Quadratic Programming Formulation Research Articles

Real-Time Topology Detection and State Estimation in Distribution Systems Using Micro-PMU and Smart Meter Data

Distribution network topology detection and state estimation in real-time are critical for modern distribution systems management and control. However, number of sensors in distribution networks are limited and communication links between switch devices and distribution management system are not well-established. In this regard, this paper proposes mixed-integer quadratic programming (MIQP) formulations to determine the topology of distribution network and estimate distribution system states simultaneously using micro-phasor measurement units (micro-PMUs) and smart meter data. Two approaches based on AC optimal power flow are proposed and analyzed: (i) polar power-voltage (PPV) formulation, and (ii) rectangular current-voltage (RIV) formulation. The proposed models include convex objective function while constraints are linearized using first-order approximation of Taylor series and Big M method. The proposed models can identify multiple simultaneous switching actions at each time instant and different topology configurations including radial and meshed networks. Only measurement data at each time interval is needed to identify topology and states of the system correctly. The proposed models are tested on a modified IEEE-33 bus system with realistic load data from Pecan Street Inc. database. The results confirm that both models can identify system topology and states with remarkable accuracy in real-time, while RIV model outperforms PPV model.

Classifying random variables based on support vector machine and a neural network scheme

ABSTRACT Support vector machine (SVM) is a supervised machine-learning method which can be used for both classification and regression models. In this paper, we introduce a new model of SVM which any of training samples containing inputs and outputs are considered the random variables with known probability functions. The SVM is first converted into equivalent quadratic programming (QP) formulations in linear and nonlinear cases. An artificial neural network for SVM learning is then proposed. The presented neural network framework guarantees to obtain the optimal solution of the SVM problem. The existence and convergence of the trajectories of the network are studied. The Lyapunov stability for the considered neural network is also shown. The efficiency of the proposed method is shown by four illustrative examples.

Coordinating transportation and pricing policies for perishable products

We study the joint pricing and transportation problem in a perishable product supply chain where products with different levels of quality are substitutable depending on the pricing policy and the customers’ quality sensitivity. We propose two frameworks to study this problem under sequential and integrated systems. Pricing and transportation decisions are made separately and then coordinated through an iterative process in the sequential model, but jointly optimized in the integrated system. The integration of transportation and pricing leads to a challenging mixed-integer quadratic programming (MIQP) formulation which is intractable for the current state-of-the-art commercial solvers. We exploit the special structure of this formulation and develop a Benders type decomposition method with a transportation master problem and a collection of pricing subproblems. To evaluate the proposed models and algorithms, we perform numerical experiments on an illustrative case study. We show that the performance of the sequential model is noticeably inferior to the integrated optimization model on two key metrics: profit and product waste. We observe, counter-intuitively, that the presence of customers with lower quality sensitivity who tolerate imperfect products increases the waste to sales ratio. Moreover, due to the interrelationship between pricing and transportation policies, the waste to sales ratio is not monotone in the cost of the freshness keeping efforts. Finally, we show the practical efficiency of our proposed decomposition-based algorithm in solving the MIQP compared to the state-of-the-art solver.

Variable stiffness locomotion with guaranteed stability for quadruped robots traversing uneven terrains.

Quadruped robots are widely applied in real-world environments where they have to face the challenges of walking on unknown rough terrains. This paper presents a control pipeline that generates robust and compliant legged locomotion for torque-controlled quadruped robots on uneven terrains. The Cartesian motion planner is designed to be reactive to unexpected early and late contacts using the estimated contact forces. Moreover, we present a novel scheme of optimal stiffness modulation that aims to coordinate desired compliance and tracking performance. It optimizes joint stiffness and contact forces coordinately in a quadratic programming (QP) formulation, where the constraints of non-slipping contacts and torque limits are imposed as well. In addition, the issue of stability under variable stiffness control is solved by imposing a tank-based passivity constraint explicitly. We finally validate the proposed control pipeline on our quadruped robot CENTAURO in experiments on uneven terrains and, through comparative tests, demonstrate the improvements of the variable stiffness locomotion.

Allocating synthetic population to a finer spatial scale: An integer quadratic programming formulation

Agent-Based Models (ABMs) are being increasingly used to evaluate urban systems, urban policies and environmental impacts. One prerequisite for using the ABM framework consists of generating a synthetic population representative of the actual population, featuring the appropriate attributes with respect to model objectives. A precise spatial positioning of the synthetic population agents is often key to ensuring ABM modeling quality. This paper considers the problem of allocating synthetic population agents to a finer spatial scale. Such an allocation process is performed from a higher-level statistical area where a synthetic population can be generated, that is, a container statistical area (CSA), to several nested non-overlapping elementary statistical areas (ESAs), where only marginals are available. This allocation step relies not only on common attributes between CSA and ESA, but also on additional discriminatory attributes, that is, attributes of interest, estimated from external data sources. The case study examined herein is based on French census and fiscal data. Common attributes include eight socio-demographic variables, totaling 17 modalities. An additional attribute of interest, that is, income, has also been added. The allocation problem at hand is modeled as an integer quadratic programming problem. An exact algorithm is first applied to solve the problem; the applicability of this algorithm proves to be limited to small-size synthetic populations. A heuristic is proposed to handle the allocation of larger-size synthetic populations. Tests carried out on the case study show that this heuristic yields near-optimal solutions; it is also computationally efficient and may fulfill the needs of a majority of users.

Fixed-time control under spatiotemporal and input constraints: A Quadratic Programming based approach

This paper presents a control synthesis framework for a general class of nonlinear, control-affine systems under spatiotemporal and input constraints. First, we study the problem of fixed-time convergence in the presence of input constraints. The relation between the domain of attraction for fixed-time stability with respect to input constraints and the required time of convergence is established. It is shown that increasing the control authority or the required time of convergence can expand the domain of attraction for fixed-time stability Then, we consider the problem of finding a control input that confines the closed-loop system trajectories in a safe set and steers them to a goal set within a fixed time. To this end, we present a Quadratic Programming (QP) formulation to compute the corresponding control input. We use slack variables to guarantee the feasibility of the proposed QP under input constraints. Furthermore, when strict complementary slackness holds, we show that the solution of the QP is a continuous function of the system states and establish the uniqueness of closed-loop solutions to guarantee forward invariance using Nagumo’s theorem. To corroborate our proposed methods, we present two case studies, an example of an adaptive cruise control problem and a two-robot motion planning problem.

The joint stochastic lot sizing and pricing problem

We consider a joint stochastic lot sizing pricing problem where period demands are price dependent and stochastic. Herein, we propose an alternative joint inventory control pricing policy for the profit-optimal (s,S,p) policy. The proposed policy – referred to as (R,S,p) is an order-up-to policy where replenishment schedule and price levels for each period are set before the planning horizon starts. As such, this policy can be considered as rather static as compared to the dynamic (s,S,p) policy. We also provide a quadratic mixed integer programming formulation in order to obtain the optimal (R,S,p) policy parameters. Furthermore, (R,S,p) policy is compared against the profit-optimal (s,S,p) policy on an extensive numerical study where several different benchmark policies are additionally used. The results show that the proposed policy exhibits a competitive profit performance with respect to the profit-optimal policy especially for low or moderate levels of demand uncertainty. The results also reveal that the relative benefit of using dynamic pricing policy, as compared to that of using dynamic inventory control, diminishes as demand uncertainty escalates.

Inverse Biot–Savart Optimization for Superconducting Accelerator Magnets

Superconducting (SC) magnets for accelerator concepts are often synthesized by numerically optimizing magnetic field waveforms, a process that requires a subsequent solution of a constrained inverse problem to identify suitable SC magnet windings. When the desired field distribution is intuitive, the inverse process is facilitated by seeding preconceived coil distributions into design optimization methods for refinement. With more complex magnetic field distributions, an initial design may be unknown, and topology optimization tools are required to synthesize current distributions without a priori guidance from a subject matter expert. In this work, we develop a constrained inverse Biot-Savart topology optimization methodology that synthesizes optimal distributions of current density in racetrack-like SC coils. The problem structure is exploited through a computationally efficient quadratic programming formulation, and the method is applied to recently published magnetic field waveforms for a recirculating proton phase shifter, a proton therapy gantry, and dipole magnets with sharp field transitions. The method and results herein identify novel winding configurations that can help magnet designers bring accelerator concepts to fruition.

Optimal Trajectory Generation for Intelligent Vehicles in Complex Traffic Based on Iteration Convex Optimization

Intelligent vehicles face considerable challenges in the complex traffic environment since they need to deal with various constraints and elements. This dissertation puts forward a novel trajectory planning framework for intelligent vehicles to generate safe and optimal driving trajectories. First, we design a spatiotemporal occupancy framework to deal with all kinds of elements in the complex driving environment based on the Frenét frame. This framework unifies various constraints on the road in the three-dimensional spatiotemporal representation and clearly describes the collision-free configuration space. Then we use the convex approximation method to construct a time-varying convex feasible region based on the above accurate temporal and spatial description. We formulate the trajectory planning problem as a standard quadratic programming formulation with collision-free and dynamics constraints. Finally, we apply the iterative convex optimization algorithm to solve the quadratic programming problem in the time-varying convex feasible region. Moreover, we design several typical experimental scenarios and have verified that the proposed method has good effectiveness and real-time.

The Bipartite Boolean Quadric Polytope

We consider the Bipartite Boolean Quadratic Programming Problem (BQP01), which generalizes the well-known Boolean quadratic programming problem (QP01). The model has applications in graph theory, matrix factorization and bioinformatics, among others. The primary focus of this paper is on studying the structure of the Bipartite Boolean Quadric Polytope (BQPm,n) resulting from a linearization of a quadratic integer programming formulation of BQP01.We present some basic properties and partial relaxations of BQPm,n, as well as some families of facets and valid inequalities. We find facet-defining inequalities including a family of odd-cycle inequalities. We discuss various approaches to obtain a valid inequality and facets from those of the related Boolean quadric polytope. The key strategy is based on rounding coefficients, and it is applied to the families of clique and cut inequalities in BQPm,n.

A Quadratic Programming Based Neural Dynamic Controller and Its Application to UAVs for Time-Varying Tasks

A quadratic programming based neural dynamic (QPND) controller design method is proposed for time-varying trajectory tracking of unmanned aerial vehicles (UAVs). Different from the traditional controller frameworks which only use pitch and roll angles to achieve the horizontal movement of the UAV, all the three attitude angles, i.e., roll, pitch and yaw angles, are applied for horizontal moving in this paper. There are at least three great superiorities of the proposed QPND controller. Firstly, three attitude angles can be coordinated control since the quadratic programming framework incorporates three attitudes. Secondly, the new framework enables the UAV to achieve the additional performance criteria or to perform subtasks. Third, the new controller can track time-varying trajectories due to neural dynamic approach. Specifically, the design process of the controller is divided into position controller design, attitude controller design and quadratic programming formulation. Finally, simulation experiments verify the feasibility and high efficiency of the proposed controller and its superiority compared with some state-of-the-art UAV controllers.

Robust extended Kalman filter estimation with moving window through a quadratic programming formulation

In this work, two new formulations for the extended (MW-EKF) and robust extended Kalman filter with moving window estimation (MW-REKF) are proposed. The MW-EKF and MW-REKF are formulated using an elegant quadratic programming problem that facilitates its implementation and decreases its computational cost. Besides that, the constrained extended Kalman filter (CEKF), constrained extended Kalman filter and smoother (CEKFS) and the moving horizon estimation (MHE) are compared in terms of computational cost and fit to the real data. The comparison is performed over a spherical-quadruple-tank model with different settings aiming to raise each approach's advantages and disadvantages. For both state and parameter estimation the MW-REKF has shown the smoothest and most robust behavior among all methodologies. This technique minimized the effect of the outliers, physical limitations, structural discrepancies, among others. The computational cost of the proposed techniques is only four times higher than CEKF and nine times smaller than the MHE.

Resilience Indices for Power/Cyberphysical Systems

An engineered system is designed to deliver certain performance related to its quality-of-service, and while doing so, it must also maintain stable operation. Resilience of a system is its ability to continue to offer system performance stably, while withstanding any adverse events. Motivated by this concept, we propose to measure the resilience level of a power system by quantifying its stability level as measured by: transient stability margin (TSM), critical clearance time (CCT), relay margin (RM), and load security margin (LSM), as well as its performance level as measured by: load loss (LL) and recovery/repair time (RT) while being exposed to adverse events. For comparability, we also propose a normalization for each of the 6 measures to a number in the unit interval [0, 1], which is scale-invariant, and further probabilistically average each of those across all possible sequences of faults (of a specified length) against their occurrence probabilities to arrive at a set of 6 unit-interval valued indices. New polynomial complexity algorithms (in the number of generators) are proposed for estimating TSM (in form of volume of region of stability) and CCT; new quadratic program formulation for precise computation of RM is developed and implemented; also, new security and stability informed notions of LSM and LL are introduced and implemented by extending continuation power flow. Such quantification of resilience levels provides a numerical measure to compare the relative abilities of different power grids to withstand the impact of sequences of adverse events. The proposed approach is illustrated by computing and comparing the resilience of three similar power system topologies differing only in the location of generators. The framework is further validated by implementing it on the IEEE 30-bus test system.

Optimal scheduling of integrated demand response-enabled integrated energy systems with uncertain renewable generations: A Stackelberg game approach

In order to balance the interests of integrated energy operator (IEO) and users, a novel Stackelberg game-based optimization framework is proposed for the optimal scheduling of integrated demand response (IDR)-enabled integrated energy systems with uncertain renewable generations, where the IEO acts as the leader who pursues the maximization of his profits by setting energy prices, while the users are the follower who adjusts energy consumption plans to minimize their energy costs. Taking into account the inherent uncertainty of renewable generations, the probabilistic spinning reserve is written in the form of a chance constraint; in addition, a district heating network model is built considering the characteristics of time delay and thermal attenuation by fully exploiting its potential, and the flexible thermal comfort requirements of users in IDR are considered by introducing a predicted mean vote (PMV) index. To solve the raised model, sequence operation theory is introduced to convert the chance constraint into its deterministic equivalent form, and thereby, the leader–follower Stackelberg game is tackled into a mixed-integer quadratic programming formulation through Karush-Kuhn-Tucker optimality conditions and is finally solved by the CPLEX optimizer. The results of two case studies demonstrate that the proposed Stackelberg game-based approach manages to achieve the Stackelberg equilibrium between IEO and users by the coordination of renewable generations and IDR. Furthermore, the study on a real integrated energy system in China verifies the applicability of the proposed approach for real-world applications.

Solving the stochastic support vector regression with probabilistic constraints by a high-performance neural network model

This paper offers a recurrent neural network to support vector machine (SVM) learning in stochastic support vector regression with probabilistic constraints. The SVM is first converted into an equivalent quadratic programming (QP) formulation in linear and nonlinear cases. An artificial neural network for SVM learning is then proposed. The presented neural network framework guarantees obtaining the optimal solution of the SVM problem. The existence and convergence of the trajectories of the network are studied. The Lyapunov stability for the considered neural network is also shown. The efficiency of the proposed method is shown by three illustrative examples.

Compositional metric learning for multi-label classification

Multi-label classification aims to assign a set of proper labels for each instance, where distance metric learning can help improve the generalization ability of instance-based multi-label classification models. Existing multi-label metric learning techniques work by utilizing pairwise constraints to enforce that examples with similar label assignments should have close distance in the embedded feature space. In this paper, a novel distance metric learning approach for multi-label classification is proposed by modeling structural interactions between instance space and label space. On one hand, compositional distance metric is employed which adopts the representation of a weighted sum of rank-1 PSD matrices based on component bases. On the other hand, compositional weights are optimized by exploiting triplet similarity constraints derived from both instance and label spaces. Due to the compositional nature of employed distance metric, the resulting problem admits quadratic programming formulation with linear optimization complexity w.r.t. the number of training examples. We also derive the generalization bound for the proposed approach based on algorithmic robustness analysis of the compositional metric. Extensive experiments on sixteen benchmark data sets clearly validate the usefulness of compositional metric in yielding effective distance metric for multi-label classification.

Tail risks in large portfolio selection: penalized quantile and expectile minimum deviation models

Accurate estimation and optimal control of tail risk is important for building portfolios with desirable properties, especially when dealing with a large set of assets. In this work, we consider optimal asset allocation strategies based on the minimization of two asymmetric deviation measures, related to quantile and expectile regression, respectively. Their properties are discussed in relation with the ‘risk quadrangle’ framework introduced by Rockafellar and Uryasev [The fundamental risk quadrangle in risk management, optimization and statistical estimation. Surv. Oper. Res. Manag. Sci., 2013, 18(1–2), 33–53], and compared to traditional strategies, such as the mean–variance portfolio. In order to control estimation error and improve the out-of-sample performance of the proposed models, we include ridge and elastic-net regularization penalties. Finally, we propose quadratic programming formulations for the optimization problems. Simulations and real-world analyses on multiple datasets allow to discuss pros and cons of the different methods. The results show that the ridge and elastic-net allocations are effective in improving the out-of-sample performance, especially in large portfolios, compared to the un-penalized ones.

Semidefinite Programming and Nash Equilibria in Bimatrix Games

We explore the power of semidefinite programming (SDP) for finding additive ɛ-approximate Nash equilibria in bimatrix games. We introduce an SDP relaxation for a quadratic programming formulation of the Nash equilibrium problem and provide a number of valid inequalities to improve the quality of the relaxation. If a rank-1 solution to this SDP is found, then an exact Nash equilibrium can be recovered. We show that, for a strictly competitive game, our SDP is guaranteed to return a rank-1 solution. We propose two algorithms based on the iterative linearization of smooth nonconvex objective functions whose global minima by design coincide with rank-1 solutions. Empirically, we demonstrate that these algorithms often recover solutions of rank at most 2 and ɛ close to zero. Furthermore, we prove that if a rank-2 solution to our SDP is found, then a [Formula: see text]-Nash equilibrium can be recovered for any game, or a [Formula: see text]-Nash equilibrium for a symmetric game. We then show how our SDP approach can address two (NP-hard) problems of economic interest: finding the maximum welfare achievable under any Nash equilibrium, and testing whether there exists a Nash equilibrium where a particular set of strategies is not played. Finally, we show the connection between our SDP and the first level of the Lasserre/sum of squares hierarchy.

Impedance control as an optimal control problem: a novel formulation of impedance controllers as a subcase of optimal control

A formulation of impedance control for redundant manipulators is presented here as a particular case of an optimal control problem. This allows the design of an impedance controller that benefits from the stability and efficiency of an optimal controller. To circumvent the high computational costs of finding optimal controllers, a suboptimal feedback controller based on state-dependent Riccati equations (SDRE) is developed. This approach is compared with the quadratic programming (QP) formulation, commonly used to solve redundancy of robotic manipulators. Numerical simulations of a redundant planar four-DOF serial-link manipulator show that the SDRE controller offers superior performance over the QP one in terms of stability, performance and required control effort.

Bidding Strategies in Energy and Reserve Markets for an Aggregator of Multiple EV Fast Charging Stations With Battery Storage

Adopting extreme fast charging for electric vehicles will significantly reduce the charging time for electric vehicle owners, which will improve the public acceptance of electric vehicle. However, under the conditions of wide spread fast charging stations, large charging power of fast charging stations will bring nonnegligible impacts to the power system. For an aggregator that owns multiple fast charging stations, installing battery storage systems within the fast charging stations can reduce the impacts and give more flexibility. It will bring extra benefit if the aggregator participates in electricity markets by utilizing the flexibility of the storage. In order to deal with the operation and market participation problem for EV fast charging stations, this paper proposes bidding strategies in both energy and reserve markets for an aggregator of multiple fast charging stations with energy storage systems. Conditional Value at Risk (CVaR) based mixed integer quadratic programming formulation is built to hedge the risks of random charging demands and volatile market prices. The proposed formulation is converted into a linear programming problem and an iterative solution method is designed to reduce the computation time. Case studies based on the trajectory data of taxis in Beijing have been carried out. Simulation results show that the aggregator can achieve higher economic benefits while satisfying the fast charging demand of electric vehicles.