Improving QC Relaxations of OPF Problems via Voltage Magnitude Difference Constraints and Envelopes for Trilinear Monomials

AC optimal power flow (AC~OPF) is a challenging non-convex optimization problem that plays a crucial role in power system operation and control. Recently developed convex relaxation techniques provide new insights regarding the global optimality of AC~OPF solutions. The quadratic convex (QC) relaxation is one promising approach that constructs convex envelopes around the trigonometric and product terms in the polar representation of the power flow equations. This paper proposes two methods for tightening the QC relaxation. The first method introduces new variables that represent the voltage magnitude differences between connected buses. Using "bound tightening" techniques, the bounds on the voltage magnitude difference variables can be significantly smaller than the bounds on the voltage magnitudes themselves, so constraints based on voltage magnitude differences can tighten the relaxation. Second, rather than a potentially weaker "nested McCormick" formulation, this paper applies "Meyer and Floudas" envelopes that yield the convex hull of the trilinear monomials formed by the product of the voltage magnitudes and trignometric terms in the polar form of the power flow equations. Comparison to a state-of-the-art QC implementation demonstrates the advantages of these improvements via smaller optimality gaps.Comment: 8 pages, 1 figur

Tightening QC Relaxations of AC Optimal Power Flow Problems via Complex Per Unit Normalization

Optimal power flow (OPF) is a key problem in power system operations. OPF problems that use the nonlinear AC power flow equations to accurately model the network physics have inherent challenges associated with non-convexity. To address these challenges, recent research has applied various convex relaxation approaches to OPF problems. The QC relaxation is a promising approach that convexifies the trigonometric and product terms in the OPF problem by enclosing these terms in convex envelopes. The accuracy of the QC relaxation strongly depends on the tightness of these envelopes. This paper presents two improvements to these envelopes. The first improvement leverages a polar representation of the branch admittances in addition to the rectangular representation used previously. The second improvement is based on a coordinate transformation via a complex per unit base power normalization that rotates the power flow equations. The trigonometric envelopes resulting from this rotation can be tighter than the corresponding envelopes in previous QC relaxation formulations. Using an empirical analysis with a variety of test cases, this paper suggests an appropriate value for the angle of the complex base power. Comparing the results with a state-of-the-art QC formulation reveals the advantages of the proposed improvements

Tightening QC Relaxations of AC Optimal Power Flow through Improved Linear Convex Envelopes

AC optimal power flow (AC OPF) is a fundamental problem in power system operations. Accurately modeling the network physics via the AC power flow equations makes AC OPF a challenging nonconvex problem. To search for global optima, recent research has developed a variety of convex relaxations that bound the optimal objective values of AC OPF problems. The well-known QC relaxation convexifies the AC OPF problem by enclosing the non-convex terms (trigonometric functions and products) within convex envelopes. The accuracy of this method strongly depends on the tightness of these envelopes. This paper proposes two improvements for tightening QC relaxations of OPF problems. We first consider a particular nonlinear function whose projections are the nonlinear expressions appearing in the polar representation of the power flow equations. We construct a convex envelope around this nonlinear function that takes the form of a polytope and then use projections of this envelope to obtain convex expressions for the nonlinear terms. Second, we use certain characteristics of the sine and cosine expressions along with the changes in their curvature to tighten this convex envelope. We also propose a coordinate transformation that rotates the power flow equations by an angle specific to each bus in order to obtain a tighter envelope. We demonstrate these improvements relative to a state-of-the-art QC relaxation implementation using the PGLib-OPF test cases. The results show improved optimality gaps in 68% of these cases

Let's talk about varying G

It is possible that fundamental constants may not be constant at all. There is a generally accepted view that one can only talk about variations of dimensionless quantities, such as the fine structure constant $\alpha_{\rm e}\equiv e^2/4\pi\epsilon_0\hbar c$. However, constraints on the strength of gravity tend to focus on G itself, which is problematic. We stress that G needs to be multiplied by the square of a mass, and hence, for example, one should be constraining $\alpha_{\rm g}\equiv G m_{\rm p}^2/\hbar c$, where $m_{\rm p}$ is the proton mass. Failure to focus on such dimensionless quantities makes it difficult to interpret the physical dependence of constraints on the variation of G in many published studies. A thought experiment involving talking to observers in another universe about the values of physical constants may be useful for distinguishing what is genuinely measurable from what is merely part of our particular system of units.Comment: 6 pages, Gravity Research Foundation essa

Correlation of bandgap reduction with inversion response in (Si)GeSn/high-k/metal stacks.

The bandgap tunability of (Si)GeSn group IV semiconductors opens a new era in Si-technology. Depending on the Si/Sn contents, direct and indirect bandgaps in the range of 0.4 eV to 0.8 eV can be obtained, offering a broad spectrum of both photonic and low power electronic applications. In this work, we systematically studied capacitance-voltage characteristics of high-k/metal gate stacks formed on GeSn and SiGeSn alloys with Sn-contents ranging from 0 to 14 at.% and Si-contents from 0 to 10 at.% particularly focusing on the minority carrier inversion response. A clear correlation between the Sn-induced shrinkage of the bandgap energy and enhanced minority carrier response was confirmed using temperature and frequency dependent capacitance voltage-measurements, in good agreement with k.p theory predictions and photoluminescence measurements of the analyzed epilayers as reported earlier. The enhanced minority generation rate for higher Sn-contents can be firmly linked to the bandgap reduction in the GeSn epilayer without significant influence of substrate/interface effects. It thus offers a unique possibility to analyze intrinsic defects in (Si)GeSn epilayers. The extracted dominant defect level for minority carrier inversion lies approximately 0.4 eV above the valence band edge in the studied Sn-content range (0 to12.5 at.%). This finding is of critical importance since it shows that the presence of Sn by itself does not impair the minority carrier lifetime. Therefore, the continuous improvement of (Si)GeSn material quality should yield longer non-radiative recombination times which are required for the fabrication of efficient light detectors and to obtain room temperature lasing action

Modulation of Sn concentration in ZnO nanorod array: intensification on the conductivity and humidity sensing properties

Tin (Sn)-doped zinc oxide (ZnO) nanorod arrays (TZO) were synthesized onto aluminum-doped ZnO-coated glass substrate via a facile sonicated sol–gel immersion method for humidity sensor applications. These nanorod arrays were grown at different Sn concentrations ranging from 0.6 to 3 at.%. X-ray diffraction patterns showed that the deposited TZO arrays exhibited a wurtzite structure. The stress/strain condition of the ZnO film metamorphosed from tensile strain/compressive stress to compressive strain/tensile stress when the Sn concentrations increased. Results indicated that 1 at.% Sn doping of TZO, which has the lowest tensile stress of 0.14 GPa, generated the highest conductivity of 1.31 S cm− 1. In addition, 1 at.% Sn doping of TZO possessed superior sensitivity to a humidity of 3.36. These results revealed that the optimum performance of a humidity-sensing device can be obtained mainly by controlling the amount of extrinsic element in a ZnO film

Dimensionless cosmology

Although it is well known that any consideration of the variations of fundamental constants should be restricted to their dimensionless combinations, the literature on variations of the gravitational constant $G$ is entirely dimensionful. To illustrate applications of this to cosmology, we explicitly give a dimensionless version of the parameters of the standard cosmological model, and describe the physics of Big Bang Neucleosynthesis and recombination in a dimensionless manner. The issue that appears to have been missed in many studies is that in cosmology the strength of gravity is bound up in the cosmological equations, and the epoch at which we live is a crucial part of the model. We argue that it is useful to consider the hypothetical situation of communicating with another civilization (with entirely different units), comparing only dimensionless constants, in order to decide if we live in a Universe governed by precisely the same physical laws. In this thought experiment, we would also have to compare epochs, which can be defined by giving the value of any {\it one} of the evolving cosmological parameters. By setting things up carefully in this way one can avoid inconsistent results when considering variable constants, caused by effectively fixing more than one parameter today. We show examples of this effect by considering microwave background anisotropies, being careful to maintain dimensionlessness throughout. We present Fisher matrix calculations to estimate how well the fine structure constants for electromagnetism and gravity can be determined with future microwave background experiments. We highlight how one can be misled by simply adding $G$ to the usual cosmological parameter set

Search for an invisible Z′Z^\prime in a final state with two muons and missing energy at Belle II

The $L_{\mu}-L_{\tau}$ extension of the standard model predicts the existence of a lepton-flavor-universality-violating $Z^{\prime}$ boson that couples only to the heavier lepton families. We search for such a $Z^\prime$ through its invisible decay in the process $e^+ e^- \to \mu^+ \mu^- Z^{\prime}$. We use a sample of electron-positron collisions at a center-of-mass energy of 10.58GeV collected by the Belle II experiment in 2019-2020, corresponding to an integrated luminosity of 79.7fb$^{-1}$. We find no excess over the expected standard-model background. We set 90$\%$-confidence-level upper limits on the cross section for this process as well as on the coupling of the model, which ranges from $3 \times 10^{-3}$ at low $Z^{\prime}$ masses to 1 at $Z^{\prime}$ masses of 8$GeV/c^{2}$

Measurement of the branching fraction and CP\it CP asymmetry of B0→π0π0B^{0} \rightarrow \pi^{0} \pi^{0} decays using 198×106198 \times 10^6 BB‾B\overline{B} pairs in Belle II data

We report measurements of the branching fraction and $\it CP$ asymmetry in $B^{0} \to \pi^{0} \pi^{0}$ decays reconstructed at Belle II in an electron-positron collision sample containing $198 \times 10^{6}$ $B\overline{B}$ pairs. We measure a branching fraction \mathcal{B}(\Bpipi) = (1.38 \pm 0.27 \pm 0.22) \times 10^{-6} and a $\it CP$ asymmetry \Acp(\Bpipi) = 0.14 \pm 0.46 \pm 0.07, where the first uncertainty is statistical and the second is systematic

Measurement of the Λc+\Lambda_c^+ lifetime

An absolute measurement of the $\Lambda^{+}_c$ lifetime is reported using $\Lambda_c^+\rightarrow pK^-\pi^+$ decays in events reconstructed from data collected by the Belle II experiment at the SuperKEKB asymmetric-energy electron-positron collider. The total integrated luminosity of the data sample, which was collected at center-of-mass energies at or near the $\Upsilon(4S)$ resonance, is 207.2~\mbox{fb}^{-1}. The result, $\tau(\Lambda^{+}_c) = 203.20 \pm 0.89 \,\mathrm{(stat)} \pm 0.77 \,\mathrm{(syst)}$ fs, is the most precise measurement to date and is consistent with previous determinations.Comment: Accepted for publication in PR