Mixed Integer Quadratically Constrained Programming Research Articles

Global optimization for large‐scale water network synthesis based on dynamic partition and adaptive bound tightening

AbstractThe synthesis of large‐scale integrated water networks is typically formulated as nonconvex mixed‐integer quadratic constrained programming (MIQCP) or QCP problems. With the complexity arising from bilinear terms in modeling mass flows of contaminants and binary variables representing the presence of units or streams, numerous local optima exist, thus presenting a significant optimization challenge. This study introduces a deterministic global optimization algorithm based on mixed‐integer programming (MIP) to tackle such problems. The approach involves dynamically strengthening the relaxed problems to converge towards the original problems. A simultaneous partition strategy is proposed combining locally uniform division with dynamic partitioned variables choosing. Furthermore, several adaptive bound contraction schemes are introduced to efficiently manage the size of the relaxed problems, assisting in accelerating the solution process. The algorithm's effectiveness and robustness are demonstrated with a large test set, showing superior performance compared to commercial solvers specifically on MIQCP problems.

Robust dynamic train regulation integrated with stop-skipping strategy in urban rail networks: An outer approximation based solution method

In dense urban rail networks with high passenger demands, uncertain disturbances occur frequently, and the resulting train delays will likely spread over the whole network rapidly, hence degrading the service quality offered to passengers. To cope with the uncertainties of frequent disturbances in urban rail networks, this paper proposes a robust train regulation strategy based on the information gap decision theory, which allows the operators to adjust the conservativeness of adjustment schemes flexibly by varying system performances but without the need for prior knowledge of uncertain disturbances. Specifically, considering the coupling relationship between train dynamic flows and passenger dynamic flows, a mixed integer quadratically constrained programming (MIQCP) model is constructed for the robust train regulation problem to generate solutions with immunity against disturbance uncertainties, in which the envelope-bound model is used to characterizing the uncertain sets of disturbances. To meet the real-time requirements of train operation adjustment, a tailored outer approximation algorithm incorporating a two-phase heuristics method is devised to effectively solve the developed robust train regulation model, thereby quickly generating high-quality solutions. Moreover, the warm start technique and domain reduction technique are carefully developed to accelerate the solving procedure. Numerical experiments based on the Beijing metro network illustrate the robustness of the proposed train regulation strategies and the effectiveness of the designed solution approach.

Impossible Boomerang Attacks Revisited

The impossible boomerang (IB) attack was first introduced by Lu in his doctoral thesis and subsequently published at DCC in 2011. The IB attack is a variant of the impossible differential (ID) attack by incorporating the idea of the boomerang attack. In this paper, we revisit the IB attack, and introduce the incompatibility of two characteristics in boomerang to the construction of an IB distinguisher. With our methodology, all the constructions of IB distinguisher are represented in a unified manner. Moreover, we show that the related-(twea)key IB distinguishers possess more freedom than the ones of ID so that it can cover more rounds.We also propose a new tool based on Mixed-Integer Quadratically-Constrained Programming (MIQCP) to search for IB attacks. To illustrate the power of the IB attack, we mount attacks against three tweakable block ciphers: Deoxys-BC, Joltik-BC and SKINNY. For Deoxys-BC, we propose a related-tweakey IB attack on 14-round Deoxys-BC-384, which improves the best previous related-tweakey ID attack by 2 rounds, and we improve the data complexity of the best previous related-tweakey ID attack on 10-round Deoxys-BC-256. For Joltik-BC, we propose the best attacks against 10-round Joltik-BC-128 and 14-round Joltik-BC-192 with related-tweakey B attack. For SKINNY-n-3n, we propose a 27-round related-tweakey IB attack, which improves both the time and the memory complexities of the best previous ID attack. We also propose the first related-tweakey IB attack on 28-round SKINNY-n-3n, which improves the previous best ID attack by one round.

Optimal multimodal multi-echelon vaccine distribution network design for low and medium-income countries with manufacturing infrastructure during healthcare emergencies

In the event of routine immunizations for containing disease outbreaks, the Expanded Programme on Immunization (EPI) led Low and Medium-Income Countries (LMICs) follow a four-tier distribution network. Such a network is suitable for LMICs importing vaccines rather than manufacturing and may not be suitable for healthcare emergencies. Recent shift from routine immunizations to pandemic-orientation, calls for large-scale multimodal distribution models. This paper proposes a cold chain network for LMICs with manufacturing hubs during health emergencies, integrating vaccine manufacturers and incorporating multiple transportation modes. The problem is formulated as a Mixed Integer Quadratically Constrained Programming (MIQCP) problem, to minimize ‘Cumulative Transportation Time’ (CTT) with the minimal number of trips. To solve large-scale problems the paper also proposes a novel algorithm that divides certain parts of the network and retains the edges that are likely to give better solutions. The proposed model extends the existing body of the literature by introducing features such as vehicle choice between locations, generalized hierarchical structure, network decomposition, and appropriateness with LMIC networks. We conduct numerical experiments to validate the proposed algorithm using data from India. The results show that the proposed model saves cumulative transportation time along the optimally generated network when compared with the actual vaccine supply network in India during healthcare emergencies. The proposed algorithm is also faster than the formulation giving comparable results while solved in a commercial solver.

Cost-efficient RAN slicing for service provisioning in 5G/B5G

Network slicing represents a substantial technological advance in 5G mobile network, greatly expanding the variety and manifoldness of network services to be supported. Additionally, 3GPP 5G New Radio (NR) has introduced novel features such as mixed numerology and mini-slots, which can be harnessed by network slicing to cater to the diverse requirements of 5G services. While however the co-existence of multiple network slices leads to a challenging resource allocation problem, these new features also severely complicate the management of radio resources. As a further point of attention, the virtualization of radio functions may exact a significant toll from the, already limited, computing resources at the network edge. It follows that a cost-efficient resource allocation across all the slices becomes crucial. In this paper, we address the above-mentioned issues by modeling a cost-efficient radio resource management in 5G NR featuring network slicing, named CERS, through a Mixed Integer Quadratically constrained Program (MIQCP). We maximize the profit of all slices simultaneously guaranteeing the target data rate and delay specified in the service level agreements (SLAs) fo the different traffic flows. To reduce the complexity of the MIQCP problem, we decompose it into two sub-problems, namely, the scheduling problem of enhanced Mobile Broadband (eMBB) user equipments (UEs) on a time-slot basis and of Ultra-Reliable Low Latency Communications (uRLLC) UEs on a mini-slot basis, while keeping the objective unchanged. To address the scheduling issue of eMBB UEs, we employ a heuristic technique, and, by leveraging the outcome of this heuristic, we derive an optimal solution for the problem of uRLLC UEs. The significance of the proposed approach over a baseline approach is evaluated through extensive numerical simulations in terms of the number of allocated uRLLC resource blocks (RBs) per mini-slot. We also assess our approach by measuring the impact of the uRLLC slice changes on the eMBB slice, and vice versa, including delay for uRLLC users and data rates for eMBB users.

Using Simplified Distflow-Based Mixed-Integer Quadratically Constrained Programming Formulation for Optimum Selection of Conductor Size in Electrical Distribution Networks

Determining conductor size is a subproblem that plays a vital role in the planning process of distribution systems. The problem of selecting the conductor cross-sectional area is usually made a model of mixed-integer non-linear programming (MINLP). It is important to note, however, that the MINLP model is not always capable of ensuring convergence to a solution that is globally optimum. In this research, a mixed-integer quadratically constrained programming (MIQCP) formulation is developed as a method for determining the conductor cross-sectional area in power distribution networks in an optimal manner. The goal function aims at minimizing the lifetime cost of lines, including initial capital cost together with operational expenses. The suggested optimization model’s constraints consist of power balance equations, thermal loading capacity of branches, nodal voltage restrictions, budget limitations for investment, and the need to have the same wire size in the main feeder. By using the simplified DistFlow approach for electrical distribution networks and precisely linearizing the product of a binary variable and a continuous variable, the MIQCP formulation is derived from the MINLP model. It is likely possible to solve the MIQCP formulation efficiently in the GAMS environment by using available commercial solvers like CPLEX. The developed MIQCP model is validated using three medium voltage distribution grids of IEEE 33 buses, IEEE 85 nodes, and the Vietnamese real 102 buses. The calculation results revealed the accuracy along with the efficacy of the proposed optimization procedure.

Maximizing EV profit and grid stability through Virtual Power Plant considering V2G

The electrification of transportation through the widespread adoption of electric vehicles (EVs) has raised substantial concerns within the realm of power grid operations. This concern predominantly stems from the elevated electricity demand brought about by the surging population of EVs, consequently exerting strain on the power grid infrastructure which can be reduced with vehicle-to-grid (V2G) technology integration. To address this issue, this paper delves further into the realm of grid integration by introducing a Virtual Power Plant (VPP) concept to enhance the synergy between EVs and power grid. This study aims to compare different realistic objectives, ranging from total active power loss and voltage drop minimization to EV profit maximization and then optimize the balance between the distribution grid power quality and VPP profit through bi-level modeling. The presented model is devised as mixed-integer quadratically constrained programming (MIQCP) and incorporates Temporal Convolutional Network (TCN) based forecasting to handle the uncertain behavior of the residential loads using historical data. The experiments are conducted in IEEE 33-Bus and real-world 240-Bus distribution networks. The results indicate that enabling bidirectional power flow between the grid and VPP can yield significant profits for EV users while only marginally impacting the active power loss, approximately around 5%. This validation underscores how V2G not only presents various advantages for power system operators but also benefits EV users simultaneously.

CVaR Quantitative Uncertainty-Based Optimal Dispatch for Flexible Traction Power Supply System

The flexible traction power supply system (FTPSS) integrating a photovoltaic (PV) generation system, an energy storage system (ESS), and a railway power conditioner (RPC) is a promising solution for future railways. However, the volatilities and random uncertainties of PV and traction load negatively impact the ability of FTPSS to operate safely and economically. A dispatch method combining conditional value-at-risk (CVaR) and probabilistic scenarios is proposed for FTPSS to quantify uncertainties into the risk cost. To reconcile economy and robustness, a flexible optimization dispatch model that considers the economic expense and risk cost is built. Finally, this optimization problem is formulated as a mixed integer quadratically constrained programming (MIQCP) model and solved by the GUROBI solver. Based on a genuine instance in China, the method’s validity has been demonstrated. The validity of the method is verified based on a real case in China. The essence of the proposed method is to mitigate the risks related to PV and traction load changes to some extent during the day-ahead dispatch stage by taking into account potential fluctuations in the real-time operating stage. The proposed method can help dispatch decision-makers create effective dispatch strategies and is conducive to FTPSS energy savings and risk avoidance.

Energy storage system coordinated with phase-shifting transformer and dynamic rating equipment for optimal operation of wind-rich smart power networks

Emergence of flexibility devices into smart power systems can assist the power system operators in making effective and economical decisions for the power system scheduling. These devices include energy storage system (ESS), phase-shifting transformer (PST), dynamic transformer rating (DTR), and dynamic line rating (DLR). In this paper, an approach is proposed for optimal day-ahead scheduling of power system using coordinated operation of ESS, PST, DTR, and DLR units under high wind power penetration situation. The aim is to minimize operational, load shedding, and wind power curtailment costs by regarding all the problem constraints. All of the model's relations as well as AC power flow equations have been convexified in order to construct a mixed-integer quadratically-constrained programming (MIQCP) model whose solution will obtain global optimum results. The developed model is implemented in GAMS and applied to the IEEE 24-bus system under various case studies, and the obtained results are investigated thoroughly. By coordinated operation of the employed flexibility devices, the loading of transformers and lines reach 159 % and 246 % compared to their nominal rating without violation of temperature limitations. Through releasing the capacity of lines and transformers, the total scheduling cost is reduced by about 52 % compared to traditional operation. Also, the whole energy of wind farms is utilized without any curtailment or constraints violation.

Efficient computational strategies for a mathematical programming model for multi-echelon inventory optimization based on the guaranteed-service approach

This paper presents a Multi-Echelon Inventory Optimization (MEIO) framework, based on the Guaranteed-Service Model (GSM), to allocate safety stocks across a supply chain with several locations and products, minimizing costs while meeting service level objectives. Extending previous work by Achkar et al. (2023), this paper enhances the Mixed-Integer Quadratically Constrained Program (MIQCP) with a highly efficient solution approach. The model introduces a piecewise linear approximation, significantly improving computational efficiency and the accuracy of the approximation for the fill rate function. It also introduces a different and more efficient approach to account for stochastic lead times using a discrete function. Moreover, an extension of the approach to account for non-normally distributed demands is proposed. The model is applied to several instances of a real-world case study from a pharmaceutical company, with more than 7300 product-location combinations, showing that optimal solutions can be obtained within few seconds of computational time.

Formulating a quadratic frequency nadir constraint in unit commitment based on energy conservation law

This paper proposes a quadratic formulation for the frequency nadir constraint (FNC) based on the energy conservation law. Incorporating the formulation into the unit commitment (UC) model, a mixed-integer quadratically constrained programming (MIQCP) model for frequency nadir constrained UC can be obtained, which can be solved efficiently by the MIQCP solver. The distinct feature of the formulation is that it considers the inertia difference among generators, which has a significant effect on the on/off schedule of generators. Besides, the formulation has a clear physical meaning and does not have to be linearized. The test on the WSCC 3-machine 9-bus benchmark system validates the accuracy of the formulation, and the case studies on a New England 10-machine 39-bus system confirm the effectiveness of the proposed formulation.

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.

Effect of climate on the optimal sizing and operation of seasonal ice storage systems

Seasonal thermal storage systems can reduce the temporal mismatch between renewable energy availability and energy demand. Ice storage systems exhibit a non-linear behaviour in the heat exchange and storage processes, complicating the formulation of optimal design and operation problems. In this work, we propose a mixed-integer quadratically-constrained programming formulation, which minimizes the Levelized Cost of Energy for space heating and cooling, including sizing of a supporting PV array. The optimization was repeated for different storage volumes, finding the system optimal operation in each case –and thereby the optimal system sizing. The heating and cooling demands were computed from an archetypal office building, placed in three reference locations with cold and semi-arid, warm and humid continental, and temperate and humid continental climates. Results show that the optimal PV size decreases with growing ice storage volume, and an ice storage works best in a temperate continental climate, covering up to 47% of the cooling demand with a 250 m 3 storage.

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%.

Techno-economic investigation of hybrid peaker plant and hydrogen refuelling station

The power and transport sectors are responsible for significant emissions of greenhouse gases. Therefore, it is imperative that substantial efforts are directed towards the decarbonisation of these industries. This study establishes a combined-solar-wind system's economic and technical practicality for producing hydrogen for an onsite hydrogen refuelling station (HRS) and electricity to meet peak demand. To minimise the levelised cost of electricity and maximise the system's reliability at different commercial locations in South Africa, the dual-objective optimisation sizing is carried out using Mixed Integer Quadratic Constrained Programming (MICQP) model and was executed with an Advanced Multi-dimensional Modelling System (AIMMS) [61], [62]. The levelised costs of electricity and hydrogen at Johannesburg, Pretoria and Cape Town for 2 MW grid export benchmark are 74.2 $/MWh/5.85 $/kg, 76.3 $/MWh/5.97 $/kg and 50 $/MWh/4.45 $/kg respectively. The CO₂ equivalent emissions (tonnes) are 54,000, 55,800, 59,000 and the corresponding carbon taxes ($) avoided for the locations are 432,100, 446,200, and 472,000 for Johannesburg, Pretoria and Cape Town respectively. The results of the framework show that it can be adopted as a viable and fossil-free replacement for conventional peaking generators.

Joint Topology and State Estimation Using TTU Monitoring Data Considering Three-Phase Imbalance

A grand challenge for operators to develop optimal controls at the distribution level lies in how to obtain the real-time network topology and operation state in lack of specialized high precision monitoring devices, especially with high penetration of distributed renewable energy. Driven by this motivation, this paper proposes a novel topology and operation state joint estimation framework based on the monitoring data from transformer terminal units (TTUs). In the framework, the bottom layer generates an equivalent network model and nodal voltage and current measurements with synchronized phase angles from the distribution network and the voltage and power measurement data from TTU. The middle layer formulates, linearizes and solves an optimization problem to fit the model with the measurements. A global piecewise linearization method is proposed to convert the original mixed integer quadratically constrained program (MIQCP) problem into a mixed integer linear program (MILP) problem. In the top layer, the line switch states and feeder power flows are estimated with the solution of the optimization problem. Case studies are conducted in test systems to verify the accuracy, efficiency and robustness of the proposed framework and show that the proposed framework yields obvious advantages over the existing approaches on the estimation accuracy when using TTU measurements.

Distributed conditional cooperation model predictive control of interconnected microgrids

In this paper, we propose a model predictive control based operation strategy that allows for power exchange between interconnected microgrids. Particularly, the approach ensures that each microgrid benefits from power exchange with others. This is realised by including a condition which is based on the islanded operation cost. The overall model predictive control problem is posed as a mixed-integer quadratically-constrained program and solved using a distributed algorithm that iteratively updates continuous and integer variables. For this algorithm, termination, feasibility and computational properties are discussed. The performance and the computational benefits of the proposed strategy are highlighted in an illustrative case study.

Flexible and sustainable scheduling of electric power grids: A dynamic line and transformer rating based approach under uncertainty condition

Integration of renewable and sustainable energies into the electric power grids has affected planning and operation problems of the power system. Voltage and congestion as well as uncertainty issues are some challenges regarding this integration. Dynamic line and transformer ratings, known as DLR and DTR, are kinds of smart-grid technologies that can help alleviation of above-mentioned problems and perform a flexible scheduling for the power system. In this paper, the DLR and DTR have been simultaneously incorporated in optimal power flow (OPF) problem in wind-energy-integrated power systems considering the presence of uncertainties. The aim is to perform an optimal day-ahead scheduling of thermal and wind power units where the uncertainty of DLR and DTR as well as that of wind power generation are considered. The uncertainty scenarios are generated by Monte-Carlo method where k-means clustering algorithm is employed to reduce the number of scenarios within a scenario-based stochastic optimization. As the AC power flow and dynamic rating relations have non-convex nature, they have been convexified in order to formulate the optimization problem as a mixed-integer quadratically-constrained programming (MIQCP). The proposed approach is applied to IEEE-24-bus system in different experiments in GAMS environment. The simulations demonstrate efficiency of the conducted approach in reducing operating costs, minimizing load shedding cost, and maximum scheduling of wind power units as sustainable and clean energy resources.

Linear model decision trees as surrogates in optimization of engineering applications

Machine learning models are promising as surrogates in optimization when replacing difficult to solve equations or black-box type models. This work demonstrates the viability of linear model decision trees as piecewise-linear surrogates in decision-making problems. Linear model decision trees can be represented exactly in mixed-integer linear programming (MILP) and mixed-integer quadratic constrained programming (MIQCP) formulations. Furthermore, they can represent discontinuous functions, bringing advantages over neural networks in some cases. We present several formulations using transformations from Generalized Disjunctive Programming (GDP) formulations and modifications of MILP formulations for gradient boosted decision trees (GBDT). We then compare the computational performance of these different MILP and MIQCP representations in an optimization problem and illustrate their use on engineering applications. We observe faster solution times for optimization problems with linear model decision tree surrogates when compared with GBDT surrogates using the Optimization and Machine Learning Toolkit (OMLT).

Integrated Power Distribution Network Synthesis for Mixed Macro Blocks and Standard Cells

Power distribution network (PDN) requires more routing resources as wire resistance increases with technology scaling but IR drop constraint does not scale. We address post-placement PDN synthesis for mixed macro blocks and standard cells. Macro blocks already have partial PDN, while standard cells do not. So the design, which has been placed, is divided into a number of partitions: each macro block in its own partition, and standard cell area into a number of partitions. Multi-layer PDN synthesis is performed in each partition one by one; the goal of this problem is to insert minimum number of metal straps (in irregular pitches) while IR drop and electromigration (EM) constraints are respected. This is formulated as mixed integer quadratically constrained programming (MIQCP) and a heuristic algorithm is proposed for practical application. Experimental results with some test circuits indicate that the proposed approach achieves on average of 26% reduction in PDN straps, which helps reduce congestion overflow of signal routing by 15%, compared to popular approach in which PDN straps are located in regular pitch period.