Mathematical programming formulations for the alternating current optimal power flow problem

Bienstock, Daniel; Escobar, Mauro; Gentile, Claudio; Liberti, Leo

doi:10.1007/s10479-021-04497-z

Cited by 11 publications

(10 citation statements)

References 95 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A power grid is as a network of buses interconnected by lines. It is modelled [18] as an oriented multi-graph N = (B, L) with sizes n = |B| and l = |L|. The orientation of a line is an arbitrary but necessary convention to uniquely define the admittance matrix (see below) modelling this line.…”

Section: A Problem Formulationmentioning

confidence: 99%

Certified and accurate SDP bounds for the ACOPF problem

Oustry

d'Ambrosio

Liberti

et al. 2022

Electric Power Systems Research

Self Cite

View full text Add to dashboard Cite

Section: A Problem Formulationmentioning

confidence: 99%

Certified and accurate SDP bounds for the ACOPF problem

Oustry

d'Ambrosio

Liberti

et al. 2022

Electric Power Systems Research

Self Cite

View full text Add to dashboard Cite

“…Notice that G is the collection of generators and N is the collection of buses. For a full description on the AC-OPF problem, the reader may refer to [7]. We will consider AC-OPF instances for which the following assumption holds.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Note that in (5.2), if we substitute the expression of S ij in |S ij | ≤ s u ij , i.e., S ij Sij ≤ (s u ij ) 2 , we will actually get a quartic constraint. To implement Shor's relaxation for QCQP, we then relax it to a quadratic constraint by using the trick described in [7]. The minimum initial relaxation step of the complex hierarchy (Q cs-ts min,1 ) for (5.2) is able to provide a tighter lower bound than Shor's relaxation, which we refer to as the 1.5th order relaxation.…”

Section: Ac-opf Assumptionmentioning

confidence: 99%

“…For the sake of brevity, we assume that there are only inequality constraints in (1.1) in the rest of this paper. CPOP (1.1) arises naturally from diverse areas, such as imaging science [15], signal processing [2,4,26], automatic control [28], quantum mechanics [17], optimal power flow [6]. By introducing real variables for the real part and the imaginary part of each complex variables respectively, CPOP (1.1) can be converted into a polynomial optimization problem (POP) involving only real variables.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Exploiting Sparsity in Complex Polynomial Optimization

Wang,

Magron

2021

Preprint

View full text Add to dashboard Cite

In this paper, we study the sparsity-adapted complex moment-Hermitian sum of squares (moment-HSOS) hierarchy for complex polynomial optimization problems, where the sparsity includes correlative sparsity and term sparsity. We compare the strengths of the sparsity-adapted complex moment-HSOS hierarchy with the sparsity-adapted real moment-SOS hierarchy on either randomly generated complex polynomial optimization problems or the AC optimal power flow problem. The results of numerical experiments show that the sparsity-adapted complex moment-HSOS hierarchy provides a trade-off between the computational cost and the quality of obtained bounds for large-scale complex polynomial optimization problems.

show abstract

“…Table 6 maps a number of published formulations to BIM/BFM categories and the variable spaces in which they are defined. We refer to [6,7] for recent in-depth reviews of mathematical formulations for the OPF problem.…”

Section: State Of the Art On Unbalanced (O)pfmentioning

confidence: 99%

Real-Value Power-Voltage Formulations of, and Bounds for, Three-Wire Unbalanced Optimal Power Flow

Geth¹,

Ergun²

2021

Preprint

View full text Add to dashboard Cite

Unbalanced optimal power flow refers to a class of optimization problems subject to the steady state physics of three-phase power grids with nonnegligible phase unbalance. Significant progress on this problem has been made on the mathematical modelling side of unbalanced OPF, however there is a lack of information on implementation aspects as well as data sets for benchmarking. One of the key problems is the lack of definitions of current and voltage bounds across different classes of representations of the power flow equations. Therefore, this tutorial-style paper summarizes the structural features of the unbalanced (optimal) power problem for three-phase systems. The resulting nonlinear complex-value matrix formulations are presented for both the bus injection and branch flow formulation frameworks, which typically cannot be implemented as-is in optimization toolboxes. Therefore, this paper also derives the equivalent real-value formulations, and discusses challenges related to the implementation in optimization modeling toolboxes. The derived formulations can be re-used easily for continuous and discrete optimization problems in distribution networks for a variety of operational and planning problems. Finally, bounds are derived for all variables involved, to further the development of benchmarks for unbalanced optimal power flow, where consensus on bound semantics is a pressing need. We believe benchmarks remain a cornerstone for the development and validation of scalable and reproducible optimization models and tools. The soundness of the derivations is confirmed through numerical experiments, validated w.r.t. OpenDSS for IEEE test feeders with 3x3 impedance matrices. NomenclatureThis article depends on the definition of a variety of scalar, vector and matrix parameters and variables related to grid buses and branches (see Tables 1-5). The core variables are current, voltage and power, whereas parameters are mainly impedance and variable bounds. Fig. 1 summarizes the variables and parameters defined in the fundamental 3×3 branch model for which the well known The Π-equivalent model used. With respect to balance networks, both series and shunt elements are represented by full complex-valued matrices including the mutual impedance coupling between the conductors. Using the Π-equivalent model, the branch current can be split into a series component s and a shunt component sh , respectively. All circuit element voltages are defined w.r.t. (local) ground voltage , = 0 . Table 1 illustrates typography and mathematical notation used throughout; Table 2 defines sets and indices; Table 3 defines parameters; Table 4 defines typical engineering variables; Table 5 defines lifted variables. Black and red colors indicate real-valued variables and parameters, respectively. Blue and brown colors are used for complex-valued variables and parameters, respectively.

show abstract

Mathematical programming formulations for the alternating current optimal power flow problem

Cited by 11 publications

References 95 publications

Certified and accurate SDP bounds for the ACOPF problem

Certified and accurate SDP bounds for the ACOPF problem

Exploiting Sparsity in Complex Polynomial Optimization

Real-Value Power-Voltage Formulations of, and Bounds for, Three-Wire Unbalanced Optimal Power Flow

Contact Info

Product

Resources

About