Characteristic vector of differential equation load model and its case study on statistics analysis

The integration of uncertain photovoltaics (PVs) and flexible loads leads to uncertainties in the power system dynamic simulation results. Furthermore, geographically close PV farms are correlated and may exhibit nonlinear correlations. This article proposes a copula-based sparse polynomial chaos expansion (PCE) framework for quantifying the impacts of uncertain dynamic PVs and loads on power system dynamic simulations and stability. The dynamics include both PV and load stochasticity and those governed by differential and algebraic equations. The copula statistics are utilized to accurately characterize the dependence structure of PVs and further used to develop the copula-PCE for quantifying the impacts of uncertain PVs and loads. A probabilistic TSI is also developed to assess the uncertainties from PVs and loads on the system stability. To address the cases, where both stable and unstable conditions coexist, a preprocessing step via sample classification is proposed. The effects of different dependence structures of PVs and different numbers of uncertain sources are investigated. Comparison results with other methods on the modified IEEE 39- and 118-bus systems, including the Monte Carlo method, Latin hypercube sampling, and traditional PCE without consideration of uncertain input correlations show that the proposed method is able to accurately quantify the uncertain dynamic simulations and transient system stability while being computationally efficient.

Fitting probabilistic models to data is often difficult, due to the general intractability of the partition function. We propose a new parameter fitting method, minimum probability flow (MPF), which is applicable to any parametric model. We demonstrate parameter estimation using MPF in two cases: a continuous state space model, and an Ising spin glass. In the latter case, MPF outperforms current techniques by at least an order of magnitude in convergence time with lower error in the recovered coupling parameters.

Neurons in vivo exhibit irregular firing, and some aspect of their firing pattern such as the degree of firing irregularity does not fluctuate greatly in time for individual neurons, while it differs among neurons [1]. By analyzing spike trains recorded from a variety of cortical areas, it has been revealed that neurons in motor-related cortical areas fire regularly while neurons in visual and prefrontal cortex fire randomly [2]. Here, the firing irregularity is defined as a variability of consecutive interspike intervals, independent of the firing rate. In those studies, the firing rate has been ignored, because it may fluctuate greatly in time depending on external stimuli or behavioral contexts of an animal. Though the neuronal firing rate may fluctuate with external stimuli, this does not necessarily mean that the rate is extrinsic and perfectly controllable; there could be some uncontrollable intrinsic aspect in the firing rate fluctuation. In this contribution, we go contrary to the above-mentioned studies, and attempt to characterize neuronal spike trains solely in terms of the firing rate, ignoring the firing irregularity. The key feature of neuronal spike trains we now pay attention to is dynamic mode of rate fluctuation, such that the rate either fluctuates smoothly in time or changes abruptly between several stationary states. We compare two different models of a fluctuating rate process for their goodness-of-fit with respect to spike trains recorded from single neurons in several cortical areas of rats. The first model is the state space model constructed under the Empirical Bayes method [3,4]. The generative inhomogeneous Poisson process is defined as the conditional distribution of the spike times ti≡t1,t2,...,tn given the rateλ(t), ptiλ(t)= ∏i=1nλ(ti)exp-∫ 0Tλ(t)dt. We give the prior distribution of λ(t) by incorporating the tendency of the rate to be flat, pλ(t)∝exp-β∫ 0Tdλ/dt2dt. The hyperparameter β is selected by the principle of maximizing the marginal likelihood, and then we obtain the maximum a posteriori (MAP) estimate for the rate λ⌢(t). The second alternative model is the Hidden Markov model (HMM) whose hidden states represent different rate states [5-7]. The number of states is selected with a variation Bayes method. In this way, two different estimations of the firing rate are obtained using the continuous state space model and the discrete HMM. We compare them for each spike train obtained from a neuron, and examine whether the continuous or discrete representation of the state transition is more plausible. Then, we examine how the different preferences are distributed in different cortical areas.

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Tuberculosis is a global disease with much significant impact. Well over two million deaths are recorded each year due to Mycobacterium tuberculosis. The work by Colijn et al. (J Theor Biol 247, 765–774, 2007) examined two mathematical models (delay differential equation and network models) that explored the role of the contact structure of the population and found that in declining epidemics, localized outbreaks may occur as a result of contact heterogeneities even in the absence of host or strain variability. In this work, we carry out several qualitative studies (including the use of Sturm sequences for obtaining conditions on bifurcations) on the differential equation model with a time delay in Ref. Colijn et al., J Theor Biol 247, 765–774 (2007), and explicitly obtain threshold conditions for disease eradication as well as existence of bifurcations depending on a critical delay value which characterize the duration of fast latency. Further analysis showed the existence of a transcritical bifurcation when the reproduction number was less than one with the disease-free equilibrium co-existing with just one endemic state that was shown to be stable; the period of fast latency and the reinfection progression rate were shown to be very significant in the dynamics of the disease viz a viz the individuals who progressed to active TB within 5 years of their infection.

Nonlinearity of power system is always one of the difficulties when dealing with dynamic simulation of power systems. Solving differential-algebraic equations representing power systems are difficult without losing nonlinearity, especially for large power systems. This thesis shows an alternative method to solve nonlinear dynamical power system by producing a purely differential representation of the power systems. This new representation converts the algebraic equations to differential equations in order to have an absolute differential system. By using Runge-Kutta algorithm to solve this differential system, the results of the power system simulations are compared to trapezoidal integration algorithm commonly used to solve the differential-algebraic equations. In this thesis, IEEE 14-bus system and IEEE 118-bus system are tested with both classical generator model generator model and two-axis generator model in MATLAB. The proposed algorithm shows significantly faster convergence comparted to trapezoidal integration method in larger power systems. It is a great improvement to shorten the simulation time in while keeping the same accuracy in large power systems.

Advertisement involves significant amount of money and remain as one of the major issues bothering companies and other investment sectors. Advertising is a marketing strategy directed to the consumers through any media in order to present and promote product, services and any other cause. Sale of products occurs with or without advertisement and may increase or decrease depending on the market situation. The question is, is there a mathematical model that can surely confirm the effects of advertising on sales of products? The aim of this paper is to develop a first order linear differential equation model that seeks to justify the impacts of advertising on the sale of products in Ghanaian market. Two basic variables, the advertisement and sales of product were considered for the model. A system of two differential equations from the two basic variables were developed and converted to first order non-homogeneous linear differential equation. The geometrical method was employed to obtain qualitative information about the solution of the first order non-homogeneous linear differential equation. The MATLAB software package was used to plots the graphs to illustrate the behaviour of the sale of products with respect to the time of advertisement. The findings revealed that differential equation model can be used to justify the advertising impacts on the sale of products.

In this article, the due date setting problem consisting of due date quoting, order insertion and rescheduling with relevant sequence-dependent setups and a frozen zone is considered. A differential equation model is introduced. The differential equation can be solved analytically and requires low computing effort. Explicit expressions are gained concerning the earliest possible due date (due date quoting problem) and the latest possible insertion position (rescheduling problem) in order to meet the customer requested due date. The differential equation model is compared with an approach based on a recently introduced hybrid binomial inverse hypergeometric probability distribution for the due date quoting problem. The analytical results of the proposed model are very close to the expectation values (these cannot be explicitly calculated, and high computation effort is required) of the recently introduced approach.

Complex intraindividual variability observed in psychology may be well described using differential equations. It is difficult, however, to apply differential equation models in psychological contexts, as time series are frequently short, poorly sampled, and have large proportions of measurement and dynamic error. Furthermore, current methods for differential equation modeling usually consider data that are atypical of many psychological applications. Using embedded and observed data matrices, a statistical approach to differential equation modeling is presented. This approach appears robust to many characteristics common to psychological time series.

Stock price analysis is one of the most important issues concerning investments and financial decision-making. Thus, stock price analysis and estimation models can be very useful in the estimation of a firm’s financial development. The aim of this paper is to propose a model of differential equations that will be able to be applied in the case of stock price analysis and estimation. The differential equations model will be based on Lanchester’s combat model, a mathematical theory of war. In the field of business, such operations research models have been used in cases such as competition analysis and resources allocation optimization. The case to be examined in this paper refers to the healthcare services index stocks of the Athens Stock Exchange. A 7 × 7 differential equations model was developed to analyze the examined firms’ stocks.

Feature selection techniques have become an apparent need for diagnostic methods such as a least squares support vector machine (LS-SVM). Most researchers use wavelet transform coefficients of the time-domain transient response data obtained from filtered analog circuits as features to train a LS-SVM classifier to diagnose faults. But wavelet coefficient features have certain disadvantages such as no physical meanings. Thus, in this paper, two new feature vectors with clearly defined meanings based on a time-domain response curve and a frequency response curve of a filter are proposed, respectively. In addition, a statistical property feature vector which represents global properties of the time-domain response curve or the frequency response curve is proposed. The results from the simulation data and real data for a biquad filter showed the following: (1) these proposed conventional time-domain and frequency features, which are already familiar to designers of filtered analog circuits, have good diagnostic accuracy—all above 91% for the example circuit; (2) the best accuracies using the proposed statistical property feature vector are 100% for time-domain simulation data, and for both real experiment data ; (3) the diagnostic accuracy using the proposed combined feature vector is more accurate than conventional feature vectors; (4) an LS-SVM can be used to diagnose faults in a real analog circuit that only has a few fault samples.

Mathematical models to study quorum sensing (QS) have become an important tool to explore all aspects of this type of bacterial communication. A wide spectrum of mathematical tools and methods such as dynamical systems, stochastics, and spatial models can be employed. In this chapter, we focus on giving an overview of models consisting of differential equations (DE), which can be used to describe changing quantities, for example, the dynamics of one or more signaling molecule in time and space, often in conjunction with bacterial growth dynamics. The chapter is divided into two sections: ordinary differential equations (ODE) and partial differential equations (PDE) models of QS. Rates of change are represented mathematically by derivatives, i.e., in terms of DE. ODE models allow describing changes in one independent variable, for example, time. PDE models can be used to follow changes in more than one independent variable, for example, time and space. Both types of models often consist of systems (i.e., more than one equation) of equations, such as equations for bacterial growth and autoinducer concentration dynamics. Almost from the onset, mathematical modeling of QS using differential equations has been an interdisciplinary endeavor and many of the works we revised here will be placed into their biological context.

Mathematical models are routinely calibrated to experimental data, with goals ranging from building predictive models to quantifying parameters that cannot be measured. Whether or not reliable parameter estimates are obtainable from the available data can easily be overlooked. Such issues of parameter identifiability have important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained. Identifiability analysis is well-established for deterministic, ordinary differential equation (ODE) models, but there are no commonly adopted methods for analysing identifiability in stochastic models. We provide an accessible introduction to identifiability analysis and demonstrate how existing ideas for analysis of ODE models can be applied to stochastic differential equation (SDE) models through four practical case studies. To assess structural identifiability, we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically motivated synthetic data and Markov chain Monte Carlo methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github.

This paper presents the system identification and validation of Synchronous Buck Converter (SBC) models using both white-box and black-box approaches. The white-box method employs difference equations derived from continuous state-space models, while the black-box approach estimates second-order state-space representations. Models are experimentally validated against laboratory measurements, with accuracy quantified using the Normalized Root Mean Square Error (NRMSE). By analysing the trade-off between model fidelity and real-time feasibility, the study identifies the most accurate model for reliable real-time simulation and controller development.

This study examines different approaches to modeling change over time, comparing growth curve models with difference equation models. Growth curve models provide a descriptive framework for capturing longitudinal trends but primarily serve as phenomenological tools that summarize observed trajectories. In contrast, difference equation models offer a process-based perspective, modeling system dynamics iteratively and allowing researchers to explore change processes. This study’s empirical analyses highlight both the advantages and limitations of such models. Simulations demonstrate the effectiveness of difference equations in capturing system dynamics, but high levels of variability in early time periods can make the models fit somewhat poorly in some cases. Findings suggest that while growth curve models remain valuable for summarizing change, difference equation models provide deeper insights into the mechanisms driving temporal processes. Future research should explore hybrid modeling approaches that integrate these frameworks, using dynamical systems theory to enhance psychological and organizational research. An R package to assist with these analyses is available at https://github.com/smspain/DifferenceEquationsInR