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Optimization-based Islanding of Power Networks using Piecewise Linear AC Power Flow

P. A. Trodden, W. A. Bukhsh, A. Grothey, K. I. M. McKinnon

arXiv:1310.6675v2math.OCcs.SYeess.SY

TL;DR

DC-based islanding can yield AC-infeasible islands with abnormal voltages, motivating an approach that models voltage and reactive power directly. The paper uses a piecewise linear AC power-flow model in an optimization-based framework, and reports that none of its PWL AC islands were infeasible, unlike 20% of DC cases.

  • Problem

    DC-based islanding can produce AC-infeasible solutions and abnormal voltages when local reactive-power shortages or surpluses are not captured.

    Source: A. Motivation: effect of topology… · p. 9

  • Method

    The framework uses piecewise linear AC power flow to design islands while accounting for voltage and reactive power.

    Source: §V · p. 17

  • Results

    20% of DC cases led to AC-infeasible islands, whereas none of the islands found using the PWL AC model were infeasible.

    Source: 3) AC feasibility · p. 16

  • Takeaways & Limitations

    The framework finds islands balanced in real and reactive power that satisfy AC power-flow and operating constraints.

    Source: §V · p. 17

Abstract

from arXiv · show

In this paper, a flexible optimization-based framework for intentional islanding is presented. The decision is made of which transmission lines to switch in order to split the network while minimizing disruption, the amount of load shed, or grouping coherent generators. The approach uses a piecewise linear model of AC power flow, which allows the voltage and reactive power to be considered directly when designing the islands. Demonstrations on standard test networks show that solution of the problem provides islands that are balanced in real and reactive power, satisfy AC power flow laws, and have a healthy voltage profile.

NOMENCLATURE

The nomenclature defines the sets, parameters, and variables used to describe buses, lines, generators, loads, and their island assignments.

Sets

  • Sets identify buses, lines, generators, loads, their connections, uncertain lines, and buses assigned to each section.The notation includes B0 and B1 for buses assigned to sections 0 and 1.Source: Sets · p. 1

Parameters

  • Parameters specify network properties, generator limits and costs, load demands, line limits, and load-loss penalties.The parameter list includes conductance, susceptance, voltage, angle, and penalty terms.Source: Parameters · p. 2

Variables

  • Variables represent voltage and phase, line and generator power, supplied load, and binary assignments or connection statuses.Binary variables mark bus section assignments and generator and line connection status.Source: Variables · p. 2

I. INTRODUCTION

Intentional islanding seeks to split a network while preserving feasibility and stability, but DC-based approaches can miss voltage and reactive-power problems. The paper proposes an AC-aware optimization framework for designing islands.

I. INTRODUCTION

  • The paper frames islanding as a corrective measure for limiting severe-disturbance effects and preventing wide-area blackouts.Networks operating closer to limits and facing greater load and distributed-generation uncertainty may be harder to protect from cascading failures –.Source: §I · p. 2
  • An islanding solution must balance static requirements such as load-generation balance and network limits with dynamic frequency and voltage stability.The split must also avoid loss of synchronism or voltage collapse.Source: §I · p. 2
  • Existing islanding approaches include coherent-machine grouping, graph and search methods, iterative stability analyses, and generator-domain partitioning,,,,,,,,,,,.The approaches differ in how they choose boundaries and handle balance, power flows, and stability.Source: §I · p. 3
  • Prior optimization can explicitly handle power balances, flows, and operating limits, but AC constraints make topology optimization a difficult MINLP,.The paper’s proposed piecewise linear AC formulation addresses this computational challenge while retaining voltage and reactive-power considerations.Source: §I · pp. 3–4
  • DC-based islanding can produce AC-infeasible islands because it omits voltage effects and local reactive-power shortages or surpluses.Such local imbalances can cause abnormal voltages even when islands have sufficient reactive generation capacity overall.Source: §I · p. 4
  • The proposed method uses a piecewise linear AC power-flow approximation in an islanding MILP, with decisions on line disconnection, load shedding, and generator adjustment.It is intended to accommodate different islanding objectives, including reducing load shed and keeping coherent synchronous machines together.Source: §I · p. 4

II. PIECEWISE LINEAR AC POWER FLOW

The paper develops a piecewise linear approximation of AC power flow that retains voltage and reactive power for use in islanding optimization. Its linear-plus-cosine form targets reactive-flow accuracy while remaining compatible with piecewise linear modeling.

A. A linear-plus-cosine model of AC power flow

  • The proposed approximation retains voltage and reactive power, unlike the standard DC approximation, which drops reactive-power variables and equations.The model expands line flows about unit voltage magnitudes and zero phase difference.Source: A. A linear-plus-cosine model of AC… · pp. 4–5
  • The approximation is developed from AC real and reactive power balances and expressions for line flows in both directions.The line-flow formulation accounts for the convention that a transformer is located at the branch’s from-end bus.Source: A. A linear-plus-cosine model of AC… · p. 5
  • The linear-plus-cosine approximation models the cosine term because it matters for reactive-flow accuracy, while the sine term remains linearized.Including the sine term as well slightly reduces real-flow error but makes no significant difference to reactive power.Source: A. A linear-plus-cosine model of AC… · p. 5
  • The cosine term can be represented by piecewise linear functions, allowing the approximation to be used in an MILP formulation.The paper motivates this form because reported DC-islanding infeasibilities were mainly associated with reactive-power and voltage limits.Source: A. A linear-plus-cosine model of AC… · p. 5

B. Piecewise linear AC OPF

The PWL AC OPF formulation represents cosine with piecewise linear segments and applies system limits. The paper’s comparison shows that the SOS formulation captures a low-load generation-cost rise that relaxed PWL and DC models miss.

B. Piecewise linear AC OPF

  • The PWL AC OPF represents cosine with N linear pieces whose segment coefficients match cosine at the breakpoints.The formulation specifies line flows and imposes system limits.Source: B. Piecewise linear AC OPF · pp. 6–7
  • The stated line-flow limits are real-power limits; an MVA limit may be converted by assuming nominal voltage.The approximation-error table is identified as covering line-flow terms, but no cell values are supplied in these passages.Source: Table I · p. 6, B. Piecewise linear AC OPF · p. 7
  • The SOS PWL formulation captures the generation-cost rise at low load, whereas relaxed PWL and DC models do not.The comparison uses the modified WSCC 9-bus network; the relaxed model permits lines to store reactive power irrespective of their end voltages and angles.Source: B. Piecewise linear AC OPF · p. 7, Figure 2 · p. 7
  • The cosine approximation can be implemented with binary variables or SOS-2, yielding an MILP or MIQP depending on the generation-cost function.Relaxing the piecewise constraints removes the need for binaries or SOS sets, but can produce a non-tight result.Source: B. Piecewise linear AC OPF · p. 7

III. A FORMULATION FOR SYSTEM ISLANDING USING PIECEWISE LINEAR AC

The formulation uses piecewise-linear AC power flow in islanding optimization so that topology choices account for voltage and reactive-power effects. Its constraints assign buses to separated sections and enforce feasible operation after switching.

III. A FORMULATION FOR SYSTEM ISLANDING USING PIECEWISE LINEAR AC

  • Prior MILP islanding selected lines, load shedding, and generator adjustments, then used AC optimal load shedding to check feasibility and refine settings.That follow-on AC solve accounts for voltage and reactive power, but some resulting islanding solutions remained AC-infeasible.Source: A. Motivation: effect of topology… · p. 8

A. Motivation: effect of topology changes on voltage profile

  • DC-based islanding can leave islands AC-infeasible despite sufficient reactive capacity, motivating topology optimization that models voltage and reactive power without a full MINLP.In the 24-bus example, relaxing voltage bounds yielded 0.6443 p.u. at bus 6 and an over-limit flow on line (2, 6); the high-shunt-capacitance cable and bus-6 reactor affected the voltage profile.Source: A. Motivation: effect of topology… · pp. 8–9

1) Sectioning constraints:

  • The model separates disjoint target bus sets B0 and B1, while optimization assigns other buses to sections to achieve balance and optimize the objective.Sectioning constraints enforce separation by switching lines without fixing the precise boundaries in advance.Source: 1) Sectioning constraints · pp. 9–10

2) Power flow:

  • After islanding, generators may make small output changes or switch off, while loads may be partly or fully shed to maintain balance.Bus voltages, angles, and line flows are modeled because they change with the topology and operating adjustments.Source: 2) Power flow · p. 10
  • Line-specific variables make disconnected-line flows zero regardless of bus voltages and angles, while connected lines recover the normal power-flow equations.The line and bus variables are linked so disconnected lines can have independent end-bus angles and minimum line-voltage values.Source: 2) Power flow · pp. 10–11
  • Generator ramping and reserve limits shape feasible post-islanding adjustments; the examples assume a two-minute ramping interval and omit fast-start units.The formulation allows other time limits, informed by post-contingency response protocols.Source: 3) Operating constraints · p. 11

3) Operating constraints:

Operating constraints bound generator outputs, permit intentional load shedding, and apply line limits.

3) Operating constraints:

  • Generator real-power changes are bounded around pre-islanding output by ramp and output limits and available reserve; the examples assume two minutes for ramping.Reactive output is allowed within a specified range, and the formulation permits other ramping time limits.Source: 3) Operating constraints · p. 11
  • The examples treat real- and reactive-power bounds independently, a stated approximation that may need coupling constraints near a generator’s real-power upper limit.The passage describes the feasible output region as a narrow slice through a convex set and says adding coupling constraints is straightforward when independence is unsuitable.Source: 3) Operating constraints · p. 11
  • Binary ζ_g represents the generator real-power on/off setting, and generators in G1 are required to remain on.Load shedding is intentional rather than automatic, and line limits are applied as constraints.Source: 3) Operating constraints · pp. 11–12

4) Rules for other component switching:

The shunt-component switching formulation is explored in the 24-bus example.

4) Rules for other component switching:

C. Objective functions for islanding

The objectives target viable islands through reduced load and generator disruption, isolation of uncertain areas, and generator coherency. They balance expected load supply against the preference to separate weakly coupled machines.

C. Objective functions for islanding

  • Minimizing load disruption and generator disturbance is framed as improving the chance that a partition remains viable until restoration.The general aim is a feasible operating state after separating sections 0 and 1.Source: C. Objective functions for islanding · p. 12

1) Isolating uncertain regions and maximizing expected load supply:

  • For a suspected cascading-failure area, B0 and L0 include its buses and lines plus uncertain components, with section 1 constrained to exclude uncertain lines.The approach seeks to isolate the troubled area while leaving the rest in a known, secure steady state.Source: C. Objective functions for islanding · p. 12
  • Expected load-supply value rewards loads in section 1 more than in potentially failing section 0, favoring a smaller impacted area.The section-0 reward is discounted by β_d, interpreted as the probability that a load placed there can be supplied.Source: 1) Isolating uncertain regions and… · p. 12

2) Promoting generator coherency:

  • Generator coherency aims to separate groups along weak connections, since weak machine connections give rise to slow, lightly damped oscillations.Slow-coherency analysis can determine the machine groupings.Source: 2) Promoting generator coherency · p. 13
  • The coherency objective uses dynamic coupling W_ij and a separation indicator η_ij, favoring weakly coupled machines in different sections and stronger coupling within each section.It can be combined with expected load supply, weighted by k > 0, to keep strongly coupled machines together while maximizing expected supply.Source: 2) Promoting generator coherency · p. 13
  • When coherent groups are preassigned to B0 and B1, sectioning constraints separate them while optimization assigns other buses to minimize load shed.Minimizing the coherency objective alone favors keeping all machines together, so additional constraints may be needed to force separation.Source: 2) Promoting generator coherency · p. 13, 3) Splitting into coherent groups · p. 13

3) Splitting into coherent groups:

The coherent-group formulation can minimize changes to generator outputs while enforcing a two-section split; small objective penalties can also guide the solver toward particular solutions and improve computation.

3) Splitting into coherent groups:

  • The formulation can seek generator outputs as close as possible to their initial values while splitting machines into two sections.If more separation is needed, the optimization can be rerun on each resulting island.Source: 3) Splitting into coherent groups · p. 14

4) Penalties:

  • Small additional objective penalties can improve computation by encouraging integral binary variables in relaxations and guiding the solver toward particular solutions.They are intended to be small enough not to significantly affect the primary objective.Source: 4) Penalties · p. 14
  • Penalizing 1 − cos θ_l helps reduce line loss, while line-cut penalties can prioritize retaining high-flow lines; reports this weighting more often leads to dynamically stable solutions.A small multiple of the pre-islanding flow can set the line-cut penalty.Source: 4) Penalties · p. 14
  • Generator-disconnection penalties can be weighted uniformly or set to discourage switching off larger units.The latter is achieved by assigning greater weights to larger units.Source: 4) Penalties · p. 14

D. Overall formulation

The study evaluates islanding objectives on generated test-system scenarios, then checks candidate islands with AC optimal load shedding. In the bus-6 example, both piecewise-linear AC solutions satisfy AC constraints, with lower expected load supply than the DC solution.

D. Overall formulation

  • The overall problem optimizes a chosen islanding objective subject to sectioning, line-switching, power-balance, and power-flow constraints.Source: D. Overall formulation · p. 14
  • The formulation also imposes the piecewise-linear approximation, line-flow limits, and load-shedding constraints.Source: D. Overall formulation · p. 14

A. Islanding to minimize expected load loss

1) AC-feasible islanding of 24-bus network:

Finding an islanding solution within a fixed time can matter, particularly when reacting after a fault. The tests examine the best feasible solution found over 5–45 seconds.

1) AC-feasible islanding of 24-bus network:

  • The best feasible solution was tested at fixed solve times from 5 to 45 seconds, since finding optimal or near-optimal solutions can take an unpredictable amount of time.Short-time feasible solutions are especially relevant when islanding follows a fault.Source: 2) Computation time · p. 15

2) Computation time:

The PWL AC approach found islanding solutions quickly and avoided the AC-infeasible islands observed with the DC model. The examples also show trade-offs between preserving coherent generators and meeting island operating requirements.

2) Computation time:

  • Good islanding solutions were found for all networks within 30 seconds, usually sooner; topology usually changed little or not at all between solutions returned at 5 and 45 seconds.All test cases with 39 or fewer buses solved to negligible % gaps within 5 seconds and are not shown.Source: 2) Computation time · p. 16

3) AC feasibility:

  • 20% of cases using the DC model led to AC-infeasible islands, whereas none found using the PWL AC model were infeasible.The returned PWL islanding solutions also produced AC solutions with a mean objective error below 0.02%.Source: 3) AC feasibility · p. 16, 2) Computation time · p. 16
  • With the joint objective at k = 100, the 24-bus solution kept buses 1 and 2 connected and had expected load supply of 2670 MW, versus 2699 MW for the load-only objective.The joint objective yielded Jcoh = 0.00; the load-only solution that switched line (1, 2) had Jcoh = 2.26.Source: 4) Promoting generator coherency · p. 16, Table II · p. 16

B. Coherency-based islanding

  • For the 39-bus coherency-based split, bus 32’s generator output fell from 671 to 373 MW and 311 MW was shed; no alternative required less total generator-output change.The groups were buses 30, 31, and 39, and all other generator buses.Source: B. Coherency-based islanding · p. 17, Table IV · p. 17

V. CONCLUSIONS AND FUTURE WORK

  • The framework supports different islanding objectives while finding islands balanced in real and reactive power and satisfying power-flow and operating constraints.The conclusions also report demonstrating objectives that promote generator coherency.Source: §V · p. 17
  • Future work includes detailed simulations on representative networks and blackout scenarios, including transient and dynamic performance; decomposition and aggregation are being explored to improve efficiency for larger networks.The paper identifies these as future work and current work, respectively.Source: §V · pp. 17–18
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