Roughness Index Research Articles

Comparative Analysis of Deep Neural Networks and Graph Convolutional Networks for Road Surface Condition Prediction

Road maintenance is essential for supporting road safety and user comfort. Developing predictive models for road surface conditions enables highway agencies to optimize maintenance planning and strategies. The international roughness index (IRI) is widely used as a standard for evaluating road surface quality. This study compares the performance of deep neural networks (DNNs) and graph convolutional networks (GCNs) in predicting IRI values. A unique aspect of this research is the inclusion of additional predictor features, such as the type and timing of recent roadwork, hypothesized to affect IRI values. Findings indicate that, overall, the DNN model performs similarly to the GCN model across the entire highway network. Given the predominantly linear structure of national highways and their limited connectivity, the dataset exhibits a low beta index, ranging from 0.5 to 0.75. Additionally, gaps in IRI data collection and discontinuities in certain highway segments present challenges for modeling spatial dependencies. The performance of DNN and GCN models was assessed across the network, with results indicating that DNN outperforms GCN when highway networks are sparsely connected. This research underscores the suitability of DNN for low-connectivity networks like highways, while also highlighting the potential of GCNs in more densely connected settings.

A high-precision two-parameter road roughness grading and identification methodology across multiple frequency intervals

Based on a dual track road shape meter, various road roughness was measured. It was discovered that the road grading method in ISO 8608 assumes a constant waviness value of W = 2 with a one-straight line approximation, but it could not accurately fit and grade the measured road roughness. A high-precision piecewise fitting function for the low-frequency and high-frequency parts of the road roughness power spectral density (PSD) is established. A high-precision two-parameter grade method in multiple frequency bands for road roughness is proposed ( G q( n0), the geometric average of the road roughness index at n0, W, the waviness, is the slope of the road roughness PSD fitting expression in the double logarithmic coordinate system). According to the vehicle driving performance obtained by simulation, the grading method is verified. For Paved and Undulating roads, compared to the road roughness classification method in ISO 8608, the performance simulated based on the grade results of the two parameters grading method are closer to the performance simulated based on the measured road roughness. The grading method proposed in this paper can more accurately fit and classify the measured road roughness. The influence of these parameters at each frequency interval on vehicle driving performance is studied. A two-parameter identification method for road roughness PSD at variable intervals over a large frequency range is proposed. In the double logarithmic coordinate system, this method divides the selected spatial frequency into three sub-intervals along the abscissa, in each sub-interval, the parameters G q( n0) and W are identified separately. This method can provide effective control inputs for the vehicle’s active or semi-active suspension system.

Reconstructing Road Roughness Profiles Using ANNs and Dynamic Vehicle Accelerations

Road networks are crucial infrastructures that play a significant role in the progress and advancement of societies. However, roads deteriorate over time due to regular use and external environmental factors. This deterioration leads to discomfort for road users as well as the generation of noise and vibrations, which negatively impact nearby structures. Therefore, it is essential to regularly maintain and monitor road networks. The International Roughness Index (IRI) is commonly used to quantify road roughness and serves as a key indicator for assessing road condition. Traditionally, obtaining the IRI involves manual or automated methods that can be time-consuming and expensive. This study explores the potential of using artificial neural networks (ANNs) and dynamic vehicle accelerations from two simulated car models to reconstruct road roughness profiles. These models include a simplified quarter-car (QC) model with two degrees of freedom, valued for its computational efficiency, and a more intricate full-car (FC) model with seven degrees of freedom, which replicates real-life vehicle behavior. This study also examines the ability of ANNs to predict the mechanical properties of the FC model from dynamic vehicle responses to obstacles. We compare the accuracy and computational efficiency of the two models and find that the QC model is almost 10 times faster than the FC model in reconstructing the road roughness profile whilst achieving higher accuracy.

Analisa Derajat Kejenuhan, Perlengkapan Jalan, dan Nilai Kerataan Pada Jalan Raya Kepandean, Kabupaten Tegal

Road safety audits are an effort to improve safety and prevent traffic accidents. The implementation of the audit can be carried out by the survey method directly in the field and can use the Hawkeye car to get the International Roughness Index (IRI) value. The results of this study show that the condition of the road on the Kepandean Highway section with the IRI value of the route from the direction of SMPN 1 Dukuhturi is 6.5 while the IRI value of the route to SMPN 1 Dukuhturi is 6.3 so that it can be concluded that the road is in the medium category (not good) and requires periodic repair and maintenance efforts

Flash flood prediction modeling in the hilly regions of Southeastern Bangladesh: A machine learning attempt on present and future climate scenarios

Flash floods are highly destructive, and their frequency and intensity are expected to escalate due to climatic changes. This study thus investigated flash flood susceptibility (FFS) by applying machine learning algorithms and climate projection to predict both present and future hazard scenarios in the southeastern hilly regions of Bangladesh. To predict FFS, we evaluated twelve flood-influencing variables: elevation (EL), slope (SL), aspect (AS), drainage density (DD), distance to stream (DS), topography roughness index (TRI), stream power index (SPI), topographic wetness index (TWI), soil permeability (SP), precipitation (PR), land use and land cover (LULC) and normalized difference vegetation index (NDVI). Earth observation data, field surveys, and past flood records were used to create a detailed flood inventory. Among the machine learning models tested, the random forest (RF) algorithm outperformed others, including support vector machine (SVC), logistic regression (LR), and extreme gradient boosting (XGBoost), and was subsequently used for flood susceptibility mapping based on future precipitation projections under two Sixth Coupled model intercomparison project (CMIP6) climate change scenarios: SSP1-2.6 and SSP5-8.5. Our findings indicated that the areas at high to very high risk of flooding are projected to increase significantly under both the SSP1-2.6 and SSP5-8.5 scenarios. Initially, around 38 % of the studied region had high to very high flood susceptibility, but this is expected to rise to 40–42 % over the projected time periods. These spatial delineations of flood-prone areas can provide guidance for developing effective mitigation and adaptation strategies to address the adverse impacts of flash flooding in the hilly river basins of Bangladesh.

Extraction of isotropic and anisotropic components for optical surface micro-metrology based on the two-dimensional power spectral density analysis

Extraction of surface characteristics is essential during surface processing and optical inspections. In this work, we propose a new extraction method by dividing the global feature within multiple directions into isotropic and anisotropic components and combining noise filtration, two-dimensional component extraction, and fast reconstruction. The mathematical descriptions of global and local features were derived. The noise by holes, scratches, and finite sampling points was restrained by Bearing Ratio analysis, Hough transform, and Welch window operation. The isotropic and anisotropic surface components were extracted in the two-dimensional power spectral density domain, reconstructed in the two-dimensional Fourier domain, and inversed in Cartesian coordinates. Five surfaces with anisotropic structural characteristics ranging from zero to two dimensions were analyzed. The general applicability was proved according to the consistency between surface processing methods and extracted results. A chemical mechanical polishing experiment was designed and accomplished to verify the sensitivity of the extraction method. The subtle variation in surface morphology was captured on the reconstructed surfaces near the polishing end-point, confirming its detectability on weak anisotropic components. This process-oriented surface extraction method can achieve qualified results without transcendental knowledge of surface conditions and offers openness to various surface evaluation criteria by statistical roughness indicators, which supports surface inspection for multiple surface processing techniques.

Effects of plantations on rainfall redistribution in a rocky mountain area of North China

AbstractRainfall redistribution plays a crucial role in the water cycle. However, the main factors affecting the redistribution of rainfall remain uncertain. We chose three different plantations—cork oak (Quercus variabilis Bl.), oriental arborvitae (Platycladus orientalis L.) and black locust (Robinia pseudoacacia L.)—to investigate the role of plantations in rainfall redistribution and to determine the main factors influencing rainfall redistribution. The results indicated that cork oak exhibited the highest stemflow (0.34%) and the lowest canopy interception (12.58%), whereas black locust had the lowest stemflow (0.21%), and oriental arborvitae displayed the greatest canopy interception (32.8%). Under different density conditions for cork oaks, the stemflow was highest (0.39%) in low‐density forests with 750 trees ha−2 and lowest (0.34%) in medium‐density forests with 1100 trees ha−2. Meanwhile, the highest canopy interception (17.68%) was observed in high‐density forests (1300 trees ha−2), while the lowest interception rate (9.22%) was found in low‐density forests. The main factors affecting rainfall redistribution and their contribution rates were as follows: bark roughness index (35%), wind speed (18.6%), tree species (14.2%), diameter at breast height (11.2%), stand density (9.6%) and rainfall amount (5.4%). Our findings suggested that structural characteristics of trees are the primary factors affecting rainfall redistribution. Planting cork oak in the rocky mountain regions of North China is recommended because of its substantial stemflow production, particularly under low‐density growth conditions. Therefore, this study has significant guiding implications for the selection of afforestation tree species in similar rocky mountain areas globally.

The Effect of Road Roughness on Travel Time

Travel time can be decreased by increasing the speed. Speed has a relationship with road roughness. In this research, there were forty different routes and travel time for each route was simulated by thirty different kinds of speed resulted from three categories of International Roughness Index : smooth, fair, and rough that range from 0.60 to 3.50. The routes were taken from daily operation – shipping activities which the starting point and the end point were from the main warehouse. The sequence of locations for each route in shipping activity was determined by the OptimoRoute software. The shortest, the mean and the longest travel time for each route was determined by Phython program. The results showed that travel time on smooth road was more efficient around 36.17% than fair road, more efficient around 57.19% than rough road, and the travel time on fair road was more efficient around 49.10% than rough road.

Characterization of alkali treated and untreated Abutilon indicum FIBERS

The primary objective of this study is to characterize Abutilon indicum stem Fibres (AIF) obtained through the water retting method. The fundamental attributes were investigated for untreated and alkali-treated AIFs, including physical and chemical properties, thermal stability as determined by thermogravimetric analysis, functional groups as identified by Fourier transform infrared spectroscopy (FTIR), crystalline properties as determined by X-ray diffraction (XRD) analysis, surface roughness as assessed by atomic force microscopy (AFM), and surface textures as determined by scanning electron microscopy (SEM) images. Results convey that, the surface-modified fibers upon alkali treatment exhibit an increase in cellulose composition from 56.12 % to 65.43 % and a reduction in hemi-cellulose, lignin, moisture and wax content. Such a higher composition of cellulose content in the fibers influenced the surface roughness, crystalline size and crystalline index (CI) which thereby indicates the presence of Cellulose III1 and other crystallites arranged in order over the alkali-treated AIFs. Alkali treatment increased the degradation temperature and thermal stability from 1750 C to 2020 C and 302.60 C to 3400 C, respectively. From the above-mentioned test results, it is evident that the surface-modified AIFs from alkali treatment provided more desired results than the untreated counterparts.

A Comparative Study of Pavement Roughness Prediction Models under Different Climatic Conditions

Predicting the International Roughness Index (IRI) is crucial for maintaining road quality and ensuring the safety and comfort of road users. Accurate IRI predictions help in the timely identification of road sections that require maintenance, thus preventing further deterioration and reducing overall maintenance costs. This study aims to develop robust predictive models for the IRI using advanced machine learning techniques across different climatic conditions. Data were sourced from the Ministry of Energy and Infrastructure in the UAE for localized conditions coupled with the Long-Term Pavement Performance (LTPP) database for comparison and validation purposes. This study evaluates several machine learning models, including regression trees, support vector machines (SVMs), ensemble trees, Gaussian process regression (GPR), artificial neural networks (ANNs), and kernel-based methods. Among the models tested, GPR, particularly with rational quadratic specifications, consistently demonstrated superior performance with the lowest Root Mean Square Error (RMSE) and highest R-squared values across all datasets. Sensitivity analysis identified age, total pavement thickness, precipitation, temperature, and Annual Average Daily Truck Traffic (AADTT) as key factors influencing the IRI. The results indicate that pavement age and higher traffic loads significantly increase roughness, while thicker pavements contribute to smoother surfaces. Climatic factors such as temperature and precipitation showed varying impacts depending on the regional conditions. The developed models provide a powerful tool for predicting pavement roughness, enabling more accurate maintenance planning and resource allocation. The findings highlight the necessity of tailoring pavement management practices to specific environmental and traffic conditions to enhance road quality and longevity. This research offers a comprehensive framework for understanding and predicting pavement performance, with implications for infrastructure management both locally and worldwide.

Evaluation of Flexible Pavement Friction Coefficients: A Case Study of East-West Highway Near Mahendranagar City, Nepal

Friction is a crucial factor in highway engineering, as it affects the traction between a vehicle's wheels and the road surface. This traction significantly influences the movement, speed, and efficiency of vehicles on highways. The coefficient of friction depends on various factors, such as pavement materials, traffic volume, road age, temperature, and weather conditions. The Skid Resistance Test Method is adopted for the study, which involves using a Portable Skid Resistance Tester to measure the British Pendulum Number (BPN) and then calculating the coefficient of friction by dividing the BPN by 10. An evaluation of a 5 km stretch of the East-West highway near Mahendranagar Bazaar revealed a longitudinal coefficient of friction of 0.354 and a lateral coefficient of friction of 0.184 under both dry and wet conditions. These findings were relevant for design speeds ranging from 60 to 80 km/h and considered an Annual Average Daily Traffic (AADT) of 10,321 Passenger Car Units (PCU) during the year 2023/2024. The assessment also indicated a variance of 0.19% and a poor International Roughness Index (IRI) value of 2.05. These aggregates had specific gravities ranging from 2.6 to 2.67 and were sized at 13.2 mm. The application rate for the aggregates ranged from 12 to 15 kg per square meter. It is important to note that a higher coefficient of friction leads to reduced vehicle efficiency. Thus, it is crucial to carefully consider the coefficient of friction values and detailed specifications to ensure traffic safety and road integrity.

Analysis of Road Roughness and Driver Comfort in 'Long-Haul' Road Transportation Using Random Forest Approach.

Global trade depends on long-haul transportation, yet comfort for drivers on lengthy trips is sometimes neglected. Rough roads have a major negative influence on driver comfort and increase the risk of weariness, distracted driving, and accidents. Using Random Forest regression, a machine learning technique well-suited to examining big datasets and nonlinear relationships, this study examines the relationship between road roughness and driver comfort. Using the MIRANDA mobile application, data were gathered from 1,048,576 rows, including vehicle acceleration and values for the International Roughness Index (IRI). The Support Vector Regression (SVR) and XGBoost models were used for comparative analysis. Random Forest was preferred because of its ability to be deployed in real time and use less memory, even if XGBoost performed better in terms of training time and prediction accuracy. The findings showed a significant relationship between driver discomfort and road roughness, with rougher roads resulting in increased vertical acceleration and lower comfort levels (Road Roughness: SD-0.73; Driver's Comfort: Mean-10.01, SD-0.64). This study highlights how crucial it is to provide smooth surfaces and road maintenance in order to increase road safety, lessen driver weariness, and promote long-haul driver welfare. These results offer information to transportation authorities and policymakers to help them make data-driven decisions that enhance the efficiency of transportation and road conditions.

Estimate non-Gaussian road roughness from international roughness index

Gaussian model is commonly used to characterize road roughness, while field measured data is usually non-Gaussian. A new method based on the amplitude modulation and International Roughness Index is proposed in this paper to estimate non-Gaussian road roughness. A modified Gamma distribution is proposed to describe the International Roughness Index and to create the amplitude modulation signal. To ensure that the high kurtosis of the amplitude-modulated non-Gaussian profile is correctly transferred to the vehicle response, the method is further developed to fulfill the specifications for the fatigue damage spectrum by controlling the spectral content of the amplitude modulation signal. Herein, a Gaussian amplitude modulation signal with a cutoff frequency is firstly generated, and then converted to a modified Gamma distribution amplitude modulation signal based on the cumulative distribution function transform. Field data are used to verify the proposed method. The results show that a good match is obtained between the estimated and measured non-Gaussian road roughness. The estimated International Roughness Index and fatigue damage potential of vehicle response are well represented using the proposed method.

New Roughness Index of Passenger Car Rotational Vibration

Current road roughness indexes are limited to reflecting the unwanted rotational vibration of vehicles. This study focused on the relationship between the international roughness index (IRI) and rotational vibration around the longitudinal axis (pitch) and horizontal axis (roll) of the vehicle body. Sinusoidal wave input of different wavelengths and phase shifts between the left and right wheel tracks were used as excitation of a full-car model. Marked pitch and roll vehicle vibration were observed at shorter wavelengths (<1 m) and lower speeds (<50 km/h) of harmonic excitation. The IRI was limited to predicting the pitch and roll vibration of the full-car model at lower wavelengths, below 1 m. A simple alternative solution based on the IRI model was proposed as a robust, sensitive tool to detect unwanted rotational vibration of a vehicle. The proposal was based on the IRI approach using an alternative IRI model speed of 30 km/h, instead of the standard 80 km/h. The IRI for 30 km/h was several times more sensitive to pitch and roll vibration in the wavelength band up to 1.5 m than the IRI for 80 km/h. The new approach is recommended as an additional roughness index for monitoring roads with an allowable speed of up to 60 km/h.

An Artificial Neural Network approach to assess road roughness using smartphone-based crowdsourcing data

Monitoring road surface conditions is a crucial task for road authorities to develop effective infrastructure maintenance programs. Despite smartphones have been introduced as cost-effective and real-time solution for this purpose, several challenges must be addressed before their real-world application. This study investigates the utilization of smartphone-based crowdsourcing data and Artificial Neural Networks (ANN) to enhance the precision of road surface condition estimation. Initially, data are collected from four different smartphone models mounted in various vehicles, including vertical acceleration, geographic location, and speed. The root mean square of the vertical acceleration data, along with vehicle speed, is then employed as input features for the ANN, while the true International Roughness Index (IRI) values serve as the corresponding output features. Comparative analysis between ANN and regression models based on statistical metrics such as Mean Squared Error (MSE) and Pearson correlation revealed that ANN outperforms regression models. The obtained MSE and Pearson correlation values for ANN (0.56 and 0.91) surpass those of regression models (0.72 and 0.88). Moreover, results indicated that utilizing crowdsourcing smartphone data yielded superior outcomes compared to using a single smartphone for this purpose.

Statistical and machine learning models for predicting spalling in CRCP

Continuously reinforced concrete pavement (CRCP), crucial for the resilience of transportation infrastructure owing to its continuous steel reinforcement, confronts a critical challenge in the form of spalling—a distress phenomenon posing a threat to pavement durability and overall structural integrity. The detachment or breakage of concrete from the surface compromises CRCP’s functionality and raises safety concerns and escalating maintenance costs. To address this pressing issue, our study investigates the multifaceted factors influencing spalling, employing a comprehensive approach that integrates statistical and machine learning techniques for predictive modeling. Descriptive statistics meticulously profile the dataset, emphasizing age, thickness, precipitation, temperature, and traffic parameters. Regression analysis unveils key relationships, emphasizing the significance of age, annual temperature, annual precipitation, maximum humidity, and the initial International Roughness Index (IRI) as influential factors. The correlation matrix heatmap guides feature selection, elucidating intricate interdependencies. Simultaneously, feature importance analysis identifies age, Average Annual Daily Traffic (AADT), and total pavement thickness as crucial contributors to spalling. In machine learning, adopting models, including Gaussian Process Regression and ensemble tree models, is grounded in their diverse capabilities and suitability for the complex task at hand. Their varying predictive accuracies underscore the importance of judicious model selection. This research advances pavement engineering practices by offering nuanced insights into factors influencing spalling in CRCP, refining our understanding of spalling influences. Consequently, the study not only opens avenues for developing improved predictive methodologies but also enhances the durability of CRCP infrastructure, addressing broader implications for informed decision-making in transportation infrastructure management. The selection of Gaussian Process Regression and ensemble tree models stems from their adaptability to capture intricate relationships within the dataset, and their comparative performance provides valuable insights into the diverse predictive capabilities of these models in the context of CRCP spalling.

From ensemble learning to deep ensemble learning: A case study on multi-indicator prediction of pavement performance

Recently, Big data analytics approaches were combined with emerging machine learning techniques, which can provide more sophisticated insights into information-intensive activity. Compared to traditional shallow-architecture machine learning algorithms, Deep learning could excavate more potential information from the raw features. However, its powerful representational capacity relies on the support of enormous samples. The ensemble trees system performs superior on small sample problems due to better generalization capacity. To merge both benefits of deep learning and ensemble tree system, this paper developed a deep ensemble algorithm applied to multiple indicators prediction of pavement performance, including International Roughness Index and pavement 3-layer modulus. The deep ensemble algorithm is developed by merging a deep neural network (DNN) with the decision manifold property of the decision trees (TabNet) into a cascade ensemble system, combined with a sliding window algorithm to extract dependency information from raw data. During the training stage, the Bayesian Optimization Algorithm (BOA) is used to search for the optimal combination of sub-decision makers in the cascade ensemble. And equipped with GPU, it can speed up by 2.6–4.0 times. In the case study of pavement engineering, with sufficient training samples, it can achieve an average accuracy of 98.74 %, higher than DNN (97.49 %) and XGBoost (96.12 %) in predicting pavement indicators. With insufficient training samples, it can achieve an accuracy improvement of 12 % than XGBoost (75 %) and 24.5 % than DNN (62.5 %).

Modeling the Influence of Internal Factors on Vehicular Mobility Performance

Traffic mobility is significantly influenced by road conditions and the internal factors governing vehicular performance. Internal factors play a crucial role in vehicular mobility, influencing vehicles operate and performance. While existing vehicular mobility models primarily address external factors, external factors such as governor, engine, gear train and differential unit remain underexplored. This study formulates a comprehensive model to address this gap, focusing on the internal components that affect vehicular mobility. The proposed model employs both analytical and numerical methods to derive transfer functions for these subsystems using Matrix Laboratory (MATLAB), aiming to capture the dynamic behavior of vehicles under various conditions. The model was evaluated by examining vehicle transaction times across different road surface qualities, measured by the International Roughness Index (IRI) values of 3.5, 3.0, and 2.0, and tractive forces ranging from 4500 N to 17750 N, with applied pedal forces of 50 N, 100 N, 150 N and 200 N. Results indicated that higher tractive forces lead to reduced transaction times, with IRI values of 3.5 m/km showing a decrease from 122.0 seconds to 31.0 seconds as tractive forces increased from 4500 N to 17750 N. Similarly, for IRI values of 2.0 m/km, the transaction times reduced from 67.5 seconds to 7.5 seconds under the same conditions. The analysis further demonstrated that increased applied pedal forces correspond to higher tractive forces, thereby enhancing vehicular mobility and performance. These findings highlight the critical role of internal factors in optimizing vehicular mobility and performance, suggesting that internal subsystem dynamics should be integrated into future mobility models for more accurate and comprehensive assessments.

Sand blasting for hydrophobic surface generation in polymers: Experimental and machine learning approaches

Wettability is a crucial surface feature of polymers due to their numerous interaction-destined applications. This study focuses on the application of sand blasting process for investigating the wettability of polymeric materials to produce hydrophobic behavior. Four different polymeric materials, Acrylonitrile Butadiene Styrene (ABS), Poly(methyl methacrylate) (PMMA), Polypropylene (PP), and Polycarbonate (PC) underwent sand blasting with varying process parameters, following a comprehensive plan for the design of experiments. Subsequent analyses included surface roughness measurement and wettability tests, supplemented by scanning electron and confocal microscopy observations to gain deeper insights into the blasted surfaces. A predictive model based on a machine learning algorithm was developed using the backpropagation technique to correlate the surface treatment parameters to surface roughness and wettability indexes. From the experimental results sand blasting proved to be efficient in creating hydrophobic surfaces on all the tested materials. The developed neural network demonstrated high fitting degrees between the predicted and measured values. ABS exhibited the most hydrophobic behavior and emerged as a strong candidate for further investigations.

Near-Ground wind field characteristics of tracking photovoltaic systems based on field measurements

The study conducts field measurements at a tracking photovoltaic power station in Zhongwei City, Ningxia Hui Autonomous Region, investigating near-ground incoming wind characteristics, spatial coherence, and their impact on the flow field of the tracking photovoltaic system. Actual measurements indicate a low dispersion degree in the roughness index of this area, with an average roughness index of α = 0.1228, which is consistent with the characteristics of open land and desert areas. The incoming wind speed profile largely complies with European wind load specifications, with turbulence intensity measured at different heights exceeding other wind load specifications, primarily due to increased sand movement contributing to turbulence in desert areas. Additionally, this study proposes calculation formulas for measured turbulence intensity, integration scale, and gust factor, offering valuable insights into the incoming wind field characteristics in this region. Furthermore, the investigation into spatial coherence and decay parameters revealed the superior performance of the Krenk model in high frequency bands and over long distances. Lastly, the tracking photovoltaic array induces noticeable shading effects, altering flow field characteristics significantly, with a diminished impact on wind flow velocity as the tilt angle of the photovoltaic panel decreases. These studies serve as a valuable reference for gaining a deeper understanding of wind field characteristics, wind load calculation, and the optimal design of tracking photovoltaic systems.