**Modelling Metro Ridership Pattern and Passenger Flow**

## Build a Transportation Model using Material Balance and Flux Analysis

Delhi Metro Rail Corporation (DMRC) is the fifth largest metro network in the world. Kashmere Gate Station on the Yellow Line is the busiest intersections among all. **We have focused on designing a model (using Python) of the ridership pattern employing material balance phenomenon.** This includes information on how the ridership varied during the day (peak and lean hours), de-boarding weightage of the station and passenger accumulation at the station.

The model could be of great use to the Metro Corporation for optimizing the train frequency during different hours of the day, providing hassle-free commute by controlling the passenger flow.

# Model Variables

The system under consideration for our model was the junction of Kashmere Gate (KG), New Delhi, India. The station building, constitutes train tracks, shops and other facilities.

DMRC, runs rides for **18 hours** every day. **3000 people** can travel at one instance in an **8-coach train**. Metro train has a **30-second lodge time** in between stations. Peak-hour timing ranges from 8 am to 12 pm and 5 pm to 9 pm (**8 hours total**). Non-peak hour timing is equal to the daily metro timing minus the peak hour timing (18–8=**10** **hours**). Kashmere Gate, one of the busiest metro stations with a footfall of about **240,000 **. Therefore, the average hourly footfall at Kashmere Gate station (Yellow Line) is approximately **13,333** (daily footfall/daily operating hours=240,000/18).

Yellow Line serves **37 stations. **The stations on this line are classified into two categories — junctions (intersections) and non-junctions (regular) stations. A total of **8 junctions** (**Figure 2** depicts them in coloured boxes on the extreme right) are the part of this line and rest a part of the regular stations. Average train frequency of this line during peak and non-peak hours:

Peak hours= 02 min 47 sec (**167 sec**)

Non- peak hours= 3 min 16 sec (**196 sec**)

# Model Assumptions

(1) The peak and non-peak hours of the day were the points of focus and their combined average values were measured.

(2) Dwell time of 30 seconds on the platform was considered.

(3) Only 8-coach trains ply on the route with a passenger capacity (C )of 3000 with total passenger capacity.

(4) Yellow Line had 8 junction stations and 29 non-junction (regular) stations but the station where the train departed was excluded since no passenger gets down at that station (therefore only 28 non-junction stations were considered). The uniform number of passengers got off at each station type. In other words, no distinction within the junction and non-junction stations was observed. The peculiarity was made between the two station classes instead. Mathematically, say, if 0.5% of train capacity got off at all junction stations and the remaining 0.5% got off at non-junction stations, then weight assigned to each junction station (wj) would be 0.5/8 (8 junctions) while weight assigned to each non-junction (wn) 0.5/28 (28 non-junctions).

(5) At every station, there would be a net number of passengers alighting the train (passengers will get off the train while some will board it. It was assumed their net effect is equal to some people getting down the train at each station resulting in the train becoming empty at the last station).

(6) Two platforms for each line with trains plying in both +ve and –ve direction were considered.

**Model**

The total mass in any system is always conserved, states mass (material) balance theory:

The technical details about how we arrived at the final model equations can be understood by visiting the following link:

For looking at the model python code, visit the following link:

*Model Evaluation*

*Model Evaluation*

The above model indicating material balance on the junction was solved using Python. The values of weights and and ratios and were treated as model-tuning parameters. Each combination of the parameters was taken as a single parameter configuration. For example, parameter configuration 1-([0.1, 0.9], [0.1, 0.9]); parameter configuration 2-([0.1, 0.9], [0.2, 0.8]) and so on. The optimum parameters were obtained using a grid search methodology by minimizing the model error.

**Results**

The optimum ratios of total capacity of passengers getting down at different station class (junction and non-junction) were 0.85 and 0.15, respectively. This meant that 85% of the total passengers got down at all the junction stations, whereas the remaining 15% alighted at all the regular stations. The optimum values of weights, and were 0.10625 and 0.00536, respectively. These were the station class weights implying that each junction station had a passenger de-boarding weightage of 10.625%. It concluded that out of 85% of the total passengers, 10.625% passengers got down at each junction station. And each regular station had a passenger de-boarding weightage of 0.536% meaning that out of 15% of the total passengers, 0.536% passengers got down at each regular station.

The optimum values of ratios and were 0.95 and 0.05, respectively. Hence, 95% of the passengers getting off at Kashmere Gate walked out of the station while remaining 5% stayed back at the station for some reason, like waiting for a connecting train or eating at food stalls. The percentage error was 0.65% (minimum) for the parameter configuration, 322- ([0.85,0.15], [0.95,0.05]) (**Figure 4**).

Average hourly footfall for optimum parameters was 13,246. This was the average hourly footfall at Kashmere Gate computed by the model after tuning all the model parameters. Actual average hourly footfall was 13,333. Total daily footfall for optimum parameters was 2,38,428.

The total daily footfall at Kashmere Gate was computed by multiplying average hourly footfall by the total number of operational hours during the day. This value essentially indicated the total crowd that could be expected on an average, at the station every day. Average hourly footfall during peak hours and non-peak hours were 14,026 and 12,466. Clearly, there was a higher footfall at the station during peak hours as compared to non-peak hours. The rise in average hourly footfall from non-peak to peak hours was 12.51%. This meant the station witnessed 12.51% more passengers during peak hours than non-peak hours. Accumulation or the average footfall was calculated as 13,246 passengers/hour at Kashmere Gate. This value was close to the actual figure (as mentioned in the materials and methods section) of 13,333 hourly footfalls at the junction

Refer to the following link for deeper understanding of the model with a closer look at dynamic adaptability in different cases:

# Flux Analysis

Flux is** **described as any effect (such as magnetic, mass, electric, etc.) that happens (apparent or real) to flow or travel through a surface. In case of transport phenomena, flux explains the magnitude and direction of flow of a substance.** **The above study can be extended to estimate the diffusion of passengers using flux analysis by Fick’s First Law. It states that, ‘Molar flux due to diffusion is proportional to the conentration gradient.’ Fick’s First Law is as follows,

Concentration gradient is the driving force in the diffusion process. Diffusion coefficient is the relative ease by which particles diffuse through a membrane. Our assumption for future analyses will consider the train-platform interface as a membrane across which there will be diffusion of passengers in the direction orthogonal to the direction of the motion of train on Line 2 (Yellow Line). Hence, the passengers will diffuse in Y-direction (either from the train to platform or vice-versa) as the train moves in X-direction.

Note that the passengers will be motivated to diffuse (or move) through the membrane if the junction is their destination or starting point to another destination.

The ease with which passengers will diffuse across the membrane will be inversely proportional to the strength of the crowd (since it will hinder their motion) and directly proportional to the number of doors available on the train (since it will facilitate fast motion). Therefore, the flux can also be written as follows:

Thus, the flux analysis will probably provide the flow pattern of passengers across the junctions. Flow pattern based on the flux of passengers depends on the crowd strength, availability of doors in the train and likelihood of destination of passengers. Using this data DMRC can calculate the probable flow of passengers at the junction at any given time to direct the flow to reduce crowding at one particular region.

# Before You Go

*Thanks** for reading! Feel free to use this model for any train network. If you have any difficulty or any doubts kindly comment below. Your support is always highly appreciated. If you want to get in touch with me, reach me on jatin.kataria94@gmail.com.*