Mixed Integer Quadratic Programming Research Articles

Energy management for scalable battery swapping stations: A deep reinforcement learning and mathematical optimization cascade approach

The rapid growth of electric vehicles is driving the expansion of scalable Battery Swapping Stations (BSSs) to meet the demand for fast charging. However, existing BSS energy management struggles to adapt to the changes in battery counts online, the uncertainty of electricity prices and battery demand, as well as the complexity of demand response (DR). To address these issues, we propose a cascading approach that combines Deep Reinforcement Learning (DRL) with Mathematical Optimization (MO). Firstly, our method employs a two-layer optimization framework and establishes a BSS model. The upper controller uses DRL to make sequential decisions on total power for each time slot, addressing sequential decision-making under uncertainty. The lower controller uses MO to allocate total power to each charging bay for a single time slot, addressing space complexity. Secondly, we tailor proximal policy optimization algorithms and mixed-integer quadratic programming algorithms for BSS DR. The DRL is universally adaptable to scalable BSS, while the MO automatically generates and solves optimization models based on the working charging bay count. The proposed method is at least 33 times faster than existing methods and performs well in scalable BSSs under uncertain conditions. It can converge within a typical BSS DR scheduling slot of 15 min based on an i7-8950H CPU, even for a BSS with 3200 charging bays.

MAGE: Microgrids with Advanced Grid Efficiency through Battery-Aware EV management

Higher penetration of Electric Vehicles (EVs) yields a positive impact by curbing pollution. However, this rapid proliferation of EVs presents several challenges to the efficient management of EVs and the existing grid infrastructure due to the very high energy requirement of newly adopted EVs, which can also act as energy resources. This work delves into the problem of intelligent charging–discharging of EVs in a single offline microgrid framework, which can partially cater to the energy needs of other EVs (V2V, vehicle-to-vehicle), grid (V2G, vehicle-to-grid), building (V2B, vehicle-to-building), etc. This approach also preserves the battery’s health by limiting the number of charging–discharging cycles. Hence, a battery degradation-aware energy scheduler named MAGE (Microgrids with Advanced Grid Efficiency through Battery-Aware EV Management) is formulated to minimize the microgrid’s operational cost (MOC). The MOC is comprised of — the energy procurement cost, the cost of surplus energy sales, and penalties for unsatisfied demands. A formulation based on Mixed-Integer Quadratic Programming is proposed to ascertain an optimal solution. Heuristic algorithms are designed to identify a reasonable solution quickly. A comprehensive experimental analysis of our proposed approach exhibits its potential in optimizing MOC and ensures grid efficiency by reducing the peak load and fattening the load curve.

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.

Accelerating Cutting-Plane Algorithms via Reinforcement Learning Surrogates

Discrete optimization belongs to the set of N P-hard problems, spanning fields such as mixed-integer programming and combinatorial optimization. A current standard approach to solving convex discrete optimization problems is the use of cutting-plane algorithms, which reach optimal solutions by iteratively adding inequalities known as cuts to refine a feasible set. Despite the existence of a number of general-purpose cut-generating algorithms, large-scale discrete optimization problems continue to suffer from intractability. In this work, we propose a method for accelerating cutting-plane algorithms via reinforcement learning. Our approach uses learned policies as surrogates for N P-hard elements of the cut generating procedure in a way that (i) accelerates convergence, and (ii) retains guarantees of optimality. We apply our method on two types of problems where cutting-plane algorithms are commonly used: stochastic optimization, and mixed-integer quadratic programming. We observe the benefits of our method when applied to Benders decomposition (stochastic optimization) and iterative loss approximation (quadratic programming), achieving up to 45% faster average convergence when compared to modern alternative algorithms.

Omnichannel product selection and shelf space planning optimization

As the retail industry is evolving toward an omnichannel paradigm, we study a product selection and shelf space planning problem faced by omnichannel retailers. We aim to maximize aggregated profits across channels by determining the optimal product offerings for the offline and online channels while optimizing the allocation of limited shelf space to these products in brick-and-mortar stores. Furthermore, the showroom effect of physical stores on online sales is also taken into account. We propose a corresponding model that jointly optimizes these decisions. To solve the nonlinear nonconvex model of practical scale, we first reformulate it into a mixed integer quadratic programming by exploring the structural properties. Next, we use the reformulation linearization technique to further improve the computational speed. Our numerical studies validate the efficiency of the proposed solution approach. In addition, we derive some managerial insights, including that retailers should adopt joint optimization and consider how the involved effects and parameters may influence their profits and decisions.

Optimal household appliances scheduling for smart energy management considering inclining block rate tariff and net-metering system

Smart grids that integrate household renewable energy sources and share information with households can help create and maintain a smarter data-driven environment. Within this environment, flexible home energy management policies that minimize household energy costs can be adopted. This paper considers a smart home with a renewable energy source that favors satisfying its energy needs at minimum cost. This is achievable by smartly scheduling the use of its domestic appliances to match a given energy grid tariff. Focusing on the case of Egypt in which an inclining block rate (IBR) tariff is imposed, this paper fills a gap in the literature regarding the load scheduling models aiming to minimize energy cost at the household level whenever such a tariff exists. A new mixed integer quadratic programming (MIQP) model is formulated for this scheduling problem, considering the adopted net metering system with installed domestic photovoltaic (PV) systems in Egypt. The model generates the optimal household load schedule and the optimal amounts of energy to exchange with the grid while considering all the system and consumer utility constraints. To assess the applicability of the proposed model, a survey is conducted to identify the diversity and characteristics of using the electrical appliances by the Egyptian households. Based on the collected survey results, the effectiveness of the proposed MIQP model is investigated. Results confirm the effectiveness of the proposed model to minimize energy cost for different categories of the Egyptian households.

MPC-based framework incorporating pre-disaster and post-disaster actions and transportation network constraints for weather-resilient power distribution networks

Enhancing power system resilience to extreme weather requires an effective coordination of pre-disaster preventive response actions and post-disaster repair and restoration services. In this context, this article presents a two-stage framework to enhance resilience of power distribution networks to severe weather events. It considers operating and resource constraints in interdependent power and transportation networks based on official guidance from transport appraisal studies. It also incorporates sources of information available in a smart city context, such as energy measurements, traffic flows, and geographical information systems. In the first stage, a pre-disaster operational planning strategy is defined to prepare the grid for a high-risk outage scenario with the objective of minimizing the value of lost loads, formulated as a mixed-integer quadratic programming model. In the second stage, a post-disaster corrective strategy is implemented over a receding time horizon to compensate for deviations between the actions computed in the first stage and the actions required to minimize power outages, embedded into a model predictive control scheme. The effectiveness of the proposed framework is demonstrated on a real-world large-scale distribution network in the United Kingdom over a range of outage scenarios. Simulation results show that the proposed framework is capable of effectively minimizing first-stage operational costs and second-stage deviations between the projected and actual load demand supply. Numerical results obtained with the proposed framework indicate that the load energy unserved is at least twice smaller than with typical practices adopted by distribution network operators and computational times take two minutes or less. Therefore, it can be effectively used by distribution network operators to ensure an appropriate level of preparedness to power outages caused by extreme weather along with prompt restoration and repair services.

Exact mixed-integer quadratic formulation and solution for large-scale thermal unit commitment

Abstract Thermal unit commitment (UC) is a nonlinear combinatorial optimization problem that minimizes total operating costs while considering system load balance, on/off restrictions and other constraints. Successfully solving the thermal UC problem contributes to a more reliable power system and reduces thermal costs. This paper presents an exact mixed-integer quadratic programming (EMIQP) method for large-scale thermal UC problems. EMIQP revolutionizes the landscape by seamlessly translating the intricate nonlinear combinatorial optimization problem of UC into an exact mixed-integer quadratic formulation. This approach also elegantly reimagines on/off constraints as mixed-integer linear equations, employing both the sum and respective approaches. Our case studies unequivocally demonstrate the exceptional prowess of the EMIQP method, consistently securing the global optimum. Moreover, the mathematical-based EMIQP method produces identical results at each run, which is extremely important for UC in the real world.

Multi-criteria Optimization of Real-time DAGs on Heterogeneous Platforms under P-EDF

This article tackles the problem of optimal placement of complex real-time embedded applications on heterogeneous platforms. Applications are composed of directed acyclic graphs of tasks, with each directed-acyclic-graph (DAG) having a minimum inter-arrival period for its activation requests and an end-to-end deadline within which all of the computations need to terminate since each activation. The platforms of interest are heterogeneous power-aware multi-core platforms with Dynamic Voltage and Frequency Scaling (DVFS) capabilities, including big.LITTLE Arm architectures and platforms with GPU or FPGA hardware accelerators with Dynamic Partial Reconfiguration capabilities. Tasks can be deployed on CPUs using partitioned EDF-based scheduling. Additionally, some of the tasks may have an alternate implementation available for one of the accelerators on the target platform, which are assumed to serve requests in non-preemptive FIFO order. The system can be optimized by minimizing power consumption, respecting precise timing constraints, maximizing the applications’ slack, respecting given power consumption constraints, or even a combination of these, in a multi-objective formulation. We propose an off-line optimization of the mentioned problem based on mixed-integer quadratic constraint programming (MIQCP). The optimization provides the DVFS configuration of all the CPUs (or accelerators) capable of frequency switching and the placement to be followed by each task in the DAGs, including the software-vs.-hardware implementation choice for tasks that can be hardware accelerated. For relatively big problems, we developed heuristic solvers capable of providing suboptimal solutions in a significantly reduced time compared to the MIQCP strategy, thus widening the applicability of the proposed framework. We validate the approach by running a set of randomly generated DAGs on Linux under SCHED_DEADLINE, deployed onto two real boards, one with Arm big.LITTLE architecture, the other with FPGA acceleration, verifying that the experimental runs meet the theoretical expectations in terms of timing and power optimization goals.

A Unified Framework Integrating Trajectory Planning and Motion Optimization Based on Spatio-Temporal Safety Corridor for Multiple AGVs

Planning safe and smooth trajectories for multiple autonomous ground vehicles (MAGVs) in a complex dynamic unstructured environment is a fundamental and challenging task. In this paper, a novel unified framework integrating trajectory planning and motion optimization (TPMO) is proposed based on spatio-temporal safety corridor (STSC), which guarantees collision avoidance and trajectory smoothness. The proposed TPMO framework consists of two parts. The first part is to establish the STSC for each AGV based on the mixed integer quadratic programming (MIQP) algorithm. The proposed STSC method ensures collision avoidance in the environment of static and dynamic obstacles, and provides a longitudinal and lateral coupled trajectory (LLCT) for trajectory planning. The second part is to design a motion optimization methodology, which considers the constraints of AGV geometry as well as longitudinal and lateral coupled motion characteristics. Moreover, our formulation provides a theoretical guarantee that the entire trajectory is optimal under collision avoidance. Finally, the proposed TPMO framework is applied to solve the optimal cooperative trajectory and motion planning problem of MAGVs in a near-natural simulation and real vehicle environments, validating the proposed framework's effectiveness and practicality.

Research on Indoor Digital Design and Experience Based on Perception and Behavioral Association

Abstract This study examines interior digital design and experience based on perceptual-behavioral associations, focusing on how users’ perceptual cognition affects the internal design experience. This study collects data on users’ physiological and psychological responses when observing interior design by combining eye movement and electroencephalography techniques. Morphological analysis was used to deconstruct and reorganize the interior design elements, while the Mixed Integer Quadratic Programming (MIQP) method was used to design the interior space layout. The experimental results show that subjects’ average perceptual imagery ratings of interior schemes in the internal design perception and behavioral experiments differed significantly, and the EEG data and eye movement data exhibited specific behavioral characteristics. For example, there were statistically significant differences in perceptual imagery ratings and physiological data across interior design schemes within a particular time window. Combining eye-movement and EEG data can capture users’ perceptual perceptions and behavioral responses.

A Hierarchical Multi-Vehicle Coordinated Motion Planning Method Based on Interactive Spatio-Temporal Corridors

Multi-vehicle coordinated motion planning has always been challenged to safely and efficiently resolve conflicts under non-holonomic dynamic constraints. Constructing spatial-temporal corridors for multi-vehicle can decouple the high-dimensional conflicts and further reduce the difficulty of obtaining feasible trajectories. Therefore, this paper proposes a novel hierarchical multi-vehicle coordinated motion planning method based on interactive spatio-temporal corridors (ISTCs). In the first layer, based on the initial guidance trajectories, Mixed Integer Quadratic Programming is designed to construct ISTCs capable of resolving conflicts in generic multi-vehicle scenarios. And then in the second layer, Non-Linear Programming is settled to generate in-corridor trajectories that satisfy the vehicle dynamics. By introducing ISTCs, the multi-vehicle coordinated motion planning problem is able to be decoupled into single-vehicle trajectory optimization problems, which greatly decentralizes the computational pressure and has great potential for real-world applications. Besides, the proposed method searches for feasible solutions in the 3-D <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(x,y,t)$</tex-math></inline-formula> configuration space, preserving more possibilities than the traditional velocity-path decoupling method. Simulated experiments in unsignalized intersection and challenging dense scenarios have been conducted to verify the feasibility and adaptability of the proposed framework.

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.

Optimal Solution for the Integrated Model of the Problems in Capacitated Vehicle Routing and Vehicle Scheduling at the Multi-door Depot

To have an efficient Supply Chain (SC), the coordination and integration of the activities in the SC are mandatory. Routing vehicles is an optimization problem in the SC. When the routed vehicles return to the depot and the doors at the depot are busy, the returned vehicles must wait to unload the accumulated shipments. Therefore, properly scheduling these vehicles to those doors at the depot to minimize the waiting time is an optimization problem in the SC. Therefore, in this study, routing vehicles to collect the shipments from suppliers and scheduling vehicles to doors at the depot, based on a first-come-first-serve basis, are simultaneously solved. Hence, the objective of this integrated vehicle routing and scheduling problem (VR&amp;SP) is to minimize the total cost which contains the following components: vehicle travelling cost between suppliers, loading cost at the suppliers, vehicle waiting cost, unloading cost at the depot and vehicle operations cost. A Mixed Integer Quadratic Programming (MIQP) model is developed to solve the integrated VR&amp;SP. The Branch and Bound algorithm method is employed to obtain the exact optimal solution to this MIQP using LINGO optimization software. The compatibility of the developed MIQP model is verified by the randomly generated small-scale instances of VR&amp;SP. Therefore, it can be concluded that, this model solves the vehicle routing to suppliers and vehicle scheduling to doors at the depot simultaneously. Since VR&amp;SP is a NP-hard problem, heuristics or meta-heuristic methods are recommended to solve the large-scale instances of VR&amp;SP. Moreover, it can be recommended for further studies to amend this model by incorporating additional constraints to make it more applicable to real-world scenarios.

Optimal design of hydrogen-based storage with a hybrid renewable energy system considering economic and environmental uncertainties

Hydrogen and electricity derived from renewable sources present feasible alternative energy options for the decarbonisation of the transportation and power sectors. This study presents the utilisation of hydrogen generated from solar and wind energy resources as a clean fuel for mobility and backup storage for stationary applications under economic and environmental uncertainties. This is achieved by developing a detailed techno-economic model of an integrated system consisting of a hydrogen refuelling station and an electric power generation system using Mixed Integer Quadratic Constrained Programming (MIQCP), which is further relaxed to Mixed Integer Linear Programming (MILP). The model is implemented in the Advanced Interactive Multidimensional Modelling Software (AIMMS) and considering the inherent uncertainties in the wind resource, solar resource, costs and discount rate, the total cost of the three configurations (Hybrid PV-Wind, Standalone PV, and Standalone wind energy system) was minimised using robust optimisation technique, and the corresponding optimal sizes of the components, levelised cost of energy (LCOE), excess energy, greenhouse emission avoided, and carbon tax were evaluated. The levelised cost of the deterministic optimisation solution for all the configuration ranges between 0.0702 $/kWh to 0.0786 $/kWh, while the levelised cost of the robust optimisation solution ranges between 0.07188 $/kWh to 0.1125 $/kWh. The proposed integration has the advantages of affordable hydrogen and electricity prices, minimisation of carbon emissions and grid export of excess energy.

Optimal cyclic scheduling of ethylene plants in backup furnace mode with separation capacity constraints

Ethylene plants (EPs) are a critical output source of ethylene. The cyclic scheduling optimization is crucial to increase the profitability of an EP. The EP comprises the ethylene cracking furnace system (ECFS), quenching, compression, separation sections, etc., which culminates in a large-scale optimization problem. In addition, the performance-decaying and semi-continuous characteristics of the ECFS inevitably lead to fluctuations in product throughput, resulting in an unstable production process and product quality decline. To address this issue, we propose a mixed-integer quadratic programming (MIQP) model for the cyclic scheduling of the EP in backup furnace mode with separation capacity constraints. This novel model can reduce fluctuations in product throughput by running a backup furnace when one furnace is shut down for decoking. Besides, the separation capacity constraints ensure that high-quality products can be separated from the cracked gas downstream. Moreover, the reformulated normalized multiparametric disaggregation technique (RNMDT), which relies on convex envelopes, is employed to relax the proposed MIQP model into a mixed-integer linear programming (MILP) model by partitioning the domain of the variables in the quadratic terms. Subsequently, an optimization framework based on the RNMDT is established to enhance the solution efficiency of the original problem. Finally, a case study from the real world reveals the effectiveness of the proposed method, which can provide instructive optimal cyclic scheduling for the EP.

Stability Verification of Neural Network Controllers Using Mixed-Integer Programming

We propose a framework for the stability verification of Mixed-Integer Linear Programming (MILP) representable control policies. This framework compares a fixed candidate policy, which admits an efficient parameterization and can be evaluated at a low computational cost, against a fixed baseline policy, which is known to be stable but expensive to evaluate. We provide sufficient conditions for the closed-loop stability of the candidate policy in terms of the worst-case approximation error with respect to the baseline policy, and we show that these conditions can be checked by solving a Mixed-Integer Quadratic Program (MIQP). Additionally, we demonstrate that an outer and inner approximation of the stability region of the candidate policy can be computed by solving an MILP. The proposed framework is sufficiently general to accommodate a broad range of candidate policies including ReLU Neural Networks (NNs), optimal solution maps of parametric quadratic programs, and Model Predictive Control (MPC) policies. We also present an open-source toolbox in Python based on the proposed framework, which allows for the easy verification of custom NN architectures and MPC formulations. We showcase the flexibility and reliability of our framework in the context of a DC-DC power converter case study and investigate its computational complexity.

Trade-off between optimal design and operation in district cooling networks

Especially in densely populated areas, district cooling represents an opportunity to reduce energy consumption and emissions. Nevertheless, this technology is characterised by large capital costs which impede its diffusion. As a consequence, optimization tools can significantly help to unleash their potential. In this paper, a methodology is proposed to combinedly optimize the design and operation of a district cooling system based on a Mixed Integer Quadratic Programming. The model is compared to the design only optimization, based on a properly tailored heuristic approach. The models, when applied to a case study characterized by seasonal demand, provide similar solutions, which differ by 0.5 % in terms of objective value for a standard scenario. The simultaneous design and operation optimization does not provide sensible savings with respect to optimizing solely the design. A sensitivity analysis is performed to prove the robustness of the results. The results showed that the simultaneous operation and design optimization would be limited to 1 % of total costs in the case of seasonal cooling demand. On the other hand, if the cooling demand persists throughout the year, as in tropical climates, the combined optimization provides significant benefits, since these savings reach 4.7 % of total costs.

National-scale bi-directional EV fleet control for ancillary service provision

Deploying real-time control on large-scale fleets of electric vehicles (EVs) is becoming pivotal as the share of EVs over internal combustion engine vehicles increases. In this paper, we present a Vehicle-to-Grid (V2G) algorithm to simultaneously schedule thousands of EVs charging and discharging operations, that can be used to provide ancillary services. To achieve scalability, the monolithic problem is decomposed using the alternating direction method of multipliers (ADMM). Furthermore, we propose a method to handle bilinear constraints of the original problem inside the ADMM iterations, which changes the problem class from Mixed-Integer Quadratic Program (MIQP) to Quadratic Program (QP), allowing for a substantial computational speed up. We test the algorithm using real data from the largest carsharing company in Switzerland and show how our formulation can be used to retrieve flexibility boundaries for the EV fleet. Our work thus enables fleet operators to make informed bids on ancillary services provision, thereby facilitating the integration of electric vehicles.

Risk-based distributionally robust optimal dispatch for multiple cascading failures in regional integrated energy system using surrogate modeling

Regional integrated energy system (RIES) with the close coupling of multiple energies faces various cascading failure risks in its operation, and formulating the risk-based optimal dispatch scheme is important to ensure the secure and economic operation of the RIES. Considering that the occurrence probability of each subsequent failure path caused by the initial failure in the cascading failure chain varies with the change of dispatch scheme, the cooling/heating and gas pipeline dynamics and the uncertainty of renewable energy, a risk-based distributionally robust optimal dispatch model for multiple cascading failures in a RIES is proposed. A kernel ambiguity set is constructed by using the maximum mean discrepancy between the real and reference probability distributions and the kernel mean embedding technology, which can obtain the more accurate worst-case probability distribution. Moreover, a surrogate modeling method based on piecewise convex quadratic functions is proposed to learn the mapping between the cascading failure risks and the decision variables, and the original large-scale mixed integer nonlinear programming model is transformed into a small-scale mixed integer quadratic constrained programming model for efficient solution. Finally, case study on an actual RIES demonstrates the effectiveness and advantages of the proposed method. Compared to optimal dispatch without considering the cascading failure risks, the proposed method can reduce the total failure risk cost by nearly 60%.