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Optimal Sizing and Placement of Distributed Storage in Low Voltage Networks

Philipp Fortenbacher, Martin Zellner, Göran Andersson

arXiv:1512.01218v2eess.SYcs.SY

TL;DR

Storage sizing and placement in LV grids is difficult when grid constraints and nonlinear AC power flow are included. The paper introduces FBS-OPF, a linearized multiperiod approach, and applies it to compare centralized and distributed storage. In the studied test system, distributed storage is generally preferable, although the authors note that how general this conclusion is remains to be investigated.

  • Problem

    Grid constraints make storage sizing and placement difficult because nonlinear AC power-flow equations render the problem intractable for long investment horizons.

    Source: §I · p. 1

  • Method

    FBS-OPF recasts nonlinear AC-OPF as a linear program for radial LV networks and iteratively updates voltages using a forward-backward-sweep load flow.

    Source: §I · p. 1

  • Results

    In the studied test system, distributed storage is generally superior in profit, with profitability points at ≈100 e/kWh for centralized storage and ≈230 e/kWh for distributed storage.

    Source: D. Economic Viability · p. 6

  • Takeaways & Limitations

    For the studied test system, distributed storage configurations are preferable because the economic impact of PV curtailment exceeds the saved network losses.

    Source: §V · p. 6

    How general the conclusion that distributed storage is preferable is remains to be investigated.

    Source: §V · p. 7

Abstract

from arXiv · show

This paper proposes a novel algorithm to optimally size and place storage in low voltage (LV) networks based on a linearized multiperiod optimal power flow method which we call forward backward sweep optimal power flow (FBS-OPF). We show that this method has good convergence properties, its solution deviates slightly from the optimum and makes the storage sizing and placement problem tractable for longer investment horizons. We demonstrate the usefulness of our method by assessing the economic viability of distributed and centralized storage in LV grids with a high photovoltaic penetration (PV). As a main result, we quantify that for the CIGRE LV test grid distributed storage configurations are preferable, since they allow for less PV curtailment due to grid constraints.

I. INTRODUCTION

The paper addresses storage sizing and placement in LV networks where grid constraints make nonlinear AC power-flow formulations difficult to solve over long horizons. It introduces FBS-OPF, a radial-network-based linear approach, and uses it to assess centralized and distributed storage viability.

I. INTRODUCTION

  • Grid-constrained storage sizing and placement is difficult because nonlinear AC power-flow equations make the problem intractable for long investment horizons.The paper frames the task around residential customers seeking to maximize self-consumption from their PV assets.Source: §I · p. 1
  • FBS-OPF recasts nonlinear AC-OPF as a linear program for radial LV networks and iteratively updates voltages using a forward-backward-sweep load flow.The paper incorporates this approach into a multiperiod optimal power flow problem with storage sizing and placement constraints.Source: §I · p. 1, (FBS-OPF) · p. 2
  • The paper compares centralized and distributed storage viability for customers with energy-market access and PV in an LV network.The introduction identifies reduced network losses and lower investment costs as possible advantages of centralized storage, and local PV absorption as a potential advantage of distributed storage.Source: §I · p. 1

(FBS-OPF)

  • For radial networks, the BIBC matrix maps nodal current injections to branch currents.The paper also defines a reduced matrix by deleting the slack-bus column.Source: (FBS-OPF) · p. 2

A. Voltage Approximation

The method approximates voltage drops and losses to obtain linear expressions suitable for optimization. Its voltage approximation relies on small voltage angles and a high R/X ratio, while its loss approximation uses piecewise-linear planes.

A. Voltage Approximation

  • The voltage approximation assumes nodal voltage angles below 10° and an R/X ratio above 2, conditions described as usual for LV networks.These assumptions allow the complex voltage-drop expression to be approximated as absolute voltage drops relative to the slack-bus voltage.Source: A. Voltage Approximation · p. 2

B. Loss Approximation

  • The method approximates quadratic active- and reactive-power loss functions with piecewise-linear functions divided into four regions.The regions are defined using branch currents i0 and i1 and branch resistances.Source: B. Loss Approximation · p. 2
  • Four planes define the piecewise-linear loss approximation, with the blue surface approximating the exact red loss surface.The illustrated example is a two-bus system with one line.Source: B. Loss Approximation · p. 2, Figure 1 · p. 2

C. Branch Flow Approximation

The branch-flow approximation uses the assumption that reactive-power injections are much smaller than active-power injections during normal LV-grid operation.

C. Branch Flow Approximation

  • Assuming reactive injections are much smaller than active injections in normal LV-grid operation allows the branch-current expression to neglect the reactive-power contribution.This approximation is applied to the exact branch-flow current expression.Source: C. Branch Flow Approximation · p. 2

D. FBS-OPF Formulation

The formulation recasts the approximated AC-OPF as a linear program with power-balance, voltage, loss, branch-flow, and generator-bound constraints.

D. FBS-OPF Formulation

  • The LP formulation includes power balance and the voltage approximation, with epigraph constraints representing the convex active- and reactive-power loss approximations.The max operators are represented by equivalent inequalities.Source: D. FBS-OPF Formulation · p. 3
  • Branch-flow limits and lower and upper bounds on voltage and active and reactive generator powers constrain the LP.The formulation uses boxed-bounded active- and reactive-power settings.Source: D. FBS-OPF Formulation · p. 3
  • The formulation notes that piecewise-linear approximation could also represent a specific power-factor range, as reported in.The stated formulation instead considers boxed-bounded active- and reactive-power settings.Source: D. FBS-OPF Formulation · p. 3

E. Forward Backward Sweep (FBS) Algorithm

The FBS algorithm iteratively solves the optimization while updating voltage-dependent approximations using a forward-backward sweep. It stops when voltage mean absolute error falls below a threshold; the sweep method typically converges in a small number of iterations.

E. Forward Backward Sweep (FBS) Algorithm

  • The algorithm repeats optimization, current calculation in the forward stage, and voltage updates in the backward stage until mean absolute error falls below a predefined threshold.The forward-backward sweep method typically uses h ≤ 4 iterations.Source: E. Forward Backward Sweep (FBS)… · p. 3
  • Setting initial voltages to slack-bus values overestimates voltage drops and branch flows, so iterative voltage updates reduce the resulting solution deviation.The approximation matrices depend on nodal voltage magnitudes.Source: E. Forward Backward Sweep (FBS)… · p. 3
  • The initialization sets v0 to the slack-bus voltage and defines two operating-current regions at 25% and 75% of the estimated maximum current.The maximum expected current is specified as M fCgpmax.Source: E. Forward Backward Sweep (FBS)… · p. 3

F. Multiperiod FBS-OPF

The multiperiod formulation extends single-step OPF across a horizon because storage transfers energy between time steps. The paper also compares FBS-OPF with AC-OPF for convergence and optimality, and compares execution times in the case study.

F. Multiperiod FBS-OPF

  • Storage transfers energy between time steps, coupling the single-step FBS-OPF problem and motivating a multiperiod formulation over horizon N.The optimization vector stacks the per-step vectors x1 through xN.Source: F. Multiperiod FBS-OPF · p. 3
  • The sample time T for the time-variable price profile cp(k) can range from minutes for intraday trading to hours for spot-market or base products.Source: F. Multiperiod FBS-OPF · p. 3

G. Comparison between AC-OPF and FBS-OPF

1) Testsystem:

The section evaluates FBS-OPF convergence and optimality against AC-OPF on a CIGRE low-voltage grid. It reports a small one-iteration voltage error and a first-iteration objective value above AC-OPF, with later iterations below it.

1) Testsystem:

2) Convergence:

  • 2.5e-3 p.u.: the voltage error at h = 1 is small relative to possible network-parameter inaccuracies, and the voltage projections remain within voltage limits.The authors consider one iteration reasonable for storage sizing and placement.Source: 2) Convergence · pp. 3–4

3) Optimality:

  • 2%: FBS-OPF's objective value is above AC-OPF at the first iteration, but is smaller at higher iterations; the later difference reflects a slight feeder branch-flow-limit violation.The comparison uses the one-iteration approximation as a reasonable sizing and placement strategy despite slight divergence from the exact solution.Source: Figure 4 · p. 4, 3) Optimality · p. 4

III. INTERTEMPORAL COUPLED MULTI-PERIOD OPF

Storage couples the multiperiod OPF across time because stored energy can be shifted between time intervals. Its energy evolution and operating limits are incorporated into the optimization to enable storage sizing and placement.

A. Incorporation of Storage

  • Storage makes the multiperiod problem intertemporally coupled because available energy can be delivered within selected time intervals.The time-varying energy level is represented with a discrete state-space equation.Source: A. Incorporation of Storage · p. 4
  • Charging and discharging powers enter the storage state evolution, with charging and discharging efficiencies represented in the input matrix.The energy-level evolution also uses the initial energy vector.Source: A. Incorporation of Storage · pp. 4–5
  • Storage energy is constrained to remain between the allowed minimum and maximum levels.These bounds are incorporated into the multiperiod optimization.Source: A. Incorporation of Storage · p. 5

B. Optimal Sizing and Placement of Storage

The sizing and placement formulation makes the maximum storage level a decision variable, then applies it in the multiperiod OPF. The case study compares centralized and distributed storage under specified grid, PV, and market conditions.

B. Optimal Sizing and Placement of Storage

  • The maximum storage limit becomes decision variable z, with the minimum energy level set to zero for the reformulated sizing constraints.The resulting formulation is included in the multiperiod optimization, whose cost vector represents storage costs.Source: B. Optimal Sizing and Placement of… · p. 5

IV. CASE STUDY

  • The case study optimizes storage sizing and placement for centralized and distributed configurations against future storage costs and PV share.The modeled prosumers buy and sell energy and minimize their energy bill using PV and storage.Source: §IV · p. 5

A. Definition

  • The CIGRE test grid has 18 households and 100% PV share over 31 days, with centralized storage at R0 or distributed storage at R1–R18.The simulation parameters include storage costs of 25–300 e/kWh and a one-hour resolution across 744 time steps.Source: A. Definition · p. 5

B. Implementation

The study finds distributed storage generally more profitable, while FBS-OPF supports longer simulation horizons than nonlinear AC-OPF. The simulations use a MATLAB framework, and the authors qualify the LP runtime guarantee by solver choice.

B. Implementation

  • The simulations use a MATLAB framework based on MATPOWER, with Pardiso/IPOPT for nonlinear AC-OPF and CPLEX or GUROBI for FBS-OPF.The LP runtime is assured polynomial only when using the interior-point method; simplex has an exponential worst case for degenerate problems.Source: B. Implementation · p. 5, C. Computation Time · p. 6

C. Computation Time

  • Both FBS-OPF and AC-OPF computation times grow approximately polynomially with horizon, but AC-OPF has a higher exponent.The conventional AC-OPF is limited to a couple of days, whereas FBS-OPF allows simulations for an entire year.Source: C. Computation Time · pp. 5–6

D. Economic Viability

  • Storage becomes profitable at ≈100 e/kWh centrally and ≈230 e/kWh in the distributed scenario; distributed storage is generally more profitable.The paper attributes lower PV utilization in the centralized case to curtailment caused by line or transformer overloads and voltage violations.Source: D. Economic Viability · p. 6

E. Placement of Distributed Storage

Optimizing multiple distributed storage units also determines how total capacity is placed across the network. In the studied grid, higher-cost storage favors middle-root nodes, and the authors caution that the result's generality remains to be established.

E. Placement of Distributed Storage

  • Optimizing the sizes of multiple distributed units also optimizes the network distribution of total storage capacity and implicitly solves where to place each unit.For this test grid, thermal line constraints are more dominant than voltage constraints.Source: E. Placement of Distributed Storage · p. 6
  • For higher storage costs, nodes 2, 3, 4, and 6 are favored over branch ends, a placement associated with less PV curtailment and lower network losses.The placement result is shown for the studied test grid.Source: E. Placement of Distributed Storage · p. 6, Figure 8 · p. 7

V. CONCLUSION

  • The paper concludes that distributed storage is preferable in the studied test system because the economic impact of PV curtailment exceeds the network-loss savings from energy-market transactions.The conclusion is scoped to the studied system, with typical LV-grid applicability presented as an anticipation.Source: §V · pp. 6–7
  • The authors leave the generality of the conclusion open and anticipate that larger networks with more storages may make the coupled problem intractable.They identify distributed optimization as future work on decomposition.Source: §V · p. 7
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