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Unformatted text preview: Time Space Diagrams Transportation System Capacity Dr. Antonio A. Trani Professor of Civil and Environmental Engineering Virginia Polytechnic Institute and State University Blacksburg Spring 2011 Virginia Tech 1 Why Time Space Diagrams? To estimate the following: • Headway between operations at various transportation facilities • Spacing between operations • Capacity of transportation systems • Determine basic level of service Virginia Tech 2 Sample Time Space Diagrams We examine time space diagram applications for the following systems: a) Rail (done in class) b) Automobile (done in class) c) Airport (air transportation) The rest of the handout applies time-space diagram principles to estimate the capacity of a single runway at an airport. Virginia Tech 3 Factors Affecting Runway Capacity There are numerous factors that affect runway capacity. Here are some of the most relevant: • Runway configuration (number of runways in use, location of runway exits, etc.) • Aircraft mix (percent of aircraft in various wake vortex categories) • Weather conditions (visibility, ceiling, wind direction and speed) • Airport equipage (type of navaids, ATC equipment) • Operating procedures (noise considerations, special approach and departure procedures) Virginia Tech 4 Sample Use of Technology to Use Multiple Runways • Radar surveillance is required at large airports to allow simultaneous use of parallel runways (shown in the Figure) Independent arrival streams Runway 1 Airport Terminal 4,300 ft. or more Runway 2 Virginia Tech 5 Independent Triple and Quadruple Approaches To Parallel Runways (IFR) • The idea behind this concept is to allow triple and quadruple parallel approaches to runways separated by 5,000 feet using standard radar systems (scan update rate of 4.8 seconds) at airports having field elevations of less than 1,000 feet • Increase to 5,300 ft. spacing between runways for elevations above 5,000 ft. p Runway 1 Runway 2 Runway 3 R 5,000 ft. or more 2 Virginia Tech 6 Independent Departures and Standard Radar • Simultaneous departures can be conducted if two parallel runways are located 2,500 ft. p Runway 1 2,500 ft. Runway 2 Virginia Tech 7 Independent Departures and Arrivals with Standard Airport Radar • Simultaneous departures and arrivals can be conducted if two parallel runways are located 2,500 ft. Departure Stream Runway 1 2,500 ft. Runway 2 Virginia Tech Arrival Stream 8 Time-Space Analysis • A simple technique to assess runway and airspace capacity if the headway between aircraft is known • The basic idea is to estimate an expected headway, E(h), and then estimate capacity as the inverse of the expected headway 1 Capacity = ----------E(h) E ( h ) is expressed in time units (e.g., seconds) Virginia Tech 9 Time-Space Analysis Nomenclature δ ij is the minimum separation matrix (miles). For this class we assume includes air traffic control buffers times. T i is the arrival time (to the runway) of the lead aircraft T j is the arrival time (to the runway) of the following aircraft T ij is the headway between two successive aircraft (s) δ is the minimum arrival-departure separation (miles) Virginia Tech 10 ROT i is the runway occupancy time for aircraft i (s) V i is the speed of aircraft i (lead aircraft) in miles per hour Virginia Tech 11 Time-Space Analysis Nomenclature V j is the trailing aircraft speed (miles per hour) γ is the common approach length (miles). This is the distance outside the runway where aircraft fly a common path aligned with the runway. Virginia Tech 12 Final Approach and Landing Processes Space Runway ROTi TDi ROTj Tj Ti V i γ Time V j Entry Gate Virginia Tech 13 Possible Outcomes of a Single Runway TimeSpace Diagram Since aircraft approaching a runway arrive in a random pattern we distinguish between two possible scenarios: • Closing case - Instance when the approach of the lead aircraft is less than that of the trailing aircraft ( V i ≤ V j ) • Opening Case - Instance when the approach speed of lead aircraft is higher than trailing aircraft ( V i > V j ) Virginia Tech 14 Closing Case (Equations) Headway ( T ij = T j – T i ) assuming control is exercised as the lead aircraft passes the entry gate (at a distance γ ) from the runway is, δ ij T ij = --Vj NOTE: the distance γ does not influence the outcome of this analysis because the following aircraft (fast) is “closing on” the lead vehicle (slow). Virginia Tech 15 Closing Case Diagram (Arrivals Only) Space Runway ROTi ROTj Ti Tj V i γ Time δij 1 V <V i j V j 1 Entry Gate Virginia Tech 16 Opening Case (Equations) Headway ( T ij = T j – T i ) is, δ 1- 1T ij = ---ij + γ --- – --- V j V i Vj assuming control is exercised as the lead aircraft passes the entry gate. NOTE: The second term in the previous equation measures the time aircraft (i) and (j) space themselves further over a distance γ . This term is important because Virginia Tech 17 Opening Case Diagram (Arrivals Only) Space Runway ROTi ROTj Ti Tj V j V i γ 1 1 Time V >V i j δij Entry Gate Virginia Tech 18 Mixed Operations (Arrivals/Departures) Space Runway TDi ROTi Ti T1 ROTj T2 Tj Time δ V i γ G V j T1 = Ti + RΟΤι T2 = T j - δ / V j Gap (G) exist if T2 - T1 > 0 Entry Gate TDi is the departure runway occupancy time Virginia Tech 19 Air Traffic Control Arrival-Arrival Wake Vortex Separations Near Runways Table 2. Typical In-trail Separations Near Runways in miles for Large Hub Airports (includes buffers added by air traffic control). Trailing Aircraft Lead Aircraft Heavy Large Small Heavy Large Small 5.00 3.00 3.00 6.00 3.00 3.00 7.00 5.00 3.00 Virginia Tech 20 Departure-Departure Aircraft Separations Table 2. Typical In-trail Departure Separations Near Runways in seconds. Trailing Aircraft LEAD ACFT. Heavy Large Small Heavy Large Small 90 60 60 120 60 60 120 60 60 Virginia Tech 21 Example Problem (1) Determine the saturation capacity of an airport serving two groups of aircraft: a) heavy (70% of the population) and b) small (30% of the population). Assume the common approach length γ to be 7 miles. The aircraft performance characteristics are given in the following table. Table 3. Aircraft Characterictics. Aircraft Group Runway Occupancy Time (seconds) Approach Speed (m.p.h.) Heavy 60 150 Small 40 70 Virginia Tech 22 Determine Aircraft Mix and Probabilities The following is a probability matrix establishing the chance that an aircraft of type (i) follows aircraft of type (j). We assume random arrivals. Table 4. Probability Matrix (Pij). Aircraft (i) follows aircraft (j). Trailing Aircraft Lead Aircraft Heavy Small Heavy =(0.7) x (0.7) = 0.49 = (0.7) x (0.3) = 0.21 Small = (0.3) x (0.7) = 0.21 = (0.3) x (0.3) = 0.09 NOTE: verify that ∑P ij = 1.0 i, j Virginia Tech 23 Compute Headways Between Successive Arrivals Closing case: Lead = small, Following = heavy aircraft TS – H δS – H 3 - = 0.02 hours = -------- = -------VH 150 Usually is convenient to express headway in seconds. 3δS – H T S – H = -------- = -------- 3600 = 72 seconds 150 VH Virginia Tech 24 Compute Headways Between Successive Arrivals Closing case (apply this case when speeds are equal): Lead = small, Following = small aircraft TS – S 3δS – S ----- 3600 = 154 seconds = ------- = 70 VS Lead = heavy, Following = heavy aircraft 5δH – H T H – H = -------- = -------- 3600 = 120 seconds 150 VH Virginia Tech 25 Compute Headways Between Successive Arrivals Opening case: Lead = heavy, Following = small aircraft TH – S δH – S 1- 1 ---- – ---- seconds = -------- + γ V S V H VS 711T H – S = ----- 3600 + 7 ----- – -------- 3600 = 552 70 70 150 seconds Virginia Tech 26 Arrival Aircraft Headway Table The following table summarizes the computed headways for all cases when an aircraft of type (i) follows aircraft of type (j). We assume random arrivals. Table 5. Headways (seconds) when aircraft (i) follows aircraft (j). Trailing Aircraft Lead Aircraft Heavy Small Heavy 120 552 Small 72 154 Virginia Tech 27 Compute Expected Value of Headway The expected value of the headway is: E ( T ij ) = ∑P T ij ij for all i,j pairs i, j E ( T ij ) = P H – H × T H – H + P S – H × T S – H + P H – S × T H – S + PS – S × TS – S E ( T ij ) = 0.49 ( 120 ) + 0.21 ( 72 ) + 0.21 ( 552 ) + 0.09 ( 154 ) E ( T ij ) = 203.7 seconds Virginia Tech 28 Compute Arrivals-Only Capacity • The capacity as the inverse of the expected headway 1Capacity = ------------- vehicles per second E ( T ij ) E ( T ij ) is expressed in time units (e.g., seconds) Using more standard units of capacity (aircraft per hour), 3600Capacity = ------------- vehicles per hour E ( T ij ) Virginia Tech 29 Arrivals-Only Capacity For the single runway example the arrivals-only capacity is, 3600 C arrivals = ------------ = 17.7 aircraft arrivals per hour 203.7 NOTE: this value is low for a busy airport. At busy airports small aircraft are generally handled at a different runway if possible to improve the capacity of a runway operated by heavy aircraft. Virginia Tech 30 Analysis of Runway Gaps Gaps can be studied for all four possible instances studied so far. For example, if a heavy aircraft is followed by a small one, there is a headway of 552 seconds between two successive arrivals. This leaves a large gap that be exploited by air traffic controllers to handle a few departures on the same runway. The gap for a heavy-small case is, δ G H – S = T 2 – T 1 = T S – ---- – ( T H + ROT H ) V S Virginia Tech 31 Gap Analysis Assume the arrival of the heavy aircraft occurs at time t=0 seconds. GH – S 2 552 – ----- 3600 – ( 0 + 60 ) = 70 G H – S = 389 seconds The expected time between successive departures at this airport is 83 seconds (see Table 2 adjusted by the probability values computed). A gap of 389 seconds is sufficient to “launch” four departures. You can do the Virginia Tech 32 same analysis for all other instances and estimate the departure capacity of the runway per hour. Gap: Lead aircraft = small, following aircraft = small GS – S 2 154 – ----- 3600 – ( 0 + 40 ) = 70 G S – S = 11 seconds One departure can be injected when a small aircraft follows another small aircraft. While 11.1 seconds is small gap, the fact is any gap > 0 will in theory result in one departure as long as the pilot responds quickly to ATC commands. Virginia Tech 33 Gap: Lead aircraft = small, following aircraft = heavy GS – H 2 72 – -------- 3600 – ( 0 + 40 ) = 150 G S – H = – 16 seconds No departures can be scheduled when a small aircraft follows a heavy aircraft. Virginia Tech 34 Gap: Lead aircraft = heavy, following aircraft = heavy GH – H 2 120 – -------- 3600 – ( 0 + 60 ) = 150 G H – H = 12 seconds One departure (on the average) can be scheduled between a heavy aircraft followed by another heavy aircraft. Virginia Tech 35 The analysis of gaps for four arrival instances is presented in Table 6. The number of departures per gap is also presented in Table 6. Table 6. Gaps (seconds) when aircraft (i) follows aircraft (j). Successive departures per gap are shown in parenthesis. Expected value of departure occupancy time is E(TDi) = 83 seconds). Trailing Aircraft Lead Aircraft Heavy Small Heavy 12 (1) 389 (4) Small -16 (0) 11 (1) Virginia Tech 36 Analysis of Arrival Gaps The final question that needs to be answered is: how many times each gap happens during the period of interest? From our analysis of arrivals only, we determined that on the average hour 17.7 arrivals could be processed at the runway. Since two successive arrivals are needed to form a gap, we can infer that around 16.7 gaps are present in one hour. The probabilty of each one of the four arrival instances is known and has been calculated in Table 4. Thus using these two pieces of information we estimate the number Virginia Tech 37 of times gaps will occur during one hour. Consider a heavy aircraft leading another heavy aircraft. Forty nine percent of the time this instance occurs at the airport. Thus for 16.7 gaps per hour this represents an equivalent number of hourly departures per arrival instance ( E D H – H ), ED H – H = TG ( P H – H ) ( DG H – H ) where: TG is the total number of gaps per hour, P H – H is the probablity that a heavy aircraft follows another heavy, and DG H – H is the number of departures per gap for each instance (numbers in parentheis in Table 6). Virginia Tech 38 ED H – H = 16.7 ( 0.49 ) ( 1 ) = 8.18 equivalent departures per hour Similarly, ED H – S = 16.7 ( 0.21 ) ( 4 ) = 14.03 ED S – H = 16.7 ( 0.21 ) ( 0 ) = 0 ED S – S = 16.7 ( 0.09 ) ( 1 ) = 1.50 equivalent departures per hour Virginia Tech 39 Departures with Arrival Priority Table 7 summarizes the number of departures per hour per instance. Table 7. Equivalent departures per hour per arrival instance when aircraft (i) follows aircraft (j). Trailing Aircraft Lead Aircraft Heavy Small Heavy 8.18 14.03 Small 0.00 1.50 Total departures per hour = 23.7 departures per hour Virginia Tech 40 Recapitulation of Results so Far C arrivals 3600 = 17.7 = -----------arrivals per hour 203.7 C departures = 23.7 departures per hour These results indicate that a single runway can process 17.7 arivals per hour and during the same period process 23.7 departures per hour using the gaps formed by the arrivals. Total operations = 41.4 aircraft per hour Virginia Tech 41 Final Note If only departures are processed at this runway (no arrivals), the departures only capacity is the reciprocal of the departure headway (83 seconds), 3600 C dep – NA = ----------- = 43.3 departures per hour 83 Airport engineers use a capacity diagram illusrated in the figure to display all three hourly capacity results in a single diagram. These diagrams represent a Pareto frontier of arrivals and departures. The airport can be operated inside the Pareto boundary. Virginia Tech 42 Arrivals per Hour Arrival-Departure Capacity Diagram 20 B (23.7,17.7) A (0,17.7) 10 0 C (43.3,0) 10 20 30 40 Departures per Hour Virginia Tech 43 Interpretation of Arrival-Departure Diagram • Line segement A-B represents a region where arrivals are given priority over departures. 17.7 arrivals per hour are processed and up to 23.7 departures per hour. • Line segment B-C represents a tradeoff region. Here we increase the separation between successive arrivals to allow more departures. In the limiting case (no arrivals), only departures and processed at a rate of 43.3 per hour. • Any operating point inside the Pareto frontier is feasible. Points outside the boundary encompassed by line segments A-B and B-C cannot be sustained for long periods of time. Virginia Tech 44 ...
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