arXiv subjects

Dongchan Lee

Publications and source records attributed to Dongchan Lee.

9 recordsLinked to original sources

arXiv2025

A biconvex method for minimum-time motion planning through sequences of convex sets

We consider the problem of designing a smooth trajectory that traverses a sequence of convex sets in minimum time, while satisfying given velocity and acceleration constraints. This problem is naturally formulated as a nonconvex program. To solve it, we propose a biconvex method that quickly produces an initial trajectory and iteratively refines it by solving two convex subproblems in alternation. This method is guaranteed to converge, returns a feasible trajectory even if stopped early, and does not require the selection of any line-search or trust-region parameter. Exhaustive experiments show that our method finds high-quality trajectories in a fraction of the time of state-of-the-art solvers for nonconvex optimization. In addition, it achieves runtimes comparable to industry-standard waypoint-based motion planners, while consistently designing lower-duration trajectories than existing optimization-based planners.

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arXiv2021

Robust Model Predictive Control for Nonlinear Systems Using Convex Restriction

We present an algorithm for robust model predictive control with consideration of uncertainty and safety constraints. Our framework considers a nonlinear dynamical system subject to disturbances from an unknown but bounded uncertainty set. By viewing the system as a fixed point of an operator acting over trajectories, we propose a convex condition on control actions that guarantee safety against the uncertainty set. The proposed condition guarantees that all realizations of the state trajectories satisfy safety constraints. Our algorithm solves a sequence of convex quadratic constrained optimization problems of size n*N, where n is the number of states, and N is the prediction horizon in the model predictive control problem. Compared to existing methods, our approach solves convex problems while guaranteeing that all realizations of uncertainty set do not violate safety constraints. Moreover, we consider the implicit time-discretization of system dynamics to increase the prediction horizon and enhance computational accuracy. Numerical simulations for vehicle navigation demonstrate the effectiveness of our approach.

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arXiv2021

Robust AC Optimal Power Flow with Robust Convex Restriction

Electric power grids regularly experience uncertain fluctuations from load demands and renewables, which poses a risk of violating operational limits designed to safeguard the system. In this paper, we consider the robust AC OPF problem that minimizes the generation cost while requiring a certain level of system security in the presence of uncertainty. The robust AC OPF problem requires that the system satisfy operational limits for all uncertainty realizations within a specified uncertainty set. Guaranteeing robustness is particularly challenging due to the non-convex, nonlinear AC power flow equations, which may not always have a solution. In this work, we extend a previously developed convex restriction to a robust convex restriction, which is a convex inner approximation of the non-convex feasible region of the AC OPF problem that accounts for uncertainty in the power injections. We then use the robust convex restriction in an algorithm that obtains robust solutions to AC OPF problems by solving a sequence of convex optimization problems. We demonstrate our algorithm and its ability to control robustness versus operating cost trade-offs using PGLib test cases.

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arXiv2020

Feasible Path Identification in Optimal Power Flow with Sequential Convex Restriction

Nonconvexity induced by the nonlinear AC power flow equations challenges solution algorithms for AC optimal power flow (OPF) problems. While significant research efforts have focused on reliably computing high-quality OPF solutions, it is not always clear that there exists a feasible path to reach the desired operating point. Transitioning between operating points while avoiding constraint violations can be challenging since the feasible space of the OPF problem is nonconvex and potentially disconnected. To address this problem, we propose an algorithm that computes a provably feasible path from an initial operating point to a desired operating point. Given an initial feasible point, the algorithm solves a sequence of convex quadratically constrained optimization problems over conservative convex inner approximations of the OPF feasible space. In each iteration, we obtain a new, improved operating point and a feasible transition from the operating point in the previous iteration. In addition to computing a feasible path to a known desired operating point, this algorithm can also be used to improve the operating point locally. Extensive numerical studies on a variety of test cases demonstrate the algorithm and the ability to arrive at a high-quality solution in few iterations.

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arXiv2019

Sequential Convex Restriction and its Applications in Robust Optimization

This paper presents a convex sufficient condition for solving a system of nonlinear equations under parametric changes and proposes a sequential convex optimization method for solving robust optimization problems with nonlinear equality constraints. By bounding the nonlinearity with concave envelopes and using Brouwer's fixed point theorem, the sufficient condition is expressed in terms of closed-form convex inequality constraints. We extend the result to provide a convex sufficient condition for feasibility under bounded uncertainty. Using these conditions, a non-convex optimization problem can be solved as a sequence of convex optimization problems, with feasibility and robustness guarantees. We present a detailed analysis of the performance and complexity of the proposed condition. The examples in polynomial optimization and nonlinear network are provided to illustrate the proposed method.

math.OC↗

arXiv2019

Convex Restriction of Power Flow Feasibility Sets

The convex restriction of the power flow feasible sets identifies the convex subset of power injections where the solution for power flow is guaranteed to exist and satisfy the operational constraints. In contrast to convex relaxations, the convex restriction provides a sufficient condition for power flow feasibility and is particularly useful for problems involving uncertainty in the power generation and demand. In this paper, we present a general framework of constructing convex restriction of an algebraic set defined by equality and inequality constraints and apply the framework to power flow feasibility problem. The procedure results in convex quadratic constraints that provide a sufficiently large region for practical operation of the grid.

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arXiv2018

Robustness against Disturbances in Power Systems under Frequency Constraints

The wide deployment of renewable generation and the gradual decrease in the overall system inertia make modern power grids more vulnerable to transient instabilities and unacceptable frequency fluctuations. Time-domain simulation-based assessment of the system robustness against uncertain and stochastic disturbances is extremely time-consuming. In this paper, we develop an alternative approach, which has its roots in the input-output stability analysis for Lur'e systems. Our approach consists of a mathematically rigorous characterization of the external disturbances that the power system is transiently stable and the frequency constraints are not violated. The derived certificate is efficiently constructed via convex optimization and is shown to be non-conservative for different IEEE test cases.

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arXiv2018

A Robustness Measure of Transient Stability under Operational Constraints in Power Systems

The aggressive integration of distributed renewable sources is changing the dynamics of the electric power grid in an unexpected manner. As a result, maintaining conventional performance specifications, such as transient stability, may not be sufficient to ensure its reliable operation in stressed conditions. In this paper, we introduce a novel criteria in transient stability with consideration of operational constraints over frequency deviation and angular separation. In addition, we provide a robustness measure of the region of attraction, which can quantify the ability of the post-fault system to remain synchronized even under disturbances. To assess this new stability specification, we adopt the notion of Input-to-State Stability (ISS) to the context of power systems and introduce a new class of convex Lyapunov functions, which will result in tractable convex-optimization-based stability certificates. As a result, we are able to quantify the level of disturbance a power system can withstand while maintaining its safe operation. We illustrate the introduced stability specification and certificate on the IEEE 9 bus system.

math.OC↗

arXiv2017

Dynamic Polytopic Template Approach to Robust Transient Stability Assessment

Transient stability assessment of power systems needs to account for increased risk from uncertainties due to the integration of renewables and distributed generators. The uncertain operating condition of the power grid hinders reliable assessment of transient stability. Conventional approaches such as time-domain simulations and direct energy methods are computationally expensive to take account of uncertainties. This paper proposes a reachability analysis approach that computes bounds of the possible trajectories from uncertain initial conditions. The eigenvalue decomposition is used to construct a polytopic template with a scalable number of hyperplanes that is guaranteed to converge near the equilibrium. The proposed algorithm bounds the possible states at a given time with a polytopic template and solves the evolution of the polytope over time. The problem is solved with linear programming relaxation based on outer-approximations of nonlinear functions, which is scalable for large scale systems. We demonstrate our method on IEEE test cases to certify the stability and bound the state trajectories.

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