Docking Stations Research Articles

Robust bike-sharing stations allocation and path network design: a two-stage stochastic programming model

ABSTRACT Nowadays, the significance of developing sustainable transport systems is being reconsidered due to the impact produced by an extensive use of private cars. In this scenario, bicycles are becoming more popular. To encourage cycling, bike-sharing systems have been developed. However, as safety is people’s main concern of cycling, bike-sharing systems should be part of an integral project that includes not only docking stations but a dedicated cycle path network connecting them. A two-stage stochastic programming model that maximizes the travel demand covered by the system is developed. Various demand scenarios in terms of different time periods of a day, travel intensities and levels of commuting demand are considered in the model, in order to obtain robust solutions. A real-world case study based on the city of Montevideo in Uruguay is conducted. The obtained solution shows that the model tends to develop bicycle lane network with good connectivity between transit stations and normal stations, and the obtained network possesses good robustness.

Evaluating the impact of implementing public bicycle share programs on cycling: the International Bikeshare Impacts on Cycling and Collisions Study (IBICCS)

BackgroundDespite rapid expansion of public bicycle share programs (PBSP), there are limited evaluations of the population-level impacts of these programs on cycling, leaving uncertainty as to whether these programs lead to net health gains at a population level or attract those that already cycle and are sufficiently physically active. Our objective was to determine whether the implementation of PBSPs increased population-level cycling in cities across the US and Canada.MethodsWe conducted repeat cross-sectional surveys with 23,901 residents in cities with newly implemented PBSPs (Chicago, New York), existing PBSPs (Boston, Montreal, Toronto) and no PBSPs (Detroit, Philadelphia, Vancouver) at three time points (Fall 2012, 2013, 2014). We used a triple difference in differences analysis to assess whether there were increases in cycling over time amongst those living in closer proximity (< 500 m) to bicycle share docking stations in cities with newly implemented and existing PBSPs, relative to those in cities with no PBSPs.ResultsLiving in closer proximity to bicycle share predicted increases in cycling over time for those living in cities with newly implemented PBSPs at 2-year follow-up. No change was seen over time for those living in closer proximity to bicycle share in cities with existing PBSPs relative to those in cities with no PBSP.ConclusionThese findings indicate that PBSPs are associated with increases in population-level cycling for those who live near to a docking station in the second year of program implementation.

A survey of underwater docking guidance systems

Autonomous underwater vehicles (AUVs) are increasingly being used for underwater survey and exploration missions. The expanding mission scope for AUVs highlights the need for a long-endurance operational capability, which mainly depends on propulsion system efficiency and battery capacity. The use of submerged docking stations permitting battery recharge and data download/upload offers a means of enabling persistence without compromising propulsion and payload power budgets, while also reducing associated deployment/recovery costs and risks. Autonomous docking with an underwater station is, however, complicated by the presence of currents and obstacles in the water, and by the relative dynamic differences in pose between the dock and the vehicle. A robust docking guidance system is identified as a core and crucial component for ensuring successful AUV docking. This paper presents a detailed literature review summarizing the current state-of-the-art in AUV docking guidance methodologies, identifying their relative merits and shortcomings, and revealing the docking guidance methodologies that seems to be the most prominent.

An Autonomous Docking Mechanism for Vertical Lift Unmanned Aircraft

Abstract Unmanned aerial vehicles are increasingly being tasked to connect to payload objects or docking stations for the purposes of package transport or recharging. However, autonomous docking creates challenges in that the air vehicle must precisely position itself with respect to the dock, oftentimes in the presence of uncertain winds and measurement errors. This paper describes an autonomous docking mechanism comprising a static ring and actuated legs, coupled with an infrared tracking device for closed-loop docking maneuvers. The dock’s unique mechanical design enables precise passive positioning such that the air vehicle slides into a precise location and orientation in the dock from a wide range of entry conditions. This leads to successful docking in the presence of winds and sensor measurement errors. A closed-loop infrared tracking system is also described in which the vehicle tracks an infrared beacon located on the dock during the descent to landing. A detailed analysis is presented describing the interaction dynamics between the aircraft and the dock, and system parameters are optimized through the use of trade studies and Monte Carlo analysis with a three degree-of-freedom simulation model. Experimental results are presented demonstrating successful docking maneuvers of an autonomous air vehicle in both indoor and outdoor environments. These repeatable docking experiments verify the robustness and practical utility of the dock design for a variety of emerging applications.

Effects of dockless bike-sharing systems on the usage of the London Cycle Hire

This paper evaluates the effects of dockless bike-sharing systems on the usage of the London Cycle Hire (LCH). A total number of 707 LCH docking stations are included for a period of 36 weeks. Covariates such as socio-demographic characteristics, traffic conditions, and the built environment are included in the data set. A difference in difference (DID) based propensity score matching (PSM) method is applied to select control docking stations and estimate the effects of the dockless bike-sharing systems on the LCH usage. The results suggest that a significant reduction of 22.42 (5.93%) in the average weekly usage of each LCH docking station is caused by the dockless bike-sharing system after its initial launch. We also investigate the effects of dockless bike-sharing systems on the characteristics of the LCH trips, including travel duration, travel distance, and departure time. The majority of trips replaced by the dockless bikes are short duration (0–15 min) and middle distance (1–3 km) trips. The reduction in the average weekly short duration trips of each station is 16.91 (7.16%), while the figure is 15.11 (6.74%) for middle distance trips. The average travel duration is increased by 0.25 min, but the travel distance shows no significant change due to data restriction. Furthermore, the LCH usage is significantly reduced by 6.85% during the weekday commuting peak, and 10.47% during the weekend leisure peak. As to weekday off-peak and other times, reductions are not statistically significant.

Association, Blockage, and Handoffs in IEEE 802.11ad-Based 60-GHz Picocells—A Closer Look

The link misalignment and high susceptibility to blockages are the biggest hurdles in realizing 60GHz based wireless local area networks (WLANs). However, much of the previous studies investigating 60GHz alignment and blockage issues do not provide an accurate quantitative evaluation from the perspective of WLANs. In this paper, we present an in-depth quantitative evaluation of commodity IEEE 802.11ad devices by forming a 60GHz WLAN with two docking stations mimicking as access points (APs). Through extensive experiments, we provide important insights about directional coverage pattern of antennas, communication range and co-channel interference and blockages. We are able to measure the IEEE 802.11ad link alignment and association overheads in absolute time units. With a very high accuracy (96-97%), our blockage characterization can differentiate between temporary and permanent blockages caused by humans in the indoor environment, which is a key insight. Utilizing our blockage characterization, we also demonstrate intelligent handoff to alternate APs using consumergrade IEEE 802.11ad devices. Our blockage-induced handoff experiments provide important insights that would be helpful in integrating millimeter wave based WLANs into future wireless networks.

Receding Horizon Control for Station Inventory Management in a Bike-Sharing System

A docking bike-sharing system (BSS) is modeled as a network representing the underlying transportation network. Mobile agents (replenishment trucks) traverse the network making routing decisions and deciding how and when to replenish station inventories so as to prevent imbalances due to users’ one-way rides as well as time-varying demand. This load balancing process entails selecting both optimal routes for the agents and the number of bikes to load/unload at a station with an objective of minimizing a user dissatisfaction metric. First, we establish a time-dependent replenishment fill-to level policy for each station based on the demand rates and station capacities. Next, we focus on developing a receding horizon controller (RHC) to find optimal routes. The controller proceeds in an event-driven manner to determine after each event the optimal routes for a fleet of agents over a finite planning horizon, with the control applied over a shorter action horizon. The proposed controller is applied to a simulated BSS with station and demand parameters taken from the public data sets of Bluebikes, the BSS in Boston, MA, USA, and a cost-benefit analysis is performed on agent shift hours. In order to demonstrate the robustness of the RHC, sensitivity analysis is also performed on the arc travel times and the demand processes. Note to Practitioners—This paper is motivated by the load balancing problem faced by BSS with finite-capacity docking stations and time-varying demand; stations become empty or full as their popularity as an origin or destination varies over the course of a day. The proposed RHC creates dynamic cooperative routes for a fleet of load balancing trucks to move bikes between stations by considering the current inventory, demand rates, and proximity of all stations and trucks. The strength of this controller is that it achieves optimality over a specified planning horizon and reacts quickly to random inventory changes which update the planning horizon on a rolling basis. Thus, the optimal routes found for a planning horizon are updated at the next decision point, i.e., an intersection. This event-driven RHC decreases the complexity of finding optimal routes so that it may be used in real time. Insights obtained from the application of this approach to the Boston BSS are that extending the length of the receding horizon provides marginal benefits beyond a certain value and that the controller is robust with respect to the stochastic behavior of the user demand and the truck travel time processes. The approach is amenable to extensions that can include incentivizing users so as to enhance load balancing beyond external truck-based interventions.

Ocean Climate Monitoring

Measuring ocean physics and atmospheric conditions at the sea-surface has been taking place for decades in our world’s oceans. Enhancing RD we discuss a “Hybrid” air-sea interaction deep-sea monitoring system that has been developed in the private sector to mirror ocean-climate community data streams and has been successfully deployed on three basin-scaled programs in the Indian Ocean (RAMA, First Institute of Oceanography, FIO, China), the Andaman Sea (MOMSEI, Monsoon Onset Monitoring, FIO) and the Pacific Ocean (China’s Institute of Oceanology, Academy of Sciences research in the western tropical Pacific). This application is a base to build upon as new sensors are developed and increased sampling at higher resolutions is required. A multidisciplinary tool to add new biological and biogeochemical sensors to study interaction with the physical processes of our oceans. This application can also be used at FLUX sites to enhance the Argo Program, telemetry docking stations for autonomous vehicles such as sail-drones, gliders and wave riders for enhancement and contribution to the Global Tropical Moored Buoy Array (GTMBA), Global Ocean Observing System (GOOS), Global Climate Observing System (GCOS), and the Global Earth Observing System of Systems (GEOS).

Effect of built environment on shared bicycle reallocation: A case study on Nanjing, China

Shared bicycles provide a convenient mobility option to commuters especially for short-distance trips. Nevertheless, it also presents a challenge to bicycle-sharing operators as they have to deal with reallocation issues, i.e. removing bicycles from parking facilities which are at or near capacity and refilling parking facilities that are in need of bicycles. Few studies in the literature have actually tried understanding why certain docking stations are prone to excessive demand or suffer from a lack of parking supply. This paper attempts to identify the demographic, built-environment and transport-infrastructure indicators that can potentially aid policy-makers and operators in identifying parking facilities prone to bicycle reallocation. In particular, we have adopted the bicycle sharing operations in Nanjing, China as a case study to understand how such indicators can be identified for appropriate parking infrastructural enhancements. To achieve this goal, this study has established zero-inflated negative binomial models using multi-source data including point-of-interest (POI), daily weather, transit stop location, demographic data and bike-share smart card data. The model results obtained from this study suggest that built environment correlates significantly to shared bicycle reallocation count. In general, bicycle docking stations with large reallocation counts are more likely to be found near residences, bus stops, metro stations, employment areas, restaurants, amenities, parks, sports facilities, and clinics/hospitals; while stations near entertainment facilities, places of attraction, hotels, shopping malls, and educational institution tend to have balanced demand and supply. Analysis on the elasticity values revealed that mean temperature and station capacity are the most influential factors in bicycle reallocation. Among all POIs, presence of restaurants and areas with high employment tend to exhibit strongly a need for morning bicycle removal and evening bicycle refilling at docked stations. Policy makers can provide actual guidelines in the planning of shared bicycle parking facilities using the findings and methodologies presented in this study.

Urban Mobility Demand Profiles: Time Series for Cars and Bike-Sharing Use as a Resource for Transport and Energy Modeling

The transport sector is currently facing a significant transition, with strong drivers including decarbonization and digitalization trends, especially in urban passenger transport. The availability of monitoring data is at the basis of the development of optimization models supporting an enhanced urban mobility, with multiple benefits including lower pollutants and CO2 emissions, lower energy consumption, better transport management and land space use. This paper presents two datasets that represent time series with a high temporal resolution (five-minute time step) both for vehicles and bike sharing use in the city of Turin, located in Northern Italy. These high-resolution profiles have been obtained by the collection and elaboration of available online resources providing live information on traffic monitoring and bike sharing docking stations. The data are provided for the entire year 2018, and they represent an interesting basis for the evaluation of seasonal and daily variability patterns in urban mobility. These data may be used for different applications, ranging from the chronological distribution of mobility demand, to the estimation of passenger transport flows for the development of transport models in urban contexts. Moreover, traffic profiles are at the basis for the modeling of electric vehicles charging strategies and their interaction with the power grid.

A spatiotemporal and graph-based analysis of dockless bike sharing patterns to understand urban flows over the last mile

The recent emergence of dockless bike sharing systems has resulted in new patterns of urban transport. Users can begin and end trips from their origin and destination locations rather than docking stations. Analysis of changes in the spatiotemporal availability of such bikes has the ability to provide insights into urban dynamics at a finer granularity than is possible through analysis of travel card or dock-based bike scheme data. This study analyses dockless bike sharing in Nanchang, China over a period when a new metro line came into operation. It uses spatial statistics and graph-based approaches to quantify changes in travel behaviours and generates previously unobtainable insights about urban flow structures. Geostatistical analyses support understanding of large-scale changes in spatiotemporal travel behaviours and graph-based approaches allow changes in local travel flows between individual locations to be quantified and characterized. The results show how the new metro service boosted nearby bike demand, but with considerable spatial variation, and changed the spatiotemporal patterns of bike travel behaviour. The analysis also quantifies the evolution of travel flow structures, indicating the resilience of dockless bike schemes and their ability to adapt to changes in travel behaviours. More widely, this study demonstrates how an enhanced understanding of urban dynamics over the “last-mile” is supported by the analyses of dockless bike data. These allow changes in local spatiotemporal interdependencies between different transport systems to be evaluated, and support spatially detailed urban and transport planning. A number of areas of further work are identified to better to understand interdependencies between different transit system components.

Who are the 'super-users' of public bike share? An analysis of public bike share members in Vancouver, BC.

Public bike share programs have been critiqued for serving those who already bicycle, or more well-off individuals who already have a multitude of transportation options. While substantial research focuses on characteristics of public bike share members, it often overlooks their intensity of use which may relate more directly to transport and health gains. In this study we link system data with member survey data to characterize “super-users” of Vancouver's public bike share system. We used system data from September 1, 2016–August 31, 2017 to calculate member-specific trip rates (trips/month). We linked system data to demographic and travel data for members who completed an online survey in 2017 (1232 members who had made 89,945 trips). We defined super-users as those who made 20 or more trips/month. We used a logistic regression to model demographic and travel characteristics associated with super-users as compared to regular users. Of the 1232 members, 204 were super-users. Super-users made 47% of the trips and had a median trip rate of 29.3 trips/month. In adjusted models, super-users were more likely to be young, male, have household incomes below $75,000, and live and work near bike share docking stations. Super-users had fewer transportation options than regular users, with lower odds of having a personal bike or car share membership. Amongst members, we found a distinct demographic profile for super-users relative to regular users, suggesting that usage is an important consideration when quantifying transport and health gains, and the resulting equity implications of public bike share programs.

Soft, Skin-Interfaced Microfluidic Systems with Passive Galvanic Stopwatches for Precise Chronometric Sampling of Sweat.

Comprehensive analysis of sweat chemistry provides noninvasive health monitoring capabilities that complement established biophysical measurements such as heart rate, blood oxygenation, and body temperature. Recent developments in skin-integrated soft microfluidic systems address many challenges associated with standard technologies in sweat collection and analysis. However, recording of time-dependent variations in sweat composition requires bulky electronic systems and power sources, thereby constraining form factor, cost, and modes of use. Here, presented are unconventional design concepts, materials, and device operation principles that address this challenge. Flexible galvanic cells embedded within skin-interfaced microfluidics with passive valves serve as sweat-activated "stopwatches" that record temporal information associated with collection of discrete microliter volumes of sweat. The result allows for precise measurements of dynamic sweat composition fluctuations using in situ or ex situ analytical techniques. Integrated electronics based on near-field communication (NFC) protocols or docking stations equipped with standard electronic measurement tools provide means for extracting digital timing results from the stopwatches. Human subject studies of time-stamped sweat samples by in situ colorimetric methods and ex situ techniques based on inductively coupled plasma mass spectroscopy (ICP-MS) and chlorodimetry illustrate the ability to quantitatively capture time-dynamic sweat chemistry in scenarios compatible with field use.

Bikesharing in Philadelphia: Do lower-income areas generate trips?

Many bikeshare programs seek to equitably serve residents from different income strata. Experience from other cities in the US and elsewhere in the developed world suggest that bikeshare systems are used mostly by more affluent populations. In order to encourage more bikeshare usage among low-income populations, the City of Philadelphia sited docking stations in low-income neighborhoods, allowed cash payments, and provided discounts to those receiving food-stamps. We examined one-year of data for Philadelphia’s Indego bikeshare system between April 2017 and March 2018. Travel patterns were examined and suggested that bikeshare trips taken from docking stations in lower-income areas are for work commute trips. Multivariate regression models confirmed that lower-income areas generate fewer trips while controlling for other factors such as transit access and whether the station is proximate to a bicycle lane. Our results suggest that despite Indego’s efforts, more work is needed to generate bikeshare trips in lower income neighborhoods.

Estimating the Impacts of Capital Bikeshare on Metrorail Ridership in the Washington Metropolitan Area

Bikeshare programs have transformed the urban transportation landscape. However, their impacts on rail transit have not been fully examined. Some researchers find shared bikes help reduce the first-mile/last-mile gaps and boost rail transit ridership, although others see bikeshare as a competitor for riders. Previous studies have mostly relied on surveys of bikeshare program users as the data source, and few have addressed this question using more rigorous methods. In this paper, the authors take the Washington metropolitan area as an example and use statistical methods to quantify the impact of the bikeshare program on rail transit ridership. Using detailed ridership data between 2010 and 2015, they break down Metrorail ridership by type (entries vs. exits) and time of the day (AM peak vs. PM peak) to analyze how Capital Bikeshare (CaBi) interacts with Metrorail. Furthermore, Metrorail stations are categorized into core stations and peripheral stations to examine the impacts of CaBi in different built environments. Regression results show that the impacts of CaBi vary by Metrorail station location. For core Metrorail stations, CaBi docking stations within ¼-mile of a Metrorail station reduce rail ridership in all measures. In particular, CaBi would reduce the number of AM-peak exits by 4,738 per station per month. However, CaBi complements Metrorail in peripheral neighborhoods. Having CaBi installed nearby would increase monthly Metrorail ridership by 1,175 AM-peak exits, 1,417 PM-peak entries, 2,284 AM-peak entries, and 2,422 PM-peak exits. Based on the findings, the authors suggest a collaboration between Metrorail and CaBi to add more bikeshare stations within ¼-mile of peripheral Metrorail stations to increase the ridership of both systems.

Predicting Demand of Distributed Product Service Systems by Binomial Parameter Mapping: A Case Study of Bike Sharing Station Expansion

Quantitative approaches for estimating user demand provide a powerful tool for engineering designers. We hypothesized that estimating binomial distribution parameters n (user population size) and p (user population product affinity) from historical user data can predict demand in new situations for distributed product service systems. Distributed product service systems allow individuals to use shared products at different geographic locations as opposed to owning them. This approach is demonstrated on a major bike-sharing system (BSS) expansion. BSSs position rental bikes around a city in docks at prescribed locations. BSS operators must predict the rider demand when sizing new docking stations, but current demand estimation methods may not be suitable for distributed systems. The main contribution of this paper is the development and application of a revealed preference demand estimation method for distributed product service systems. While much current research seeks to solve distributed system operational problems, we estimate the user population characteristic to provide insight into the initial installation design problem. We introduce the use of spatial surface plots to extrapolate binomial parameters n and p over the service area. These surfaces allow more accurate prediction of relative ridership levels at new station locations. By utilizing Spearman's rho as a comparison benchmark, our approach yields a stronger correlation between our prediction and the observed new station utilization (rho = 0.83, stations = 46, p &lt; 0.01) than the order implemented by the BSS operator (rho = 0.59, stations = 46, p &lt; 0.01).

Spatiotemporal Clustering Analysis of Bicycle Sharing System with Data Mining Approach

The main objective of this study is to explore the spatiotemporal activities pattern of bicycle sharing system by combining together temporal and spatial attributes variables through clustering analysis method. Specifically, three clustering algorithms, i.e., hierarchical clustering, K-means clustering, expectation maximization clustering, are chosen to group the bicycle sharing stations. The temporal attributes variables are obtained through the statistical analysis of bicycle sharing smart card data, and the spatial attributes variables are quantified by point of interest (POI) data around bicycle sharing docking stations, which reflects the influence of land use on bicycle sharing system. According to the performance of the three clustering algorithms and six cluster validation measures, K-means clustering has been proven as the better clustering algorithm for the case of Ningbo, China. Then, the 477 bicycle sharing docking stations were clustered into seven clusters. The results show that the stations of each cluster have their own unique spatiotemporal activities pattern influenced by people’s travel habits and land use characteristics around the stations. This analysis will help bicycle sharing operators better understand the system usage and learn how to improve the service quality of the existing system.

Mobile phone GPS data in urban bicycle-sharing: Layout optimization and emissions reduction analysis

As a representation of smart and sustainable city development, bicycle-sharing system is one of the hottest topics in the domains of transportation, public health, urban planning, and so on. In this paper, a model is proposed for analyzing the potential reduction in emissions associated with the adoption of a bicycle-sharing system. Methods are proposed for extracting human travel modes from mobile phone GPS trajectories, together with a geometry-based probability model, to support particle swarm optimization. A comparison study is implemented to analyze the model’s computational efficiency. Based on the resulting optimal layout for the network of bicycle docking stations, and considering demand uncertainty, a multi-scenario integer linear programming model is proposed to optimize rebalancing procedures (i.e., moving bicycles between docking stations according to demand), to determine the detailed design-scale information required. Mobile phone GPS trajectories from approximately 3.7 million local mobilities are used to construct a case study for Setagaya Ward, Tokyo. The results show that, compared with the previous methods, the optimal layout solved by the proposed method could reduce emissions by a further 6.4% and 4.4%. With an increase from 30 to 90 bicycle stations, the adoption of bicycle-sharing can reduce CO2 emissions by approximately 3.1–3.8 thousand tonnes. However, emission reduction will maximally decrease by 21.26% after offset by bicycles production and rebalancing-generated emission.

Improving the technology of operating electric locomotives using electric power storage device

The article discussed the questions of the railway transportation organization along the transport corridors passing through the territory of Russia using the innovative locomotives of new generation with the polygon’s structure of the transportation process’ organization. There was carried out an analysis of the railway car standing at the docking stations of the single-phase alternating and direct current. The results of simulation modeling of the transportation process on the railway sections, which have a docking station, are given. There are shown the positive effects of the perfection of the polygon’s exploitation technology of the locomotive fleet due to introduction of the double feed electric locomotives to the transport corridors, manifesting in the cargo delivery’s time reduction due to reduction of the ready train’s standing time at docking stations twice, in comparison with the time during exploitation of the only single-system electric locomotives, and also in increasing of the area’s speed of trains in the all polygon of the heavy trains’ turnover. The usage of the double feed electric locomotives in areas, which contains different kinds of the current, allows increasing the quality parameters of the traction electric rolling stock’s exploitation. There were considered the ways to increase the energy efficiency of the double feed electric locomotives, particularly by applying the electric energy storage unit for connection to the DC link of the electric locomotive’s power circuit. The economic assessment of exploitation of the two-systems electric locomotives was carried out.

Future UAV-Based ITS: A Comprehensive Scheduling Framework

Multi-rotor unmanned aerial vehicles (UAVs) have been recently recognized as one of the top emerging technologies to be utilized in various smart city domains such as intelligent transportation systems (ITS). They represent an innovative mean to complement existing technologies to surveil transportation network, control traffic, and monitor incidents. The UAVs usually operate for time-limited missions due to their limited battery capacity. Hence, they need to frequently return to their docking stations to recharge their batteries, which handicaps their mission coverage and performance. When designing a UAV-based ITS infrastructure, it is crucial to leverage the UAV fleet effectively. In this paper, a generic management framework of UAVs for ITS applications is developed. The problem of docking/charging station placement is first investigated to find optimized locations for a given number of stations to be installed by the ITS operator. To this end, two fundamental criteria are taken into account: i) the flying time required by the UAV to reach the mission/incident location and ii) the risk of battery failure during the UAV operation. The two algorithms, namely a penalized weighted k-means algorithm and the particle swarm optimization algorithm, are designed for this purpose. Once the docking stations are placed, a UAV scheduling program is formulated to optimally cover the pre-known missions while minimizing the total energy consumption of the fleet and respecting a coverage efficiency targeted by the ITS operator. The proposed proactive scheduling approach employs multiple UAVs in sequential and parallel ways to cover spatially and temporally distributed events in the geographical area of interest over a long period.