Mixed Integer Quadratic Programming Research Articles

Hierarchical Model Predictive Control for Autonomous Collision Avoidance of Distributed Electric Drive Vehicle with Lateral Stability Analysis in Extreme Scenarios

This paper proposes an active collision avoidance controller based on a hierarchical model predictive control framework for distributed electric drive vehicles (4IDEV) considering extreme conditions. In this framework, a two-layer strategy is developed. The upper layer is the path replanning controller based on nonlinear MPC (nMPC), from which a collision-free path including the optimal lateral displacement and yaw angle can be obtained in real-time while encountering the obstacles. The lower layer is the path tracking controller based on hybrid MPC (hMPC), and the coordinated control inputs (yaw moment and the front wheel steering angle) are solved by a Mixed-Integer Quadratic Programming (MIQP) with the piecewise affine (PWA) tire model considering tire saturation region. Moreover, to improve the lateral stability when tracking, the stable zone of lateral stability in the high-risk condition is analyzed based on the phase portrait method, by which the constraints of vehicle states and inputs are derived. The verification is carried out on the MATLAB and CarSim co-simulation platform, and the simulation results show that the proposed active collision avoidance controller can track the reference path accurately and prevent vehicle instability in extreme scenarios.

A Decentralized B&B Algorithm for Motion Planning of Robot Swarms With Temporal Logic Specifications

In this letter, we study the problem of decentralized motion planning of robot swarms under high-level temporal logic specifications with a top-down approach. We use Swarm Signal Temporal Logic (SwarmSTL) to express swarm-level specifications. By encoding SwarmSTL formulas as mixed binary-integer constraints on the swarm features, the motion planning problem is formulated as a mixed-integer quadratic programming (MIQP) problem. We develop a decentralized Branch and Bound (B&B) algorithm with a node decentralization scheme such that the nodes in the B&B tree can be processed in parallel with communication among the agents and the agents can achieve consensus on the solution. Also, several search strategies to accelerate the decentralized B&B algorithm are proposed, and the performance improvements are presented. We evaluate the proposed algorithm using a supply transportation example with different formulas.

Learning the Optimal Strategy of Power System Operation With Varying Renewable Generations

Optimal dispatch of modern power systems often entails efficiently solving large-scale optimization problems, especially when generators have to respond to the fast fluctuation of renewable generation. This paper develops a method to learn the optimal strategy from a mixed-integer quadratic program with time-varying parameters, which can model many power system operation problems such as unit commitment and optimal power flow. Different from existing machine learning methods that learn a map from the parameter to the optimal action, the proposed method learns the map from the parameter to the optimal integer solution and the optimal basis, forming a discrete pattern. Such a framework naturally gives rise to a classification problem: the parameter set is partitioned into polyhedral regions; in each region, the optimal 0-1 variable and the set of active constraints remain unchanged, and the optimal continuous variables are affine functions in the parameter. The outcome of classification is compared with analytical results derived from multi-parametric programming theory, showing interesting connections between traditional mathematical programming theory and the interpretability of the learning-based method. Tests on a small-scale problem demonstrate the partition of the parameter set learned from data meets the theoretical outcome. More tests on the IEEE 57-bus system and a real-world 1881-bus system validate the performance of the proposed method with a high-dimensional parameter for which the analytical method is intractable.

Target-oriented robust location–transportation problem with service-level measure

We study a target-oriented, multi-period location–transportation problem where customer demands are uncertain. This problem is to determine the facility locations, production quantities, capacities, and shipment quantities, where the objective is to achieve the desired profit and fill rate to the full extent when it is impossible to reach both. To achieve the goal, we propose a target-oriented framework for the location–transportation problem, where a service-level measure is constructed to guarantee the desired fill rate and a hard constraint on profit is imposed to ensure a decent profit. This framework gets rid of the issues arising from estimating the weights of different objectives in a multi-objective optimization approach. To capture the characteristics of a multi-period decision-making process, an affine decision rule is introduced. Our method not only ensures that the transportation decisions of each period can adapt to realized demands wisely, but also prevents the high complexity of the model resulting from uncertainty and adaptation. Specifically, to tackle challenges of problem intractability, we reformulate the robust counterpart of the problem into a conservative approximation in the form of a mixed-integer quadratic program and propose a Benders decomposition approach to produce effective solutions. Finally, the performance of the target-oriented framework is assessed through computational experiments based on realistic instances.

Price‐based unit commitment with decision‐dependent uncertainty in hourly demand

The price-based unit commitment (PBUC) problem aims to optimise the power generating units' schedules to meet the system demand with the objective to maximise the generation companies' (GENCOs') profit. State-of-the-art PBUC models have taken into account exogenous uncertainties in renewable generation, demand, and price signals. This study proposes a novel PBUC problem formulation with endogenous or decision-dependent uncertainty (DDU) in the elastic portion of the demand. The proposed PBUC model is formulated as a mixed-integer non-linear programming (MINLP) problem with non-convex continuous relaxation. A concavification approach is developed to reformulate the non-convex MINLP model as an equivalent mixed-integer quadratic programming (MIQP) model whose continuous relaxation is convex. Case studies considering GENCOs owning and operating 3, 12, 19, and 40 generating units demonstrate the efficacy of the proposed DDU-aware PBUC formulation on the GENCOs' anticipated profits.

Energy management of multi-microgrids based on game theory approach in the presence of demand response programs, energy storage systems and renewable energy resources

In the deregulated power market, decentralized scheduling of microgrids is a complex problem with many technical and economic barriers. Achieving a comprehensive model that satisfies all technical, security and economic constraints and prevents market power is one of the concerns of distribution system operators. Therefore, this paper presents a two-stage framework in which DFR is implemented to satisfy technical and security constraints and microgrids day-ahead scheduling is done by a game-theoretic approach to avoid market power. The proposed model is implemented on a modified 69-bus distribution network by implementing a shiftable-load demand response (DR) program. The problem is formulated as a two-stage model: In the first stage, the distribution feeder reconfiguration (DFR) problem is solved to reduce losses, while in the second stage, the Disco and microgrids day-ahead scheduling problem is solved by a multi-objective optimization method based on the game theory with regard to the obtained topology. The optimization problem is also modeled as a mixed-integer quadratic programming (MIQCP) problem, solved using the CPLEX solver in GAMS software. The results show the high impact of the optimal topology on the decrease in loss, voltage deviation, and operation costs. In addition, the operation results indicate that the multi-objective optimization based on the game-theoretic method not only reduced market power, but also reduced operating costs by 23.7%.

Optimal dispatching strategy and real-time pricing for multi-regional integrated energy systems based on demand response

With the penetration of multiple distributed energy sources, demand side management (DSM) of the regional integrated energy system (RIES) becomes more complicated in the energy market. Real-time pricing (RTP) is an effective method for DSM, which can flexibly guide the supply and demand sides to adjust their behavior to participate in demand response (DR). In this paper, a hierarchical energy system is studied including multiple RIESs with multiple energy dispatch and supplement. To maximize the social welfare, a bilevel programming model is developed, in which the upper level aims at maximizing the profits of the supplier, and the lower level aims at maximizing the RIESs' welfare. Then, the proposed bilevel model is transformed into a mixed integer quadratic programming model using duality theory and Karush-Kuhn-Tucker conditions. Furthermore, the RTP strategy is obtained, and the optimal energy scheme of RIES is given in the solution. Compared simulations in different scenarios, the total social welfare is increased by about 14.12%, the peak-to-valley difference of power load and carbon emissions are reduced by 16.99% and 5.7% respectively after DR. The results show that the proposed bilevel model under the RTP is conducive to social economy and environment.

Optimal Sizing of Battery Energy Storage System in a Shipboard Power System with considering Energy Management Optimization

Due to the increasing concerns about the environmental and economic issues of traditional ships, all-electric ships with energy storage and renewable energy integration have become more and more appealing for the forthcoming future. In this paper, an optimal energy storage system (ESS) capacity determination method for a marine ferry ship is proposed; this ship has diesel generators and PV panels. ESSs sizing optimization and power system scheduling optimization are simultaneously conducted and it is converted to a mixed-integer quadratic programming (MIQP) model with special modeling techniques. The case study shows that the proposed method is flexible and effective, and the relationships between the ESSs size and the discharge rate, life cycle times, or initial investment cost are investigated.

SDP-quality bounds via convex quadratic relaxations for global optimization of mixed-integer quadratic programs

We consider the global optimization of nonconvex mixed-integer quadratic programs with linear equality constraints. In particular, we present a new class of convex quadratic relaxations which are derived via quadratic cuts. To construct these quadratic cuts, we solve a separation problem involving a linear matrix inequality with a special structure that allows the use of specialized solution algorithms. Our quadratic cuts are nonconvex, but define a convex feasible set when intersected with the equality constraints. We show that our relaxations are an outer-approximation of a semi-infinite convex program which under certain conditions is equivalent to a well-known semidefinite program relaxation. The new relaxations are implemented in the global optimization solver BARON, and tested by conducting numerical experiments on a large collection of problems. Results demonstrate that, for our test problems, these relaxations lead to a significant improvement in the performance of BARON.

A heuristic algorithm for medical staff's scheduling problems with multiskills and vacation control.

The main issue related to the duty schedule is to allocate medical staff to each medical department by considering personnel skills and personal vocation preferences. However, how to effectively use staff's multiskill characteristics and how to execute vocation control have not been well investigated. This article aims to develop duty scheduling and vacation permission decisions to minimize the sum of customers' waiting costs, the overtime cost of medical staff, the cost of failing to meet medical staff' vacation requirements, and the cost of mutual support between departments. This study formulated the problem as a multiperiod mixed integer nonlinear programming model and developed a hybrid heuristic based on evolutionary mechanism of genetic algorithm and linear programming to efficiently solve the proposed model. Five types of problems were solved through Lingo optimization and the proposed approach. For small-scale problems, both methods can find the optimal solutions. For a slightly larger problem, the solutions found by the proposed approach are superior those of Lingo. This research discusses the complex decision-making problem of on-duty arrangement and vacation control of medical staff in a multidepartmental medical center. This research formulates the medical staff's scheduling and vacation control problems as constrained mixed integer quadratic programming problems. Computational results indicate that the proposed approach can efficiently produce compromise solutions that outperform the solutions of the Lingo optimization software.

Station-grid Co-planning of Integrated Energy System with Consideration of Multi-energy storage and Reliability Cost

In order to improve the reliability of integrated energy system, the station-grid co-planning model and its solution method are proposed. The planning model considers the constraints of distribution network, gas distribution network, heat distribution network and energy station, and aims at minimizing the total planning cost within the planning period including investment cost, maintenance cost and reliability cost. It realizes the optimal planning for the types and locations of energy supply sources, distribution pipelines, energy stations and energy storage devices, such as electricity storage devices, gas storage devices, heat storage devices and cooling storage devices. The model is a mixed integer quadratic programming problem, which is solved by Gurobi. The case study uses the modified electricity-gas-heat integrated energy system to analyse the system planning scheme and planning cost under two scenarios. The results show that the proposed model can reasonably plan the integrated energy system and further improve the reliability of the integrated energy system.

Post-Event restoration strategy for coupled distribution-transportation system utilizing spatiotemporal flexibility of mobile emergency generator and mobile energy storage system

In this paper, a mixed integer quadratic programming (MIQCP) based post-event distribution restoration framework in response to high impact low probable events is propounded to maintain the power supply continuity of critical loads particularly in emergency conditions. Mobile power sources (MPSs) including mobile energy storage system (MESS) and mobile emergency generator (MEG) are considered as grid-support services in resiliency enhancement strategy due mainly to their spatiotemporal flexibility. Power system constraints are modelled according to the linearized DistFlow equations; on the other hand, traffic congestion is also considered while directing MPSs from staging locations to damaged areas. Repair crew dispatching schedule for maximizing the objective function is provided in a deterministic fashion. Numerical results based on the conducted case studies on modified version of the 15-bus distribution system show the capabilities of the model. Hereby power demand of all critical loads is served after an optimal allocation of MPS even though the damages are not completely cleared in the branches. Further, the decisions are close to what can be realistically expected in practical implementation, that fast-responded resources are exploited depending on traffic-related constraints and economic issues.

Strategic Interactions between Liquefied Natural Gas and Domestic Gas Markets: A Bilevel Model

This paper presents a bilevel programming model to aid decision-making for two players who interact strategically in the liquefied natural gas (LNG) supply chain, possibly with conflicting interests. In the proposed model, an LNG operator is the leader and a natural gas (NG) producer is the follower. The LNG operator attempts to optimally locate LNG export terminals, purchase gas from the NG producer, and export it as LNG. The NG producer aims to optimize production, pipeline investments, and sales to domestic consumers and the LNG operator. To solve the bilevel problem, we first reformulate it as a single-level problem by exploiting the convexity of the lower-level problem. Then, we use disjunctive reformulations of complementarity constraints and piecewise linear approximations of objective function terms to convert the problem into a convex quadratic mixed-integer program (QMIP). Computational experiments confirm that the QMIP is tractable and can be solved efficiently. We apply our bilevel framework to a case study of the Gulf-Southwest region of the United States and evaluate several decision-making scenarios. The scenario results emphasize the importance of using the bilevel methodology to anticipate the effects of new LNG export facilities on NG prices across the domestic gas network. Adding these large gas-consuming facilities at specific locations puts upward pressure on domestic gas prices, although this is somewhat mitigated by the NG producer's optimal production response to the increased demand.

Piecewise affine modeling and hybrid optimal control of intelligent vehicle longitudinal dynamics for velocity regulation

In this paper, a novel piecewise affine (PWA) modeling framework and a hybrid optimal control scheme based on the developed PWA model for the intelligent vehicle longitudinal dynamics are proposed. The proposed methodology approximates the nonlinear characteristics of the system major components, i.e. the engine, the automatic transmission and the tire, using the PWA representation method firstly. Due to the affine submodel switching behaviors of the PWA model and the fact that the intelligent vehicle must be worked in two discrete modes (driving and braking) for regulating the longitudinal velocity automatically, the control procedure of the intelligent vehicle longitudinal dynamics actually shows a hybrid nature with both continuous variables and discrete events. Thus, to further solve the system hybrid control problem, the intelligent vehicle longitudinal dynamics whose major components are modelled in the PWA form is further transformed into a computational mixed logical dynamical (MLD) model in this work, based on which a hybrid model predictive control (HMPC) strategy, which can simultaneously calculate the switching sequences of the intelligent vehicle longitudinal working modes (binary control inputs) and the throttle angle and braking pressure (continuous control inputs) during autonomous velocity regulation, is designed by solving a mixed-integer quadratic programming (MIQP) problem. Simulation and experimental results are finally provided to verify the superior control performance of the designed hybrid controller in longitudinal velocity regulation under typical driving conditions.

Optimal distributed transaction of multiple microgrids in grid-connected and islanded modes considering unit commitment scheme

This paper addresses multi-directional energy trading problems (ETPs) for multiple microgrids (MMGs) in grid-connected and islanded modes considering the unit commitment (UC) scheme. Firstly, a mixed integer quadratic programming (MIQP) model is formulated to describe unit commitment problems (UCPs) for each microgrid (MG) and energy transactions among MMGs. Then binary variables from UCPs are relaxed to convert the original non-convex model into a tractable convex one. Furthermore, an alternating optimization method (AOM) composed of alternating direction method of multipliers (ADMM) and MIQP block is proposed to solve continuous and binary variables alternately. Especially, a rounding-off procedure with thresholds is designed to treat out-of-limit problems of generators and smooth the parameter passing between ADMM and MIQP block with corresponding convergence proof. Finally, comprehensive comparisons are conducted to demonstrate the effectiveness of proposed method. Simulation results indicate that AOM can be directly applied to solve ETPs for MMGs in a distributed manner, and thresholds are effective for operators to adjust the operating range of generators flexibly under the condition that ETPs are completely solvable.

Exact solution approaches for the workload smoothing in assembly lines

In this paper, the problem of minimizing the smoothness index for an assembly line given a fixed cycle time and the number of workstations is studied. This problem which is known as the workload smoothing line balancing problem (WSLBP) is a mixed-integer quadratic programming problem. Until recently, this problem has only been tackled using heuristic approaches. Recently, there have been some attempts to solve this problem exactly using mixed-integer linear programming (MILP). The MILP formulations, however, are not usually capable of solving large size problem instances. In this paper, the aim is to solve the WSLBP using mathematical programming formulations by using off-the-shelf solvers. Differently from the literature some non-MILP formulations are also considered for the problem. For this purpose, three MILP formulations, one from the literature, and two non-MILP formulations are compared. The two non-MILP formulations include a mixed-integer second order cone programming formulation and a constraint programming model. The superiority of the non-MILP formulations over the considered MILP formulations is experimentally shown.

Centralised coordination of EVs charging and PV active power curtailment over multiple aggregators in low voltage networks

Widespread application of residential photovoltaic (PV) systems and electric vehicles (EVs) has led the distribution system operators (DSOs) to face new technical challenges such as overloading and significant voltage variations, especially on low voltage (LV) grids. This adverse effect may be mitigated by employing the aggregators as intermediary actors to coordinate the operation at the distribution level. Therefore, this paper proposes a centralised coordination strategy to mitigate both PV and EV impacts, such as voltage rising/dropping at noon or evening, respectively, by defining the optimal export limit of PV power and EVs charging among multiple aggregators at the DSO level. The latter is in charge to dictate the optimal aggregated signals to every aggregator by employing a mixed-integer quadratic programming (MIQP) approach. The aggregated PV power is evenly managed for each aggregator by weighting. Two convex optimisation models are defined to satisfy both power and voltage constraints of the LV network. Each proposed optimisation approach can be utilised when there is or no detailed information about the LV network topology. The concepts discussed in this paper are tested on a real low voltage network considering a critical penetration level of EVs and PVs.

Bounds on efficient outcomes for large-scale cardinality-constrained Markowitz problems

When solving large-scale cardinality-constrained Markowitz mean–variance portfolio investment problems, exact solvers may be unable to derive some efficient portfolios, even within a reasonable time limit. In such cases, information on the distance from the best feasible solution, found before the optimization process has stopped, to the true efficient solution is unavailable. In this article, I demonstrate how to provide such information to the decision maker. I aim to use the concept of lower bounds and upper bounds on objective function values of an efficient portfolio, developed in my earlier works. I illustrate the proposed approach on a large-scale data set based upon real data. I address cases where a top-class commercial mixed-integer quadratic programming solver fails to provide efficient portfolios attempted to be derived by Chebyshev scalarization of the bi-objective optimization problem within a given time limit. In this case, I propose to transform purely technical information provided by the solver into information which can be used in navigation over the efficient frontier of the cardinality-constrained Markowitz mean–variance portfolio investment problem.

Energy Storage Planning for Profitability Maximization by Power Trading and Ancillary Services Participation

One of the main applications of energy storage systems (ESSs) is transmission and distribution systems cost deferral. Further, ESSs are efficient tools for localized reactive power support, peak shaving, and energy arbitrage. This article proposes an ESSs planning algorithm that includes all previous services. The proposed algorithm increases the distribution company profit and minimizes its future system upgrade cost. For a comprehensive planning algorithm, other options, such as including static VAR compensators (SVCs), feeders upgrade, or adding distributed generators, are considered along with ESSs. Different scenarios are utilized to model the load variation, the renewable resources’ intermittency, and the market price fluctuation. The problem constraints include an ESS dynamic model that reflects the capacity and lifetime limitations. Different battery technologies with different costs and dynamic characteristics are compared. The network power flow model is added to account for the voltage level and feeders’ capacity. Power limits constraints are included for the DGs and SVCs as well. The optimization problem is formulated as a mixed-integer quadratic programming problem. To validate the proposed technique, a case study is conducted on a real 41-bus radial feeder in Ontario, Canada using real data for renewable sources, loads, and market prices.

Optimal operation of integrated energy system considering virtual heating energy storage

The integrated energy system (IES) is the physical carrier of the Energy Internet, whose optimal operation has become a hot topic because of its effectiveness in improving energy utilization efficiency. This paper proposes an optimal operation model of integrated energy system, where the internal energy flow transmission characteristic of the heating network is modeled as virtual heating energy storage by the node method to enhance the operating flexibility of IES. The second-order cone relaxation method and the incremental formulation for the piecewise linearization method are utilized to transform the proposed nonlinear model into mixed integer quadratic constraint programming model, which can be effectively solved by the commercial solver. Case studies show that the proposed model effectively reduces the operating costs and improves the renewable energy penetration level of IES.