Bilinear Terms Research Articles

An IGDT-WDRCC based optimal bidding strategy of VPP aggregators in new energy market considering multiple uncertainties

This study addresses the volatility and uncertainty challenges in managing renewable energy within electricity markets, particularly focusing on the role of Virtual Power Plant (VPP) aggregators. Recognizing the risks these uncertainties pose to the revenue and stability of power systems, the paper presents a novel information gap decision theory (IGDT)-Wasserstein metric based distributionally robust chance constraint (WDRCC) approach to devise an optimal bidding strategy for VPP operators. It involves a data-driven distributionally robust optimization framework, leveraging the worst-case scenario from the distributed resource uncertainties, guided by an ambiguity set rooted in the Wasserstein metric. Furthermore, the distributionally robust chance constraint modeling is introduced ensuring that uncertainty constraints of distributed resources meet a predefined risk level. Although this method shows promising out-of-sample performance, it relies on forecasted energy prices, a notable limitation given the price volatility and information inadequacy in the newly-opened market. To address this, the risk-averse bidding strategy, grounded in IGDT, is proposed simulataneously to safeguard the operator’s expected returns against price uncertainties, implementing an advanced piecewise linear approximation technique, ”nf4l,” for linearizing the bi-linear term from IGDT. The effectiveness of this approach is empirically validated through a comprehensive case study and sensitivity analysis.

Distributionally Robust Portfolio Optimization under Marginal and Copula Ambiguity

AbstractWe investigate a new family of distributionally robust optimization problem under marginal and copula ambiguity with applications to portfolio optimization problems. The proposed model considers the ambiguity set of portfolio returns in which the marginal distributions and their copula are close—in terms of the Wasserstein distance—to their nominal counterparts. We develop a cutting-surface method to solve the proposed problem, in which the distribution separation subproblem is nonconvex and includes bilinear terms. We propose three approaches to solve the bilinear formulation, namely (1) linear relaxation via McCormick inequalities, (2) exact mixed-integer linear program reformulation via disjunctive inequalities, and (3) inner approximation method via a novel iterative procedure that exploits the structural properties of the bilinear optimization problem. We further carry out a comprehensive set of computational experiments with distributionally robust portfolios featuring Conditional Value-at-Risk (CVaR) measures. These tests aim to compare the accuracy of the proposed algorithms, analyze the impact of the radius of the Wasserstein ambiguity ball on the portfolio, and assess portfolio performance. We use a rolling-horizon approach to conduct the out-of-sample tests, which show the superior performance of the portfolios under marginal and copula ambiguity over the equally weighted and ambiguity-free Mean-CVaR benchmark portfolios.

Robust wind power capacity planning under fuel price uncertainty using conic duality theory and piecewise McCormick relaxation

Increasing the proportion of renewable energy sources (RESs) in power generation is crucial due to fossil fuel depletion and rising environmental pollution. In this regard, identifying the most lucrative sites and sizes for installing wind farms, as one of the fastest-growing types of RESs, is imperative. This paper presents a robust profit-oriented wind power capacity planning (WPCP) considering long- and short-term uncertainty. The model forms a two-stage min-max-min hierarchical structure. The first stage minimizes the investment cost plus maximum regret, while the second stage maximizes the profit under the worst-case uncertainty realization. Unlike the existing approaches where uncertainty in fossil fuel prices is neglected, we model fuel price uncertainty using both polyhedral and ellipsoidal uncertainty sets. In this respect, the third level is formulated as a bi-level program, with the upper level being the profit maximization and the lower level being the locational marginal price (LMP) calculation. In the case of the ellipsoidal set, the conic duality theory is employed to dualize the lower level. The piecewise McCormick relaxation (PMR) technique linearizes the bilinear terms. The nested column-and-constraint generation (NCCG) technique solves the formulated problem. A clarifying case study is employed to demonstrate the efficacy of the proposed model.

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.

Solving a bilevel strategic offering model for a generation company in a hydrothermal energy market: A new convexification approach

This paper proposes a bilevel strategic offering (SO) model suitable for a strategic company (SC) that operates in a hydro-dominated day-ahead energy market. The upper-level model of the proposed SO model maximizes the SC’s profits, while the lower-level model embodies a hydrothermal market clearing (MC). This MC incorporates detailed hydro and thermal constraints for units of both the SC and non-strategic companies (NSC), in addition to the network constraints. The proposed solution approach entails replacing the lower-level model with its corresponding primal and dual constraints, along with a strong duality equality constraint. This leads to a non-convex mathematical programming with equilibrium constraints (MPEC) model, which is then reformulated to feature only a single bilinear term in its structure. Finally, the recast MPEC model is convexified around an initial point and solved iteratively until a convergence criteria is achieved. The model and solution approach proposed are evaluated on an adapted version of the IEEE 24-bus system, exploring scenarios where the SC acts strategically or non-strategically (perfect competition).

A bilinear modeling in counts time series with applications

This paper introduces a modified bilinear model for integer-valued time series in which thinning operators are applied in bilinear terms involving the product of the input and state process separately. The proposed model is able to consider overdispersion. Furthermore, it connects a feature of the integer-valued autoregressive conditional heteroskedasticity and integer-valued autoregressive processes. Important properties of the proposed model as well as the sufficient condition for stationarity are derived. After considering some estimation methods based on time domain and frequency domain approaches, simulation studies are conducted to check the performance of the estimates. The analysis of practical cases in social science is accomplished to highlight the usefulness of the proposed model in applications, and the model’s adequacy is provided. Further, the suggested model discusses the problem of data forecasting.

Efficient separation of RLT cuts for implicit and explicit bilinear terms

Efficient separation of RLT cuts for implicit and explicit bilinear terms

A comprehensive stochastic-based adaptive robust model for transmission expansion planning

To address various uncertainties appearing in the transmission expansion planning (TEP) problem, appropriate tools such as stochastic programming (SP) or robust optimization (RO) are required. On the other hand, both long-term (LT) and short-term (ST) uncertainties must be well captured to acquire the most reliable and robust solution. The current works model the LT uncertainty by RO and the ST uncertainty by SP, forming a two-stage adaptive robust optimization problem. In contrast, this paper presents a novel approach in which the SP and RO are simultaneously utilized to cope with each type of uncertainty, i.e., LT and ST uncertainty. Unlike the current methods with bilinear terms appearing in the subproblems, the trilinear terms appear in the objective function of the proposed approach, which are linearized accordingly. In addition, we propose a mixed-integer bilinear programming (MIBP) model to specify the rectangles used as uncertainty sets, which is further solved using Konno's algorithm. A standard column-and-constraint (CCG) generation solves the established tri-level structure. Two case studies are used to implement the formulation developed in this work. Results signify the effectiveness of the proposed method.

Hyper-Rayleigh Scattering and Third-Harmonic Scattering in Chiral Liquids: Basic Evidences and Differences with Linear Chiroptical Techniques.

Nonlinear chiroptical methods like hyper-Rayleigh optical activity (HROA) and third-harmonic optical activity (THOA) have a great potential in characterizing structure-chiroptical properties of molecular systems. Here, with these methods, we bring for the first time experimental and theoretical evidence of nonlinear chiroptical differences between enantiomers of simple chiral molecules, using exclusively linearly polarized incident light. The origin of these unexpected nonlinear chiroptical contributions comes from a new nonlinear source term of the form βOA(∇×μ(nω)), which is a bilinear coupling term including the linear chirality parameter, βOA, and the curl of the nonlinear induced electric dipole moment, μ(nω), both involving dipolar magnetic interactions. Through a simple nonlinear chiroptical model that we propose, we show that specific nonlinear chiroptical parameters can be quantified, thus bringing new insights into stereochemical and electronic structural features of molecular and supramolecular systems.

An extended incremental technique for solving economic dispatch with practical considerations

This work develops a mixed-integer incremental model to solve economic dispatch (ED) with practical challenges of prohibited operating zones (POZs), transmission losses, and valve loading effects (VLE). The first challenge, the POZ, disconnects ED feasible space. The second challenge arises from transmission losses and imposes bilinear terms on ED. The VLE, in the third challenge, renders the ED objective function non-smooth and nonconvex. Due to the practical challenges, a disconnected and multimodal solution space arises, requiring a tailored solver. This work presents an extended incremental (Ext-Inc) technique to fulfill the requirement. The Ext-Inc enforces the POZs by using auxiliary variables and logical constraints. Moreover, the Ext-Inc uses linear combinatorial terms to represent the nonconvex VLEs and bilinear terms. The Ext-Inc is evaluated through ED case studies. The Ext-Inc finds the optimal cost of 1,016,279 $/h, 24,512 $/h, 62,456 $/h, 32,704 $/h, 121,412 $/h, 653,645 $/h in 10-unit, 13-unit, 15-unit, 20-unit, 40-unit, and 200-unit respectively. These findings either replicate or enhance the solution documented in previous studies, demonstrating the effectiveness of the Ext-Inc in addressing real-world challenges. Moreover, the proposed method demonstrates remarkable efficiency, solving single-period ED case studies in under 3 seconds and the multi-period case study within one minute.

On bilinear estimates and critical uniqueness classes for Navier-Stokes equations

We are concerned with bilinear estimates and uniqueness of mild solutions for the Navier-Stokes equations in critical spaces. For that, we construct general settings in which estimates for the bilinear term of the mild formulation hold true without using auxiliary norms such as Kato time-weighted ones. We first obtain necessary conditions in abstract critical spaces and then consider further structures to obtain the estimates in general classes of Besov, Morrey and Besov-Morrey spaces based on Banach spaces. Examples of applications are provided in different spaces as well as for other PDEs. In particular, as far as we know, the bilinear estimate and the uniqueness property obtained in the framework of Besov-weak-Herz spaces are not available in the existing literature. The proofs are mainly based on characterizations and estimates on the corresponding predual spaces.

Convexification of Bilinear Terms over Network Polytopes

It is well-known that the McCormick relaxation for the bilinear constraint z = xy gives the convex hull over the box domains for x and y. In network applications where the domain of bilinear variables is described by a network polytope, the McCormick relaxation, also referred to as linearization, fails to provide the convex hull and often leads to poor dual bounds. We study the convex hull of the set containing bilinear constraints [Formula: see text] where xi represents the arc-flow variable in a network polytope, and yj is in a simplex. For the case where the simplex contains a single y variable, we introduce a systematic procedure to obtain the convex hull of the above set in the original space of variables, and show that all facet-defining inequalities of the convex hull can be obtained explicitly through identifying a special tree structure in the underlying network. For the generalization where the simplex contains multiple y variables, we design a constructive procedure to obtain an important class of facet-defining inequalities for the convex hull of the underlying bilinear set that is characterized by a special forest structure in the underlying network. Computational experiments conducted on different applications show the effectiveness of the proposed methods in improving the dual bounds obtained from alternative techniques. Funding: This work was supported by Air Force Office of Scientific Research [Grant FA9550-23-1-0183]; National Science Foundation, Division of Civil, Mechanical and Manufacturing Innovation [Grant 2338641]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2023.0001 .

Disorder Operator and Rényi Entanglement Entropy of Symmetric Mass Generation.

The "symmetric mass generation" (SMG) quantum phase transition discovered in recent years has attracted great interest from both condensed matter and high energy theory communities. Here, interacting Dirac fermions acquire a gap without condensing any fermion bilinear mass term or any concomitant spontaneous symmetry breaking. It is hence beyond the conventional Gross-Neveu-Yukawa-Higgs paradigm. One important question we address in this Letter is whether the SMG transition corresponds to a true unitary conformal field theory. We employ the sharp diagnosis including the scaling of disorder operator and Rényi entanglement entropy in large-scale lattice model quantum MonteCarlo simulations. Our results strongly suggest that the SMG transition is indeed an unconventional quantum phase transition and it should correspond to a true (2+1)d unitary conformal field theory.

Supersymmetry with scalar sequestering

Supersymmetric models with a strongly interacting superconformal hidden sector (HS) may drive soft supersymmetry (SUSY) breaking scalar masses, bilinear soft term Bμ and Higgs combinations mHu,d2+μ2 to small values at some intermediate scale, leading to unique sparticle mass spectra along with possibly diminished fine-tuning in spite of a large superpotential μ parameter. We set up a computer code to calculate such spectra, which are then susceptible to a variety of constraints: (1) possible charge-or-color breaking (CCB) minima in the scalar potential, (2) unbounded from below (UFB) scalar potential, (3) improper electroweak symmetry breaking, (4) a charged or sneutrino lightest SUSY particle (LSP), (5) generating mh∼125 GeV, (6) consistency with LHC sparticle mass limits, and (7) naturalness. We find this bevy of constraints leaves little or no viable parameter space for the case where hidden sector dynamics dominates minimal supersymmetric standard model (MSSM) running, even for the case of nonuniversal gaugino masses. For the case with moderate HS running with comparable MSSM running, and with universal gaugino masses, then the fine-tuning is ameliorated, but nonetheless remains high. Viable spectra with moderate HS running and with low fine-tuning and large μ can be found for nonuniversal gaugino masses. Published by the American Physical Society 2024

Optimal Corrective Maintenance Policies via an Availability-Cost Hybrid Factor for Software Aging Systems

Availability is an important index for the evaluation of the performance of software aging systems. Although the corrective maintenance increases the system availability, the associated cost may be very high; therefore, the balancing of availability and cost during the corrective maintenance phase is a critical issue. This paper investigates optimal corrective maintenance policies via an availability-cost hybrid factor for software aging systems. The system is described by a group of coupled differential equations, where the multiplier effect of the repair rate on a system variable is bilinear term. Our aim is to drive an optimal repair rate that ensures a balance between the maximal system availability and the minimal repair cost. In a finite time interval [0,T], we rigorously discuss the state space of the system and prove the existence of the optimal repair rate, and then derive the first-order necessary optimality conditions by applying a variational inequality with the adjoint variables.

De Broglie-Bohm analysis of a nonlinear membrane: From quantum to classical chaos.

Within the de Broglie-Bohm theory, we numerically study a generic two-dimensional anharmonic oscillator including cubic and quartic interactions in addition to a bilinear coupling term. Our analysis of the quantum velocity fields and trajectories reveals the emergence of dynamical vortices. In their vicinity, fingerprints of chaotic behavior such as unpredictability and sensitivity to initial conditions are detected. The simultaneous presence of the off-diagonal -kxy and nonlinear terms leads to robust quantum chaos very analogous to its classical version.

Bilinear R-parity violating supersymmetry under the light of neutrino oscillation, Higgs and flavor data

In this work, we explore a well motivated beyond the Standard Model scenario, namely, R-parity violating Supersymmetry, in the context of light neutrino masses and mixing. We assume that the R-parity is only broken by the lepton number violating bilinear term. We try to fit two non-zero neutrino mass square differences and three mixing angle values obtained from the global χ2 analysis of neutrino oscillation data. We have also taken into account the updated data of the standard model (SM) Higgs mass and its coupling strengths with other SM particles from LHC Run-II along with low energy flavor violating constraints like rare b-hadron decays. We have used a Markov Chain Monte Carlo (MCMC) analysis to constrain the new physics parameter space. While doing so, we ensure that all the existing collider constraints are duly taken into account. Through our analysis, we have derived the most stringent constraints possible to date with existing data on the 9 bilinear R-parity violating parameters along with μ and tan β. We further explore the possibility of explaining the anomalous muon (g - 2) measurement staying within the parameter space allowed by neutrino, Higgs and flavor data while satisfying the collider constraints as well. We find that there still remains a small sub-TeV parameter space where the required excess can be obtained.

A novel approach for optimizing the natural gas liquefaction process

The conversion of natural gas into liquefied natural gas (LNG) requires substantial energy consumption. This study proposes a deterministic mathematical model to find the optimal operation conditions for an LNG plant to minimize the energy consumption for turbomachinery and the total thermal conductance of the heat exchangers. General Algebraic Modeling System (GAMS) linked to a Dynamic Link Library was used to calculate the thermodynamic properties of the working fluids. A derivative-based optimization algorithm is used. Results indicate that the novel optimization approach allows the satisfactory management of the model nonlinearities associated, for example, with the bilinear terms involved in the energy balances and the mathematical functions used to calculate the thermodynamic properties. A preprocessing phase for initializing process variables is developed to facilitate model convergence. In comparison to an optimal design reported in the literature, which was obtained by integrating a well-established evolutionary optimization approach with the Aspen HYSYS simulator, the results indicated that the net electrical power could be reduced by up to 10% when the derivative-based optimization algorithm is used. The proposed deterministic approach, consisting of a mathematical model, an initialization phase, and an optimization algorithm, can help process engineers overcome the challenges associated with LNG process optimization.

Dynamic nuclear polarization pulse sequence engineering using single-spin vector effective Hamiltonians.

Dynamic nuclear polarization (DNP) has proven to be a powerful technique to enhance nuclear spin polarization by transferring the much higher electron spin polarization to nuclear spins prior to detection. While major attention has been devoted to high-field applications with continuous microwave irradiation, the introduction of fast arbitrary waveform generators is gradually increasing opportunities for the realization of pulsed DNP. Here, we describe how static-powder DNP pulse sequences may systematically be designed using single-spin vector effective Hamiltonian theory. Particular attention is devoted to the intricate interplay between two important parts of the effective first-order Hamiltonian, namely, linear field (single-spin) terms and Fourier coefficients determining scaling of the bilinear coupling terms mediating polarization transfer. We address two cases. The first case operates in the regime, where the microwave field amplitude is lower than the nuclear Larmor frequency. Here, we illustrate the predictive strength of a single-spin vector model by comparing analytical calculations with experimental DNP results at 9.8 GHz/15 MHz on trityl radicals at 80 K. The second case operates in the high-power regime, where we combine the underlying single-spin vector design principles with numerical non-linear optimization to optimize the balance between the linear terms and the bilinear Fourier coefficients in a figure of merit function. We demonstrate, numerically and experimentally, a broadband DNP pulse sequence PLATO (PoLarizAtion Transfer via non-linear Optimization) with a bandwidth of 80 MHz and optimized for a microwave field with a maximum (peak) amplitude of 32 MHz.

Optimizing admissible region of nodal net load for active distribution network based on cost/benefit analysis

As the proportion of renewable energy sources (RESs) increases, active distribution network (ADN) operation becomes a more challenging problem due to unpredictable nature of RESs. To address this problem, a new dynamic economic dispatch (DED) for ADN is proposed which can determine the optimal scheduling results based on cost/benefit analysis. Intervals with variable endpoints are applied to describe the nodal net load, which is also named as admissible region in this paper. The admissible region of nodal net load and other scheduling results are simultaneously determined by striking a balance between operation cost when net load varies within the corresponding intervals and benefit represented by reduction of operation risk when net load varies out of their intervals. To facilitate the solution, a new injection shift factor (ISF) of the linearized DistFlow branch equations is proposed, and the voltage magnitude and reactive power can be well addressed, which especially fits for ADN. A surrogate affine policy is applied to avoid the introduction of bilinear terms. And a new step-superimposing linearization method for risk cost is proposed by exploring the specific characteristics of risk cost expression, then binary variables can be eliminated in the model. Finally, this model is recast as a linear programming problem and the computation efficiency can be guaranteed. The validity and effectiveness of the proposed model are illustrated on the IEEE 33-node and 69-node distribution systems.