Mixed-integer Linear Optimization Research Articles

Sustainable development of the water-energy-Co2 nexus in the refining sector: A stochastic multi-objective optimization under emissions trading systems

Refineries consume substantial energy and water while emitting significant CO2. Solvent-based Carbon Capture (CC) offers a viable CO2 mitigation solution but increases water and energy consumption and requires considerable investments. This research aims to overcome these barriers and improve the Iranian refining sector towards Sustainable Development Goals (SDGs) 6, 7, and 13 by mitigating CO2 emissions, producing low-carbon hydrogen, and utilizing unconventional water resources for CC plants. A comprehensive nexus approach was incorporated to account for the intricate interconnections between SDGs and refineries' water, energy, and CO2 networks. We evaluate the implementation of CO2 market regulations corresponding to seven international Emissions Trading Systems (ETSs) of the EU, UK, New Zealand (NZ), Regional Greenhouse Gas Initiative (RGGI), South Korea, California, and China in the Iranian refining sector to incentivize CC integration. The problem was formulated using Stochastic Multi-Objective Mixed-Integer Linear Programming optimization, with K-means Clustering applied to account for ETS CO2 price fluctuations and uncertainties. Results showed intense competition between economic and environmental objectives without ETS regulations, but appropriate ETS implementation could transform this conflict into a cooperative compromise. Under EU and UK scenarios, 60.3% and 67% of CO2 emissions could be avoided with fully covered capture costs, potentially leading to net economic profit through surplus allowance sales. Implementation of the other five ETSs could reduce capture costs by 18–67%, with a 16.4% CO2 avoidance.

Design of closed-loop manufacturing-remanufacturing system network considering uncertain defect and waste rate: Scenario-based chance-constrained programming

To facilitate sustainability by extending the life-cycle and residual value of production materials, a collaborate scheduling scheme for closed-loop manufacturing system with disturbance uncertainty is studied. Based on the features of the closed-loop manufacturing system under multi-product and multi-period scenarios, a mixed-integer linear optimization model is developed, where manufacturing and remanufacturing units are coordinated by recycling the production materials from work-in-process and final products. Based on this, a novel closed-loop manufacturing-remanufacturing scheduling framework is proposed. Additionally, in response to variations in product quality and processing capabilities, a scenario-based chance-constrained programming is introduced to address potential sources of uncertainty. Subsequently, a sequential optimization method is devised to reduce computational complexity. Finally, an industrial case from the electronic factory is investigated to demonstrate its applicability by implementing the proposed approach with comparison to several other strategies.

Benchmark of mixed-integer linear programming formulations for district heating network design

Optimal routing and investment decisions are key design criteria to reduce the high investment costs of district heating systems. However, these optimization problems have prohibitively high computational costs for large districts. Four different mixed-integer linear optimization frameworks are benchmarked in this study in order to compare their computational scaling. The frameworks exhibit significant differences in solving times for synthetic benchmarks and real-world urban districts of up to 9587 potential edges. The new open-source framework topotherm, developed for this work, exhibits the best computational performance when only one time step is optimized. The comparison between the models Résimont, DHmin, DHNx, and topotherm shows two main trends. First, fewer integer variables do not necessarily translate to lower solving times, and second, using redundant binary variables, which introduce symmetries into the constraints, leads to higher solving times. None of the considered optimization frameworks is able to solve the largest benchmark problems for five time steps within the allowed time limit and tolerance. These findings highlight the challenges of and pressing need to develop efficient models for simultaneous optimization of district heating network topology, pipe sizing, and operation.

On the relationship between the value function and the efficient frontier of a mixed integer linear optimization problem

On the relationship between the value function and the efficient frontier of a mixed integer linear optimization problem

Mixed-Integer Linear Optimization Formulations for Feature Subset Selection in Kernel SVM Classification

Mixed-Integer Linear Optimization Formulations for Feature Subset Selection in Kernel SVM Classification

Evolving scientific discovery by unifying data and background knowledge with AI Hilbert

The discovery of scientific formulae that parsimoniously explain natural phenomena and align with existing background theory is a key goal in science. Historically, scientists have derived natural laws by manipulating equations based on existing knowledge, forming new equations, and verifying them experimentally. However, this does not include experimental data within the discovery process, which may be inefficient. We propose a solution to this problem when all axioms and scientific laws are expressible as polynomials and argue our approach is widely applicable. We model notions of minimal complexity using binary variables and logical constraints, solve polynomial optimization problems via mixed-integer linear or semidefinite optimization, and prove the validity of our scientific discoveries in a principled manner using Positivstellensatz certificates. We demonstrate that some famous scientific laws, including Kepler’s Law of Planetary Motion and the Radiated Gravitational Wave Power equation, can be derived in a principled manner from axioms and experimental data.

Spatial analysis of thermal groundwater use based on optimal sizing and placement of well doublets

This paper proposes an approach to optimize the technical potential of thermal groundwater use by determining the optimal sizing and placement of extraction–injection well doublets. The approach quantifies the maximum technically achievable volume of extracted groundwater in a given area and, hence, the amount of heat exchanged with the aquifer, considering relevant regulatory and hydraulic constraints. The hydraulic constraints ensure acceptable drawdown and rise of groundwater in extraction and injection wells for sustainable use, respectively, prevention of internal hydraulic breakthroughs, and adequate spacing between neighboring doublets. Analytical expressions representing these constraints are integrated into a mixed-integer linear optimization framework allowing efficient application to relatively large areas. The applicability of the approach is demonstrated by a real case study in Munich, where the geothermal potential of each city block is optimized independently. Six optimization scenarios, differing in terms of required minimum installed doublet capacity and spacings between doublets, underline the adaptability of the approach. The approach provides a comprehensive and optimized potential assessment and can be readily applied to other geographic locations. This makes it a valuable tool for thermal groundwater management and spatial energy planning, such as the planning of fourth and fifth generation district heating systems.

Data-Driven and Domain Adaption Framework for Optimizing Energy Usage Forecasting in Smart Buildings Using ARIMA Model

Building energy demand and energy system supply are increasingly balanced by energy storage and short-term DSM. This is necessary due to fluctuating renewable energy supplies and rising building power use. Worldwide, structures utilize tons of energy. Greenhouse gas emissions and operational expenses decrease with construction energy efficiency. Forecasting and optimization algorithms can solve challenges, including supply chains (inventory optimization), traffic, and sustainable energy system battery/load/production scheduling for carbon-free energy generation. We often solve optimization problems that need forecasting due to uncertain future values. Predicting and optimizing are challenging; therefore, little research has been done. Our method uses building energy modeling professionals' data to forecast neighboring building types for new or unknown building types. After training, we utilize the models to estimate energy usage for the k-closest building types and combine the predictions using a weighted average. We used time-series decomposition to detect uncertainty and a hybrid model to close this gap: The concepts encode static features and predictable patterns in time-series simulation results. The model learns latent performance differences and calibrates output using outcomes and history records. Historical data predicts public building energy ARIMA use. Our method covers data processing, training, validation, and forecasting. We measured our method. Mixed integer linear optimization and ARIMA projected most accurately.

Fast robust optimization of ORC based on an artificial neural network for waste heat recovery

Uncertainty in the organic Rankine cycle (ORC) system and the highly complex ORC models pose challenges to optimal operation. A data-driven robust parametric optimization for the ORC system is proposed to ensure high and stable performance under uncertainty. The uncertainty of the cyclic variables is estimated by their distribution parameters, and the average (expected) thermodynamic performance (the net output power of the ORC) is maximized as the optimization objective while minimizing its variance. An artificial neural network with the rectified linear unit is used as a surrogate model for optimization. Then the robust parametric optimization problem can be transformed into a mixed-integer linear optimization problem with a chance-constrained form, and the robust optimal solution can be solved quickly. Case studies show that the solution time of the robust optimization problem using the proposed method is about 1.4 s, which is much less than obtaining the optimal solution based on ORC mechanism models. Meanwhile, the optimal operating condition derived by the proposed robust optimization approach outperforms that obtained by the traditional deterministic strategy. The proposed approach not only improves the robustness of the system but also demonstrates the importance of considering uncertainty in parametric optimization.

An optimal expansion planning of power systems considering cycle-based AC optimal power flow

This paper presents a novel mixed-integer linear optimization formulation of the AC network-constrained, cost-based, integrated expansion planning problem. The formulation is used to determine the investment needs per technology including the location and sizing of new generation, energy storage, and transmission network assets in a future low-carbon power system. To reduce the size of the resulting problem, the AC optimal power flow (AC-OPF) model is represented in a compact way using cycle constraints. A bound tightening procedure is also considered to reduce the search space and improve the solver performance by adjusting the voltage bounds within the AC-OPF. Contrary to typically used formulations of the integrated expansion planning problem, the constraints considered here include all main aspects of system operation, namely unit commitment, energy storage system management, AC-OPF, and reactive power compensation. Thus, in this paper, we examine how both the proposed transmission expansion modeling developments and the interrelation of the integrated planning constraints affect the computation of the solution to the expansion planning problem. The performance of this formulation is assessed on the RTS-GMLC test system by computing the expansion plan and comparing it with the results of three other expansion planning formulations most frequently employed in the recent literature to address the integrated expansion planning problem for medium to large-scale systems. Expansion plans are computed and compared for different case studies and multiple scenarios. According to the comparative analysis, neglecting the AC-OPF or the unit commitment constraints can increase the total system costs by 7.10%–9.57% or 6.29%–8.39%, respectively. Unlike other modeling approaches, the proposed approach does not rely on simplifications that impact the quality of the solution. Thanks to the incorporated cycle-based AC-OPF constraints and the consideration of a bound tightening procedure, the computation time is reduced by 17.67%–27.21%.

A machine learning approach to analysis of Canadian provincial power system decarbonization

Canada committed to reducing national greenhouse gas emissions by 40–45 % by 2030 and achieving net-zero by 2050, relying on low-carbon electrification across economic sectors. This study investigates the machine learning’s potential to analyze least-cost optimization of Canadian provincial power systems to 2050, to identify robust pathways for net-zero electricity system greenhouse gas emissions, and to provide policy insights respecting wind and solar generation, reducing emissions from natural gas generation, and expanding interprovincial transmission. Surrogate modeling is applied as sensitivities to generation technology capital costs, electricity demand, and carbon pricing to results from an existing capacity expansion mixed-integer linear optimization model. Under almost all future scenarios, six provinces develop new wind generation, with substantial deployment in Alberta (mean 27.3 GW, max. 62.7 GW), Ontario (mean 6.8 GW, max. 19.3 GW), Saskatchewan (mean 3.6 GW, max 8.9 GW) and Manitoba (mean 1.5 GW, max 3.5 GW). With the potential exception of BC, where higher-cost wind generation results in solar generation becoming the primary low-carbon electrical energy source (mean 25.2 GW), future solar generation across Canada is more limited (mean 37.1 GW) and less certain than wind generation in all other Canadian provinces. Existing natural gas generation is maintained in almost all scenarios in all provinces, providing support for maintaining these resources for reliability and reserve requirements. Very limited natural gas is developed under any scenarios, with the exception of British Columbia (mean 4.3 GW) where more than 80 % of scenarios include new natural gas capacity. Additional transmission transfer capacity is developed in most scenarios between BC-Alberta (∼60 % of scenarios), Saskatchewan-Manitoba (∼85 %), and Ontario-Quebec (∼75 %). Carbon prices, along with wind and solar capital costs, act as primary determinants of least-cost decarbonization pathways. These findings offer insights for Canadian policymakers into net-zero pathways for provincial electricity systems that remain robust despite variability in inputs.

Stochastic optimization framework for hybridization of existing offshore wind farms with wave energy and floating photovoltaic systems

Due to the complementarity of renewable energy sources, there has been a focus on technology hybridization in recent years. In the area of hybrid offshore power plants, the current research projects mostly focus on the combinational implementation of wind, solar, and wave energy technologies. Accordingly, considering the already existing offshore wind farms, there is the potential for the implementation of hybrid power plants by adding wave energy converters and floating photovoltaics. In this work, a stochastic sizing model is developed for the hybridization of existing offshore wind farms using wave energy converters and floating photovoltaics considering the export cable capacity limitation. The problem is modeled from an investor perspective to maximize the economic profits of the hybridization, while the costs and revenues regarding the existing units and the export cable are excluded. Furthermore, to tackle the uncertainties of renewable energy generation, as well as the energy price, a scenario generation method based on copula theory is proposed to consider the dependency structure between the different random variables. Altogether, the hybridization study is modeled in a mixed integer linear programming optimization framework considering the net present value of the project as the objective function. The results showed that hybrid-sources-based energy generation provided the highest economic profit in the studied cases in the different geographical locations. Furthermore, the technical specifications of the farms have also been considerably improved providing more stable energy generation, guaranteeing a minimum level of power in a high share of the time, and with a better utilization of the capacity of the cable while the curtailment of energy is maintained within the acceptable range.

Impact of tariff structures on energy community and grid operational parameters

This paper presents a mixed-integer linear programming optimization model of a renewable energy community comprised of members with local generators, battery energy storage systems, electric vehicles, and heat pumps and thermal energy storage, thus representing a local multi-energy system. The goal of the paper is to analyze the impact of different tariff structures for incentivizing energy sharing within energy communities on the distribution grid operational parameters. To achieve this, a comprehensive analysis of a hypothetical renewable energy community is conducted using the presented optimization model. Various tariff structures are evaluated, including flat and dynamic tariffs, reflecting the day-ahead market prices for energy and supply, as well as various network tariff designs, such as regulated charge reduction, time-of-use network tariffs and capacity payments. The results indicate that sharing energy within communities reduces energy losses in the grid, peak loads and reverse power flows and results in a smoother voltage profile. It is also found that tariffs that include dynamic electricity and supply costs can further increase these benefits, while including capacity payments as an incentive mechanism yields most favourable results for all analyzed criteria.

Standalone hybrid power-hydrogen system incorporating daily-seasonal green hydrogen storage and hydrogen refueling station

In light of the recent developments in the green hydrogen industry, it is essential to investigate its compatibility and integration with existing electrical grids. This task becomes more challenging when considering its implementation in off-grid sites that solely rely on renewable energy. In cases where off-grid sites should provide electricity and hydrogen to a variety of consumers, it is compulsory to have electrical and hydrogen storage devices with long-term storage capabilities. Consequently, a precise programming technique is necessary to manage the storage and consumption of hydrogen and electricity. To investigate such a model, this paper presents a connected electricity-hydrogen grid in which all the required energy is supplied by solar and hydroelectric units. This off-grid system supplies the necessary energy for industrial and residential loads, electric vehicle charging stations, fuelcell car refueling stations, and natural gas pipelines. The power-to-hydrogen (P2H) and hydrogen-to-power (H2P) systems are used to facilitate energy flow between the electrical and hydrogen sectors. A daily-seasonal hydrogen storage is integrated as well. The proposed method contributes by integrating and improving the modeling, management, optimization, and operation of the daily-seasonal hydrogen storage and electrical storage. Its primary objective is to supply electricity and hydrogen to various consumers using solely renewable energy, even in the face of renewable energy fluctuations and outages. The model is mathematically expressed as a mixed integer linear optimization, implemented in GAMS software, and solved by the CPLEX solver. It aims at maximizing profit from selling electricity and hydrogen to the users. Results demonstrate that P2H and H2P units have a complementary operation with the solar system. When there is abundant solar energy, the P2H converts excess clean energy to green hydrogen and H2P operates when solar energy is not available. Seasonal storage also stores hydrogen from spring to autumn and discharges it in winter because of higher hydrogen prices in this season. Such an operation increases the annual profit by about 9%. When the hydroelectricity unit faces an outage in hours 8 to 13, the fuelcell uses the stored hydrogen and comes into operation to compensate hydropower outage. The net present value of profit amounts to $275,179 per year. The total hydrogen required is approximately 49,870 kg per year, with a total electricity requirement of 1190 MWh per year. The levelized revenue from hydrogen is achieved at around $4 per kilogram, while the levelized revenue from electricity is about $0.058 per kilowatt-hour. It is concluded that the integration of seasonal storage effectively reduces costs, boosts profits, and efficiently manages renewable energy fluctuations across different seasons.

Quantum Circuit Reconstruction from Power Side-Channel Attacks on Quantum Computer Controllers

The interest in quantum computing has grown rapidly in recent years, and with it grows the importance of securing quantum circuits. A novel type of threat to quantum circuits that dedicated attackers could launch are power trace attacks. To address this threat, this paper presents first formalization and demonstration of using power traces to unlock and steal quantum circuit secrets. With access to power traces, attackers can recover information about the control pulses sent to quantum computers. From the control pulses, the gate level description of the circuits, and eventually the secret algorithms can be reverse engineered. This work demonstrates how and what information could be recovered. This work uses algebraic reconstruction from power traces to realize two new types of single trace attacks: per-channel and total power attacks. The former attack relies on per-channel measurements to perform a brute-force attack to reconstruct the quantum circuits. The latter attack performs a single-trace attack using Mixed-Integer Linear Programming optimization. Through the use of algebraic reconstruction, this work demonstrates that quantum circuit secrets can be stolen with high accuracy. Evaluation on 32 real benchmark quantum circuits shows that our technique is highly effective at reconstructing quantum circuits. The findings not only show the veracity of the potential attacks, but also the need to develop new means to protect quantum circuits from power trace attacks. Throughout this work real control pulse information from real quantum computers is used to demonstrate potential attacks based on simulation of collection of power traces.

Closed loop model predictive control of a hybrid battery-hydrogen energy storage system using mixed-integer linear programming

The derivation of an efficient operational strategy for storing intermittent renewable energies using a hybrid battery-hydrogen energy storage system is a difficult task. One approach for deriving an efficient operational strategy is using mathematical optimization in the context of model predictive control. However, mathematical optimization derives an operational strategy based on a non-exact mathematical system representation for a specified prediction horizon to optimize a specified target. Thus, the resulting operational strategies can vary depending on the optimization settings.This work focuses on evaluating potential improvements in the operational strategy for a hybrid battery-hydrogen energy storage system using mathematical optimization. To investigate the operation, a simulation model of a hybrid energy storage system and a tailor-made mixed integer linear programming optimization model of this specific system are utilized in the context of a model predictive control framework. The resulting operational strategies for different settings of the model predictive control framework are compared to a rule-based controller to show the potential benefits of model predictive control compared to a conventional approach. Furthermore, an in-depth analysis of different factors that impact the effectiveness of the model predictive controller is done. Therefore, a sensitivity analysis of the effect of different electricity demands and resource sizes on the performance relative to a rule-based controller is conducted. The model predictive controller reduced the energy consumption by at least 3.9 % and up to 17.9% compared to a rule-based controller. Finally, Pareto fronts for multi-objective optimizations with different prediction and control horizons are derived and compared to the results of a rule-based controller. A cost reduction of up to 47 % is achieved by a model predictive controller with a prediction horizon of 7 days and perfect foresight.

Comparative modeling of cost-optimal energy system flexibility for Swedish and Austrian regions

This study develops a reproducible method for estimating the cost-efficient flexibility potential of a local or regional energy system. Future scenarios that achieve ambitious climate targets and estimate the cost-efficient flexibility potential of demonstration sites were defined. Flexible potentials for energy system assessment are upscaled from the demonstration sites in Eskilstuna (Sweden) and Lower Austria (Austria). As heat pumps (HPs) and district heating (DH) are critical for future heat demand, these sites are representative types of DH networks in terms of size and integration with the electricity grid. In both regions a TIMES model is used for energy system optimization, while for upscaling, Eskilstuna uses the building-stock model ECCABS, whereas Lower Austria uses a mixed integer linear programming optimization model, and the BALMOREL power system model. According to the modeling, HPs will dominate Eskilstuna’s heating sector by 2040. In Lower Austria, DH becomes more prevalent, in combination with wood biomass and HPs. These findings are explained by the postulated technological-economic parameters, energy prices, and CO2 prices. We conclude that future electricity prices will determine future heating systems: either a high share of centralized HPs (if electricity prices are low) or a high share of combined heat-and-power (if electricity prices are high). Large-scale energy storage and biomass can be essential solutions as may deliver increased cost-effectiveness, if available and under certain conditions.

Fertilizer Logistics in Brazil: Application of a Mixed-Integer Programming Mathematical Model for Optimal Mixer Locations

Background: Brazil is one of the largest consumers of fertilizers and is highly dependent on the international market to meet its demand for agricultural production inputs. The complexity of the fertilizer supply chain motivated us to carry out this study on redesigning the fertilizer logistics chain and evaluate strategies for reducing logistics costs by redesigning the fertilizer mixing network in Brazil, a country that is heavily dependent on imported fertilizers for agriculture. Methods: We introduce a multi-product mixed-integer linear programming optimization model encompassing the logistics network, from import ports to mixing factories and agricultural fertilizer supply centers. This model includes logistics infrastructure and taxes, accounting for greenhouse gas emissions (specifically carbon dioxide) in fertilizer logistics. Results: The results indicate that expanding the port capacity for fertilizer importation can significantly reduce logistics costs and greenhouse gas emissions by up to 22.5%, decreasing by 23.9% compared to the baseline. We also observed that removing taxes on fertilizer importation can reduce logistics costs by approximately 11%, but it increases greenhouse gas emissions by 2.25% due to increased reliance on road transport. We identified 15 highly resilient regions for establishing mixing factories, evaluated various scenarios and determined the importance of these locations in optimizing the fertilizer supply network in the country. Moreover, the results suggest a significant potential to enhance the role of Brazil’s Northern Arc region in fertilizer import flows. Conclusions: Public policies and private initiatives could be directed toward encouraging the establishment of mixing factories in the identified regions and increasing transport capacity in the Northern Arc region. Improving the logistical conditions of the fertilizer network would contribute to food security by reducing the costs of essential inputs in food production and promoting sustainability by reducing greenhouse gas emissions.

Lua APIs for mathematical optimization

In this paper we present our Lua programming language libraries that enable modeling and solving mixed integer linear and nonlinear mathematical optimization problems. On one side, these libraries provide a framework for developing algorithms written in Lua that use certain well known solvers within more complex procedures. On the other side, they facilitate the transformation of both input and output data of a mathematical programming problem, in order to compute the standardized coefficients that have to be transmitted to the solvers or prepare solution of solved problems for further data processing. Both of these use cases, together with Lua programming language simplicity, versatility and performance, make Lua a suitable programming language for use in scientific and engineering researches.

Energy Management System for a Smart Green Nanogrid feeding a Research Laboratory with Autonomous Mobile Robots

This paper proposes a mixed-integer linear programming optimization model used to define an energy management system tailored for nanogrids in buildings, integrating renewable energy sources, battery energy storage systems and task-executing autonomous mobile robots. Focused on a nanogrid to be realised at the Savona Campus of the University of Genoa, the energy management system optimizes power flows and robot task scheduling in order to minimize the operating costs, the curtailment of the photovoltaic source and the number of unperformed tasks. Its novelty lies in combining energy and task planning constraints, offering significant potential for sustainable building energy management.