Richards Model Research Articles

A robust optimization approach for dynamic traffic signal control with emission considerations

We consider an analytical signal control problem on a signalized network whose traffic flow dynamic is described by the Lighthill–Whitham–Richards (LWR) model (Lighthill and Whitham, 1955; Richards, 1956). This problem explicitly addresses traffic-derived emissions as constraints or objectives. We seek to tackle this problem using a mixed integer mathematical programming approach. Such class of problems, which we call LWR-Emission (LWR-E), has been analyzed before to certain extent. Since mixed integer programs are practically efficient to solve in many cases (Bertsimas et al., 2011b), the mere fact of having integer variables is not the most significant challenge to solving LWR-E problems; rather, it is the presence of the potentially nonlinear and nonconvex emission-related constraints/objectives that render the program computationally expensive.To address this computational challenge, we proposed a novel reformulation of the LWR-E problem as a mixed integer linear program (MILP). This approach relies on the existence of a statistically valid macroscopic relationship between the aggregate emission rate and the vehicle occupancy on the same link. This relationship is approximated with certain functional forms and the associated uncertainties are handled explicitly using robust optimization (RO) techniques. The RO allows emissions-related constraints and/or objectives to be reformulated as linear forms under mild conditions. To further reduce the computational cost, we employ a link-based LWR model to describe traffic dynamics with the benefit of fewer (integer) variables and less potential traffic holding. The proposed MILP explicitly captures vehicle spillback, avoids traffic holding, and simultaneously minimizes travel delay and addresses emission-related concerns.

On the Braess Paradox with Nonlinear Dynamics and Control Theory

We show the existence of the Braess paradox for a traffic network with nonlinear dynamics described by the Lighthill–Whitham–Richards model for traffic flow. Furthermore, we show how one can employ control theory to avoid the paradox. The paper offers a general framework applicable to time-independent, uncongested flow on networks. These ideas are illustrated through examples.

A non-phenomenological model of competition and cooperation to explain population growth behaviors.

This paper is an extension of a previous work which proposes a non-phenomenological model of population growth that is based on the interactions among the individuals of a population. In addition to what had already been studied—that the individuals interact competitively—in the present work it is also considered that the individuals interact cooperatively. As a consequence of this new consideration, a richer dynamics is observed. For instance, besides getting the population models already reached from the original version of the model (as the Malthus, Verhulst, Gompertz, Richards, Bertalanffy and power-law growth models), the new formulation also reaches the von Foerster growth model and also a regime of divergence of the population at a finite time. An agent-based model is also presented in order to give support to the analytical results. Moreover, this new approach of the model explains the Allee effect as an emergent behavior of the cooperative and competitive interactions among the individuals. The Allee effect is the characteristic of some populations of increasing the population growth rate in a small-sized population. Whereas the models presented in the literature explain the Allee effect with phenomenological ideas, the model presented here explains this effect by the interactions between the individuals. The model is tested with empirical data to justify its formulation. Another interesting macroscopic emergent behavior from the model proposed is the observation of a regime of population divergence at a finite time. It is interesting that this characteristic is observed in humanity's global population growth. It is shown that in a regime of cooperation, the model fits very well to the human population growth data from 1000 AD to nowadays.

A new approach to modelling water flooding in a polymer electrolyte fuel cell

The distribution and migration of liquids in various layers of a PEFC is commonly modelled by the 3D flow equations. Given the fact these layers are very thin, there are major problems with such models, including heavy computational efforts and doubts in their applicability to the gas diffusion layer (GDL). Recently, a new approach for modelling multiphase flow through a stack of thin porous layers has been developed [Qin and Hassanizadeh, Int. J. Heat Mass Transfer 70 (2014) 693–708]. In this approach, which is called “reduced continua model”, each layer is modelled as a 2D domain with governing equations formulated in terms of thickness-averaged properties. The mass exchange between layers is prescribed by a new constitutive equation. The aim of this paper is to illustrate the implementation of the reduced continua model and show its advantages in modelling liquid water flooding in the GDL and micro porous layer (MPL) of a PEFC. We find that, in comparison to the Richards model, the reduced continua model predicts quite similar water dynamics in the MPL, but a lower steady-state water saturation in the GDL, particularly under the channel area. We provide a quantitative indication of the enormous computational efficiency of the reduced continua model as compared to the Richards model. Finally, the sensitivity studies of major material parameters of the reduced continua model have been provided.

Multiscale Traffic Flow Model Based on the Mesoscopic Lighthill–Whitham and Richards Models

This article reviews the multiscale modeling concept as an adaptive strategy in traffic flow modeling. The central objective for the multi-scale model is to describe and to predict traffic phenomena (e.g., individual accelerations, queuing) that occur at different scales by switching between modeling paradigms. The emphasis in this paper is placed on resolving inconsistencies that arise in the process of integrating models, which are addressed with the mesoscopic Lighthill–Whitham and Richards models—together, widely known as the Lighthill–Whitham–Richards (LWR) model—as the coarse-scale model in the framework. The mesoscopic model is fully consistent at the macroscopic scale with the classic LWR model but does facilitate tracking of individual vehicles. Because of this characteristic, this model can be combined with microscopic models, because no disaggregation or aggregation process is needed. It is thus implied, for example, that travel information (e.g., destination) can be carried over this model without loss of information. Also, vehicle heterogeneity can be incorporated in the mesoscopic model and, as a result, is preserved in the process of switching between models. A brief overview of the mesoscopic LWR model and its solution by the variational theory is presented. General boundary conditions, including internal boundaries (e.g., moving bottlenecks) and interfaces between models, are analyzed and evaluated. The solution of some cases, in which the stationary and moving bottlenecks interact, is provided. The paper concludes with a case study that demonstrates the integration of microscopic and mesoscopic models in a multiscale model and investigates the influence of various boundary conditions on traffic features.

Model Selection and Evaluation Based on Emerging Infectious Disease Data Sets including A/H1N1 and Ebola.

The aim of the present study is to apply simple ODE models in the area of modeling the spread of emerging infectious diseases and show the importance of model selection in estimating parameters, the basic reproduction number, turning point, and final size. To quantify the plausibility of each model, given the data and the set of four models including Logistic, Gompertz, Rosenzweg, and Richards models, the Bayes factors are calculated and the precise estimates of the best fitted model parameters and key epidemic characteristics have been obtained. In particular, for Ebola the basic reproduction numbers are 1.3522 (95% CI (1.3506, 1.3537)), 1.2101 (95% CI (1.2084, 1.2119)), 3.0234 (95% CI (2.6063, 3.4881)), and 1.9018 (95% CI (1.8565, 1.9478)), the turning points are November 7,November 17, October 2, and November 3, 2014, and the final sizes until December 2015 are 25794 (95% CI (25630, 25958)), 3916 (95% CI (3865, 3967)), 9886 (95% CI (9740, 10031)), and 12633 (95% CI (12515, 12750)) for West Africa, Guinea, Liberia, and Sierra Leone, respectively. The main results confirm that model selection is crucial in evaluating and predicting the important quantities describing the emerging infectious diseases, and arbitrarily picking a model without any consideration of alternatives is problematic.

Modeling dominant height growth including site attributes in the GADA approach for Quercus faginea Lam. in Spain

Aim of the study: To develop a site index model for Quercus faginea Lam. stands.Area of study: SpainMaterial and Methods: Data from 81 growth series collected in plots where Q. faginea was the main species were used for modelling. Different generalized algebraic difference equations (GADA) were fitted from traditionally used models. Richards model was selected and used to expand the parameters with environmental variables.Research highlights: Winter rainfall (WR), annual potential evapotranspiration (PET) and pH were introduced increasing the prediction ability of the GADA. It is strongly recommended to apply the model with ages lower than 80 years because the lack of data above that age makes bias increase and efficiency decrease.Keywords: Site index; Lusitanian oak; environmental variables.

Scaling and Pedotransfer in Numerical Simulations of Flow and Transport in Soils

Flow and transport parameters of soils in numerical simulations need to be defined at the support scale of computational grid cells. Such support scale can substantially differ from the support scale in laboratory or field measurements of flow and transport parameters. The scale dependence of flow and transport parameters essentially precludes the direct use of measured or pedotransfer-estimated parameter values in numerical simulations. The hypothesis of this work was that a support-based scaling law can be introduced that can convert pedotransfer-estimated saturated hydraulic conductivity values into values to be used over grid cells for finite-element-based simulations of water flow and tracer transport in variably saturated soils. A 4-month-long experiment was conducted at the USDA–ARS experimental site where Cl− as a tracer was applied with a pulse of irrigation water and its transport in groundwater and variably saturated shallow coarse-textured soils was monitored in two rows of wells on a daily basis. The HYDRUS-3D software was used to set and calibrate the Richards model for flow simulations and the convective–dispersive equation for transport simulations. Saturated hydraulic conductivity values were estimated with class pedotransfer functions derived from the USDA database containing results of about 1000 measurements in soils of different textures and bulk densities. A power law scaling for the saturated hydraulic conductivity was suggested based on literature data. When only two parameters of the scaling law rather than nine values of hydraulic conductivity from nine soil materials were calibrated, using the scaled saturated hydraulic conductivity values resulted in an accuracy of simulations that was similar to the accuracy of the calibrated model results. Upscaling of pedotransfer-estimated saturated hydraulic conductivities can provide reasonable estimates for numerical flow and transport modeling in variably saturated soils.

Ability of non-linear mixed models to predict growth in laying hens

In this study, the Von Bertalanffy, Richards, Gompertz, Brody, and Logistics non-linear mixed regression models were compared for their ability to estimate the growth curve in commercial laying hens. Data were obtained from 100 Lohmann LSL layers. The animals were identified and then weighed weekly from day 20 after hatch until they were 553 days of age. All the nonlinear models used were transformed into mixed models by the inclusion of random parameters. Accuracy of the models was determined by the Akaike and Bayesian information criteria (AIC and BIC, respectively), and the correlation values. According to AIC, BIC, and correlation values, the best fit for modeling the growth curve of the birds was obtained with Gompertz, followed by Richards, and then by Von Bertalanffy models. The Brody and Logistic models did not fit the data. The Gompertz nonlinear mixed model showed the best goodness of fit for the data set, and is considered the model of choice to describe and predict the growth curve of Lohmann LSL commercial layers at the production system of University of Antioquia.

Control of a lane-drop bottleneck through variable speed limits

The discharging flow-rate of a lane-drop bottleneck can drop when its upstream is congested, and such capacity drop leads to additional traffic congestion as well as safety threats. Even though many studies have demonstrated that variable speed limits (VSL) can effectively delay and even avoid the occurrence of capacity drop, there lacks a simple approach for analyzing the performance of a VSL control system. In this study, we formulate the VSL control problem for the traffic system in a zone upstream to a lane-drop bottleneck based on two traffic flow models: the Lighthill–Whitham–Richards (LWR) model, which is an infinite-dimensional partial differential equation, and the link queue model, which is an ordinary differential equation and approximates the LWR model. In both models, the discharging flow-rate is determined by a recently developed model of capacity drop, and the upstream in-flux is regulated by the speed limit in the VSL zone. We first analytically study the properties of the control system with the link queue model. For an open-loop control system with a constant speed limit, we prove that a constant speed limit can introduce an uncongested equilibrium state, in addition to a congested one with capacity drop, but the congested equilibrium state is always exponentially stable. Then we apply a feedback proportional-integral (PI) controller to form a closed-loop control system, in which the congested equilibrium state and, therefore, capacity drop can be removed by both I- and PI-controllers. Both analytical and numerical results show that, with appropriately chosen controller parameters, the closed-loop control system is stable, effect, and robust. Finally, we show that the VSL strategies based on I- and PI-controllers are also stable, effective, and robust for the LWR model. Since the properties of the control system are transferable between the two models, we establish a dual approach for studying the control problems of nonlinear traffic flow systems. We also confirm that the VSL strategy is effective only if capacity drop occurs. The obtained method and insights can be useful for future studies on other traffic control methods and implementations of VSL strategies in lane-drop bottlenecks, work zones, and incident areas.

A POD-based reduced-order FD extrapolating algorithm for traffic flow

A traffic flow Lighthill, Whitham, and Richards (LWR) model is studied by means of a proper orthogonal decomposition (POD) technique. A POD-based reduced-order finite difference (FD) extrapolating algorithm (FDEA) with lower dimension and fully second-order accuracy is established. Two numerical experiments are used to show that the POD reduced-order FDEA is feasible and efficient for finding numerical solutions of traffic flow LWR model.

Saturation overshoot and hysteresis for twophase flow in porous media

Saturation overshoot and hysteresis for two phase flow in porous media are briefly reviewed. Old and new challenges are discussed. It is widely accepted that the traditional Richards model for twophase flow in porous media does not support non-monotone travelling wave solutions for the saturation profile. As a concequence various extensions and generalizations have been recently discussed. The review highlights different limits within the traditional theory. It emphasizes the relevance of hysteresis in the Buckley–Leverett limit with jump-type hysteresis in the relative permeabilities. Reviewing the situation it emerges that the traditional theory may have been abandoned prematurely because of its inability to predict saturation overshoot in the Richards limit.

Parameter estimations of sigmoidal models of cancer

Background Tumor growth, a relationship between tumor size and time, is of special interest since growth estimation is very critical in a clinical practice. There are some mathematical models which describe tumor growth and have prediction capabilities. Typically, there are three ways to model non-complex growth behavior: exponential, logistic and sigmoidal. In 1825, Benjamin Gompertz introduced the Gompertz function, a sigmoid function, which is found to be applicable to various growth phenomena, in particular tumor growth. Besides the Gompertz model which includes three parameters, the Weibull and Richards models with four parameters are known as sigmoidal models. The aim of this project is to introduce continuous fractional and discrete fractional models of the tumor growth and also estimate parameters of these models in order to have better data fitting.

Proposals for Nothofagus antarctica diameter growth estimation: simple vs. global models

Tree growth is one of the main variables needed for forest management planning. The use of simple models containing traditional equations to describe tree growth is common. However, equations that incorporate different factors (e.g. site quality of the stands, crown classes of the trees, silvicultural treatments) may improve their accuracy in a wide range of stand conditions. The aim of this work was to compare the accuracy of tree diameter growth models using (i) a family of simple equations adjusted by stand site quality and crown class of trees, and (ii) <br /> a unique global equation including stand and individual tree variables. Samplings were conducted in 136 natural even-aged Nothofagus antarctica (Forster f.) Oersted stands in Southern Patagonia (Argentina) covering age (20–200 years), <br /> crown class and site quality gradients. The following diameter growth models were fitted: 16 simple equations using two independent variables (age and one equation for each stand site quality or crown class) based on Richards model, plus a unique global equation using three independent variables (age, stand site quality and crown class). Simple equations showed higher variability in their accuracy, explained between 54% and 92% of the data variation. The global model presented similar accuracy like the better equations of the simple growth models. The unification of the simple growth models into a unique global equation did not greatly improve the accuracy of estimations, but positively influenced the biological response of the model. Another advantage of the global equation is the simple use under a wide range of natural stand conditions. The proposed global model allows to explain the tree growth of N. antarctica trees along the natural studied gradients.    

On the micro-macro limit in traffic flow

We investigate the relations between a macroscopic Lighthill-Whitham and Richards model and a microscopic follow-the-leader model for traffic flow. Solutions to the microscopic model are proved to tend to those to the macroscopic one in a sort of kinetic limit, i.e. as the number of individuals tends to + \infty while their total mass is constant. Based on this convergence result, we approximately compute the solutions to a conservation law by means of the integration of an ordinary differential system.

Modeling the individual growth of the Gulf corvina, Cynoscion othonopterus (Pisces: Sciaenidae), using a multi-model approach

The Gulf corvina, Cynoscion othonopterus, is an endemic sciaenid of the Upper Gulf of California and Colorado River Delta biosphere reserve. It is a high-value fishery resource in this region. To sustainably manage this resource, it is necessary to address its individual growth pattern. Previous studies on C. othonopterus growth have fitted the von Bertalanffy growth model (VBGM) without examining alternative models. In this study, the main objective was to analyze the individual growth of the Gulf corvina via a multi-model inference (MMI) approach rather than simply applying the VBGM. Five growth models—VBGM, logistic, Gompertz, Schnute, and Schnute–Richards—were tested. The parameters of each model and their confidence intervals (CI) were computed using the maximumlikelihood method. The MMI was used to average the asymptotic length (L∞). The best-fitting model was selected using the Akaike information criterion, and the Schnute–Richards growth model best described the growth of the Gulf corvina. The L∞ values obtained via MMI averaged 735.0 mm (95% CI: 730.4–739.5 mm). The conclusion is that the growth of C. othonopterus exhibits a biphasic pattern that is better described by a higher-parameter model such as the Schnute–Richards model.

Predictive modeling of β-carotene accumulation by Dunaliella salina as a function of pH, NaCI, and irradiance

Predictive modeling of β-carotene accumulation by Dunaliella salina as a function of NaCI, pH, and irradiance was studied. Modified Logistic, Gompertz, Schnute, Richards, and Stannard models were fitted to describe β-carotene accumulation by the alga under various environmental conditions. Lag time (λ, days), maximum accumulation (A, pg/cell), and the maximum production rate (μ, 1/day) for β-carotene accumulation were calculated by modified Logistic and Gompertz models. Values of λ, A, and μ for β-carotene accumulation varied between 0.26 and 20.14 days, 57.48 to 198.76 pg β-carotene/cell, and 1.80 to 3.68 1/day, respectively. Results revealed that Logistic and Gompertz models could be used to describe the accumulation of β-carotene by D. salina as a function of salt concentrations, pH, and irradiance. The highest asymptotic value was predicted from Logistic and Gompertz models at pH 9.0, 48 kerg/(cm2 s) light intensity, and 20% NaCl concentration.

On the continuum approximation of the on-and-off signal control on dynamic traffic networks

In the modeling of traffic networks, a signalized junction is typically treated using a binary variable to model the on-and-off nature of signal operation. While accurate, the use of binary variables can cause problems when studying large networks with many intersections. Instead, the signal control can be approximated through a continuum approach where the on-and-off control variable is replaced by a continuous priority parameter. Advantages of such approximation include elimination of the need for binary variables, lower time resolution requirements, and more flexibility and robustness in a decision environment. It also resolves the issue of discontinuous travel time functions arising from the context of dynamic traffic assignment.Despite these advantages in application, it is not clear from a theoretical point of view how accurate is such continuum approach; i.e., to what extent is this a valid approximation for the on-and-off case. The goal of this paper is to answer these basic research questions and provide further guidance for the application of such continuum signal model. In particular, by employing the Lighthill–Whitham–Richards model (Lighthill and Whitham, 1955; Richards, 1956) on a traffic network, we investigate the convergence of the on-and-off signal model to the continuum model in regimes of diminishing signal cycles. We also provide numerical analyses on the continuum approximation error when the signal cycles are not infinitesimal. As we explain, such convergence results and error estimates depend on the type of fundamental diagram assumed and whether or not vehicle spillback occurs to the signalized intersection in question. Finally, a traffic signal optimization problem is presented and solved which illustrates the unique advantages of applying the continuum signal model instead of the on-and-off model.

Assessment of malaria incidence using the Richards model in Arunachal Pradesh, India

Monitoring of malaria intensity in terrain regions of Arunachal Pradesh, India is very difficult as the dynamics of mosquito populations varies to a large extent due to altitude and frequent changes in climatic conditions. There is a scarcity of information on the influence of climatic factors on malaria morbidity in Arunachal Pradesh. Hence, a pilot study was conducted from 2006 to 2010 to understand malaria transmission dynamics, seasonal distribution and disease morbidity. Plasmodium vivax and P. falciparum are the two major parasites for malaria transmission in Arunachal Pradesh. Out of 142 558 malaria cases analysed from 2006 to 2010, P. vivax infection contributed 72·1% followed by P. falciparum (27·9%). However, the overall morbidity of malaria declined from 37/1000 in 2006 to 18/1000 population in 2010. From this study it was observed that the temporal distribution of malaria cases varied between districts and high morbidity rates were reported mostly during the wet season. To understand malaria transmission dynamics in the study area, the Richards model was used to predict malaria cases. The output of the results from this model predicted a higher number of malaria cases (K) during 2006 and a gradual decline in subsequent years. Similarly, the growth rate r, and exponential deviation α, were almost identical for all the years, which shows that the Richards model is the most suitable model for the prediction of malaria cases.

Spatial and temporal distributions of migration in bio-retention systems

Urban bio-retention system is meaningful in reducing rainfall runoff and enhancing infiltration capacity. But the moisture migration in bio-retention systems are not clear under climate change. The spatial and temporal distribution of moisture under different rainfall events in bio-retention systems are studied in this paper based on experimental data in Beijing. Richards model is introduced to simulate the spatial and temporal distribution of moisture including pressure head, hydraulic head and water content under different initial and boundary conditions. As a result, we found that from the depth of the node to the lower boundary, the values of pressure head and hydraulic head increase with depth and decrease with time, while the values of water content represent opposite trends relative to the distribution of pressure head and hydraulic head in bio-retention systems.