Constructing Convex Inner Approximations of Steady-State Security Regions

We propose a scalable optimization framework for estimating convex inner approximations of the steady-state security sets. The framework is based on Brouwer fixed point theorem applied to a fixed-point form of the power flow equations. It establishes a certificate for the self-mapping of a polytope...

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Bibliographic Details
Published in:IEEE transactions on power systems 2019-01, Vol.34 (1), p.257-267
Main Authors: Nguyen, Hung D., Dvijotham, Krishnamurthy, Turitsyn, Konstantin
Format: Article
Language:English
Subjects:
Approximation
Dependence
ENGINEERING
Feasibility
Fixed points (mathematics)
Flow equations
inner approximation
Jacobian matrices
Mapping
Mathematical analysis
Mathematical model
nonconvexity
Nonlinear equations
OPF
Optimization
Polytopes
Power flow
Power systems
Security
solvability
Steady state
Uncertainty
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Summary:We propose a scalable optimization framework for estimating convex inner approximations of the steady-state security sets. The framework is based on Brouwer fixed point theorem applied to a fixed-point form of the power flow equations. It establishes a certificate for the self-mapping of a polytope region constructed around a given feasible operating point. This certificate is based on the explicit bounds on the nonlinear terms that hold within the self-mapped polytope. The shape of the polytope is adapted to find the largest approximation of the steady-state security region. While the corresponding optimization problem is nonlinear and non-convex, every feasible solution found by local search defines a valid inner approximation. The number of variables scales linearly with the system size, and the general framework can naturally be applied to other nonlinear equations with affine dependence on inputs. Test cases, with the system sizes up to 1354 buses, are used to illustrate the scalability of the approach. The results show that the approximated regions are not unreasonably conservative and that they cover substantial fractions of the true steady-state security regions for most medium-sized test cases.
ISSN:0885-8950
1558-0679
DOI:10.1109/TPWRS.2018.2868752