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Convexity of Energy-Like Functions: Theoretical Results and Applications to Power System Operations

Krishnamurthy Dvijotham, Steven Low, Michael Chertkov

arXiv:1501.04052v3math.OC

TL;DR

Modernizing power systems requires tools to assess operational safety under changing generation, demand response, uncertainty, and tighter reserves. The paper generalizes energy functions by characterizing their convexity domain and using convex optimization to analyze and solve power-flow equations. The resulting domain covers many operationally relevant cases, while IEEE 14- and 118-bus studies illustrate solution-finding and operational applications.

  • Problem

    Power-system transformations create a need for tools that assess whether the current system state remains operationally safe.

    Source: §I · p. 1

  • Method

    The paper characterizes energy-function convexity in voltage magnitudes and phases and uses the resulting structure to formulate convex power-flow analysis and algorithms.

    Source: §I · p. 2, B. Convex Relaxations of OPF · p. 9, B. Theoretical versus Numerical Domain… · p. 11

  • Results

    The convex approach finds power-flow solutions in almost all tested cases when a solution exists, with a potential gap near the insolvability boundary.

    Source: C. Power Flow In Larger Networks:… · p. 14

  • Takeaways & Limitations

    Energy-function convexity can certify power-flow solution existence, uniqueness, or non-existence within the characterized domain and support operational analyses.

    Source: §VI · p. 14

Abstract

from arXiv · show

Power systems are undergoing unprecedented transformations with the incorporation of larger amounts of renewable energy sources, distributed generation and demand response. All these changes, while potentially making power grids more responsive, efficient and resilient, also pose significant implementation challenges. In particular, operating the new power grid will require new tools and algorithms capable of predicting if the current state of the system is operationally safe. In this paper we study and generalize the so-called energy function as a tool to design algorithms to test if a high-voltage power transmission system is within the allowed operational limits. In the past the energy function technique was utilized primarily to access the power system transient stability. In this manuscript, we take a new look at energy functions and focus on an aspect that has previously received little attention: Convexity. We characterize the domain of voltage magnitudes and phases within which the energy function is convex. We show that the domain of the energy function convexity is sufficiently large to include most operationally relevant and practically interesting cases. We show how the energy function convexity can be used to analyze power flow equations, e.g. to certify solution uniqueness or non-existence within the domain of convexity. This and other useful features of the generalized energy function are described and illustrated on IEEE 14 and 118 bus models.

I. INTRODUCTION

The paper generalizes energy functions into a convexity-based framework for analyzing power-flow and other operational problems in changing power systems. It characterizes when convexity holds and uses that structure to support power-flow algorithms and solution analysis.

I. INTRODUCTION

  • The generalized energy-function approach is intended to address operational challenges created by renewable generation, distributed generation, demand response, uncertainty, and reduced operating reserves.The paper positions the approach as a tool for assessing operational safety and margins beyond linearized analysis.Source: §I · p. 1
  • The paper treats energy functions as a unifying tool for power-flow analysis, critical loadability, and optimal power flow because their stationary points correspond to power-flow solutions in lossless systems.The framework extends earlier work to more general energy-like functions and to lossy networks with a fixed resistance-to-reactance ratio.Source: §I · p. 2
  • Convexity is studied because its exact domain had not been characterized beyond the local vicinity of a stable equilibrium.The resulting condition reflects the intuition that neighboring voltage magnitudes and phases should not differ too much.Source: §I · pp. 2–3
  • A nonlinear but convex matrix inequality in voltage phases and magnitudes guarantees energy-function convexity.For acyclic networks, the condition exactly characterizes the convexity domain; for mesh networks, it generally gives an inner approximation.Source: §I · pp. 2–3
  • The paper analyzes convexity of the energy function jointly in voltage magnitudes and phases, uses convexity to design a provably convergent algorithm for PF equations, and illustrates the approach on IEEE 14 and 118-bus networks.Source: §I · p. 3

II. MODELING POWER SYSTEMS

The model represents a transmission network as a graph of buses and lines, using lossless, constant-injection assumptions to define the energy-function setting. It distinguishes bus types and the voltage variables used in the analysis.

II. MODELING POWER SYSTEMS

  • The modeling assumptions take transmission lines to be lossless and nodal active and reactive injections or consumptions to be constant.These are presented as standard assumptions for high-voltage transmission structure-preserving models.Source: §II · p. 3

A. Notation

  • The transmission network is represented as a graph whose vertices are buses and whose edges are transmission lines, supporting a framework for varied load and generator models.An incidence matrix and line admittances provide the network representation.Source: A. Notation · p. 3
  • The notation uses voltage magnitudes Vi, phases θi, injections Pi and Qi, and logarithmic voltage variables ρi = log(Vi).Phase and logarithmic-magnitude differences are denoted θij and ρij.Source: A. Notation · p. 4
  • (P, V) buses fix active power and voltage magnitude, whereas (P, Q) buses fix active and reactive power and leave voltage phase and magnitude variable.A slack bus fixes voltage magnitude and phase while its injections adjust to balance the network.Source: A. Notation · p. 4

B. Background

Power-flow equations are coupled nonlinear balance equations for bus voltages and power injections. The paper introduces them as the operational equations to which the energy-function framework is applied.

1) Power Flow Equations:

  • The nonlinear power-flow equations arise because power is the product of voltages and currents, despite linear underlying circuit laws.They are used for tasks such as determining generation, specifying demand, and minimizing line losses.Source: 1) Power Flow Equations · p. 4
  • The equations assume a balanced quasi-steady state in which dynamic disturbances have been resolved by automatic control actions.The balance condition is appropriate to the roughly five-minute power-flow time scale described in the paper.Source: 1) Power Flow Equations · p. 4
  • Power-flow equations consist of 2n coupled equations in 2n variables, with two of voltage magnitude, phase, active power, and reactive power fixed at each bus.The remaining two variables are solved according to the bus type.Source: 1) Power Flow Equations · pp. 4–5
  • At (P, V) buses, active power and voltage magnitude are fixed, while phase and reactive power adjust; generators are typically modeled this way.Voltage regulators maintain the specified voltage values.Source: 1) Power Flow Equations · p. 5
  • At (P, Q) buses, active and reactive injections are fixed, so voltage phase and magnitude are the variables determined by the equations.Loads are typically represented as constant active and reactive power demands.Source: 1) Power Flow Equations · p. 5
  • The slack bus absorbs network power imbalances, with its injections free and its phase and logarithmic voltage fixed at zero.This supplies the balancing reference for the remaining bus equations.Source: 1) Power Flow Equations · p. 5

2) Energy Functions:

The paper reframes the energy-function potential as a variational representation of power-flow equations and characterizes a convex region containing operationally relevant states. Within this region, convex optimization can certify power-flow existence or non-existence, while equilibria satisfying the conditions are asymptotically stable.

2) Energy Functions:

  • Energy-function potential stationary points correspond to solutions of the power-flow equations, enabling their variational analysis.The paper focuses on this property rather than directly analyzing swing-system dynamics.Source: 2) Energy Functions · pp. 5–6

III. ENERGY FUNCTION AND CONVEXITY

  • The paper defines the convexity domain by requiring convexity along every segment from the origin to a state, selecting the region containing the origin.This handles the possibility of multiple disconnected convex regions.Source: §III · p. 6

A. Why is Convexity Interesting?

  • Convexity enables an optimization problem that either finds a power-flow solution within the domain or certifies that none exists there.The justification is that power-flow solutions are energy-function stationary points.Source: A. Why is Convexity Interesting? · p. 6

1) Asymptotic Stability:

  • An equilibrium satisfying the algebraic power-balance equations and lying in the convexity domain is asymptotically stable under the swing dynamics.The result is stated in terms of stability of the linearization around the equilibrium.Source: 1) Asymptotic Stability · pp. 6–7

2) Existence of Solutions:

The paper characterizes a convexity domain for the energy function and studies its implications for power-flow solution existence. The conditions are exact for trees, sufficient for meshed networks, and numerically encompass most practical solutions.

2) Existence of Solutions:

  • For trees, the power-flow equations have a solution if and only if a solution lies in the designated domain D.Numerical evidence suggests this observation may extend to mesh networks.Source: 2) Existence of Solutions · p. 7
  • Solutions within D are described as desirable because they have small neighboring phase differences and sufficiently large voltage magnitudes at (P, Q) buses.These properties motivate restricting attention to D in practical operation.Source: 2) Existence of Solutions · p. 7

B. Characterization of Domain of Convexity of the Energy Function

  • The energy function is jointly convex on a convex domain C, and for tree networks C equals the full convexity domain with strict convexity in its interior.This is the central characterization underlying the existence results.Source: B. Characterization of Domain of… · p. 7
  • The convexity result also implies asymptotic stability when the equilibrium lies inside the convexity domain.The equilibrium is then located where the energy function has the required convexity property.Source: B. Characterization of Domain of… · p. 8
  • When only (P, V) buses are present, the condition reduces to phase differences below π; mixed (P, Q) connections require additional semidefinite constraints.The mixed-bus condition does not simplify, although numerical results indicate that most practical solutions lie within the convexity domain.Source: B. Characterization of Domain of… · pp. 7–8
  • For tree networks, the convexity conditions are necessary and sufficient within the region whose line phase differences are smaller than π.For meshed networks, the conditions remain sufficient but may not be necessary because edge phase variables are treated independently.Source: B. Characterization of Domain of… · p. 8

C. Solving Power Flow Equations via Convex Optimization

The paper converts power-flow solution search into convex optimization over the energy-function convexity domain. Strict convexity yields uniqueness there, and the formulation can certify whether constrained solutions exist.

C. Solving Power Flow Equations via Convex Optimization

  • Power-flow equations are equivalent to finding a stationary point of the energy function, so convex optimization can search for solutions within C.The paper emphasizes this as one application of the generalized energy-function approach.Source: C. Solving Power Flow Equations via… · p. 8
  • A power-flow solution exists in S ⊂ Int(C) if and only if the corresponding optimization problem has its unique optimum in S.The equivalence follows from the energy-function gradient and KKT conditions.Source: C. Solving Power Flow Equations via… · p. 8
  • Strict convexity permits at most one stationary point in Int(C), so any power-flow solution there is unique.This uniqueness applies within the stated convexity domain, not necessarily globally.Source: C. Solving Power Flow Equations via… · p. 9

A. Principal Singular Surfaces

  • The approach constructs convex inner approximations to the principal region and differs from prior singular-surface analyses in its treatment of general settings.The related principal-region results were developed for lossless networks with all buses of type (P, V).Source: A. Principal Singular Surfaces · p. 9
  • Extending the principal-region analysis and verifying its conjectures for networks with (P, Q) nodes remains future work.The paper notes that the principal region can be nonconvex and may contain only unstable power-flow solutions in some networks.Source: A. Principal Singular Surfaces · p. 9

B. Convex Relaxations of OPF

The paper relates its energy-function formulation to convex OPF relaxations and to prior power-flow existence tests. Its method provides necessary-and-sufficient solution tests and polynomial-time algorithms within the convexity domain, including operational constraints.

B. Convex Relaxations of OPF

  • For lossless networks, the approach applies to arbitrary topologies and could potentially extend from power flow to OPF.The paper primarily addresses power-flow equations rather than OPF.Source: B. Convex Relaxations of OPF · p. 9
  • When S ⊂ C, the proposed OPF-related formulation is a convex-concave saddle-point problem that can be solved using duality.The formulation includes convex injection costs and convex operational constraint sets.Source: B. Convex Relaxations of OPF · pp. 9–10
  • The paper leaves further developments and conditions for equivalence between its minimax formulation and OPF to subsequent work.This marks the boundary of the OPF discussion.Source: B. Convex Relaxations of OPF · p. 10

C. Conditions on Existence of solutions to PF equations

  • Within C, the method finds a power-flow solution if and only if one exists there, unlike approaches that provide only sufficient conditions.The approach is algorithmic and guaranteed to find the solution efficiently in polynomial time.Source: C. Conditions on Existence of… · p. 10
  • The method can additionally test for a power-flow solution satisfying operational constraints such as line-protection or line-flow limits when those constraints define S ⊂ C.This targets solutions that satisfy both the equations and the operational restrictions.Source: C. Conditions on Existence of… · p. 10
  • The relationship between injection-space conditions and whether the corresponding state lies in C remains an open problem, even for all-(P, V) networks.Extending non-conservative existence conditions to settings with (P, Q) buses is identified as future work.Source: C. Conditions on Existence of… · p. 10

V. NUMERICAL ILLUSTRATIONS

Numerical illustrations examine power-flow solutions in a 2-bus system and compare theoretical and numerical convexity domains in a 3-bus system. The results show solutions disappearing at critical loading and theoretical convexity predictions matching actual regions.

A. Existence of Solutions: 2-bus Network

  • In the 2-bus system, two power-flow solutions exist at low load: a stable solution inside the convexity domain and an unstable solution outside it.As load increases, the solutions approach one another and the convexity boundary.Source: A. Existence of Solutions: 2-bus Network · p. 11
  • At a critical loading level, the two 2-bus solutions reach the convexity-domain boundary and vanish, leaving no power-flow solution beyond that level.Source: A. Existence of Solutions: 2-bus Network · p. 11

B. Theoretical versus Numerical Domain of Convexity

C. Power Flow In Larger Networks: Existence of Solutions

The larger-network experiments test whether searching within the energy-function convexity domain can identify power-flow solutions and signal insolvability. On IEEE 14- and 118-bus cases, the convex approach generally agrees closely with an existing insolvability test, with a small boundary gap.

D. Operational Constraints and Convexity

The paper derives simple voltage-ratio and phase-difference bounds that guarantee inclusion in the energy-function convexity domain. It concludes that practical power-flow solutions generally lie within this domain and identifies extensions to broader operational applications and lossy networks.

D. Operational Constraints and Convexity

  • Voltage magnitude-ratio and neighboring phase-difference bounds define a sufficient operational region contained in the energy-function convexity domain.The sufficient set requires exp(|ρ_i − ρ_j|) ≤ b_ρ and |θ_i − θ_j| ≤ b_θ for every network edge.Source: D. Operational Constraints and Convexity · p. 14
  • When |ρi −ρj| ≤ 1.5, phase differences can still be as large as 50 degrees; these bounds are reasonable for practical power systems and seldom exceeded.Source: D. Operational Constraints and Convexity · p. 14
  • With fairly lax voltage-ratio and phase-difference constraints, practical power-flow solutions are guaranteed to lie within the convexity domain.Source: D. Operational Constraints and Convexity · p. 14

VI. CONCLUSIONS AND PATH FORWARD

  • The generalized energy-function framework supports convex optimization for solving power-flow equations and applications including optimal power flow, state and topology estimation, distance to insolvability, and optimal load shedding.Source: §VI · p. 14
  • The reported results focus on lossless networks; extending the convexity and solvability results to lossy networks remains future work.Source: §VI · p. 15

VIII. APPENDIX

The appendix proves convexity of the energy function by reformulating its Hessian as matrix-positive-semidefinite conditions. It establishes that the sufficient convexity domain is exact under the stated tree-network argument.

A. Proof of Theorem III.3

  • The proof reduces energy-function convexity to positive semidefiniteness of Hessian blocks and a Schur-complement condition.The relevant requirements are R ⪰ 0 and M − NR^-1N^T ⪰ 0.Source: A. Proof of Theorem III.3 · pp. 16–17
  • The constructed set C is convex, contains the origin, and satisfies C ⊂ D under the first part of the proof.At the origin, the relevant matrix is positive semidefinite because it is symmetric and diagonally dominant.Source: A. Proof of Theorem III.3 · p. 18
  • For connected tree networks, edge-phase variables correspond one-to-one with non-slack bus phases, allowing the converse inclusion C ⊇ D.Schur complements and invertibility of the reduced incidence matrix connect the edge-variable and bus-variable formulations.Source: A. Proof of Theorem III.3 · p. 18
  • Combining both inclusions yields the exact characterization D = C for the convexity domain.The singular-block cases are handled by a continuity argument.Source: A. Proof of Theorem III.3 · p. 18
  • A matrix-valued convexity lemma applies when phase differences satisfy |y| ≤ π, enabling convexity of the nonlinear Hessian terms.The proof uses scalarization against positive-semidefinite matrices and composition rules.Source: A. Proof of Theorem III.3 · pp. 18–19

B. Lossy Networks

The lossy-network extension assumes a common reactance-to-resistance ratio and rewrites the power-flow equations variationally. Under these assumptions, the resulting energy function is convex over the stated domain.

B. Lossy Networks

  • The lossy-network result assumes transmission lines share a fixed reactance-to-resistance ratio and all buses except the slack bus are (P,Q) buses.Source: B. Lossy Networks · p. 19
  • The power-flow equations are transformed through a linear combination into a variational form suitable for energy-function analysis.Source: B. Lossy Networks · p. 19
  • Theorem VIII.1 states that the energy function for the constant-R/X lossy system is convex over the specified domain.Source: B. Lossy Networks · p. 19
  • For the IEEE 14-bus case, the figure compares the load-scaling point where the convex solver fails with the point where insolvability is detected.Source: Figure 3 · p. 20
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