Nonlinear Equality Constraints Research Articles

State Estimation Using SCADA and PMU Measurements for Networks Containing Classic HVDC Links

• Hybrid state estimation (HSE) algorithm based on SCADA and PMUs measurements for AC power systems with embedded classic high voltage direct current (HVDC) links. • AC and DC measurements are modeled independently as functions of AC and DC states, respectively. • AC and DC measurements (states) are processed (estimated) simultaneously. • Coupling of AC and DC states is expressed by a set of three nonlinear equality constraints. This paper proposes a hybrid state estimation (HSE) algorithm which employs conventional measurements received from supervisory control and data acquisition (SCADA) system and synchrophasors measured from phasor measurement units (PMUs) for AC power systems with embedded classic high voltage direct current (HVDC) links. DC measurements obtained from the HVDC link are assumed to be sampled at the same rate as AC measurements. Zero injections and AC/DC coupling equations are modeled as equality constraints. By applying the nonlinear weighted least squares (WLS) algorithm with equality constraints, the HSE calculates both AC and DC states simultaneously. The effectiveness of the proposed algorithm is verified by simulations on the IEEE standard 14 and 30 bus test systems. The test results reveal the benefits of the proposed approach in terms of estimation accuracy and convergence speed.

Improved Technique for Autonomous Vehicle Motion Planning Based on Integral Constraints and Sequential Optimization

The study is dedicated to elaborating and analyzing a technique for autonomous vehicle (AV) motion planning based on sequential trajectory and kinematics optimization. The proposed approach combines the finite element method (FEM) basics and nonlinear optimization with nonlinear constraints. There were five main innovative aspects introduced in the study. First, a 7-degree polynomial was used to improve the continuity of piecewise functions representing the motion curves, providing 4 degrees of freedom (DOF) in a node. This approach allows using the irregular grid for roadway segments, increasing spans where the curvature changes slightly, and reducing steps in the vicinity of the significant inflections of motion boundaries. Therefore, the segment length depends on such factors as static and moving obstacles, average road section curvature, camera sight distance, and road conditions (adhesion). Second, since the method implies splitting the optimization stages, a strategy for bypassing the moving obstacles out of direct time dependency was developed. Thus, the permissible area for maneuvering was determined using criteria of safety distance between vehicles and physical limitation of tire–road adhesion. Third, the nodal inequality constraints were replaced by the nonlinear integral equality constraints. In contrast to the generally distributed approach of restricting the planning parameters in nodes, the technique of integral equality constraints ensures the disposition of motion parameters’ curves strictly within the preset boundaries, which is especially important for quite long segments. In this way, the reliability and stability of predicted parameters are improved. Fourth, the seamless continuity of both the sought parameters and their derivatives is ensured in transitional nodes between the planning phases and adjacent global coordinate systems. Finally, the problem of optimization rapidity to match real-time operation requirements was addressed. For this, the quadrature integration approach was implemented to represent and keep all the parameters in numerical form. The study considered cost functions, limitations stipulated by the vehicle kinematics and dynamics, as well as initial and transient conditions between the planning stages. Simulation examples of the predicted trajectories and curves of kinematic parameters are demonstrated. The advantages and limitations of the proposed approach are highlighted.

Solving Combined Optimal Transmission Switching and Optimal Power Flow sequentially as convexificated Quadratically Constrained Quadratic Program

A novel approach to efficiently solve a combined optimal transmission switching and optimal power flow problem is derived and evaluated. The approach is based on a sequentially solved quadratically constrained quadratic program with a new way of convexification. This convexification requires adapted modeling and uses two procedural steps: First, nonlinear equality constraints are eliminated by quadratically approximating their inverse system of functions and inserting it into the objective function and into the inequality constraints. The resulting objective function and inequality constraints are approximated quadratically. Second, the remaining nonconvex parts of the objective function and the inequality constraints are identified by eigenvalue analysis of their Hessian matrices and eliminated by using piecewise linear approximations. Issues of accuracy and convergence are examined and countermeasures are presented. Additionally, the avoidance of islanding and the sequentiality of parallel circuits are considered. Case studies show fast und stable convergence of the approach.

A υ-Constrained Matrix Adaptation Evolution Strategy With Broyden-Based Mutation for Constrained Optimization.

To solve the nonconvex constrained optimization problems (COPs) over continuous search spaces by using a population-based optimization algorithm, balancing between the feasible and infeasible solutions in the population plays an important role over different stages of the optimization process. To keep this balance, we propose a constraint handling technique, called the υ -level penalty function, which works by transforming a COP into an unconstrained one. Also, to improve the ability of the algorithm in handling several complex constraints, especially nonlinear inequality and equality constraints, we suggest a Broyden-based mutation that finds a feasible solution to replace an infeasible solution. By incorporating these techniques with the matrix adaptation evolution strategy (MA-ES), we develop a new constrained optimization algorithm. An extensive comparative analysis undertaken using a broad range of benchmark problems indicates that the proposed algorithm can outperform several state-of-the-art constrained evolutionary optimizers.

A quadratic voltage model with modifications for optimal power flow of meshed networks

A quadratic voltage (QV) model with modifications is proposed to formulate the meshed networks’ optimal power flow (OPF) problem. In the QV model, convex relaxations are imposed without approximated terms. The modifications aim to drive the convex relaxations to be exact so that the obtained solution will be feasible to OPF. In the modifications, the set of nonlinear equality constraints with quadratic and bi-linear terms of voltages’ variables are convexified with the convex–concave procedure (CCP). An iterative algorithm is imposed, of which the convergence and its applicability are proved with the mathematical analysis. The superiority of the QV model with modifications is tested over several systems, which indicates the practicality of the proposed model and algorithm.

Data assimilation with soft constraints (DASC) through a generalized iterative ensemble smoother

This work investigates an ensemble-based workflow to simultaneously handle generic, nonlinear equality and inequality constraints in reservoir data assimilation problems. The proposed workflow is built upon a recently proposed umbrella algorithm, called the generalized iterative ensemble smoother (GIES), and inherits the benefits of ensemble-based data assimilation algorithms in geoscience applications. Unlike the traditional ensemble assimilation algorithms, the proposed workflow admits cost functions beyond the form of nonlinear-least-squares, and has the potential to develop an infinite number of constrained assimilation algorithms. In the proposed workflow, we treat data assimilation with constraints as a constrained optimization problem. Instead of relying on a general-purpose numerical optimization algorithm to solve the constrained optimization problem, we derive an (approximate) closed form to iteratively update model variables, but without the need to explicitly linearize the constraint systems. The established model update formula bears similarities to that of an iterative ensemble smoother (IES). Therefore, in terms of theoretical analysis, it becomes relatively easy to transit from an ordinary IES to the proposed constrained assimilation algorithms, and in terms of practical implementation, it is also relatively straightforward to implement the proposed workflow for users who are familiar with the IES, or other conventional ensemble data assimilation algorithms like the ensemble Kalman filter (EnKF). Apart from the aforementioned features, we also develop efficient methods to handle two noticed issues that would be of practical importance for ensemble-based constrained assimilation algorithms. These issues include localization in the presence of constraints, and the (possible) high dimensionality induced by the constraint systems. We use one 2D and one 3D case studies to demonstrate the performance of the proposed workflow. In particular, the 3D example contains experiment settings close to those of real field case studies. In both case studies, the proposed workflow achieves better data assimilation performance in comparison to the choice of using an original IES algorithm. As such, the proposed workflow has the potential to further improve the efficacy of ensemble-based data assimilation in practical reservoir data assimilation problems.

An Augmented Lagrangian Method Exploiting an Active-Set Strategy and Second-Order Information

In this paper, we consider nonlinear optimization problems with nonlinear equality constraints and bound constraints on the variables. For the solution of such problems, many augmented Lagrangian methods have been defined in the literature. Here, we propose to modify one of these algorithms, namely ALGENCAN by Andreani et al., in such a way to incorporate second-order information into the augmented Lagrangian framework, using an active-set strategy. We show that the overall algorithm has the same convergence properties as ALGENCAN and an asymptotic quadratic convergence rate under suitable assumptions. The numerical results confirm that the proposed algorithm is a viable alternative to ALGENCAN with greater robustness.

Warping-Inclusive Kinematic Coupling in Mixed-Dimension Macro Models for Steel Wide Flange Beam Columns

A warping-inclusive kinematic coupling method to be used in finite-element analysis of members featuring wide-flange cross sections is proposed in this paper. This coupling method is used in mixed-dimension macromodels that combine continuum and beam-column elements to reduce the computational cost of purely continuum finite-element models. The proposed coupling method, utilizing either linear or nonlinear constraint equations, is implemented and validated in a commercial finite-element software; the source code is made publicly available. Case studies indicate that including warping in the coupling formulation is critical for components that may experience coupled local and lateral-torsional buckling. Also highlighted is the potential of macromodels to reduce the total degrees of freedom by up-to about 60%, and computational memory use by up to around 80%, while retaining solution fidelity for beam, column, and panel zone components in steel moment-resting frames. The case studies show that the linear constraint equation formulation may not be suitable for all problems; however, it may still yield acceptable results as long as the level of twisting is insignificant and lateral-torsional buckling is not critical.

A Relaxed and Bound Algorithm Based on Auxiliary Variables for Quadratically Constrained Quadratic Programming Problem

Quadratically constrained quadratic programs (QCQP), which often appear in engineering practice and management science, and other fields, are investigated in this paper. By introducing appropriate auxiliary variables, QCQP can be transformed into its equivalent problem (EP) with non-linear equality constraints. After these equality constraints are relaxed, a series of linear relaxation subproblems with auxiliary variables and bound constraints are generated, which can determine the effective lower bound of the global optimal value of QCQP. To enhance the compactness of sub-rectangles and improve the ability to remove sub-rectangles, two rectangle-reduction strategies are employed. Besides, two ϵ-subproblem deletion rules are introduced to improve the convergence speed of the algorithm. Therefore, a relaxation and bound algorithm based on auxiliary variables are proposed to solve QCQP. Numerical experiments show that this algorithm is effective and feasible.

An affine-scaling interior-point filter line-search algorithm for constrained optimization

This paper presents and analyzes an affine-scaling interior-point algorithm with a filter line-search method for solving nonlinear optimization problems with nonlinear equality constraints and nonnegative variables. In our scheme, we require that a damped Newton’s method is applied to the perturbed first-order necessary conditions to produce a search direction. Some filtered rules for a fixed barrier parameter are used to determine step acceptance. Second-order correction technique is used to reduce infeasibility and overcome the Maratos effect. The global convergence and fast local convergence rate of the proposed algorithm are established under some suitable conditions.

Hybrid Satellite-UAV-Terrestrial Networks for 6G Ubiquitous Coverage: A Maritime Communications Perspective

In the coming smart ocean era, reliable and efficient communications are crucial for promoting a variety of maritime activities. Current maritime communication networks (MCNs) mainly rely on marine satellites and on-shore base stations (BSs). The former generally provides limited transmission rate, while the latter lacks wide-area coverage capability. Due to these facts, the state-of-the-art MCN falls far behind terrestrial fifth-generation (5G) networks. To fill up the gap in the coming sixth-generation (6G) era, we explore the benefit of deployable BSs for maritime coverage enhancement. Both unmanned aerial vehicles (UAVs) and mobile vessels are used to configure deployable BSs. This leads to a hierarchical satellite-UAV-terrestrial network on the ocean. We address the joint link scheduling and rate adaptation problem for this hybrid network, to minimize the total energy consumption with quality of service (QoS) guarantees. Different from previous studies, we use only the large-scale channel state information (CSI), which is location-dependent and thus can be predicted through the position information of each UAV/vessel based on its specific trajectory/shipping lane. The problem is shown to be an NP-hard mixed integer nonlinear programming problem with a group of hidden non-linear equality constraints. We solve it suboptimally by using Min-Max transformation and iterative problem relaxation, leading to a process-oriented joint link scheduling and rate adaptation scheme. As observed by simulations, the scheme can provide agile on-demand coverage for all users with much reduced system overhead and a polynomial computation complexity. Moreover, it can achieve a prominent performance close to the optimal solution.

A Dynamic Alternating Direction of Multipliers for Nonconvex Minimization with Nonlinear Functional Equality Constraints

A Dynamic Alternating Direction of Multipliers for Nonconvex Minimization with Nonlinear Functional Equality Constraints

Hidden invariant convexity for global and conic-intersection optimality guarantees in discrete-time optimal control

Non-convex discrete-time optimal control problems in, e.g., water or power systems, typically involve a large number of variables related through nonlinear equality constraints. The ideal goal is to find a globally optimal solution, and numerical experience indicates that algorithms aiming for Karush–Kuhn–Tucker points often find solutions that are indistinguishable from global optima. In our paper, we provide a theoretical underpinning for this phenomenon, showing that on a broad class of problems the objective can be shown to be an invariant convex function (invex function) of the control decision variables when state variables are eliminated using implicit function theory. In this way, optimality guarantees can be obtained, the exact nature of which depends on the position of the solution within the feasible set. In a numerical example, we show how high-quality solutions are obtained with local search for a river control problem where invexity holds.

Prioritized optimization by Nash games : towards an adaptive multi-objective strategy

This work is part of the development of a two-phase multi-objective differentiable optimization method. The first phase is classical: it corresponds to the optimization of a set of primary cost functions, subject to nonlinear equality constraints, and it yields at least one known Pareto-optimal solution xA*. This study focuses on the second phase, which is introduced to permit to reduce another set of cost functions, considered as secondary, by the determination of a continuum of Nash equilibria, {x̅ε} (ε≥ 0), in a way such that: firstly, x̅0=xA* (compatibility), and secondly, for ε sufficiently small, the Pareto-optimality condition of the primary cost functions remains O(ε2), whereas the secondary cost functions are linearly decreasing functions of ε. The theoretical results are recalled and the method is applied numerically to a Super-Sonic Business Jet (SSBJ) sizing problem to optimize the flight performance.

Deep Conservation: A Latent-Dynamics Model for Exact Satisfaction of Physical Conservation Laws

This work proposes an approach for latent-dynamics learning that exactly enforces physical conservation laws. The method comprises two steps. First, the method computes a low-dimensional embedding of the high-dimensional dynamical-system state using deep convolutional autoencoders. This defines a low-dimensional nonlinear manifold on which the state is subsequently enforced to evolve. Second, the method defines a latent-dynamics model that associates with the solution to a constrained optimization problem. Here, the objective function is defined as the sum of squares of conservation-law violations over control volumes within a finite-volume discretization of the problem; nonlinear equality constraints explicitly enforce conservation over prescribed subdomains of the problem. Under modest conditions, the resulting dynamics model guarantees that the time-evolution of the latent state exactly satisfies conservation laws over the prescribed subdomains.

Set-valued state estimation of nonlinear discrete-time systems with nonlinear invariants based on constrained zonotopes

This paper presents new methods for set-valued state estimation of discrete-time nonlinear systems whose trajectories are known to satisfy nonlinear equality constraints, called invariants (e.g., conservation laws). Set-valued estimation aims to compute tight enclosures of the possible system states in each time step subject to unknown-but-bounded uncertainties. Most existing methods employ a standard prediction-update framework with set-based prediction and update steps based on various set representations and techniques. However, achieving accurate enclosures for nonlinear systems remains a significant challenge. This paper presents new methods based on constrained zonotopes that improve the standard prediction-update framework for systems with invariants by adding a consistency step. This new step uses invariants to reduce conservatism and is enabled by new algorithms for refining constrained zonotopes based on nonlinear constraints. This paper also presents significant improvements to existing prediction and update steps for constrained zonotopes. Specifically, new update algorithms are developed that allow nonlinear measurement equations for the first time, and existing prediction methods based on conservative approximation techniques are modified to allow a more flexible choice of the approximation point, which can lead to tighter enclosures. Numerical results demonstrate that the resulting methods can provide significantly tighter enclosures than existing zonotope-based methods while maintaining comparable efficiency.

Constraint-oriented integrated longitudinal and lateral robust control for connected and automated vehicle platoons

The coupling, nonlinearity, and uncertainty characteristics of vehicle dynamics make the accurate longitudinal and lateral control of an automated and connected vehicle platoon a tough task. Little research has been conducted to fully address the characteristics. By using the ideology of constraint-following control this article proposes an integrated longitudinal and lateral adaptive robust control methodology for a vehicle platoon with a bidirectional communication topology. The platoon control objectives contain the path tracking stability, the platoon internal stability, and the string stability. First, we establish the nonlinear kinematics path tracking model and the coupled vehicle longitudinal and lateral dynamical model that contains time-varying uncertainties. Second, we design a series of nonlinear equality constraints that directly guarantee the control objectives based on the kinematic relations. On this basis, an adaptive robust constraint-following control is proposed. It is shown that the control guarantees the uniform boundedness and the uniform ultimate boundedness of the constraint-following error and the uncertainty estimation error. Finally, simulation results are provided to validate the effectiveness of the proposed methodology.

Complexity of Proximal Augmented Lagrangian for Nonconvex Optimization with Nonlinear Equality Constraints

We analyze worst-case complexity of a Proximal augmented Lagrangian (Proximal AL) framework for nonconvex optimization with nonlinear equality constraints. When an approximate first-order (second-order) optimal point is obtained in the subproblem, an $$\epsilon $$ first-order (second-order) optimal point for the original problem can be guaranteed within $${\mathcal {O}}(1/ \epsilon ^{2 - \eta })$$ outer iterations (where $$\eta $$ is a user-defined parameter with $$\eta \in [0,2]$$ for the first-order result and $$\eta \in [1,2]$$ for the second-order result) when the proximal term coefficient $$\beta $$ and penalty parameter $$\rho $$ satisfy $$\beta = {\mathcal {O}}(\epsilon ^\eta )$$ and $$\rho = \varOmega (1/\epsilon ^\eta )$$ , respectively. We also investigate the total iteration complexity and operation complexity when a Newton-conjugate-gradient algorithm is used to solve the subproblems. Finally, we discuss an adaptive scheme for determining a value of the parameter $$\rho $$ that satisfies the requirements of the analysis.

Strong and weak measurable optimal controls

<abstract> Sufficient conditions for weak and strong minima in an optimal control problem of Lagrange are provided. This sufficiency theory is applicable for problems containing fixed end-points, nonlinear dynamics, nonlinear isoperimetric inequality and equality constraints together with nonlinear mixed time-state-control pointwise inequality and equality restrictions. The presence of purely measurable optimal controls is a fundamental component of this theory. </abstract>

A Nonmonotone Matrix-Free Algorithm for Nonlinear Equality-Constrained Least-Squares Problems

Least squares form one of the most prominent classes of optimization problems, with numerous applications in scientific computing and data fitting. When such formulations aim at modeling complex systems, the optimization process must account for nonlinear dynamics by incorporating constraints. In addition, these systems often incorporate a large number of variables, which increases the difficulty of the problem, and motivates the need for efficient algorithms amenable to large-scale implementations. In this paper, we propose and analyze a Levenberg-Marquardt algorithm for nonlinear least squares subject to nonlinear equality constraints. Our algorithm is based on inexact solves of linear least-squares problems, that only require Jacobian-vector products. Global convergence is guaranteed by the combination of a composite step approach and a nonmonotone step acceptance rule. We illustrate the performance of our method on several test cases from data assimilation and inverse problems: our algorithm is able to reach the vicinity of a solution from an arbitrary starting point, and can outperform the most natural alternatives for these classes of problems.