Control, and Optimization for Urban Mobility Anuradha Annaswamy - PowerPoint PPT Presentation

Socio-Technical Modeling, Control, and Optimization for Urban Mobility Anuradha Annaswamy Active-adaptive Control Laboratory Massachusetts Institute of Technology Joint Work with Thao Phan, Yue Guan, Eric Tseng, Eric Wingfield, Ling Zhu, and Crystal Wang Sponsor: Ford-MIT Alliance

Empowered Consumers + Urban Mobility ? Transactive Control ≥ Strategies Consumers Assets (Prices and Fees) Empowered Highway Occupancy Drivers/Riders Shuttle Occupancy Parking spaces Efficient Resource Utilization Taxis Example 1: Dynamic Toll-pricing for congestion reduction Example 2: Shared Mobility on Demand using Dynamic Routing and Pricing CNTS Workshop, July 8-9, 2019

EXAMPLE 1: DYNAMIC TOLL PRICING CNTS Workshop, July 8-9, 2019

Motivation: Alleviate Traffic Congestion Virginia London average speeds of 60 mph 33% reduction in inbound car Stockholm maintained traffic, 30% decrease in minutes of time spent in traffic dropped by 33% delay experienced (morning peak) and 50% (evening peak) Minneapolis San Diego Florida average speeds of 50 mph maintained 95% of the time, with drivers save up to 20 minutes avoiding 8.8 to 13.3% reduction in travel 85% driver satisfaction delay in the worst congestion times Varying toll prices aids Urban Mobility! CNTS Workshop, July 8-9, 2019

Empowered Consumers and Urban Mobility (MnPass, Minneapolis, MN) Transactive Control Varying Toll Price Empowered Drivers Congestion Dynamics Traffic Density CNTS Workshop, July 8-9, 2019

A Socio-Technical Model Driver Probability of evaluation acceptance Infrastructure D U (𝜈, λ, 𝜄) 𝑞 𝑏 response Driver Preference + Traffic Model Decision Making D U = 𝛽 ∗ 𝑢𝑗𝑛𝑓 𝑡𝑏𝑤𝑗𝑜𝑕𝑡 + 𝛾 ∗ 𝑄𝑠𝑗𝑑𝑓 + 𝛿 • Traffic model: Accumulator based 𝑞 𝑏 Low 1 • Utility function: Cost and time savings Probability of Risk averse • Probability of Acceptance – population acceptance model High 0 D U Value function 1 𝑞 𝒃 = 1 + 𝑓 −𝝇∆𝑽 CNTS Workshop, July 8-9, 2019

Toll-pricing controller: Nonlinear PI 𝑔𝑚𝑝𝑥 𝑝𝑣𝑢 dynamic toll lanes zero toll lanes 𝑔𝑚𝑝𝑥 𝑗𝑜 actual desired dynamic dynamic lane $$$ lane density density road Transactive driver behavior dynamics Controller • Logistic Function probability of consumer purcha 1 • Identify parameters X: 2.525 Y: 0.5126 0.5 • Use inverse nonlinearity in the price-controller 0 0 5 10 15 20 price CNTS Workshop, July 8-9, 2019

Response to High Input Flow High input flow is introduced in the middle of the operating period to test the systems’ ability to prevent congestion. Our model-based control (blue) is successful in keeping the HOT density low compared to MnPASS (red). Dynamic Toll Lane: PID Dynamic Toll Pricing in the Morning Peak 45 (Veh/mile/lane) 8 MnPASS Pricing Ford-MIT Pricing Critical density 7 40 6 35 5 Density Price 30 4 3 25 2 20 1 15 6 6.5 7 7.5 8 8.5 9 0 6 6.5 7 7.5 8 8.5 9 Time by hour Time by hour Dynamic Toll Pricing in the Morning Peak Dynamic Toll Pricing in the Morning Peak 1600 65 1550 60 1500 55 1450 Flow (cars/hr) 50 1400 Speed 1350 45 1300 40 1250 35 1200 1150 30 6 6.5 7 7.5 8 8.5 9 6 6.5 7 7.5 8 8.5 9 Time by hour Time by hour CNTS Workshop, July 8-9, 2019

EXAMPLE 2: SHARED MOBILITY ON DEMAND CNTS Workshop, July 8-9, 2019

A Shared Mobility on Demand (SMoDS) Solution 1. Request: passengers request shuttle rides with 2. Offer: the shuttle server distributes offers to specified pickup/drop-off locations, maximum passengers with ride details including pickup locations, distances willing to walk. walking distances, pickup times, drop-off locations, drop-off times, and prices. 4. Operate : the shuttle server sends out ride details to 3. Decide: passengers decide whether to accept or passengers. decline the offers. Leads to a Constrained Optimization Problem CNTS Workshop, July 8-9, 2019

Dynamic Routing Determine optimal sequence 𝑇 of routing points 𝑆 𝑻,𝑺 ∈𝑻 𝒈 ×𝑺 𝒈 𝑫(𝑻, 𝑺) min CNTS Workshop, July 8-9, 2019

Numerical Results (Dynamic Routing; all passengers accept the ride-offer ) new requests received (a) 1 st batch (b) 2 nd batch (c) Original route of the 1 nd batch Clustering pattern Clustering pattern 1 st batch: Before update new requests received new requests received 𝑫 = 𝑫 = , , , , , , 𝑫 = 𝑫 = 𝑫 = 𝑫 = , , 𝑫 = 𝑫 = , , 𝑫 = 𝑫 = , , 𝑫 = 𝑫 = 2 nd batch After update , , , 𝑫 = 𝑫 = , , , , , , , , 𝑫 = 𝑫 = , , , , 𝑫 = 𝑫 = , 𝑫 = 𝑫 = 𝑫 = 𝑫 = 𝑫 = 𝑫 = 𝑫 = 𝑫 = (e) Dynamic routing (d) Static routing CNTS Workshop, July 8-9, 2019

A Schematic of the SMoDS Solution Alternative Transportation Options Reference R travel times Dynamic Routing via Desired AltMin Algorithm Probability Passenger of Behavioral Model Acceptance 𝑡 𝑞 𝑆 𝑊(∙) and 𝜌(∙) 𝑞 ∗ tariff 𝛿 Dynamic Pricing via CPT 𝑆 𝑔 𝑌 (𝑦) tari 𝑡 : subjective probability of acceptance framed by 𝑆 𝑞 𝑆 CNTS Workshop, July 8-9, 2019 47

Conventional Utility Theory • Several alternatives with utilities 𝑉 𝑏 1 , …, 𝑉 𝑏 𝑜 • Corresponding probabilities 𝑞 1 , …, 𝑞 𝑜 𝑛 Utility function of ride-sharing 𝑘 𝑞 𝑗𝑘 𝑣 𝑗 = ෍ 𝑉 𝑏𝑗 𝑘=1 2 𝑢 𝑞 1 , 𝑢 𝑞 2 ] 𝑣 𝑗 = න 𝑉 𝑏 𝜐 𝑞 𝑗 𝜐 𝑒𝜐 𝑉 𝑏𝑙 = 𝑔 𝜐 , 𝜐 ∈ [𝑢 𝑞 1 𝑢 𝑞 𝑣 1 : Utility function of taking a private car; 𝑣 𝑜 : Utility function of taking a bus • Not adequate if uncertainty is large CNTS Workshop, July 8-9, 2019

Behavioral Dynamics of Human Beings: Prospect Theory In prospect theory*, the utility of the 𝑗 𝑢ℎ option • 𝑛 𝑘 )𝜌(𝑞 𝑗𝑘 ) 𝑣 𝑗 = ෍ 𝑊(𝑣 𝑗 𝑘=1 • Human beings are irrational in two ways: 𝑘 ) : loss aversion - losses hurt more than 1. How do we perceive utility 𝑊(𝑣 𝑗 the benefit of gains 2. How do we assess probability 𝜌(𝑞 𝑗𝑘 ) : overreact to small probability events and underreact to large probability events * Kahneman and Tversky, 1992 CNTS Workshop, July 8-9, 2019

Irrationality – Loss Aversion • Loss aversion: losses hurt more than gains feel good 𝛾 + 𝑣 𝑗𝑘 − 𝑆 if 𝑣 𝑗𝑘 > 𝑆 , 𝑘 ) = ቐ 𝑊(𝑣 𝑗 −𝜇 𝑆 − 𝑣 𝑗𝑘 𝛾 − if 𝑣 𝑗𝑘 < 𝑆 , • Framing effects: 𝑆 is the reference point of the framing, where people feel neutral, differentiate gain from loss (𝜇 > 1) • Example: it is better to not have a $5 loss than to gain $5. 𝒌 ) 𝑾(𝒗 𝒋 𝜸 + 𝒗 𝒋 𝒌 − 𝑺 El Rahi et al., Prospect 𝑺 Theory for Smart Grid, 2017. 𝒗 𝒋𝒌 −𝝁 𝑺 − 𝒗 𝒋 𝒌 𝜸 − CNTS Workshop, July 8-9, 2019

Irrationality – Overreact to Small Probability • Overreact to small probability events and underreact to large probability events 𝜌 𝑞 𝑗𝑘 = exp −(−𝑚𝑜𝑞 𝑗𝑘 ) 𝛽 , 𝛽 < 1 El Rahi et al., Prospect Theory for Smart Grid, 2017. • Example: people would not play a lottery with a 1% chance to win $100K and a 99% chance to lose $1K CNTS Workshop, July 8-9, 2019

Prospect Theory for Shared Mobility • The utility function is a combination of time and price: 𝑣 = 𝑏 + 𝑐 𝑞 𝑈 𝑥𝑏𝑚𝑙 + 𝑐 𝑥 𝑈 𝑥𝑏𝑗𝑢 + 𝑐 𝑠 𝑈 𝑠𝑗𝑒𝑓 + 𝛿𝜍 2 , 𝑣: 𝑣(𝜐) • 𝜐 ∈ 𝑢 𝑞 1 , 𝑢 𝑞 𝑆 ∞ 𝑊(𝑣) 𝑒 𝑊(𝑣) 𝑒 𝑡 = න 𝑉 𝑆 𝑒𝑣 𝜌 𝐺 𝑉 (𝑣) 𝑒𝑣 + න 𝑒𝑣 −𝜌 1 − 𝐺 𝑉 (𝑉) 𝑒𝑣 −∞ 𝑆 • 𝑆: reference 𝜐 𝑒𝑔(𝜐) - Cumulative Distribution Function (CDF) • 𝐺 𝜐 = ׬ −∞ – Extract from demand pattern and historical data – 𝐺 𝜐 exists but unknown Objective probability of acceptance Subjective probability of acceptance 𝑓 𝑉 𝑝 𝑡 𝑓 𝑉 𝑆 𝑞 𝑝 = 𝑡 = 𝑞 𝑆 𝑓 𝑉 𝑝 + 𝑓 𝐵 𝑝 𝑡 + 𝑓 𝐵 𝑆 𝑡 𝑓 𝑉 𝑆 𝑉 𝑝 and 𝐵 𝑝 : objective utility of the SMoDS 𝑡 and 𝐵 𝑆 𝑡 : subjective utility of the SMoDS 𝑉 𝑆 and the alternative and the alternative CNTS Workshop, July 8-9, 2019

Implication 1 – Fourfold Pattern of Risk Attitudes Example: Two outcomes, probabilities of 0.95,0.05 Fourfold pattern of risk attitudes (a) 𝑺 = + and 𝒈 = (c) 𝑺 = + and 𝒈 = a) Risk averse over high probability gains Gains ) − 𝑺 b) Risk seeking over high probability losses = 𝑽 − − (𝑽 𝑺 CPT Non-CPT c) Risk seeking over low probability gains CPT Non-CPT (b) 𝑺 = + and 𝒈 = (d) 𝑺 = + and 𝒈 = d) Risk averse over low probability losses Losses Conclusions: CPT Non-CPT CPT Non-CPT Tariff [$] Tariff [$] High Probability Low Probability Quantification of the qualitative • statements Truncated Poisson distribution with two outcomes 𝑦 + 𝑐𝛿 and 𝑦 + 𝑐𝛿 1. the presence of risk seeking passengers gives flexibility in • Relative Attractiveness increasing tariffs; RA = 𝑉 𝑝 − 𝐵 𝑝 − (𝑉 𝑆 𝑡 − 𝐵 𝑆 𝑡 ) 2. the presence of risk averse passengers requires additional constraints on tariffs. CNTS Workshop, July 8-9, 2019 47