Mixed Integer Quadratically Constrained Programming Research Articles

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.

Assessing the role of hybrid energy storage in generation expansion planning for enhanced frequency stability

As the power grid transforms towards a renewable energy dominant grid, which are associated with frequency stability challenges, it is important to combat these challenges through proper power system planning. This paper therefore investigates the impact of hybrid energy storage (HES) in enhancing the frequency stability of the modern grid. The optimization problem is formulated as a mixed integer quadratic constrained programming model to minimize the system cost, while maximizing system inertia. The proposed model is then implemented on an IEEE 6-bus system, and solved using CPLEX solver in GAMS, with its performance evaluated using two study cases. Case 1 considers only the economic objective (cost), while case 2 considers both the economic objective (cost) and the technical objective (system inertia). The result show that the incorporation of HES in the generation planning model improved the overall system inertia of the system by 11% (from 6.199s in case 1, to 6.901s in case 2), however the total system cost increased by 3% (from 218M$ in case 1 to 2.25M$ in case 2).

Airgap Insertion and Layer Reassignment Under Setup and Hold Timing Constraints

Airgap formed in intermetal dielectric (IMD) reduces coupling capacitance, and thus can be utilized for timing optimization. Metal layers with airgap are limited due to high cost of airgap formation. Layer reassignment is to relocate some timing critical wires in nonairgap layers to airgap layers while noncritical wires in airgap layers are reassigned to nonairgap layers. Airgap insertion is to determine the amount of airgaps that are inserted for each critical wires in airgap layers. The two problems are solved in unified fashion with a goal of maximizing setup total negative slack (TNS) while satisfying hold constraints and design rules. They can be formulated as mixed-integer quadratically constrained programming (MIQCP). So, for practical application, a heuristic algorithm is presented and is experimentally compared to MIQCP with small examples. The experiments demonstrate that setup TNS and setup worst negative slack (WNS) are improved by 37% and 8%, respectively; they are improved by 26% and 5% with a simple-minded approach. The algorithm is also parallelized for application to larger circuits; runtime is decreased by 69% with eight threads.

Combined Generation and Transmission Expansion Planning Model for Improved Modern Power System Resilience

In recent times, the power grid composition is changing as more renewable energy generators and energy storage systems are being integrated. This in turn influences the dynamics of the new grid as frequency instability sets in due to declining system inertia. It is therefore important to properly select renewable energy generators and energy storage units to mitigate frequency instabilities, particularly during times of system contingencies. This study therefore, proposes a new deterministic generation and transmission expansion planning model that enhances the frequency stability of the modern grid. Thus, a mixed integer quadratic constrained programming model is developed to minimize emissions, and cost, while maximizing the overall system inertia. The proposed multi-objective model was validated using three case scenarios and tested on a modified IEEE 9-bus test system. The results obtained from the model simulation reveal that the proposed model (case 3) achieves better frequency stability, higher system inertia, and minimized emissions compared to the models in case 1(only economic objective is considered) and case 2(only economic and environmental objectives are considered), however, at a marginally higher cost which justifies the aim of the study.

Distribution Network Topology Identification Using Asynchronous Transformer Monitoring Data

The topology of a distribution network with a high penetration of fluctuating renewable energy is usually adjusted frequently for the reliable and economical operation. This paper proposes an automatic topology identification method based on measurements from widely distributed transformer terminal units (TTUs) to align the topology model in the management system with the actual topology of the distribution network. The proposed method formulates the topology identification problem as an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L2$</tex-math></inline-formula> -norm minimization problem. The challenge lies in the absence of phase angle measurements and the sampling errors caused by the non-synchronous sampling time of different TTUs. Adding variables into the minimization problem for the unavailable phase angles introduces nonlinearity and non-convexity, and the non-synchronous sampling time can cause significant sampling errors when measuring the fluctuating power injections. The non-convexity is addressed by proposing a novel linearization and convex relaxation approach to convert the formulated minimization problem into a convex mixed integer quadratically constrained program (MIQCP) problem. The significant sampling error is addressed by generating pseudo measurements from Lagrange interpolation to be used as alternatives to the actual power measurements having significant errors. Case studies conducted in test systems demonstrate the effectiveness and efficiency of the proposed method.