

arXiv:1711.01627v5math.OC

TL;DR

Real-time DER optimization must handle changing operating conditions, engineering constraints, and limitations of conventional open-loop optimization. This paper develops a feedback-based online algorithm for individual DERs and aggregations using a time-varying bi-level formulation. In real-system experiments, it tracks requested substation power changes, reduces swings associated with PV variation using storage, and regulates voltage under a modified test setting.

  • Problem

    Conventional optimization may be too slow for second-scale operation, produce infeasible setpoints under modeling or measurement errors, and require impractical pervasive metering of non-controllable assets.

    Source: §I · p. 1

  • Method

    The framework uses a time-varying bi-level formulation and a measurement-adjusted projected primal-dual gradient algorithm for individual DERs and jointly controlled groups.

    Source: §I · p. 2, §VI · p. 11

  • Results

    In the real-system experiments, substation power closely tracks ramping signals and setpoint steps, storage lowers swings from PV variation, and voltage regulation is demonstrated with PV capacity tripled and the upper limit reduced to 1.01 pu.

    Source: §V · pp. 10–11

  • Takeaways & Limitations

    The framework supports DER objectives and grid services while maintaining electrical quantities within engineering limits, with analytical claims framed as tracking the time-varying optimization solution.

    Source: §I · p. 2, §VI · p. 11

    The test feeder is stiff and models only the primary side of distribution transformers; secondary-side models, where violations are more likely, were unavailable.

    Source: §V · p. 9

Abstract

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This paper develops an algorithmic framework for real-time optimization of distribution-level distributed energy resources (DERs). The proposed framework optimizes the operation of both DERs that are individually controllable and groups of DERs (i.e., aggregations) at an electrical point of connection that are jointly controlled. From an electrical standpoint, wye and delta single- and multi-phase connections are accounted for. The algorithm enables (groups of) DERs to pursue given performance objectives, while adjusting their (aggregate) powers to respond to services requested by grid operators and to maintain electrical quantities within engineering limits. The design of the algorithm leverages a time-varying bi-level problem formulation capturing various performance objectives and engineering constraints, and an online implementation of primal-dual projected-gradient methods. The gradient steps are suitably modified to accommodate appropriate measurements from the distribution network and the DERs. By virtue of this approach, the resultant algorithm can cope with inaccuracies in the distribution-system modeling, it avoids pervasive metering to gather the state of non-controllable resources, and it naturally lends itself to a distributed implementation. Analytical stability and convergence claims are established in terms of tracking of the solution of the formulated time-varying optimization problem. The proposed method is tested in a realistic distribution system with real data.

I. INTRODUCTION

The paper targets real-time DER optimization that accommodates changing conditions and system constraints, while addressing the computational and metering limits of conventional approaches. It proposes a feedback-based online method for individual devices and DER groups, with a bi-level formulation and support for varied connections and controls.

I. INTRODUCTION

  • Conventional optimization can be too slow for second-scale operation, may yield physically infeasible setpoints under model and measurement errors, and often requires impractical pervasive metering.These limitations motivate real-time decision making as DER integration increases and ambient conditions and non-controllable assets vary.Source: §I · p. 1
  • The framework formulates time-varying bi-level optimization over DER trajectories and embeds dynamic operating and engineering constraints, including voltage, ampacity, device-region, and interconnection power-flow limits.The interconnection power-flow targets can support grid services or market operations.Source: §I · p. 1
  • A projected primal-dual gradient algorithm uses grid and device measurements to track the time-varying solution, cope with model inaccuracies, avoid pervasive metering of non-controllable resources, and support distributed implementation.The paper establishes analytical stability and convergence claims in terms of tracking that solution.Source: §I · p. 2
  • For DER groups behind a meter, the bi-level formulation optimizes net power while accounting for individual constraints, using inner approximations of aggregate feasibility and a systematic cost-gradient computation.For discrete commands, the method convexifies operating sets for setpoint computation and uses error-diffusion techniques to obtain implementable setpoints,.Source: §I · p. 2
  • The framework extends prior work, to wye and delta multi-phase systems, non-convex and discrete commands, and testing on a California distribution network with real data.The introduction also notes that general bi-level problems are NP-hard even when both levels are convex, while this application can solve the inner level through its KKT conditions,.Source: §I · p. 2

II. PRELIMINARIES AND SYSTEM MODEL

The preliminaries establish notation for vectors, matrices, norms, gradients, and set operations used in the system model.

II. PRELIMINARIES AND SYSTEM MODEL

  • Uppercase and lowercase boldface denote matrices and column vectors, respectively; superscripts T, *, and H denote transpose, complex conjugate, and conjugate transpose.The notation also defines real and imaginary parts, j as the imaginary unit, and absolute value or set cardinality using vertical bars.Source: §II · p. 2
  • The text defines vector and matrix 2-norms, entry-wise absolute value, diagonal matrices, and the gradient of a function with respect to a vector.The all-ones and all-zeros vectors are denoted 1_N and 0_N.Source: §II · pp. 2–3
  • For sets, ⊕ denotes the Minkowski sum, ch denotes the convex hull, and proj denotes a closest-point projection with ties broken arbitrarily.The projection minimizes the Euclidean distance to the set.Source: §II · p. 3

A. DER Model

The DER model covers individually controlled devices and jointly controlled aggregations, with setpoint regions that capture device constraints and support both continuous and discrete commands. For aggregations, the paper develops analytical inner approximations of combined operating regions for real-time use.

A. DER Model

  • The model includes individually controllable DERs and groups controlled as a whole, with devices supporting wye or delta connections and single- or three-phase operation.Groups represent, for example, homes, buildings, multi-inverter renewable systems, and EV parking garages.Source: A. DER Model · p. 3
  • An individual DER's convex, compact set of power setpoints captures hardware and operating constraints, and its controller is assumed to drive output powers to commanded setpoints.The inverter set varies over time as its operating parameters change with ambient conditions or internal state.Source: A. DER Model · p. 3
  • For non-convex devices such as HVAC systems with discrete commands, the setpoint calculation uses the convex hull of implementable commands and a randomization procedure recovers implementable setpoints.A device locked in a state by operational constraints can have a singleton implementable set during that interval.Source: A. DER Model · p. 3
  • An aggregation's net-power feasibility region is represented by the exact Minkowski sum of its devices' operating regions or an inner approximation, with the paper seeking a simple analytical approximation for real-time algorithms.The aggregation setpoint is the sum of the setpoints of its member devices.Source: A. DER Model · p. 3
  • For two operating regions, Proposition 2 gives an inner approximation contained in their Minkowski sum, and the stated inner approximation is convex and compact.It is expressed in the same form as the paper's prototypical set in (1).Source: A. DER Model · p. 4
  • The propositions support building an aggregation's feasible net-power region by successively combining the operating sets of its member devices.The paper illustrates this procedure for a home with PV, a battery, and an EV.Source: A. DER Model · pp. 3–4

B. Network Model

The network model represents a multi-phase distribution grid with wye- and delta-connected devices and uses approximate linear models of electrical quantities to support real-time algorithm design and analysis.

B. Network Model

  • The network has multiphase nodes and distribution lines, with node 0 as the three-phase slack bus connecting the distribution grid to the rest of the electrical system.Controllable and non-controllable devices at multiphase nodes may be wye- or delta-connected.Source: B. Network Model · p. 4
  • The AC power-flow description collects phase-to-ground voltages, phase net-current injections, delta-connection phase-to-phase currents, and complex powers injected by wye- and delta-connected devices.The equations use admittance-matrix submatrices formed from network topology and distribution-line models.Source: B. Network Model · p. 4
  • Approximate linear models express monitored voltage magnitudes, line-current magnitudes, and substation real powers as functions of DER setpoints and model parameters.The fixed-point linearization parameters capture connection types and, when that method is used, do not require non-controllable powers for their computation.Source: B. Network Model · p. 4
  • The approximate models facilitate algorithm design and performance analysis, while grid and device measurements are used to cope with their inaccuracies.The text identifies voltage and current quantities at points of interest and permits line-current measurements or pseudo-measurements.Source: B. Network Model · pp. 4–5

III. FEEDBACK-BASED DYNAMIC OPTIMIZATION

The framework updates DER power setpoints on a second timescale to optimize operational objectives as conditions change. A time-varying optimization formalism represents changing objectives, constraints, non-controllable assets, and ambient conditions.

III. FEEDBACK-BASED DYNAMIC OPTIMIZATION

  • DER power setpoints are updated on a second timescale to maximize operational objectives while coping with changing ambient conditions and non-controllable assets.The iteration interval h reflects communication delays and utility or aggregator operational considerations.Source: §III · p. 5
  • The time-varying optimization formalism models optimal DER trajectories under possibly changing objectives and operational constraints.Source: §III · p. 5
  • The formulation also accounts for variability in non-controllable assets and ambient conditions when modeling DER operational trajectories.Source: §III · p. 5

A. Formalizing Optimal Operational Trajectories

The section formulates a time-varying convex optimization problem for DER operation, including network limits and power-flow targets. It motivates an online feedback-based algorithm by identifying why solving the problem in batches can be impractical.

A. Formalizing Optimal Operational Trajectories

B. Real-time Algorithm

The real-time algorithm uses a regularized time-varying saddle-point formulation and modified projected-gradient updates to generate DER power setpoints. Feedback measurements support operation despite model inaccuracies, while local DER and aggregation steps enable distributed execution.

B. Real-time Algorithm

  • Theorem 1 establishes that an aggregation’s cost function is convex and Lipschitz continuous, with its gradient given by the optimal dual variable for the aggregation power-sum constraint.Source: B. Real-time Algorithm · p. 6
  • Regularization makes the Lagrangian strongly convex in primal variables and strongly concave in dual variables, supporting modified online projected-gradient tracking.The regularized optimizer may differ from the original problem’s optimizer, with that discrepancy bounded as stated in the paper.Source: B. Real-time Algorithm · p. 6
  • At each time step, the algorithm updates power setpoints for DERs using a projected-gradient method based on the regularized time-varying saddle-point formulation.Source: B. Real-time Algorithm · p. 6
  • Voltage, current, and point-of-common-coupling power measurements replace corresponding analytical expressions, avoiding measurement of uncontrollable-device states.Measurements can be replaced by estimated state when a real-time state-estimation procedure is available.Source: B. Real-time Algorithm · p. 7
  • The utility or aggregator broadcasts dual variables, while individual DERs and aggregations implement their update steps locally.The substeps for voltage, current, and point-of-common-coupling measurements can run in parallel.Source: B. Real-time Algorithm · p. 7
  • For discrete setpoints, error diffusion uses accumulated error between continuous and discrete commands to obtain the next implementable setpoint.The modified projection steps include a localized nonconvex program for computing implementable commands.Source: B. Real-time Algorithm · p. 7

IV. PERFORMANCE ANALYSIS

The analysis bounds tracking error for the time-varying optimizer and establishes Q-linear convergence up to an asymptotic error bound under stated conditions. That bound reflects measurement errors and variation in the optimal trajectory.

IV. PERFORMANCE ANALYSIS

  • The analysis assumes synchronous updates; asynchronous analysis is possible with heavier notation and additional assumptions.Source: §IV · p. 7
  • Theorem 3 bounds the distance between the algorithm’s primal-dual iterates and the time-varying primal-dual optimizer.Source: §IV · p. 8
  • If c(α) < 1, the iterates converge Q-linearly to the time-varying optimizer up to an asymptotic error bound.Source: §IV · p. 8
  • The asymptotic bound depends on trajectory variation σ and measurement errors e; when both are zero, the algorithm converges to the static optimization problem’s solution.Source: §IV · p. 8

V. EXPERIMENTS ON A REAL SYSTEM

Experiments on a real SCE feeder test one-second optimization with real load and irradiance data. The algorithm tracks substation power setpoints while respecting voltage limits, including during fast PV variability.

V. EXPERIMENTS ON A REAL SYSTEM

  • The test uses a real SCE feeder with 126 multiphase nodes, 366 single-phase connection points, and load and irradiance data interpolated to one-second resolution.The feeder includes wye and delta connections, and the optimization problem changes every second.Source: §V · p. 9
  • The feeder is stiff and the model includes only transformer primary sides, making voltage violations less visible than they may be on secondary sides.The paper identifies the lack of secondary-side models as a downside of this real-feeder test case.Source: §V · p. 9
  • Tests evaluate the algorithm during sunny and cloudy periods, controlling DERs to track substation power setpoints while keeping voltages within limits; non-controllable devices introduce most variability.PV and battery inverters are modeled with a 0.25-second time constant (approximately 1.25-second settling time), and communication delays are 0.1 seconds per link.Source: §V · pp. 9–10
  • Substation power closely tracks ramping signals and step changes; tripling battery capacity enables complete accommodation of PV variability in the cloudy case.The comparison reports power swings up to 1.8 MW with PV at maximum output and batteries unused; the tripled-capacity comparison shows phase c.Source: §V · pp. 10–11, Figure 7 · p. 10, Figure 6 · p. 10
  • With PV capacity increased 3× and the upper voltage limit tightened to 1.01 pu, the algorithm regulates voltages while driving substation power to setpoints.The plotted voltage trajectories compare the 1.01 pu limit with 1.05 pu.Source: §V · p. 11, Figure 8 · p. 11

VI. CONCLUDING REMARKS

The framework performs real-time optimization for individually controlled DERs and jointly controlled groups, balancing performance objectives, grid services, and engineering limits. Its feedback-based primal-dual method tracks a time-varying optimization solution, with analytical stability and convergence claims.

VI. CONCLUDING REMARKS

  • The framework optimizes both individually controllable DERs and groups of DERs jointly controlled at an electrical point of connection.It adjusts individual or aggregate powers to pursue performance objectives, respond to grid-operator services, and maintain electrical quantities within engineering limits.Source: §VI · p. 11
  • A time-varying bi-level formulation captures performance objectives and engineering constraints, and the online algorithm uses primal-dual projected-gradient methods.Source: §VI · p. 11
  • The feedback-based algorithm can cope with distribution-model inaccuracies, avoids pervasive metering of non-controllable resources, and lends itself to distributed implementation.Source: §VI · p. 11
  • Analytical stability and convergence claims are established in terms of tracking the solution of the formulated time-varying optimization problem.Source: §VI · p. 11
  • The paper identifies extending the technical findings to time-varying nonconvex problems as future work.Source: §VI · p. 11

A. Proof of Proposition 1

The appendices provide proofs and an illustration concerning the aggregation of DER feasibility sets, properties of the associated dual problems, and tracking of the time-varying optimization solution.

A. Proof of Proposition 1

  • The proof of Proposition 1 uses a description of the Minkowski sum of a circle and an interval, with its vertical boundary expressed using a concave function.Source: A. Proof of Proposition 1 · p. 11

B. Proof of Proposition 2

  • For Proposition 2, the outer approximation contains sums of feasible DER points, but its converse need not hold; condition (6) supports the stated inner approximation.Source: B. Proof of Proposition 2 · p. 11

C. Illustration of Assumption 2

  • The example shows that the dual function is not globally strongly concave, while interior aggregate points have a unique optimum in a locally strongly concave region; boundary points can have infinitely many optima.Source: C. Illustration of Assumption 2 · p. 12

D. Proof of Lemma 1

  • The proof rewrites constraint (13c) as H_j x − x̄_j = 0, with H_j full row rank; Assumption 1 and the compact gradient domain bound the optimal dual variable, whose uniqueness follows from Assumption 2.Source: D. Proof of Lemma 1 · p. 12

E. Proof of Theorem 1

  • Theorem 1 uses convexity of the optimal value function and identifies its subdifferential with the optimal dual variables to establish the stated value-function properties.Source: E. Proof of Theorem 1 · p. 12

F. Proof of Theorem 2

  • Theorem 2’s proof invokes local strong concavity of the dual function when relating optimal dual variables for interior feasible aggregate points.Source: F. Proof of Theorem 2 · p. 12

G. Proof of Theorem 3

  • Theorem 3’s proof uses a strongly monotone, Lipschitz map and bounds tracking error using feedback-gradient error and changes in the optimal solution.Source: G. Proof of Theorem 3 · p. 13

H. Data for the Simulations

Tables I and II document the PV and energy-storage systems in the distribution network used for the simulations.

H. Data for the Simulations

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