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A Linear-Programming Approximation of AC Power Flows

Carleton Coffrin, Pascal Van Hentenryck

arXiv:1206.3614v3cs.AImath.OC

TL;DR

The paper addresses the inability of active-power-only LDC approximations to represent reactive power and voltage magnitudes needed for broader AC-feasibility applications. It proposes LPAC linear programs using coupled angle–magnitude modeling and linearized nonlinear terms, and reports high accuracy against AC solutions with proof-of-concept applications in restoration and capacitor placement.

  • Problem

    The LDC model omits reactive power and voltage magnitudes, limiting applications such as capacitor placement and voltage stability.

  • Method

    LPAC models approximate AC power flow by jointly modeling voltage phase angles and magnitudes, using piecewise linear cosine terms and Taylor approximations for remaining nonlinear terms.

  • Results

    Comparisons with AC solutions on standard IEEE and MatPower benchmarks show high accuracy for active and reactive power, phase angles, and voltage magnitudes.

  • Takeaways & Limitations

    The LPAC models can serve as building blocks for optimization problems involving reactive-power flow and voltage-magnitude constraints, including restoration and capacitor placement.

Abstract

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Linear active-power-only DC power flow approximations are pervasive in the planning and control of power systems. However, these approximations fail to capture reactive power and voltage magnitudes, both of which are necessary in many applications to ensure voltage stability and AC power flow feasibility. This paper proposes linear-programming models (the LPAC models) that incorporate reactive power and voltage magnitudes in a linear power flow approximation. The LPAC models are built on a convex approximation of the cosine terms in the AC equations, as well as Taylor approximations of the remaining nonlinear terms. Experimental comparisons with AC solutions on a variety of standard IEEE and MatPower benchmarks show that the LPAC models produce accurate values for active and reactive power, phase angles, and voltage magnitudes. The potential benefits of the LPAC models are illustrated on two "proof-of-concept" studies in power restoration and capacitor placement.

NOMENCLATURE

The paper motivates LPAC models as linear-programming approximations that bridge the LDC model and AC power flow by retaining voltage magnitudes and reactive power.

  • The LDC model is computationally attractive but omits reactive power and voltage magnitudes needed for capacitor placement and voltage-stability applications.
  • The paper introduces LPAC linear programs to approximate AC power flow while bridging the gap between LDC and AC formulations.
  • LPAC models jointly represent voltage phase angles and magnitudes, coupled through active- and reactive-power equations.
  • Their approximations use piecewise linear cosine terms and Taylor series for the remaining nonlinear terms.
  • Experiments on standard IEEE and MatPower benchmarks report high accuracy for active and reactive power, phase angles, and voltage magnitudes.
  • The models are presented alongside applications in power restoration and capacitor placement as potential optimization uses.

B. An Alternate Representation

The paper favors explicit bus and line equations over the concise Y-Bus representation because they make power-flow reasoning more transparent while preserving directional asymmetry.

  • The Y-Bus formulation is concise but makes it difficult to reason about the power-flow equations.
  • The paper therefore uses explicit bus and line equations to represent the power-flow relationships.
  • The resulting equations remain asymmetric, and the line admittance values are not modified.

C. Extensions for Practical Power Networks

The AC formulation is extended to practical network components, while the traditional LDC model is retained as a benchmark derived under normal-operating assumptions.

  • The formulation supports line charging, transformers, and bus shunts found in nearly all AC system benchmarks.
  • Line Charging: Line charging is modeled by assigning equal portions of a line charge to both ends through the Y-Bus matrix.
  • Transformers: Transformer models use tap ratio and phase shift, with transformer direction important for representing both properties correctly.
  • Transformers: Line charging must be incorporated before the transformer calculation.
  • Bus Shunts: Bus shunts are fixed admittances to ground and are unaffected by transformers because they apply to buses rather than lines.
  • The Linearized DC Power Flow: The LDC approximation assumes voltage magnitudes remain close to 1.0 and vary little under normal operating conditions.
  • The Linearized DC Power Flow: This simple linear formulation serves as the baseline in the experimental results.

III. LINEAR-PROGRAMMING APPROXIMATIONS

The LPAC approach approximates AC power flows linearly by modeling phase angles and voltage magnitudes, using convex piecewise-linear cosine terms and Taylor approximations. Hot-start models use an AC base-point solution and capture reactive power while retaining a linear-programming formulation.

  • Hot-Start LPAC Model: Hot-start LPAC models use voltage magnitudes from a solved AC base-point solution.This context provides additional information, including base-point voltage magnitudes, for the approximation.
  • AC Power Flow Behavior: Phase angle differences primarily determine active power flow, while voltage changes primarily determine reactive power flow.The experiments report that voltage has limited impact on active power, whereas voltage changes can cross as many as seven reactive-power contour lines.
  • Hot-Start LPAC Model: The hot-start formulation replaces AC power equations with linear equations that approximate reactive power.Its linear program maximizes the cosine approximation and includes slack-bus and bus-level power-balance constraints.
  • Hot-Start LPAC Model: Fixing base-point voltage magnitudes can be too restrictive, motivating models that remove this restriction.The paper identifies this limitation immediately after presenting the hot-start model.

C. The Warm-Start LPAC Model

The warm-start LPAC model replaces hot-start voltages with target voltages and linearizes reactive power around those targets. It is designed for networks that must remain close to specified operating voltages.

  • The Warm-Start LPAC Model: Warm-start LPAC assumes target voltage magnitudes are available and operation must remain close to them.The paper gives ±0.1 Volts p.u. as an example of the required operating proximity to avoid hardware damage or voltage collapse.
  • The Warm-Start LPAC Model: The active-power approximation remains the hot-start formulation with target voltages replacing base-point voltages.Reactive power is instead approximated by reasoning about voltage-magnitude changes, identified as its primary flow factor.
  • The Warm-Start LPAC Model: The warm-start model defines voltage change as the difference between target voltage and true voltage.This change is used to derive the reactive-power approximation around zero voltage deviations and zero angle difference.
  • The Warm-Start LPAC Model: The reactive-power expression is linearized after separating terms represented by target voltages from terms containing voltage-change products.The remaining component includes nonlinear, non-convex terms such as φnφm cos(θ°n − θ°m).
  • The Warm-Start LPAC Model: Model 2 is a linear program whose inputs include the network, target voltage magnitudes, and cosine approximation segment count.Its constraints represent the slack bus and voltage-controlled generators in addition to the power-flow approximation.

D. The Cold-Start LPAC Model

The cold-start LPAC model handles cases without target voltages by approximating non-generator voltage magnitudes at 1.0. It is derived from the warm-start model with generator voltages retained when known.

  • The Cold-Start LPAC Model: Cold-start LPAC applies when no target voltages are available.Voltage magnitudes are approximated by 1.0 except at voltage-controlled generators with specified voltages.
  • The Cold-Start LPAC Model: The cold-start model is derived from the warm-start model by fixing target voltage magnitudes to 1 for all buses.Under this substitution, the warm-start reactive-power expression reduces to the cold-start form.
  • The Cold-Start LPAC Model: Cold-start generator voltage magnitudes are fixed through constraints on the voltage-change variables.Figure 3 notes that constraints M3.3 use φi to fix generator voltage magnitudes.

E. Extensions to the LPAC Model

LPAC models can be extended to operational and planning applications by adding generator limits, removing the comparison-only slack bus, modeling load shedding, and incorporating feasibility constraints.

  • Extensions to the LPAC Model: LPAC models extend to disaster management, reactive voltage support, transmission planning, and vulnerability analysis.The extensions are illustrated with the warm-start model and can similarly be applied to the cold-start model.
  • Extensions to the LPAC Model: Generator operating ranges are added by removing a generator from G and placing limits on its p and q variables.Voltage-controlled generators can instead be accommodated by fixing φn to zero at the relevant bus.
  • Extensions to the LPAC Model: The slack bus is included for meaningful AC-versus-LPAC comparisons but is not inherently required by the LPAC model.Generator operating ranges can remove the need for a slack bus when total power balance is modeled without it.
  • Extensions to the LPAC Model: Load shedding is modeled by making loads bounded decision variables and maximizing served load.The cosine-maximization objective remains included with a smaller weight in the restoration formulation.
  • Extensions to the LPAC Model: Because Model 2 is linear, it can incorporate voltage, reactive-injection, and line-flow feasibility constraints.Quadratic constraint functions can be approximated with piecewise-linear constraints.

IV. ACCURACY OF THE LPAC MODEL

The LPAC models are evaluated against nonlinear AC power flow on nine IEEE and MATPOWER benchmarks, covering active and reactive power, phase angles, and voltage magnitudes. Results show improved active-power and phase-angle accuracy, generally accurate reactive-power flows, and voltage accuracy that depends on network size and warm-start information.

  • Active power: LPAC models uniformly improve active-power flows, especially on the large IEEE118, IEEEdd17, and MP300 benchmarks.The model captures line losses through asymmetrical power-flow equations and the cosine approximation.
  • Phase angles: LPAC phase-angle accuracy improves significantly, particularly on larger benchmarks, although correlations remain lower than for active power.Phase angles are numerically challenging to approximate accurately.
  • Reactive power: Reactive-power flows are generally accurate and highly precise in warm-start contexts; cold-start correlations are shown for IEEEdd17m and MP300.Figure 2 highlights the two worst cold-start benchmarks.
  • Voltage magnitudes: Voltage magnitudes are very accurate on small benchmarks, but accuracy decreases with network size; warm starts significantly improve larger-network results.Cold-start errors accumulate over many lines and are associated with distance from a load point to the nearest generator.
  • Alternative linear models: Combining conductance and cosine approximations yields larger accuracy gains than integrating either approximation independently, especially as networks grow.The comparison covers LPAC-C, LPAC-G, and LPAC-CG variants against the nonlinear solutions.

V. CASE STUDIES

The case studies evaluate LPAC-based optimization models for power restoration and capacitor placement, offering preliminary evidence of efficiency and accuracy in challenging operating conditions.

  • A. Power Restoration: Power restoration uses linear models to generate starting points for AC power-flow solutions after severe network disruptions.The study samples 1,000 outage cases across contingency classes from N−3 through N−20 on IEEE30.
  • A. Power Restoration: LPAC restoration models maximize served load while enforcing generation, reactive-power, and voltage-related operating constraints.The model allows continuous load shedding while maintaining the same power factor for active and reactive load.
  • A. Power Restoration: 99.76% of the 17,000 contingencies were solved by LPAC-R-V, which failed only 40 cases.LPAC-R-V adds reactive-generation and voltage-limit constraints; successive LPAC refinements solved more contingencies than the traditional LDC model.
  • A. Power Restoration: LPAC-R-V produced load shedding only slightly larger than the overly optimistic LDC model for large contingencies.The results support using LPAC-based formulations in restoration models that currently embed LDC approximations.

B. The Capacitor Placement Problem

The capacitor placement case study embeds the cold-start LPAC model in a mixed-integer formulation that places capacitors while satisfying voltage and reactive-power constraints.

  • B. The Capacitor Placement Problem: The capacitor placement problem minimizes the number of capacitors needed to improve voltage stability under injection and voltage constraints.The studied version requires voltages to meet a lower bound while respecting capacitor and generator reactive-power limits.
  • B. The Capacitor Placement Problem: The MIP adds capacitor reactive-injection variables and binary placement indicators to the cold-start LPAC model.Capacitor injections are linked to placement decisions through standard mixed-integer constraints.
  • B. The Capacitor Placement Problem: The modified IEEE57-C benchmark creates significant voltage problems, with several bus voltages below 0.9 before capacitor placement.Transformer taps are set to 1.0 and existing synchronous condensers are removed to create the test instance.
  • B. The Capacitor Placement Problem: The CPP model was extremely accurate, with only minor lower-voltage-bound violations, while often finding the best solution value within seconds.Proving optimality could take as long as five minutes, and the voltage lower bound approached 0.985.
  • B. The Capacitor Placement Problem: Results remained consistent across voltage bounds and demonstrated LPAC benefits for decision-support applications involving reactive power and voltages.The model is presented as a proof-of-concept rather than a comprehensive solution to capacitor placement.

VI. RELATED WORK

Related work includes iterative and convex AC power-flow linearizations; LPAC emphasizes declarative linear programming while retaining reactive-power and voltage reasoning.

  • VI. RELATED WORK: Existing AC power-flow linearizations broadly fall into iterative methods and convex models.The paper contrasts LPAC with fast-decoupled load flow, polynomial approximations, SDP relaxations, and voltage-difference models.
  • VI. RELATED WORK: Iterative methods can be accurate and fast but are difficult to integrate into MIP solvers because MIP requires purely declarative models.LPAC instead forms one large linear system and can be embedded in mixed-integer optimization.
  • VI. RELATED WORK: LPAC permits decision variables and bounds for power, line capacities, voltage magnitudes, and phase angles within a linear-programming formulation.These capabilities distinguish it from iterative load-flow approaches for decision-support applications.
  • VI. RELATED WORK: Some competing convex approaches have unresolved scalability or accuracy limitations, including rapidly growing polynomial formulations and less mature SDP solvers.The cited polynomial approach does not quantify general power-flow accuracy relative to AC solutions, while SDP scalability remains open on practical examples.
  • VI. RELATED WORK: The paper concludes that LPAC models are highly accurate across standard benchmarks and can support optimization applications involving reactive power and voltage magnitudes.The conclusion reports applications in restoration and capacitor placement as evidence for this potential.
  • VI. RELATED WORK: Further study should compare LPAC against AC solvers with distributed slack buses, which may better represent real power systems.The authors note that LPAC can be extended to flexible load and generation at all buses.

APPENDIX A A LINEAR PROGRAMMING APPROXIMATION OF COSINE

The cosine approximation replaces the nonlinear cosine term with evenly spaced tangent inequalities over a chosen convex domain, producing a linear-programming representation.

  • APPENDIX A A LINEAR PROGRAMMING APPROXIMATION OF COSINE: The approximation selects a domain (l, h) and segment count s, then places s tangent inequalities on the cosine function.Figure 4 illustrates the construction with seven linear inequalities.
  • APPENDIX A A LINEAR PROGRAMMING APPROXIMATION OF COSINE: The domain should remain within (−π/2, π/2) so the cosine function is convex for the approximation.In practice, the relevant phase-angle domain is typically very small, making a narrower interval preferable.
  • APPENDIX A A LINEAR PROGRAMMING APPROXIMATION OF COSINE: Each tangent inequality uses the tangent slope −sin(a) and cosine value cos(a) at an evenly spaced point a.The tangent relation is y = −sin(a)(x−a)+cos(a) within the convex domain.
  • APPENDIX A A LINEAR PROGRAMMING APPROXIMATION OF COSINE: The algorithm treats x as the cosine argument and x^c_cos as its approximate cosine value, linking both through linear constraints.Figure 5 provides the procedure for generating the evenly spaced inequalities.
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