Mixed-integer Optimization Algorithm Research Articles

A hybrid quantum–classical algorithm for mixed-integer optimization in power systems

Mixed Integer Linear Programming (MILP) can be considered the backbone of the modern power system optimization process, with a large application spectrum, from Unit Commitment and Optimal Transmission Switching to verifying Neural Networks for power system applications. The main issue of these formulations is the computational complexity of the solution algorithms, as they are considered NP-Hard problems. Quantum computing has been tested as a potential solution towards reducing the computational burden imposed by these problems, providing promising results, motivating the can be used to speedup the solution of MILPs. In this work, we present a general framework for solving power system optimization problems with a Quantum Computer (QC), which leverages mathematical tools and QCs’ sampling ability to provide accelerated solutions. Our guiding applications are the optimal transmission switching and the verification of neural networks trained to solve a DC Optimal Power Flow. Specifically, using an accelerated version of Benders Decomposition , we split a given MILP into an Integer Master Problem and a linear Subproblem and solve it through a hybrid “quantum–classical” approach, getting the best of both worlds. We provide 2 use cases, and benchmark the developed framework against other classical and hybrid methodologies, to demonstrate the opportunities and challenges of hybrid quantum–classical algorithms for power system mixed integer optimization problems.

An optimization method of electrostatic sensor array based on Kriging surrogate model and improved non-dominated sorting genetic algorithm with elite strategy algorithm

Array-type electrostatic monitoring is gradually becoming an effective tool for aero-engine fault diagnosis. In this paper, we innovatively apply the surrogate optimization method to the optimization of the sensor array structure in order to meet the need of improving the particle information recognition capability of the electrostatic sensor array (ESA). A structure optimization method of ESA based on the Kriging surrogate model and improved NSGA-II algorithm is proposed. In this paper, a finite element simulation model of ESA is established, and the array optimization problem is abstracted as the solution of a mixed-integer optimization problem. This paper reduces the large-scale numerical simulations in the full-variable space with the help of the Kriging surrogate model. In addition, an improved NSGA-II algorithm for mixed-integer optimization is proposed. The simulation experiment verified that the average absolute error of the sensor before and after optimization for the identification of particle position and charge quantity was reduced by 69.88% and 49.68%, respectively. The array structure optimization method proposed in this paper facilitates the acceleration of the design process of electrostatic sensors and provides a scientific design method for their specific design for airborne application.

An efficient and reliable scheduling algorithm for unit commitment scheme in microgrid systems using enhanced mixed integer particle swarm optimizer considering uncertainties

The use of an electrical energy storage system (EESS) in a microgrid (MG) is widely recognized as a feasible method for mitigating the unpredictability and stochastic nature of sustainable distributed generators and other intermittent energy sources. The battery energy storage (BES) system is the most effective of the several power storage methods available today. The unit commitment (UC) determines the number of dedicated dispatchable distributed generators, respective power, the amount of energy transferred to and absorbed from the microgrid, as well as the power and influence of EESSs, among other factors. The BES deterioration is considered in the UC conceptualization, and an enhanced mixed particle swarm optimizer (EMPSO) is suggested to solve UC in MGs with EESS. Compared to the traditional PSO, the acceleration constants in EMPSO are exponentially adapted, and the inertial weight in EMPSO decreases linearly during each iteration. The proposed EMPSO is a mixed integer optimization algorithm that can handle continuous, binary, and integer variables. A part of the decision variables in EMPSO is transformed into a binary variable by introducing the quadratic transfer function (TF). This paper also considers the uncertainties in renewable power generation, load demand, and electricity market prices. In addition, a case study with a multiobjective optimization function with MG operating cost and BES deterioration defines the additional UC problem discussed in this paper. The transformation of a single-objective model into a multiobjective optimization model is carried out using the weighted sum approach, and the impacts of different weights on the operating cost and lifespan of the BES are also analyzed. The performance of the EMPSO with quadratic TF (EMPSO-Q) is compared with EMPSO with V-shaped TF (EMPSO-V), EMPSO with S-shaped TF (EMPSO-S), and PSO with S-shaped TF (PSO-S). The performance of EMPSO-Q is 15%, 35%, and 45% better than EMPSO-V, EMPSO-S, and PSO-S, respectively. In addition, when uncertainties are considered, the operating cost falls from $8729.87 to $8986.98. Considering BES deterioration, the BES lifespan improves from 350 to 590, and the operating cost increases from $8729.87 to $8917.7. Therefore, the obtained results prove that the EMPSO-Q algorithm could effectively and efficiently handle the UC problem.

Optimal design for vibration energy harvesters based on quasi-periodic structures

In this paper, the design of large-scale quasi-periodic Vibration Energy Harvesters (VEH) is optimized to enhance the harvested power of an electromagnetic mode localized structure. This work aims to optimize the output power by employing the energy localization phenomenon in a large-scale periodic configuration by introducing the minimum number of perturbations. The harvested power, number and location of perturbations are among the objectives that need to be optimized. A genetic-based mixed-integer optimization algorithm is used to meet the objective functions within a constraint on the system kinetic energy. Numerical simulations for quasi-periodic systems with 20 and 100 Degrees of Freedom (DOF) are performed. It is shown that the ratio of harvested power increases as the number of perturbations rises and it exceeds 80% of the total output power by perturbing almost one-third of the total DOFs. The proposed methodology is a decision-making aid to provide an optimal design in a generalized quasi-periodic VEH in order to reduce the number of harvesting transducers while providing a significantly high amount of harvested power.

Best Subset, Forward Stepwise or Lasso? Analysis and Recommendations Based on Extensive Comparisons

In exciting recent work, Bertsimas, King and Mazumder (Ann. Statist. 44 (2016) 813–852) showed that the classical best subset selection problem in regression modeling can be formulated as a mixed integer optimization (MIO) problem. Using recent advances in MIO algorithms, they demonstrated that best subset selection can now be solved at much larger problem sizes than what was thought possible in the statistics community. They presented empirical comparisons of best subset with other popular variable selection procedures, in particular, the lasso and forward stepwise selection. Surprisingly (to us), their simulations suggested that best subset consistently outperformed both methods in terms of prediction accuracy. Here, we present an expanded set of simulations to shed more light on these comparisons. The summary is roughly as follows: •neither best subset nor the lasso uniformly dominate the other, with best subset generally performing better in very high signal-to-noise (SNR) ratio regimes, and the lasso better in low SNR regimes; •for a large proportion of the settings considered, best subset and forward stepwise perform similarly, but in certain cases in the high SNR regime, best subset performs better; •forward stepwise and best subsets tend to yield sparser models (when tuned on a validation set), especially in the high SNR regime; •the relaxed lasso (actually, a simplified version of the original relaxed estimator defined in Meinshausen (Comput. Statist. Data Anal. 52 (2007) 374–393)) is the overall winner, performing just about as well as the lasso in low SNR scenarios, and nearly as well as best subset in high SNR scenarios.

Rejoinder: Best Subset, Forward Stepwise or Lasso? Analysis and Recommendations Based on Extensive Comparisons

In exciting recent work, Bertsimas, King and Mazumder (Ann. Statist. 44 (2016) 813–852) showed that the classical best subset selection problem in regression modeling can be formulated as a mixed integer optimization (MIO) problem. Using recent advances in MIO algorithms, they demonstrated that best subset selection can now be solved at much larger problem sizes than what was thought possible in the statistics community. They presented empirical comparisons of best subset with other popular variable selection procedures, in particular, the lasso and forward stepwise selection. Surprisingly (to us), their simulations suggested that best subset consistently outperformed both methods in terms of prediction accuracy. Here, we present an expanded set of simulations to shed more light on these comparisons. The summary is roughly as follows: •neither best subset nor the lasso uniformly dominate the other, with best subset generally performing better in very high signal-to-noise (SNR) ratio regimes, and the lasso better in low SNR regimes; •for a large proportion of the settings considered, best subset and forward stepwise perform similarly, but in certain cases in the high SNR regime, best subset performs better; •forward stepwise and best subsets tend to yield sparser models (when tuned on a validation set), especially in the high SNR regime; •the relaxed lasso (actually, a simplified version of the original relaxed estimator defined in Meinshausen (Comput. Statist. Data Anal. 52 (2007) 374–393)) is the overall winner, performing just about as well as the lasso in low SNR scenarios, and nearly as well as best subset in high SNR scenarios.

Data-Driven Identification of Characteristic Real-Driving Cycles Based on k-Means Clustering and Mixed-Integer Optimization

Virtual powertrain analysis is widely applied in the automotive industry to cope with the increasing complexity and variance of future vehicle propulsion technologies. Since the vehicle-usage behavior has a strong impact on component loads, realistic computation results require accurate assumptions about these boundaries. In this context, driving cycles (DCs) are used to represent the system boundaries in vehicle operation. The aim of this article is to identify multiple characteristic driving cycles (CDCs) from extensive vehicle measurement data which represent the full variety of possible real-driving scenarios. Vehicle measurements are segmented and consumption-relevant features are extracted from each segment. These features are then used to apply clustering and classification techniques to identify characteristic groups that are consequently assigned to different driving environments and driving styles. In order to obtain even more realistic driving scenarios, a data-fusion approach is used to incorporate a road slope signal from a NASA digital elevation model for each segment. Lastly, a genetic mixed-integer optimization algorithm is proposed to efficiently generate representative DCs for each characteristic group of driving segments. The main contribution of this article is a data-driven identification of the parameter space of real-driving scenarios from extensive vehicle measurements including the implementation of road slope information. The scenarios are represented via a constrained number of compact CDCs which enables comprehensive investigations of new powertrain technologies under average as well as extreme real-driving conditions to develop efficiency-robust powertrain systems.

A two-level hierarchical discrete-device control method for power networks with integrated wind farms

Power systems depend on discrete devices, such as shunt capacitors/reactors and on-load tap changers, for their long-term reliability. In transmission systems that contain large wind farms, we must take into account the uncertainties in wind power generation when deciding when to operate these devices. In this paper, we describe a method to schedule the operation of these devices over the course of the following day. These schedules are designed to minimize wind-power generation curtailment, bus voltage violations, and dynamic reactive-power deviations, even under the worst possible conditions. Daily voltage-control decisions are initiated every 15 min using a dynamic optimization algorithm that predicts the state of the system over the next 4-hour period. For this, forecasts updated in real-time are employed, because they are more precise than forecasts for the day ahead. Day-ahead schedules are calculated using a two-stage robust mixed-integer optimization algorithm. The proposed control strategies were tested on a Chinese power network with wind power sources; the control performance was also validated numerically.

DEVELOPING GREY-BOX DYNAMIC PROCESS MODELS

This paper presents a methodology for developing grey models of process systems, that is, models that, being based on fundamental principles and laws of nature, combine them with sub-models obtained from experimental data. The method follows two steps: the first one takes advantage of what is known, while the second uses the data and mixed-integer optimization algorithms to identify the structure and parameters of the remaining parts of the model. The method is illustrated in a challenging biotechnological process: the Acetone-Butanol-Ethanol (ABE) fermentation process.

GAMBIT: A Parameterless Model-Based Evolutionary Algorithm for Mixed-Integer Problems.

Learning and exploiting problem structure is one of the key challenges in optimization. This is especially important for black-box optimization (BBO) where prior structural knowledge of a problem is not available. Existing model-based Evolutionary Algorithms (EAs) are very efficient at learning structure in both the discrete, and in the continuous domain. In this article, discrete and continuous model-building mechanisms are integrated for the Mixed-Integer (MI) domain, comprising discrete and continuous variables. We revisit a recently introduced model-based evolutionary algorithm for the MI domain, the Genetic Algorithm for Model-Based mixed-Integer opTimization (GAMBIT). We extend GAMBIT with a parameterless scheme that allows for practical use of the algorithm without the need to explicitly specify any parameters. We furthermore contrast GAMBIT with other model-based alternatives. The ultimate goal of processing mixed dependences explicitly in GAMBIT is also addressed by introducing a new mechanism for the explicit exploitation of mixed dependences. We find that processing mixed dependences with this novel mechanism allows for more efficient optimization. We further contrast the parameterless GAMBIT with Mixed-Integer Evolution Strategies (MIES) and other state-of-the-art MI optimization algorithms from the General Algebraic Modeling System (GAMS) commercial algorithm suite on problems with and without constraints, and show that GAMBIT is capable of solving problems where variable dependences prevent many algorithms from successfully optimizing them.

Constructing minimum-width confidence bands

We construct minimum-width confidence bands with a predefined nominal coverage from a finite set of sample paths. In contrast to the several heuristics suggested in the literature, our mixed-integer optimization algorithm calculates global minimum-width pathwise confidence bands. For very large problems, our solution can be used as an approximation with a bounded and known gap to the globally optimal solution.

Constructing Optimal Pathwise Confidence Bands Using Mixed-Integer Optimization

We address the question of how to construct optimal confidence bands with a predefined nominal coverage from a finite set of sample paths. We introduce a mixed-integer optimization algorithm to calculate globally optimal pathwise confidence bands. We illustrate the method with an example of realistic size.

Least quantile regression via modern optimization

We address the Least Quantile of Squares (LQS) (and in particular the Least Median of Squares) regression problem using modern optimization methods. We propose a Mixed Integer Optimization (MIO) formulation of the LQS problem which allows us to find a provably global optimal solution for the LQS problem. Our MIO framework has the appealing characteristic that if we terminate the algorithm early, we obtain a solution with a guarantee on its sub-optimality. We also propose continuous optimization methods based on first-order subdifferential methods, sequential linear optimization and hybrid combinations of them to obtain near optimal solutions to the LQS problem. The MIO algorithm is found to benefit significantly from high quality solutions delivered by our continuous optimization based methods. We further show that the MIO approach leads to (a) an optimal solution for any dataset, where the data-points $(y_i,\mathbf{x}_i)$'s are not necessarily in general position, (b) a simple proof of the breakdown point of the LQS objective value that holds for any dataset and (c) an extension to situations where there are polyhedral constraints on the regression coefficient vector. We report computational results with both synthetic and real-world datasets showing that the MIO algorithm with warm starts from the continuous optimization methods solve small ($n=100$) and medium ($n=500$) size problems to provable optimality in under two hours, and outperform all publicly available methods for large-scale ($n={}$10,000) LQS problems.

Antenna Design With a Mixed Integer Genetic Algorithm

Antenna design variables, such as size, have continuous values while others, such as permittivity, have a finite number of values. Having both variable types in one problem requires a mixed integer optimization algorithm. This paper describes a genetic algorithm (GA) that works with real and/or binary values in the same chromosome. The algorithm is demonstrated on designing low side-lobe phase tapers, circularly polarized patch antennas, and identically thinned subarrays

Automated Simultaneous Assembly for Multistage Testing

This article offers some solutions used in the assembly of the computerized Uniform Certified Public Accountancy (CPA) licensing examination as practical alternatives for operational programs producing large numbers of forms. The Uniform CPA examination was offered as an adaptive multistage test (MST) beginning in April of 2004. Examples of automated assembly using mixed integer programming (MIP) solutions in Optimization Programming Language software (OPL Studio 3.6.1©, ILOG Inc., Gentilly, France) illustrate MST design features and implementation strategies that can be generalized to other testing programs. This article affords a compromise between the best possible combination of test content and sustainability and security over time by linear programming techniques that make use of mixed integer optimization algorithms. Relative optimization functions are formulated to ensure a variety of constraints are always met for "testlets" of differing difficulties to form panels for the MST design. Technical information is discussed to assist other practitioners and researchers with test assembly optimization problems. Some practical consequences from the selection of statistical targets on testlet exposure are illustrated. These results are described with respect to the fundamental principles of automated test assembly and continuous administration based on MST when testlet or item exposure and inventory rotation are critical concerns.

Elimination of voltage violations in the Spanish electricity market: Part I: Theory

The solution of voltage violations in the Spanish electricity market must consider: (a) the connection of off-line units (considering the active and reactive power injection effect), (b) the adjustment of the voltage control resources (generator voltages, transformer taps and shunt reactors and capacitors), (c) the preventive solution of the voltage violations under the occurrence of the postulated contingencies and (d) the start-up cost internalized in the fixed income term of the generator offer. This start-up cost plays a key role in the solution of voltage violations since the coupled solution of the 24-h scenarios is different from the individual solution of each hourly scenario. This is the first part of a two-part paper. The objective of Part I is two-fold. On one hand, the paper analyses the complexity of eliminating voltage violations proposing a simple methodology that includes the following steps: (a) contingency analysis, (b) decoupled solution of voltage violations for each hourly scenario and (c) daily-coupled solution of voltage violations. On the other hand, the paper describes the mathematical formulation of a set of mixed-integer optimization algorithms designed to obtain the hourly-decoupled solution and the daily-coupled solution of voltage violations in the Spanish electricity market. In Part II, the methodology and performance of the algorithms will be illustrated and compared using an actual example of the Spanish electricity market.

Elimination of voltage violations in the Spanish electricity market: Part II: Case study

The solution of voltage violations in each hourly scenario of the Spanish electricity market must consider: (a) the connection of off-line units (considering the active and reactive power injection effect); (b) the adjustment of the voltage control resources (generator voltages, transformer taps and shunt reactors and capacitors) and (c) the preventive solution of the voltage violations under the occurrence of the postulated contingencies. This is the second part of a two-part paper. Part I has analyzed the complexity of eliminating voltage violations proposing a simple methodology that includes the following steps: (a) contingency analysis, (b) decoupled solution of voltage violations for each hourly scenario and (c) daily-coupled solution of voltage violations. In addition, Part I has described the mathematical formulation of a set of mixed-integer optimization algorithms designed to obtain the hourly decoupled solution and the daily-coupled solution of voltage violations in the Spanish electricity market. In Part II, the methodology and performance of the algorithms will be illustrated and compared using an actual example of the Spanish electricity. The hourly solution of the algorithms is compared from an economic and technical point of view and the system cost reduction of the daily solution is demonstrated.

Optimal Design of Isolated Network Systems Operated by Renewable Energies with Mixed-integer Optimization Algorithms

In this article, a method to solve isolated rural networks facilities supported by renewable energies by mixed-integer linear programming is described. The algorithm is applied on a specific location and includes renewable energy sources like windpower, hydroelectric and biomass, electrical loads from populations and irrigation needs, as well as a storage system based on a pumping-turbinate unit.

An algorithm for mixed integer optimization

This paper introduces a new algorithm for solving mixed integer programs. The core of the method is an iterative technique for changing the representation of the original mixed integer optimization problem.

A Parametric Mixed-Integer Optimization Algorithm for Multiobjective Engineering Problems Involving Discrete Decisions

An iterative, decomposition-based algorithm is proposed in this paper, for the solution of convex parametric MINLP problems and, in extension, the identification of the noninferior solution set in multiobjective problems involving continuous and discrete decisions. The parametric optimal solution is constructed via an upper- and lower-bound procedure. Parametric upper bounds are identified with the solution of parametric NLP problems, while the parametric lower bound is updated in each iteration via the solution of a parametric MILP Master problem, which involves only the binary variables of the initial problem. Convergence properties and computational requirements are discussed in example problems from process synthesis under uncertainty and simultaneous product/process design.