Power grid simulation using matrix exponential method with rational Krylov subspaces

Deterministic cell-growth models describe the growth of cell populations using fixed mathematical rules, assuming no randomness in the system. These models are often based on differential equations that account for the rates of cell division, death, and other biological processes. The solution to the system is obtained via numerical methods. Most of the developed approaches are based on fixed step sizes. However, fixed step size implementation failed to offer the optimal solutions when dealing with stiff challenges. Fixed step size methods can be unstable for stiff equations, where some components of the solution change much more rapidly than others. The step size, required to maintain stability can become impractically small. Thus, the adaptive step size method is required. Adaptive step size methods adjust the step size dynamically based on the behavior of the solution, aiming to maintain a desired level of accuracy while optimizing computational efficiency. These methods are particularly useful for solving ordinary differential equations (ODEs) where the solution can vary rapidly in some regions and slowly in others. This study is devoted to comparing the implementation of fixed step size and adaptive step size in solving ordinary differential equations (ODEs). The fixed step size and adaptive step size numerical method are solved in this study via the fourth order Runge-Kutta method (RK4) and Runge-Kutta Fehlberg 45 method (RKF45). The performance of both numerical methods used will be analyzed by comparing the numerical results approximated with the actual data. Subsequently, the absolute error, relative error, and rounding-off error will be calculated to compare both approaches. Based on the more precise findings, this work has shown that adaptive step size is predicted to be the optimal representation for solving ODEs. As a result, this may help mathematicians to choose the most effective numerical approach for solving ODEs.

This study presents a numerical verification of the physical validity of using a quadratic dependence of the laser beamradius in modeling stimulated Raman scattering (SRS) in self-focusing media, such as toluene. The influence of various powerlawdependencies of the gain coefficient on the beam radius (n = 1, n = 2, n = 3) is analyzed in terms of numerical stability andresult accuracy. Two numerical approaches are developed: the Adams method with adaptive step size and the Euler iterativemethod with fixed step size, both based on energy conservation laws and the Kerr effect. The simulations are performed usingparameters typical for toluene (initial beam radius of 113 μm, refractive index of 1.49, laser wavelength of 694.3 nm, and criticalpower of 25 kW). The results show that the quadratic model (n = 2) provides the optimal combination of stability and physicalcorrectness, accurately capturing the Stokes generation threshold (zf ≈ 0.083 m) and power stabilization. The linear model (n =1) underestimates the self-focusing effect, violating energy conservation, while the cubic model (n = 3) exhibits numericalinstability. This work has practical significance for optimizing laser systems in spectroscopy, optoelectronics, and biomedicine.Moreover, an important aspect of developing numerical programs was to improve computational efficiency. Implementing themodeling in the form of standalone programs allows for a significant reduction in calculation time compared to manual analysisor the use of general-purpose software packages. This enables the investigation of complex nonlinear effects under realisticconditions with high spatial resolution

Forced-choice staircases with fixed step sizes: asymptotic and small-sample properties.

Back Propagation (BP) is commonly used algorithm that optimize the performance of network for training multilayer feed-forward artificial neural networks. However, BP is inherently slow in learning and it sometimes gets trapped at local minima. These problems occur mailnly due to a constant and non-optimum learning rate (a fixed step size) in which the fixed value of learning rate is set to an initial starting value before training patterns for an input layer and an output layer. This fixed learning rate often leads the BP network towrds failure during steepest descent. Therefore to overcome the limitations of BP, this paper introduces an improvement to back propagation gradient descent with adapative learning rate (BPGD-AL) by changing the values of learning rate locally during the learning process. The simulation results on selected benchmark datasets show that the adaptive learning rate significantly improves the learning efficiency of the Back Propagation Algorithm

As the size and complexity of current VLSI circuits grows, faster power grid simulation is becoming more and more desirable. In this article, we present a parallel iterative method for static VLSI power grid simulation. In the proposed enlarged-partition-based preconditioned conjugate gradient (EPPCG) power grid solver, the power grid is divided into disjoint partitions that are subsequently enlarged to obtain accurate solution within each partition. The global solution obtained by solving enlarged partition problems concurrently acts as a highly effective parallel preconditioner. The combination of effective preconditioning and efficient parallelization helps achieve very high performance. The experiments show that our parallel implementation can achieve significant speed improvement [61X--142X] over a state-of-the-art direct solver.

Dynamic load model is of great value in analysis and simulation of power grid, while the composition of load has an important influence on the characteristic and accuracy of load model. There is mass of load equipment in power grid, so it is usually necessary to classify the load equipment according to their different characteristic first, then establish a suitable load model separately, and synthesize them into a composite model according to their proportion. Only the power value of load can be measured when power grid operates in a normal and steady state, and their dynamic response characteristics are unknown. Therefore, it is difficult to obtain sufficient information for load classification. However, frequency characteristics of load are distinguished and easy to obtain, so it is useful for load classification indirectly. Firstly, several elector-magnetism models of typical load are established, and their harmonic curves and frequency characteristic are simulated and analyzed separately. Then, cluster analysis is carried out to obtain the typical characteristics and the membership degree which can represent the contents of load overall. Finally, a classification method combined with Neural Network is adopted, and the computation speed is also improved.

The engine blade is one of the important parts of the aero engine, vibration can lead to engine blade off or fracture. Active Vibration Control (AVC) has a higher flexibility is increasingly used to suppress the vibration of the blade. The filtered-x least mean square (FxLMS) algorithm is an adaptive algorithm commonly used in active vibration control, but its fixed step size has a contradiction between convergence speed and steady-state error. A new variable-step-size FxLMS (VSSFxLMS) algorithm based on hyperbolic secant function was proposed, and the sech function ensures the control signal is bounded. In the early stage, a larger step size enables the error to decrease rapidly. The step size decreases as the error decreases, and a smaller step size ensures a smaller steady-state error. Firstly, this paper summarizes three VSSFxLMS algorithms and proposes an improved algorithm; Secondly, it compares and analyses the mentioned algorithms; Then, a joint simulation platform using ADAMS and Simulink is built to verify the effectiveness of the proposed algorithms; Finally, an experimental platform was built to complete the blade vibration active control experiments using several VSSFxLMS algorithms, and the results of the experiment verify the better vibration suppression effect of the proposed algorithm.

The multiple-sets split feasibility problem, a generalization and extension of the split feasibility problem, has a variety of specific applications in real world, such as medical care, image reconstruction and signal processing. It can be a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in the operator's range. Censor et al (2005 Inverse Problems 21 2171–84) proposed a method for solving the multiple-sets split feasibility problem, whose efficiency depends heavily on step size, a fixed constant related to the Lipschitz constant of ∇p(x) (see the definition in section 1). To estimate the Lipschitz constant is a very difficult, if not an impossible task. On the other hand, even if we know the Lipschitz constant, a method with fixed step size may be slow. In this paper, we propose a new method for solving the multiple-sets split feasibility problem by adopting variable step sizes, which chooses suitable step sizes self-adaptively, based on the information from the current iterate. It thus avoids the difficult task of estimating the Lipschitz constant, while the efficiency is enhanced greatly. We prove the global convergence of the new method and report our numerical results, which are promising.

A variable step size total least squares affine-projection-like algorithm: Formula derivation and performance analysis

Delta Sigma Analog to Digital Converters (ADCs) are superior to their counterparts in achieving high resolution for low to medium bandwidth applications. This paper discusses an improved digital calibration technique for enhancing Signal to Noise ratio (SNR) of Delta Sigma ADC which is directly related to Effective Number of Bits (ENOB). The proposed method uses the Adaptive Line Enhancer (ALE) for improving the SNR of the modulator. Adaptive Line Enhancer filters out the estimate of signal of interest from noisy input signal by updating filter coefficients using Least Mean Square (LMS) Algorithm. Variable step size (VSS) LMS is used for updating filter coefficients in Adaptive Line Enhancer instead of Fixed step size (FSS) LMS to reduce the required number of iterations. The simulation results for second order single loop Delta Sigma modulator show approximately 20dB improvement in SNR after calibration and substitution of VSS LMS with FSS LMS has reduced approximately 4000 iterations.

Current open-source solutions do not include the possibility of simultaneous simulation of a data network and a power grid. The purpose of this paper is to describe the design of a simulation platform that will provide this type of simulation. The created virtualization platform that consists of several open-source tools is described in detail. The paper further deals with the description of data network virtualization and interconnection with power grid simulation. The architecture described aims to solve the missing solution of the simulation of data network and power grid interconnection.

Fixed step size random search for minimization of functions of several parameters is described and compared with the fixed step size gradient method for a particular surface. A theoretical technique, using the optimum step size at each step, is analyzed. A practical adaptive step size random search algorithm is then proposed, and experimental experience is reported that shows the superiority of random search over other methods for sufficiently high dimension.

A single current sensor based adaptive step size MPPT control of a small scale variable speed wind energy conversion system

Quantum state tomography (QST) is one of the key steps in determining the state of the quantum system, which is essential for understanding and controlling it. With statistical data from measurements and Positive Operator-Valued Measures (POVMs), the goal of QST is to find a density operator that best fits the measurement data. Several optimization-based methods have been proposed for QST, and one of the most successful approaches is based on Accelerated Gradient Descent (AGD) with fixed step length. While AGD with fixed step size is easy to implement, it is computationally inefficient when the computational time required to calculate the gradient is high. In this paper, we propose a new optimal method for step-length adaptation, which results in a much faster version of AGD for QST. Numerical results confirm that the proposed method is much more time-efficient than other similar methods due to the optimized step size.

In generalized Nash equilibrium (GNE) seeking problems over physical networks such as power grids, the enforcement of network constraints and time-varying environment may bring high computational costs. Developing online algorithms is recognized as a promising method to cope with this challenge, where the task of computing system states is replaced by directly using measured values from the physical network. In this paper, we propose an online distributed algorithm via measurement feedback to track the GNE in a time-varying networked resource sharing market. Regarding that some system states are not measurable and measurement noise always exists, a dynamic state estimator is incorporated based on a Kalman filter, rendering a closed-loop dynamics of measurement-feedback driven online algorithm. We prove that, with a fixed step size, this online algorithm converges to a neighborhood of the GNE in expectation. Numerical simulations validate the theoretical results.