Jun 6, 2017 - Trains in WMATA Metrorail system typically accelerate to ... segment from one stop to the next. ... segment speed trajectories are accepted when meeting target requirements of ... segment speed trajectories will serve as inputs to an energy ... tractive effort give the power demand of the train motor or engine.
An Algorithm to Simulate Segment Speed Trajectories of a Metrorail Segment for Energy Consumption Estimation A&WMA’s 111th Annual Conference & Exhibition Hartford, Connecticut June 25-28, 2018 Paper #411075 Weichang Yuan and H. Christopher Frey* Department of Civil, Construction and Environmental Engineering, North Carolina State University, Raleigh, NC 27695
ABSTRACT Energy consumption of metro trains is a function of motor power demand, weight, and time. Motor power demand is an indicator of motor load. Motor power demand can be estimated based on the product of tractive effort and speed of the trains. Tractive effort can be estimated by summing resistive forces on a train. Speed, acceleration, or both are needed to estimate resistive forces. Acceleration can be derived from speed. Thus, speed is a critical input for modeling energy consumption of metro trains. To estimate second-by-second energy consumption, second-bysecond speed trajectories are required. Speed trajectories can be measured using Global Positioning System (GPS) receivers. To estimate total train energy consumption of a rail system, measurements are required for each segment, which is defined as a one-way rail track between adjacent stations. Moreover, to obtain representative speed trajectories for each segment, multiple measurements for the same segment may be necessary. Such measurements are intensive and time-consuming. The loss of GPS signals in tunnels also raises doubts of the feasibility of such measurements. The purpose of this study is to demonstrate the algorithm of a segment speed trajectory simulator based on a relatively small sample size of measured speed trajectories. The algorithm is illustrated using the Washington Metropolitan Area Transit Authority (WMATA) Metrorail Red Line as an example. Speed trajectories on 27 selected aboveground segments were measured using GPS receivers. Trains in WMATA Metrorail system typically accelerate to cruising speed, maintain a typical cruising speed, and decelerate in a similar manner for a given segment from one stop to the next. The speed trajectory simulator aims to simulate segment speed trajectories following this typical pattern. A segment speed trajectory can be divided into microtrajectories in three operation modes (OMs), namely acceleration operation mode (AOM), cruise operation mode (COM), and deceleration operation mode (DOM). The speed trajectory simulator imputes micro-trajectories of the three OMs based on first-order Markov Chain transition process using the measured speed trajectories as inputs. Simulated segment speed trajectory for a segment was created by concatenating simulated micro-trajectories of the three OMs. The simulated segment speed trajectories are accepted when meeting target requirements of segment distance, average segment travel time, and segment average speed for the segment. Peak speeds of simulated segment speed trajectories are constrained to fall in a predicted region. The simulated 1
segment speed trajectories will serve as inputs to an energy consumption estimator for the WMATA Metrorail.
INTRODUCTION In the United States (U.S.) in 2016, transportation contributed 29% of the total energy consumption.1 The transportation sector in the U.S. primarily relies on fossil fuels.1 Consumption of fossil fuel leads to emissions of criteria air pollutants and greenhouse gases (GHG). Rail transport consumed 569.4 trillion Btu energy and emitted 43.6 million metric tons of carbon dioxide equivalent in 2015 in the United States.2 To reduce rail transport energy consumption, the energy consumption characteristics for each system, route, and passenger trip need to be quantified as a basis for identifying ways to improve energy efficiency. Quantifying and optimizing energy consumptions of passenger rail systems can be achieved at trip-based levels, which requires train speed trajectories.3–6 The product of train speed and train tractive effort give the power demand of the train motor or engine. Train speed trajectories also affect the work done by tractive effort. Specifically, to estimate the energy consumption of a train, resistive forces such as starting resistance, running resistance, acceleration resistance, curvature resistance, and grade resistance will need to be quantified.7–9 Second-by-second train speed trajectories are needed to estimate second-by-second train energy consumption due to tractive force. To quantify real-world train energy consumption, real-world train speed trajectories are needed. Typically, real-world train speed trajectories can be obtained via train control unit output or measurements using Global Positioning System (GPS) receivers. Train control unit is typically not accessible other than train staff. Boarding a train with GPS receivers is more practical. However, to estimate energy consumption for a rail system, measurements are required for each segment, which is defined as a one-way rail track between adjacent stations. To obtain representative speed trajectories for each segment, multiple measurements for the same segment may be necessary. Such measurements are intensive and time-consuming, thus not practical. For subway systems, the loss of GPS signals in tunnels also raises doubts of the feasibility of such measurements. Simulations of train speed trajectories using computer programs can be an alternative to real-world measurements. However, there are also concerns of the representativeness of simulated speed trajectories to real-world situations. To better represent real-world train speed trajectories for a specific train system, speed trajectory simulators need to be calibrated to measured speed trajectories of the target train system. Thus, the objective of this study is to demonstrate the algorithm of a segment speed trajectory simulator based on a relatively small sample size of measured speed trajectories. The algorithm is illustrated using the Washington Metropolitan Area Transit Authority (WMATA) Metrorail Red Line as an example. The simulation of speed trajectories of one segment (Rockville – Twinbrook) of the Red Line will be demonstrated.
APPROACH The approach includes data acquisition and processing, and speed trajectory simulation. Data acquisition and processing covers the input data to the speed trajectory simulator, including 2
distance and average travel time of the target segment (i.e. Rockville – Twinbrook) and empirical speed trajectories of selected aboveground segments. Speed trajectory simulation covers the algorithm to simulate multiple speed trajectories for the target segment based on input data.
Data acquisition and processing Segment distance, segment average travel time, and empirical speed trajectories were obtained from the District of Columbia Geographic Information System (DCGIS) service, WMATA realtime General Transit Feed Specification (GTFS-RT) service, and real-world measurements, respectively. These data were found to be accurate and precise in a previous study.10 Segment distance The shape files for Metro Stations and Lines of the WMATA Red Line were obtained via the DCGIS Service. These shape files were last modified on 6/6/2017 and 4/12/2017, respectively. The Metro Stations and Metro Lines shape files were imported into ArcMap 10.4.1 and projected on the Maryland State Plane Coordinates North American Datum of 1983 (NAD83) in feet. The station shape file gives stop locations. The line shape file was used to estimate the distances for the target segment. The Split Line at Point tool in ArcMap 10.4.1 was used to split line shape files into segments by station coordinates. The Calculate Geometry tool was used to calculate the distance of the target segment. Segment average travel time WMATA developed Application Programming Interfaces (APIs) for developers to acquire GTFSRT data for the Metrorail system. The “Train Positions” API was used to acquire live train positions. For WMATA Metrorail, real-time positions of trains were reported every 7 to 10 seconds. WMATA Metrorail system tracks are divided into blocks. Each block has a unique circuit ID. Train positions are reported based on the occupancy of specific track blocks. Total trip time between stations, dwell time, and travel time were estimated using GTFS-RT. The total trip time of each segment includes dwell time and travel time. Dwell time is defined as the time of a train idling at a station. The time of a train occupying the track blocks that represent stations is estimated as dwell time. The time of a train occupying the track blocks that represent non-stations is estimated as travel time. To estimate segment travel time, the timestamp of a train entering a current station was compared to the timestamp at which the train left a prior station. However, since the train occupancy of a track circuit ID includes the time during which the train is decelerating before stopping, accelerating after stopping, or both, dwell time is typically overestimated by GTFS-RT data. The overestimation was found to be 20 seconds on average.10 Bias in dwell time was corrected by subtraction of 20 seconds from GTFS-RT dwell time. The segment travel time was estimated by subtracting bias-corrected dwell time from travel time. The average travel time for the target segment was based on GTFS-RT data for 149 trips over 17 days (including 12 weekdays) from 7 a.m. to 11 a.m.10 Empirical segment speed trajectories A field trip was made to Washington DC during 12/13/2016 to 12/17/2016 to collect speed trajectory data by taking GPS receivers onboard. Due to GPS signal loss in tunnels, speed trajectories were collected only on selected aboveground segments as shown in Table 1. The GPS 3
Table 1. Summary of measured aboveground segments of WMATA Metrorail Red Line. Segment No. of Measurements Avg. GPS Distance* (mi) Rhode Island Ave – Brookland-CUA 3 0.87 Rhode Island Ave – NoMa Gallaudet U 3 1.06 Brookland-CUA – Fort Totten 3 1.36 Fort Totten – Brookland-CUA 3 1.36 Takoma – Silver Spring 3 1.41 Silver Spring – Takoma 1 1.43 Takoma – Fort Totten 3 1.9 Fort Totten – Takoma 3 1.91 Rockville – Twinbrook 3 2.01 Twinbrook – Rockville 3 2.01 Shady Grove – Rockville 3 2.72 Rockville – Shady Grove 3 3.56 TOTAL 34 21.60 *Average distances refer to the average distance of the segment measured by GPS receivers. receivers were placed on window seats to minimize signal loss. However, GPS-based speed trajectories have typical errors, such as sudden signal loss, extraneous or outlying data points, speed drifting, and signal white noise.11 These errors were corrected by applying cubic-spline interpolation and Savitzky-Golay filter.12 Savitzky-Golay filter is a digital filter that can be used for the purpose of smoothing digital data points. It is based on fitting successive ranges of adjacent data points with a low-degree polynomial by the method of linear least squares.
Speed trajectory simulation The trains in WMATA Metrorail system typically accelerate to cruising speed, maintain at cruising speed, and decelerate in a similar manner for a given segment from one stop to the next.12 The empirical speed trajectories can be divided into micro-trajectories that represent acceleration, cruise, and deceleration operation modes (i.e. AOM, COM, and DOM). The micro-trajectories can be rearranged to create multiple combinations of acceleration, cruise, and deceleration, which can be concatenated into segment speed trajectories. However, due to the small sample size of empirical speed trajectories, it is not feasible to create multiple combinations by simply rearranging the micro-trajectories. Trajectory data need to be simulated for all segments within the WMATA Metrorail system in future studies. Therefore, the simulator needs to simulate microtrajectories that correspond to a range of cruising speeds. The typical pattern of acceleration, cruise, and deceleration on segments between stations is the basis for the speed trajectory simulator. The speed trajectory simulator can be separated into three major parts, including operation mode analysis, micro-trajectory simulation, and segment trajectory simulation. Operation mode analysis categorizes segment speed trajectories into microtrajectories of three OMs. Micro-trajectory simulation allows simulation of micro-trajectories for 4
each OM. Segment trajectory simulation concatenates micro-trajectories of each OM into an approximation of the trajectory for the entire segment. Operation mode analysis The simulator is based on three OMs, namely AOM, COM, and DOM. AOM is defined as operation in which a train spend most of the time increasing speed. DOM is defined as operation in which a train spends most of the time decreasing speed. AOM allows for occasional minor speed reduction, which is sometimes observed in real-world acceleration speed traces, and DOM allows for occasional minor speed increase, which is also sometimes observed in real-world deceleration speed traces. In COM, speed is not truly a constant; however, speed in COM is not subjected to substantial change. The operation mode analysis is based on centered moving averaged values of second-by-second acceleration. The use of a moving average allows for occasional minor speed reduction in AOM and minor speed increase in DOM. Second-by-second acceleration is given by: 𝑎𝑖 = 𝑣𝑖+1 − 𝑣𝑖
(1)
Where, ai
= acceleration in i second;
i
= 0, 1, 2, 3, …, T, and T is the segment travel time in second;
vi
= speed in i second;
vi+1
= speed in i+1 second.
The centered moving averaged second-by-second accelerations are given by: 1
𝑎̅𝑖 = 7 ∑𝑖+3 ̅𝑗 , 𝑖 = 3, 4, … , 𝑇 − 4, 𝑇 − 3 𝑖−3 𝑎 { 𝑎̅𝑖 = 𝑎𝑖 , 𝑖 = 0, 1, 2, 𝑇 − 2, 𝑇 − 1, 𝑇
(2)
Where, 𝑎̅𝑖
= centered moving averaged acceleration in i second;
i
= 0, 1, 2, 3, …, T, and T is the segment travel time in second;
j
= from i-3 to i+3.
Based on the centered moving averaged accelerations, a segment speed trajectory is categorized into the three OMs according to the definitions in Table 2. The criteria acceleration values listed in Table 2 were chosen based on WMATA design criteria13 and sensitivity analysis.
5
Table 2. Definition of three operation modes based on 7-point moving averaged acceleration Operation Mode Acceleration Deceleration Cruise
̅𝒊 Centered moving averaged acceleration, 𝒂 𝑎̅𝑖 > 0.1 mph/s OR 𝑎̅𝑖−1> 0.05 mph/s and 𝑎̅𝑖 > 0.05 mph/s and 𝑎̅𝑖+1 > 0.05 mph/s 𝑎̅𝑖 < -0.1 mph/s OR 𝑎̅𝑖−1< -0.05 mph/s and 𝑎̅𝑖 < -0.05 mph/s and 𝑎̅𝑖+1 < -0.05 mph/s All the others
Micro-trajectory simulation The simulation of micro-trajectories in AOM, COM, and DOM are based on mimicking the firstorder Markov Chain transition process. The first-order Markov Chain transition process identifies transitions of combinations of speeds and accelerations based on second-by-second measured speed trajectories. These transitions are defined such that the resulting simulated trajectories maintain properties found in the original measured data regarding the sequence of speeds typical of each OM, while allowing for random variations in simulated trajectories. First-Order Markov Chain Transition Process Second-by-second speeds and accelerations are correlated in the measured speed trajectories. Markov Chain transitions were used to account for the correlations between adjacent seconds. At a given second, the combination of speed and acceleration is defined as an object. An object falls in a state, which is defined as a particular speed and acceleration range in the Marko Chain transition algorithm. For example, speed from 0 to 2 mph and acceleration from 0 to 0.2 mph/s can be regarded as a state; and speed from 0 to 2 mph and acceleration from 0.2 mph/s to 0.4 mph/s can be regarded as another state. The transition from on state to another is the transition from one speed-acceleration range to another speed-acceleration range. Such transitions can be simplified as the first-order Markov Chain transition process, which satisfies the Markov property that each state is dependent only on the previous state. Therefore, for example, the probability of transitioning from state x1 to state x2 is given by conditional probability: 𝑃(𝑥2|𝑥1) =
∑𝑇−1 𝑡=0 |𝑜𝑏𝑗𝑠(𝑥1 ,𝑡)∩𝑜𝑏𝑗𝑠(𝑥2 ,𝑡+1)|
(3)
∑𝑇−1 𝑡=0 |𝑜𝑏𝑗𝑠(𝑥1 ,𝑡)|
Where, Objs
= a function that returns the number of objects in a particular state at a particular time;
x1
= the current state;
x2
= the next state;
t
= time ranges from 1 to T-1.
6
Figure 1. Illustration of transition from one state to another. The black solid dots and black empty squares are used to symbolize samples in states. The black dots are transitions from state #1-3. The black empty squares are transitions form state #1-4.
Thus, Objs(x1,t) returns the number of objects in state x1 at time t, and Objs(x2,t+1) returns the number of objects in state x2 at time t+1. The numerator gives the total number of objects transit from state x1 to state x2 from time 0 to time T-1. That is, if there are p objects in state x1 in second 1, and there are q (q < p) objects in state x2 in second 2 that are from state x1, there are q transitions from state x1 to state x1 from second 1 to second 2. The number of transitions is summed at a second-by-second basis (i.e. from second 1 to 2, from second 2 to 3, etc.) to give the total number of objects transit from state x1 to state x2. The denominator gives the total number of objects at state x1 from time 0 to time T-1. Figure 1 illustrates the transition process from one state to another. At time t (t is from 0 to T, and T is the travel time), five objects are in state #1-3 and state #1-4, respectively. At time t+1, three samples transit from state #1-3 to state #2-3. Two samples transit from state #1-4 to state #2-3. The transition probability from state #1-3 to state #2-3 is 0.6. The transition probability from state #1-4 to state #2-3 is 0.4. Define “Bins” and “Cells” A target two-dimensional space which includes a speed-acceleration distribution envelope is defined. The space is partitioned with respect to speed of every 2 mph and acceleration of every 0.2 mph/s. The partitions of speed and acceleration are called “speed bins” and “acceleration bins”, respectively. The intersections of speed bins and acceleration bins are called “cells.” A cell can be regarded as a state, which is defined in the previous section. For example, 35 speed bins are divided for a speed range from 0 mph to 70 mph, and 55 acceleration bins are divided for an acceleration range from -5 mph/s to 6 mph/s. The 35 speed bins and 55 acceleration bins have 1,925 intersected cells. The partition of the two-dimensional space facilitates the construction of multiple Markov Chain processes to simulate micro-trajectories. The size of each cell is a constant, 7
determined by increments of speed and accelerations (i.e. 2 mph and 0.2 mph/s). The size of each cell was chosen so as to allow for at least 5 samples for most of the non-empty cells. Sensitivity analysis was conducted for different cell sizes. Larger cell sizes result in noisy simulated microtrajectories. Smaller cell sizes reduce variabilities regard to time, distance, and peak speed of simulated micro-trajectories. Simulated Micro-Trajectories The simulation of micro-trajectories consists of multiple Markov Chain transitions on a secondby-second basis. The Markov Chain transitions will be terminated when meeting stopping criteria. The flowchart of simulation of AOM micro-trajectory is shown in Figure 2. Initially, the speed and acceleration at time 0 are randomly chosen from 1 Hz data for speeds and accelerations at time 0 of the measured speed trajectories. The AOM speed at time 0 is defined as zero mph. However, acceleration at time 0 varies and about 85% of the initial accelerations are below 1 mph/s. All others are less than 2 mph/s. The speed and acceleration data for t0, (v0, a0), are binned to the corresponding cells based on the boundaries set for each cell. The speed at time 1 is given by: 𝑣1 = 𝑣0 + 𝑎0
(4)
Where, a0
= the acceleration at time 0;
v0
= the speed at time 0;
v1
= the speed at time 1.
With v1 calculated, the speed bin of v1 can be identified. Within the speed bin of v1, the cell at time 1 is chosen by creating a Markov Chain transition process. The chosen cell does not necessarily have the highest transitional probability; however, the cell is chosen with respect to its transition probability based on Equation 3. For example, at time 1, there are multiple cells that can be chosen, and cell j is one of them. Assuming the transition probability from the cell at time 1 to cell j is P, then the probability of cell j to be chosen is P. Within the chosen cell, an acceleration value is randomly chosen from the samples within the chosen cell, named as a1. For example, if cell j is chosen, and in cell j, there are multiple objects. Each object is a combination of speed and acceleration at a given second. The acceleration of each object within cell j is randomly chosen to be a1. Then speed at time 2 can be calculated in a similar way to v1, given by: 𝑣2 = 𝑣1 + 𝑎1
(5)
Where, a1
= the acceleration at time 1;
v1
= the speed at time 1; 8
Figure 2. Simulation flowchart of one micro-trajectory in AOM.
v2
= the speed at time 2.
The transition loop is repeated until the randomly chosen acceleration is less than or equal to zero or the chosen cell is empty. The stopping criteria of a complete micro-trajectory is given by: 𝑎𝑖+1 ≤ 0
(6)
Where, ai+1
= the acceleration at time i+1, i = 1, 2, …
For example, if the next chosen acceleration is less than or equal to 0 mph/s, the loop is terminated, and the resulted micro-trajectory is a complete micro-trajectory. For the situation of an empty cell, the simulation is “forced” to end rather than ended by meeting the acceleration criteria. Therefore, 9
when the empty cell situation happens, the simulated micro-trajectory is discarded. For example, if the target is 100 micro-trajectories, and 10 are rejected as incomplete, then 110 simulations are attempted of which 100 results in complete trajectories. DOM can be treated as a “reverse” of AOM. In AOM, speed increases from 0 to cruising speed, while in DOM, speed decreases from cruising speed to zero. Thus, for DOM micro-trajectory simulation, the algorithm is the “reverse” of AOM micro-trajectory simulation (as shown in Figure 3). Specifically, DOM simulations start from the speed at time T, the last second of DOM microtrajectories, and works backwards. Initially, the speed at time T is defined as zero mph. Acceleration at time T-1 is randomly chosen from 1 Hz data for speeds and accelerations at time T-1 of the measured speed trajectories. The speed at time T-1 is given by: 𝑣 𝑇−1 = 𝑣 𝑇 − 𝑎 𝑇−1
(7)
Where, aT-1
= the acceleration at time T;
vT-1
= the speed at time T - 1;
vT
= the speed at time T.
The speed and acceleration data for tT-1, (vT-1, aT-1), are binned to the corresponding cells based on the boundaries set for each cell. Then, a Markov Chain transition process is created to choose a former cell. Within the chosen cell, an acceleration at time T-2 is randomly chosen. The speed at time T-2 is given by: 𝑣 𝑇−2 = 𝑣 𝑇−1 − 𝑎 𝑇−2
(8)
Where, aT-2
= the acceleration at time T-2;
vT-2
= the speed at time T - 2;
vT-1
= the speed at time T - 1.
Similarly, the transition loop is repeated until the randomly chosen acceleration is greater than or equal to zero or the chosen cell is empty. The simulation algorithm of one micro-trajectory in COM is the same as AOM except for parameter values. Initial speed and acceleration are chosen from speeds and accelerations at time 0 of the measured COM micro-trajectories. First-order Markov Chain processes are created simulate transitions from current cell to the next. The transitions are terminated when the stopping criteria is met. The stopping criteria for COM simulations is when the next chosen acceleration is between -0.5 mph/s and 0.5 mph/s. The stopping criteria is chosen because more than 90% of the acceleration are between -0.5 mph/s and 0.5 mph/s based on measured COM micro-trajectories 10
Figure 3. Simulation flowchart of one micro-trajectory in DOM.
(will be shown in the result section). The differences of inputs, initial speed and acceleration, and stopping criteria of simulations in AOM, DOM, and COM are summarized in Table 3. Segment trajectory simulation A simulated segment speed trajectory consists of one simulated micro-trajectory of each OM. The simulated speed trajectory for a given segment is accepted if it conforms to a target mean average speed and satisfies multiple constraints, including AOM time and distance, DOM time and distance, and peak speed. The target average travel time for each segment is derived from GTFS-RT data. The segment distance is determined from DCGIS shape files for the rail tracks. Segment distance divided by average travel time of each segment give the mean segment average speeds. Target mean average speed for each segment is a fixed value for each segment. However, the target mean average speeds for all segment are not necessarily the same. 11
Table 3. Inputs, initial speed and acceleration, and stopping criteria for simulations of microtrajectories in three operation modes. Operation mode
Inputs
AOM
Measured microtrajectory in AOM
DOM
Measured microtrajectory in DOM
COM
Measured microtrajectory in COM
Initial speed & acceleration speeds and accelerations at time 0 of the measured speed trajectories in AOM speeds and accelerations in the last second (time T) of the measured speed trajectories in DOM speeds and accelerations at time 0 of the measured speed trajectories in COM
Stopping criteria ai + 1 ≤ 0 mph/s
ai - 1 ≥ 0 mph/s
|ai + 1|> 0.5 mph/s
Constraints were probabilistic based on 95 percent prediction intervals. Constraints regarding distance and travel time for AOM and DOM were developed based on linear regression models of measured speed trajectories. The segment peak speed is predicted based on analysis of the regression between peak speed and distance for the empirically observed segments. Unlike target mean average speeds, which are fixed values, constraints are 95% prediction intervals based on inputs of target segment distance or travel time. Specifying the constraint as a 95% prediction interval allows for simulation of real-world variability in time and distance in acceleration, time and distance in deceleration, and peak speeds. As an example, the calculation of 95% prediction interval of peak speed is given by: 1
̅)2 (𝐷−𝐷
𝑣̂𝑚𝑎𝑥,𝑖 ± 𝑡0.025,𝑛−1 √𝑀𝑆𝐸√1 + 𝑛 + ∑(𝐷 −𝐷̅)2
(10)
𝑖
Where, t0.025, n-1=
a value of the t-distribution with n-1 degrees of freedom;
MSE =
mean squared error, 𝑀𝑆𝐸 = 𝑛(𝑛−1) ∑𝑛1(𝑣̂𝑚𝑎𝑥,𝑖 − 𝑣̅𝑚𝑎𝑥 ) , 𝑣̅𝑚𝑎𝑥 is the average peak
1
2
speed, mph; D
=
target segment distance, mile;
𝑣̂𝑚𝑎𝑥,𝑖 =
point estimated peak speed using regression model, mph;
Di
=
segment distances of measured segment speed trajectories, i = 1, 2, …, n;
̅ 𝐷
=
mean of segment distances of measured segment speed trajectories.
Figure 4 shows a flowchart of the detailed algorithm to choose one simulated micro-trajectory of each operation mode to form a segment speed trajectory. The nomenclature used in Figure 4 are 12
summarized in Table 4. First, the algorithm reads the simulated micro-trajectories for the three OMs. Users input the target segment distances and travel time for a segment. Micro-trajectories that have values within the range of the prediction intervals are selected to create temporary libraries. A random selection process is used to select micro-trajectories of acceleration and deceleration modes from the temporary libraries. Based on the selected acceleration mode trajectory and deceleration mode trajectory, the distance, time duration, and average speed requirements in COM are calculated. However, the micro-trajectory in COM is not selected at this point. The selection of COM will not proceed until the random selection process of AOM and DOM micro-trajectories meet the following two criteria: 1. The maximum speed in the selected acceleration mode trajectory and the maximum speed in the selected deceleration mode trajectory have an absolute difference less than or equal to a critical value εMS ; and 2. The maximum speed in the selected acceleration mode trajectory and the average speed in cruise mode have an absolute difference less than or equal to a critical value εMS. The two criteria ensure the peak speeds in AOM and DOM will not deviate from the average speed in COM at a certain tolerance level. The selection of COM micro-trajectory is proceeded only after the two criteria are met. Unlike the selection of AOM and DOM micro-trajectories, the selection of COM micro-trajectory is not a random selection process, which reduces computational effort. Specifically, the selection of COM micro-trajectories happens with the distance and travel time in COM are known. The COM micro-trajectory is selected according to the above-mentioned COM requirements of distance, time duration and average speed. The selected AOM microtrajectory, COM micro-trajectory, and DOM micro-trajectory are concatenated to form a segment speed trajectory. The raw segment speed trajectory is then smoothed by a Savitzky-Golay filter to reduce noise. Seven-point range and third-degree polynomial were used for smoothing the raw segment speed trajectories. After smoothing, the distance and travel time of the smoothed segment speed trajectory can differ slightly from target segment distance and travel time. If the calculated distance and travel time meet the target distance and travel time with two significant figures, respectively, the smoothed speed trajectory is accepted. For this study, this process is repeated to obtain 100 accepted simulated speed trajectories for each segment.
13
Figure 4. Flowchart of the algorithm to create a segment speed trajectory from simulated sub-libraries of the three operation modes.
14
Table 4. Nomenclature used in the algorithm of Figure 4 Parameter Target segment distance AOM distance 95% PI COM distance 95% PI DOM distance 95% PI Target travel time AOM time 95% PI COM time 95% PI DOM time 95% PI Peak speed 95% PI Temporary library of AOM microtrajectories Temporary library of DOM microtrajectories The distance of an AOM trajectory The distance of a DOM trajectory The distance of a COM trajectory The time duration of an AOM trajectory The time duration of a DOM trajectory The time duration of a COM trajectory The maximum speed of an AOM trajectory The maximum speed of a DOM trajectory The average speed of a COM trajectory Absolute difference between va,max,i and vd,max,j Absolute difference between va,max,i and 𝑣̅ 𝑐,𝑘 Allowable difference in maximum speed among AOM, COM, and DOM Distance of the simulated speed trajectory Travel time of the simulated speed trajectory
Symbol D Da Dc Dd T Ta Tc Td vmax
Unit mile mile mile mile second second second second mph
index1
N/A
Index2
N/A
Da,i Dd,j Dc,k
mile mile mile
Attribute Constant Prediction interval Prediction interval Prediction interval Constant Prediction interval Prediction interval Prediction interval Prediction interval A set of acceleration mode trajectories A set of deceleration mode trajectories Constant Constant Constant
Ta,i
second
Constant
Td,j
second
Constant
Tc,k
second
Constant
va,max,i
mph
Constant
vd,max,j
mph
Constant
𝑣̅ 𝑐,𝑘
mph
Constant
Er1
mph
Constant
Er2
mph
Constant
εMS
mph
Constant
d
mile
Constant
t
second
Constant
15
Figure 5. An example operation mode analysis of a speed trajectory for the segment of Brookland – Fort Totten. Categorization based on (a) 1 second average, (b) 7 second centered moving average.
RESULTS AND DISCUSSION The results for operation mode analysis, micro-trajectory simulation, and segment speed trajectory simulation are discussed.
Operation mode Moving average method help conserve continuity in three operation modes by allowing for minor decrease of speed in AOM and increase of speed in DOM. Figure 5a and Figure 5b shows two examples of the operation mode analysis based on original accelerations (window size of 1 second) and centered moving averaged accelerations (window size 7 seconds), respectively. The main difference of the two categorization methods is the continuity of acceleration mode from about 20 s to 25 s (red circle). Since the train has not yet reached its cruising speed during 20s to 25s, the overall pattern of the train is accelerating, in spite of the slight reduction of speed. Thus, the moving average method can be used to reduce the discontinuity of AOM such that AOM can be represented by one micro-trajectory that begin at the start of the segment. A similar strategy is used in categorizing DOM. The operation mode analysis is applied to all measured segment trajectories of the WMATA Metrorail Red Line. Figures 6a, 6b, 6c, and 6d are the speed-acceleration distributions of three OMs, AOM, COM, and DOM, respectively. The three OMs cover a speed range from 0 to roughly 65 mph. Most acceleration values for AOM are positive. Some negative acceleration values in AOM are due to the allowance for minor speed reduction that can occur during an overall period of acceleration, as illustrated in Figure 4. Likewise, there are some positive acceleration values in DOM. About 65% of the COM speeds are greater than 40 mph. A train typically cruises at about 40 mph if the segment distance is less than a mile. However, in some cases, a train can cruise in
16
Figure 6. Categorization of measured segment speed trajectories in (a) three operation modes, (b) acceleration operation mode (AOM), (c) cruise operation mode (COM), and (d) deceleration operation mode (DOM) of the WMATA Metrorail Red Line. Each point represents a second-by-second speed-acceleration pair of 34 measured speed trajectories.
17
a lower speed range (< 40 mph) due to various factors (e.g. congested traffic conditions, curves, and crossing a bridge). A cruising speed below 5 mph is due to the train moving slowly when approaching a station, which does not happen frequently. The measured micro-trajectories of the three OMs are used as input into the speed trajectory simulator.
Simulated micro-trajectory Multiple simulations can be run to create libraries of simulated micro-trajectories for each OM. For example, the use of the method to simulate 10,000 trajectories is discussed here. Figure 7a-c show the distributions of speeds and accelerations of 10,000 accepted simulations in each operation mode. The envelope of the simulated speed and acceleration distributions is comparable to that of the measured ones in Figure 6b-d. The locations of the speed-acceleration clusters in Figure 6b-d were reproduced in the simulated results in Figure 7a-c. For example, speeds of 3 – 10 mph and 15 – 20 mph in Figure 6b typically have accelerations greater than 1 mph/s, which was reproduced by the micro-trajectory simulations, as shown in Figure 7a. Similarly, COM tends to happen when speed is greater than 30 mph, as shown in Figure 6c. In the low speed range, COM happens mostly between 10 mph and 20 mph. This pattern was also reproduced in the simulations, as shown in Figure 7b. Figure 6c shows most of the DOM falls in an acceleration range between -3 mph/s and 0 mph/s with few accelerations less than -3 mph/s, which was reproduced in the simulated DOM micro-trajectories.
Simulated segment trajectory One hundred simulated segment speed trajectories were simulated for the Red Line. Figure 8a and Figure 8b show an example of the segment from Rockville to Twinbrook. The 100 simulated segment trajectories generate variabilities in the three OMs while such variabilities encompass the measured example trajectory (Figure 8a). Figure 8b shows one of the one-hundred simulated segment trajectories. This simulated trajectory has similar peak speed as the measured one in Figure 8a but different acceleration and deceleration patterns. The AOM is less aggressive and the DOM is smoother in the simulated trajectory. Figure 9a-c show the cumulative frequency distributions of time, distance, and average speed in each OM and for the entire segment from Rockville to Twinbrook based on 100 simulated segment speed trajectories. This segment has a segment distance of 2.03 mi and mean average travel time of 172 seconds, which yields an average speed of 42.5 mph. The 100 accepted speed trajectories are of this same segment distance, travel time, and average speed because these three are constant target requirements for each segment. The time, distance, and average speed of the three OMs have variability because the constraints for these are specified as prediction intervals. The variations of time, distance, and average speed of the three OMs help in characterizing potential variations in energy consumption of this segment.
18
Figure 7. Speed and acceleration distributions of accepted simulations of (a) AOM microtrajectories, (b) COM micro-trajectories, and (c) DOM micro-trajectories. Sample size of each library is based on 10,000 accepted micro-trajectories.
19
Figure 8. Segment speed trajectories for Rockville to Twinbrook: (a) 100 simulated trajectories and one measured trajectory, and (b) one simulated segment trajectory.
20
Figure 9. Cumulative frequency distribution of (a) Time, (b) Distance, and (c) Average speed in each operation mode and the entire segment for the segment from Rockville to Twinbrook. Sample size = 100 simulated segment speed trajectories.
SUMMARY A speed trajectory simulator was developed for simulating multiple 1 Hz speed trajectories using the Rockville – Twinbrook segment of the WMATA Metrorail system Red Line as an example. The speed trajectory simulator requires inputs of segment distance, travel time, and empirical speed trajectory data. The simulator algorithm is based on trains following typical pattern of AOM, COM, and DOM on segments between stations. The operation mode analysis algorithm 21
was used to categorize speed trajectories into micro-trajectories of three OMs. A first-order Markov Chain process was used to enlarge the sample size of micro-trajectories and to impute micro-trajectories that correspond to a wider range of cruising speeds. Segment trajectories were concatenated using one micro-trajectory of each OM. The simulated segment trajectories conform to target mean average speeds and satisfy multiple constraints, including AOM times and distances, DOM times and distances, and peak speeds. The simulated speed trajectories will be used to the estimate energy use of each segment.
ACKNOWLEDGEMENTS The work presented herein was funded in part by the Advanced Research Projects Agency-Energy (ARPA-E), U.S. Department of Energy, under Award Number DE-AR33339-Z7218002. Tongchuan Wei assisted with collection of 1 Hz location data during the field trip. Disclaimer: The information, data, or work presented herein was funded in part by an agency of the United States Government. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.
REFERENCES 1. U.S. Energy Information Administration. Monthly Energy Review; Washington DC, 2018. 2. Stacy C. Davis; Susan E. Williams; Robert G. Boundy. Transportation Energy Data Book, 36th ed.; U.S. Department of Energy: Oak Ridge, 2017. 3. Miyatake, M.; Ko, H. Optimization of Train Speed Profile for Minimum Energy Consumption. IEEE J Trans. Electr. Electron. Eng. 2010, 5 (3), 263–269. 4. Su, S.; Li, X.; Tang, T.; Gao, Z. A Subway Train Timetable Optimization Approach Based on Energy-Efficient Operation Strategy. IEEE Trans. Intell. Transp. Syst. 2013, 14 (2), 883–893. 5. Gu, Q.; Tang, T.; Cao, F.; Song, Y. d. Energy-Efficient Train Operation in Urban Rail Transit Using Real-Time Traffic Information. IEEE Trans. Intell. Transp. Syst. 2014, 15 (3), 1216–1233. 6. Zhao, N.; Roberts, C.; Hillmansen, S.; Nicholson, G. A Multiple Train Trajectory Optimization to Minimize Energy Consumption and Delay. IEEE Trans. Intell. Transp. Syst. 2015, 16 (5), 2363–2372. 7. Hay, W. W. Railroad Engineering, vol. 1.; John Wiley & Sons, 1982. 22
8. Profillidis, V. A. Railway Management and Engineering; Ashgate Publishing, Ltd., 2014. 9. Mittal, R. K. . Energy Intensity of Intercity Passenger Rail; No. DOT/RSPD/DPB-5078/7Final Rpt., 1977. 10. Yuan, W.; Frey, H. C. Evaluation of General Transit Feed Specification Data for Electric Train Energy-Consumption Estimation. In 2018 Transportation Research Board Annual Conference; Washington DC, 2018. 11. Duran, A.; Earleywine, M. GPS Data Filtration Method for Drive Cycle Analysis Applications. In SAE 2012 World Congress & Exhibition; SAE International, 2012. 12. Yuan, W.; Frey, H. C.; Sun, Y. Quantification of Transit Train Activity Data for Energy Consumption Estimation. In Proceedings of the Air and Waste Management Association’s Annual Conference and Exhibition; Pittsburgh, 2017. 13. WMATA. WMATA Manual of Design Criteria for Maintaining and Continued Operation of Facilities and Systems; Washington DC, 2014.
KEYWORDS Speed trajectory, simulation, algorithm, train
23
