Mixed Integer Quadratic Programming Problem Research Articles

Optimised operation of integrated community energy system considering integrated energy pricing strategy: A two-layer Stackelberg game approach

Aiming at the problems of new energy absorption and inter-subject interest conflict in the integrated community energy system, this study proposed an integrated energy pricing strategy that considered the interconversion of various energy sources and built a two-layer optimisation model of the electric-gas-heat‑hydrogen interconnection integrated community energy system. The upper-layer operator model maximises the revenue by setting energy price and selling energy to users. The lower-layer users model minimises the energy cost by actively adjusting energy usage strategy according to energy price. The two sides rationally pursue the maximisation of their respective interests, and the process of bargaining can be described by Stackelberg game. The model is also considered for combining the characteristics of elastic load and vehicle-to-grid on the demand side and operates jointly with the bus battery swapping station and two-stage power-to-gas for optimizing system operation. Through theoretical derivation, it is proved that there is a unique Stackelberg equilibrium solution for the proposed game model. Further, the two-layer optimisation model is transformed into a mixed integer quadratic programming problem by applying the Karush-Kuhn-Tucher optimal condition, linear relaxation technique, and duality theorem. Finally, the global optimal energy price is obtained by solving the converted model. The optimisation results in different scenarios show that the proposed model and method not only protect the interests of the operator and users but also improve the wind power absorption capacity of the community and reduce the load fluctuation.

Achieving Emission-Reduction Goals: Multi-Period Power-System Expansion Under Short-Term Operational Uncertainty

Stochastic adaptive robust optimization is capable of handling short-term uncertainties in demand and variable renewable-energy sources that affect investment in generation and transmission capacity. We build on this setting by considering a multi-year investment horizon for finding the optimal plan for generation and transmission capacity expansion while reducing greenhouse gas emissions. In addition, we incorporate multiple hours in power-system operations to capture hydropower operations and flexibility requirements for utilizing variable renewable-energy sources such as wind and solar power. To improve the computational performance of existing exact methods for this problem, we employ Benders decomposition and solve a mixed-integer quadratic programming problem to avoid computationally expensive big-M linearizations. The results for a realistic case study for the Nordic and Baltic region indicate which investments in transmission, wind power, and flexible generation capacity are required for reducing greenhouse gas emissions. Through out-of-sample experiments, we show that the stochastic adaptive robust model leads to lower expected costs than a stochastic programming model under increasingly stringent environmental considerations.

Daily peak shaving operation of mixed pumped-storage hydro plants considering cascade hydraulic coupling

Growing peaking regulation pressure of the thermal-dominant power grid in China caused by increasing peak–valley differences is of concern in recent years. As the second largest power supply in China, hydropower is an important supporting resource for daily peak shaving operations. It is effective to install new pumped-storage hydro turbines (PSHT) based on the constructed cascade reservoirs to make full use of water resources, which is become a popular measure recently. This paper investigates the peak shaving of cascade hydropower with mixed pumped-storage (CHMPS) to reduce the variance of the residual load of the external grid. The hydraulic constraints, unit operation constraints, and impact of water head on unit generation are considered to formulate a precise model. The nonlinear and nonconvex problem is linearized and converted into a mixed-integer quadratic programming (MIQP) problem to enhance computational efficiency. Simulation results based on a hydropower system in China verify the effect of the proposed model on daily peak shaving.

Mean-Variance-VaR portfolios: MIQP formulation and performance analysis

AbstractValue-at-risk is one of the most popular risk management tools in the financial industry. Over the past 20 years, several attempts to include VaR in the portfolio selection process have been proposed. However, using VaR as a risk measure in portfolio optimization models leads to problems that are computationally hard to solve. In view of this, few practical applications of VaR in portfolio selection have appeared in the literature up to now. In this paper, we propose to add the VaR criterion to the classical Mean-Variance approach in order to better address the typical regulatory constraints of the financial industry. We thus obtain a portfolio selection model characterized by three criteria: expected return, variance, and VaR at a specified confidence level. The resulting optimization problem consists in minimizing variance with parametric constraints on the levels of expected return and VaR. This model can be formulated as a mixed-integer quadratic programming (MIQP) problem. An extensive empirical analysis on seven real-world datasets demonstrates the practical applicability of the proposed approach. Furthermore, the out-of-sample performance of the more binding optimal Mean-Variance-VaR portfolios seems to be generally better than that of the Equally Weighted and of the Mean-Variance-CVaR portfolios.

Minimal Detectable and Isolable Faults of Active Fault Diagnosis

This article aims to compute guaranteed minimal detectable and isolable faults of set-based active fault diagnosis methods for discrete linear time-invariant systems. First, guaranteed detectable and isolable faults of set-based active fault diagnosis methods are defined in this article. Second, an augmented vector is defined to integrate the effect of both inputs and faults such that the couplings of inputs and faults can be handled effectively. Third, guaranteed minimal detectable and isolable faults are computed by solving a mixed integer quadratic fractional programming problem under a proposed theoretic framework. At the end of this article, an example is used to illustrate the effectiveness of the proposed solution.

Accelerated and Adaptive Power Scheduling for More Electric Aircraft Via Hybrid Learning

In recent studies, the power scheduling in more electric aircraft (MEA) has been formulated as a mixed-integer quadratic programming problem. Many model-driven methods, such as branch-and-bound algorithms, are advocated to solve it. However, these methods are often prone to considerably high complexities, which makes real-time processing a problem. In this article, a two-stage hybrid learning-based optimization approach is proposed to address this challenging issue. In the first stage, the task of optimizing integer variables is considered as a multilabel classification problem, which is solved by a data-driven method, i.e., ensemble deep neural networks (EDNNs). In the second stage, with the obtained integer solutions, the problem is transformed into a quadratic programming problem that can be quickly solved by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">model-driven</i> numerical optimization methods. Compared to the state-of-the-art power scheduling algorithms, the model-driven and data-driven methods can compensate each other so that this two-stage hybrid learning-based approach can achieve orders of magnitude speedup in computational time, while guaranteeing the optimal scheduling performance. The structures of EDNNs are well designed so that the proposed approach can adapt to varying operating conditions in MEA. An offline simulation test and an online hardware-in-the-loop test validate the above advantages of the proposed approach.

Resilient closed-loop supply chain network design considering quality uncertainty: A case study of stone quarries

According to the important role of the mining industry in the economic growth of developing countries, this industry is taken into consideration significantly. However, few studies have been conducted on developing mining supply chains that take into account the unique conditions of this industry, such as disruptions and product quality. This paper proposes a multi-period, multi-product mixed-integer quadratic programming problem to optimize a closed-loop stone supply chain network design. Compared to previous studies, we consider resilience in the mining supply chain for the first time. Furthermore, the quality characteristics of extracted and final stones are considered in a developed reliable model, and two-stage stochastic programming is applied to handle the quality uncertainty in the mining supply chain. Additionally, a price-dependent demand is considered in this research to investigate the competition between local and imported stone prices. A real-world case study on Iranian stone mining is analyzed to demonstrate the efficiency and applicability of the proposed model. Finally, some managerial insights are presented as guidance to managers of the mentioned supply chain to help them make better decisions. The results show that the transportation costs impact profit more than the operational costs. Moreover, West Azerbaijan and Isfahan have great potential for being the poles of stone in Iran. West Azerbaijan is also appropriate for exporting stones to neighboring countries, geographically.

An approximation algorithm for indefinite mixed integer quadratic programming

In this paper, we give an algorithm that finds an \(\epsilon \)-approximate solution to a mixed integer quadratic programming (MIQP) problem. The algorithm runs in polynomial time if the rank of the quadratic function and the number of integer variables are fixed. The running time of the algorithm is expected unless P = NP. In order to design this algorithm we introduce the novel concepts of spherical form MIQP and of aligned vectors, and we provide a number of results of independent interest. In particular, we give a strongly polynomial algorithm to find a symmetric decomposition of a matrix, and show a related result on simultaneous diagonalization of matrices.

A novel preview control for MLD models and its neural network approximation for real‐time implementation: Application to semi‐active vibration control of a vehicle suspension

Advances in image processing technology have made it possible to measure the surface shape of the road ahead while driving. A new semi-active suspension control method considering the forward road surface shape is proposed. A vehicle model equipped with a semi-active suspension can be expressed as an mixed logical dynamical model. When the shape of the road ahead can be measured, the information on future disturbances is available before the vehicle undergoes. In this paper, the finite time optimization problem for the mixed logical dynamical model is formulated to consider the future disturbances as a mixed integer quadratic programming problem in the same way as the conventional control problem without future disturbance. However, the mixed integer quadratic programming problem is hard to obtain the control action within the control cycle period required in the real-time vibration control with general computers for vehicles. In this paper, the reduction of the computational load is achieved by constructing an approximation function of the designed controller. A neural network is adopted for the approximation. The performance evaluation of the proposed method is evaluated by simulations. In the simulation study, the proposed method achieves better ride comfort with the equivalent suspension stroke compared to the traditional methods.

Stackelberg game-based optimal scheduling of integrated energy systems considering differences in heat demand across multi-functional areas

Demand-side management is very critical in China’s energy systems because of its high fossil energy consumption and low system energy efficiency. A building shape factor is introduced in describing the architectural characteristics of different functional areas, which are combined with the characteristics of the energy consumed by users to investigate the features of heating load in different functional areas. A Stackelberg game-based optimal scheduling model is proposed for electro-thermal integrated energy systems, which seeks to maximize the revenue of integrated energy operator (IEO) and minimize the cost of users. Here, IEO and users are the Stackelberg game leader and followers, respectively. The leader uses real-time energy prices to guide loads to participate in demand response, while the followers make energy plans based on price feedback. Using the Karush–Kuhn–Tucker (KKT) condition and the big-M method, the model is transformed into a mixed-integer quadratic programming (MIQP) problem, which is solved by using MATLAB and CPLEX software. The results demonstrate that the proposal manages to balance the interests of IEO and users. Furthermore, the heating loads of public and residential areas can be managed separately based on the differences in energy consumption and building shape characteristics, thereby improving the system operational flexibility and promoting renewable energy consumption.

Cybersecurity Enhancement for Multi-Infeed High-Voltage DC Systems

Composed of multiple two-terminal high-voltage DC (HVDC) transmission systems, a multi-infeed HVDC (MIDC) system exchanges massive power among multiple asynchronous AC systems. However, as an intrinsically cyber-physical system, an MIDC system could suffer from cyber-attacks, leading to massive power mismatches in multiple AC systems, and resulting in catastrophic consequences. Since the sequential responses of an MIDC system and interconnected AC systems are in different timescales, this paper first establishes a two-timescale model to evaluate the sequential impacts caused by cyber-attacks. Then, an event-triggered cyber-defense strategy is proposed to enhance the cybersecurity of an MIDC system by mitigating multiple non-simultaneous cyber-attacks. Whenever new cyber-attack events occur, the proposed cyber-defense strategy, which is mathematically modeled as a mixed-integer quadratic programming problem, is executed on-line and updated in an event-triggered manner. Simulation results on an MIDC system demonstrate that the low-cost and almost blind cyber-attacks can cause severe frequency deviations, and the proposed strategy can mitigate multi-cyber-attacks effectively.

A Smart Cross-System Framework for Joint Allocation and Scheduling With Vehicle-to-Grid Regulation Service

The growing popularity and penetration of electric vehicles (EVs) bring in challenges and opportunities to the power grid. They represent a huge aggregate electricity consumption from the city power grid, while it is possible to make use of their batteries for supporting vehicle-to-grid (V2G) regulation service. Besides, there is a need to coordinate a collection of EVs in an intelligent manner through the smart transportation system in a smart city. In this paper, we propose a novel smart cross-system framework to support V2G regulation service in a smart city. Specifically, by considering electric buses (EBs) as a particular group of EVs with large battery packs installed and predictable operation schedules, we jointly solve the problems of EB allocation and scheduling with V2G regulation service. First of all, by using the deep learning approach, the short-term city traffic conditions can be predicted via the gated recurrent units (GRUs) neural network. With the predicted traffic conditions, we then formulate the EB allocation and V2G scheduling as a mixed-integer quadratic programming (MIQP) problem. In order to solve this problem effectively, we utilize the Lagrangian dual decomposition method to decouple the main problem into subproblems for EBs and devise a distributed algorithm to solve each subproblem. The simulation results show that our proposed approach is effective in EB allocation and V2G scheduling for the regulation service. Through efficient EB routing strategies under accurate predicted traffic conditions, the power fluctuations of the city power grid can be well flatten by providing the V2G regulation service.

A three-stage multi-objective optimization framework for day-ahead interaction between microgrids in active distribution networks considering flexible loads and energy storage systems

This paper provides a three-stage framework in which the microgrid strategy for participation in the electricity market is determined by considering economic, technical, and environmental indicators. In the first stage, microgrids do their day-ahead scheduling and submit offers/bids to the independent system operator (ISO). In the second stage, ISO determines the hourly market clearing price (MCP) according to the microgrids' offers/bids and finally, in the third stage, ISO solves the distribution feeder reconfiguration (DFR) problem subject to the power flow constraints. The conditional value at risk (CVaR) method is utilized to deal with uncertainties of demand and renewable energy resources (RER), and the final problem is formulated as a mixed-integer quadratic programming (MIQCP) problem. The proposed model is implemented on a 69-bus distribution system and the results show that the simultaneous presence of electrical energy storage (EES) systems and demand response (DR) program reduces the marginal prices of microgrids and consequently leads to a 19.3% reduction in MCP. The results also illustrate that the reconfigurable topology increases the ISO flexibility and thus leads to an 8.22% reduction in MCP.

A railcar re-blocking strategy via Mixed Integer Quadratic Programming

This article presents a novel solution namely Pre-blocking through Re-waybilling to minimize railcar cuts in freight railways. The Pre-blocking through Re-waybilling Problem (PRP) can be described as follows: A freight train carrying m railcars, loaded or empty, arrives at a local classification yard where, based on the destination on their electronic waybills, railcars will be physically switched to n outbound trains. Number of railcar cuts depends on the order of railcars on train. The objective is to minimize the number of railcar cuts by means of changing the electronic waybills of empty railcars, i.e. re-assignment of empty railcars to customers, while respecting supply, demand, and feasibility constraints. In order to solve this problem, a supply–demand network is constructed first, where supply nodes and demand nodes represent railcars and destinations, respectively. The problem is then formulated as a modified transportation problem. Unlike the conventional transportation problem, a linear arc cost function is proposed to describe the adjacent cars’ impact on railcar cut. The objective is converted to a quadratic function, leading to a Mixed-Integer Quadratic Programming (MIQP) problem. We improve proposed MIQP to a Convex Mixed-Integer Quadratic Programming (C-MIQP) problem by adding a quadratic term to the objective function. Illustrative and realistic examples are presented to validate the feasibility and efficiency of proposed pre-blocking optimizer.

A Classifier to Decide on the Linearization of Mixed-Integer Quadratic Problems in CPLEX

Despite modern solvers being able to tackle mixed-integer quadratic programming problems (MIQPs) for several years, the theoretical and computational implications of the employed resolution techniques are not fully grasped yet. An interesting question concerns the choice of whether to linearize the quadratic part of a convex MIQP: although in theory no approach dominates the other, the decision is typically performed during the preprocessing phase and can thus substantially condition the downstream performance of the solver. In “A Classifier to Decide on the Linearization of Mixed-Integer Quadratic Problems in CPLEX,” Bonami, Lodi, and Zarpellon use machine learning (ML) to cast a prediction on this algorithmic choice. The whole experimental framework aims at integrating optimization knowledge in the learning pipeline and contributes a general methodology for using ML in MIP technology. The workflow is fine-tuned to enable online predictions in the IBM-CPLEX solver ecosystem, and, as a practical result, a classifier deciding on MIQP linearization is successfully deployed in CPLEX 12.10.0.

Global optimization on non-convex two-way interaction truncated linear multivariate adaptive regression splines using mixed integer quadratic programming

Multivariate adaptive regression splines (MARS) has been adopted as a popular surrogate function to model the unknown relationships between input and output variables in complex systems. Searching optimal solutions on the complicated surrogate response surface of MARS serves as important tasks in various applications. In this paper, we present an efficient and effective approach to find a global optimal value of MARS models that incorporate two-way interaction terms which are products of truncated linear univariate functions (TITL-MARS). Specifically, with a MARS model consisting of linear and quadratic structures, we can reformulate the optimization problem on TITL-MARS into a mixed integer quadratic programming problem (TITL-MARS-OPT), which can be further solved in a more principled way. To illustrate the effectiveness of our proposed approach TITL-MARS-OPT, we come up with a genetic algorithm and a gradient descent algorithm to solve the original version of TITL-MARS, and we compared the performance of the proposed approach with the benchmark algorithms. Numerical experiments are conducted on a spectrum of examples with different levels of complexity, including 6 existing models and a real world application in the wind farm optimization. Our proposed approach can find a global optimal successfully and efficiently while the comparison algorithms fail to find a global optimal solution. In the end, Python code for TITL-MARS and TITL-MARS-OPT is made available on GitHub ( https://github.com/JuXinglong/TITL-MARS-OPT).

Enhanced-decision energy trading for island renewable microgrids considering large interruptible refrigeration load intra-hour switching

Large interruptible refrigeration load (LIRL) can be incorporated into the scheduling model of microgrid (MG) to optimize the operational cost. However, based on a classic 1-hour step, the LIRL is either turned on or off, resulting in unsatisfactory results. Although a smaller scheduling step can adjust the LIRI more finely, the computational burden will be increased simultaneously. To address the problem, this paper proposes a novel enhanced-decision energy trading strategy for renewable MGs. First, enhanced decisions are proposed, where on-state and off-state durations of LIRL in an hour instead of on or off state would be determined. On the basis, based on a 1-hour scheduling step, an enhanced-decision trading model is presented, where the energy operator maximizes the electricity sale profit, and load aggregators (LA) minimize the electricity usage cost. This optimization model is first converted into a classic mixed-integer quadratic programming (MIQP) problem, and then solved by a distributed method. Simulation studies indicate that the proposed energy trading strategy can reduce the electricity price fluctuation range by 0.43 yuan (20.6%), increase the social cost by 104 yuan (3.5%), and benefit the participators.

Vibration control of semi-active suspension by the neural network that learned the optimal preview control of MLD model

Advances in image processing technology have made it possible to measure the surface shape of the road ahead while driving. We propose a new semi-active suspension control method that considers the forward road surface shape. A vehicle model equipped with a semi-active suspension can be expressed as an MLD (Mixed Logical Dynamical) model. When the shape of the road ahead can be measured accurately, that the information on the future disturbances is available before the vehicle undergoes. In this paper, we formulate the finite time optimization problem of the MLD model in consideration of the future disturbances as a MIQP (Mixed Integer Quadratic Programming) problem in the same way as the conventional optimal control problem without future disturbance. The solution to the MIQP problem can be obtained by a commonly available solver software. However, because the MIQP problem is classified as NP-hard, it is hard to obtain the control action within the control cycle period required by the vibration control problem with general computers for vehicles. In this paper, we achieve this reduction in computational load by constructing an approximation function for the designed controller. A multilayer neural network is adopted for the approximation. The performance evaluation of the proposed control method was evaluated by a simulation. In the simulation study, the proposed method was able to achieve better ride comfort with the equivalent suspension stroke compared to the traditional MLD predictive control and the Skyhook approximation methods. Moreover, the proposed method enables to generate the control signal within the control cycle period.

Modeling of Electric Vehicles as Mobile Energy Storage Systems Considering Multiple Congestions

To realize the optimal operation of urban coupled transportation power systems underthe road, charging facilities, and transmission line congestions, a dynamic optimal traffic power flow (DOTPF) model was formulated under congestions. Based on the time space network (TSN) approach, a novel TSN with queues was proposed, considering the moving, parking, charging, and queueing state transitions. A vehicle routing problem was formulated for electric vehicles (EVs) and further incorporated into the dynamic traffic assignment problem (DTAP), reducing the traffic demand losses. With security and reserve constraints, a dynamic security-constrained carbon dioxide-oriented optimal power flow (OPF) problem was formulated to reduce the carbon emission and generation cost, by optimizing the scheduling of thermal units and energy storage systems. A multi-objective DOTPF problem was formulated, and further reformulated into a convex mixed-integer quadratic programming problem. The effectiveness of the proposed DOTPF was verified based on the simulation results on coupled IEEE-30 and Sioux Falls system.

Passive Fault Tolerant Control of Motorboat System

A hybrid passive control strategy is developed for a class of hybrid systems modeled by Mixed Logical Dynamical (MLD) approach. It allows to model different operating modes of the system and constraints. We proposed using the MPC for control system .The passive controller is used to take in account the actuator failure and the optimization problem is transformed into a mixed-integer quadratic programming problem (MIQP). The considering fault-tolerance capabilities are developed and discussed. The proposed method is illustrated by a motorboat system.