1. Lab 2: Feedback Control of Bus Operations
Henri Az´elart, Dirk Lauinger, Yann Martinson, Youssef Kitane
December 10, 2018
Abstract
We evaluate the impact of integral, proportional-integral and holding spacing control on the distri-
bution of simulated headways of buses in a single-line network with Kb buses and Ks stops. Further, we
suggest receding horizon control for centralized spacing control.
Introduction
Bus bunching occurs naturally because passenger flows will exacerbate small disturbances in bus arrival
times stemming from variations in passenger demand (related to dwell time) and cruising speed which may
be due to congestion or different driver behaviors. In this lab, we only consider the fluctuations in dwell
time and cruising speed without the positive feedback from passenger flows. Further, we assume that buses
can overtake each other.
1 I-Control
As expected, the integral control reduces the variation in headways and increases the mean headway. This
means that the buses are more equally spaced in time and that the interarrival between buses is greater as
compared to the case with no control. The greater interarrival time is due to the fact that buses always go
at maximum speed in the case with no control, whereas the integral control slows them down to equalize
their spacing.
Figure 1: Headway distributions of the no-control (blue) and I-control (red) case (with KI = 1)
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2. 2 PI-Control
Again as expected, the PI-Control is even more effective at reducing the dispersion of headways than the
I-Control. In addition, buses run faster than with the I-Control, but still slower as compared to the scenario
with no control. Fundamentally, the PI-Control reacts faster than the I-Control because it is not only sensitive
to the spacing error itself, but also to changes in the spacing error. It can be thought of as accomplishing two
objectives: (i) stabilize the spacing error (which further reduces its variance) and (ii) bringing the spacing
error to zero. Whether or not the PI or the I control yields a smaller mean headway depends on the choice
of the parameters KI, KP .
Figure 2: Headway distributions of the no-control (blue), the I-control (red) with KI = 1, and the PI-control
(yellow) with KP = 10.
3 Bus holding
We implemented a threshold holding controller that holds a bus at a station if the spacing error ei(t) is
smaller than a threshold KHth
, i.e. the bus is too close to the previous bus. Formally:
ei(t) ≤ KHth
=⇒ hold the bus from time tT to (t + 1)T
Figure 3: Headway distributions of the no-control (blue), the I-control (red) with KI = 1, the PI-control
(yellow) with KP = 10, and holding with Hth = −50 (purple).
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3. The holding control reduces the variance in headways although less so than the PI-control. It results in a
smaller mean headway than both the I- and the PI-control. As with all the control approaches, the trade-off
between bus speed and headway variance depends on the choice of the control parameters.
4 Coordinated Control
One way of coordinating the control of the buses in the network is to opt for a centralized receding horizon
approach such as model predictive control. The underlying principle is to gather information from all buses
at the same time, choose the speed of each individual bus so that an objective is optimized subject to some
constraints, communicate these decisions to each individual bus and repeat the procedure after a while.
The objective could be a trade-off between maximizing bus speed and minimizing the spacing error
between buses. The constraints may express the physical limitations on bus speeds.
Conclusion
We saw that I-, PI-, and holding-control are all effective at reducing the variation in the headways between
buses. For the tuning parameters we tried, holding control had the smallest mean headway. In general, all
control approaches trade-off bus speed (and thus a small mean headway) and headway variance. How this
trade-off is done depends on the tuning parameters of each controller. We should note the PI-Controller
is more expressive than the I-Controller. In fact, setting KP = 0 would make it work like an I-Controller.
In theory, the PI-Controller should thus be able to strike a better trade-off between mean headway and
headway variance. In fact, this is what we observe, the PI-controller has both a lower mean headway and
a lower variance than the I-controller. The theoretical comparison to the holding control is more difficult
because the holding control is bang-bang control (that is a control action is taken if the spacing error exceeds
a threshold) whereas I- and PI-Control are continuous control. Further, holding control only applies when
the bus is at a stop whereas I- and PI-Control only apply when the bus is moving.
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