1 Iterative Quadractic Optimization for the Bus Holding Control Problem Luiz Alberto Koehler, Werner Kraus Jr., Member, IEEE, Eduardo Camponogara, Member, IEEE Abstract—A multiple control-point strategy for the holding control of a bus transit system is presented. The model developed is deterministic and assumes availability of real-time information and historical data of the system. Stochastic effects are distur- bances to be compensated by the feedback nature of the control. The objective is to minimize total user delay which is modeled by a non-convex cost function and non-linear constraints. In order to solve the problem efficiently, simplifications of the original model are introduced together with an iterative quadratic programming optimization procedure. A numerical example illustrates the application of the method indicating its feasibility for real-time applications and the good approximation of the global optimum provided by the heuristic solution. I. I NTRODUCTION BRT (Bus Rapid Transit) systems represent a viable alter- native and low-cost solution to the problem of urban public transportation when compared with other systems such as light rail [9], particularly in developing countries. In many cases, BRT operation comprises high-frequency, round-trip lines with headways of less than 10 min. In such contexts, a typical problem is the bunching phenomenon first described by [14], whereby small variations in stoppage time add up to the grouping of consecutive buses along a corridor. Osuna and Newell [15] concluded that the waiting time of passengers at stops decreases with increasing headway regularity and that the value is minimal for zero variation in relation to the nominal headway. These two conclusions are the basis for the headway control strategies. Among the various existing strategies, holding control stands out as the most used [3]. It consists of retaining the bus at specific control points for some time, seeking to reduce the headway variation between the buses. Because of its practical importance, many solutions have been proposed in the literature for the holding control. The reader is referred to [17] and [6] for a review of earlier contributions. Recently, there has been a renewed interest in headway control appearing in the literature. This may be due to the increasing adoption of BRT systems around the world and the consequent need for efficient operational-level management actions. Implicit in this development is the need for better models and algorithms than those previously reported. Del- gado et al. [6] formulate a capacity constrained model with L. A. Koehler is with the Department of Electrical and Telecommunication Engineering, Regional University of Blumenau, Santa Catarina State, Brazil. e-mail: {luiz}@furb.br. W. Kraus Jr. and E. Camponogara are with the Department of Automation and Systems Engineering, Federal University of Santa Catarina, Florian´ opolis, SC, Brazil. e-mail: {werner;fas;camponog}@das.ufsc.br. Manuscript received ... two main assumptions: the prevalence of boarding times over alighting and the ability of the bus driver to halt boarding when so instructed by the control system in order to handle capacity, which may not be always applicable. Still, the resulting formulation of the optimization problem can be solved with off-the-shelf packages and the outcome is very promising, with simulations showing a 22% improvement in travel times over the no control case; pure holding (with no capacity control) yields an 11% improvement. Saez et al. [4] formulate a holding and stop skipping control strategy as a multi-objective optimization solved with a genetic algorithm. This restricts the admissible holding times to be in the {0, 30, 60, 90} set (in seconds). Yang et al. [16] study holding control for one bus also using a genetic algorithm, leaving open the question of applicability of the method in the simultaneous control of multiple buses. Bellei and Gkoumas [1] adopt a Monte Carlo simulation approach in the comparison of threshold- based and information-based headway control in one, two and three arbitrarily selected stops, concluding that performance is improved with more control points. The question of model complexity is ever present in the above mentioned studies. Some works resort to Artificial In- telligence type heuristics (multi-agents, genetic algorithms) to solve the optimization problem with less restrictive hypothesis, while others make assumptions that simplify the models so that standard optimization tools can be applied in a reasonably fast calculation. A different approach is taken by Daganzo [5]. Instead of formulating a mathematical programming model for op- timization purposes, headway control is done via a simple proportional feedback controller whose reference is the ser- vice headway. Stability conditions in a stochastic framework are established analytically which provide many interesting insights into operational considerations such as the amount of slack that should be allocated and the corresponding advantage of dynamic headway control as opposed to schedule-based control. Since it is not an optimization procedure, design parameters (including the service headways that provide the reference for the control) are defined beforehand as in classic feedback control methods. This leaves open the question of how to vary reference headways dynamically in order to accommodate temporary and localized disturbances along the route in order to minimize passenger delay, as done by mathematical programming methods. The contribution of this paper is methodological. A work- able optimization method is presented that has low computa- tional complexity, thus allowing for real-time use. As stated previously, other works typically resort to simplifying assump-