# On the Control of Transportation Networks with Delays

**Martin Heyden**, Department of Automatic Control, LTH

**Abstract:**

In this thesis, a general model for transportation on directed tree graphs is studied. The nodes in the graph correspond to different storage locations, and the edges describe between which storage locations transportation is possible. The transportation is assumed to be subject to delay. Furthermore, nodes at the top of the network are allowed to produce more of the studied quantity. As an example, this setup can model an irrigation network, consisting of several pools that are connected via gates. The gates allow water to be transported from the upstream to the downstream pool. Each pool can be described by a node, and the edges describe which pools are connected by a gate. The production corresponds to taking water out from a reservoir and into a pool.

A common approach for control of large-scale networks is to stabilize the system around the optimal equilibrium point. However, as the operating conditions of the network change, the optimal equilibrium point will also change. In this thesis, the dynamic performance of the network is optimized, where the cost associated with deviations from the nominal levels is minimized. The transportation variations are not penalized, as it is assumed that this cost is negligible (for example, in the case of irrigation networks, gravity is responsible for the movement).

The optimal controller is shown to be highly structured, without imposing any structural constraints on the controller that normally limit performance. This structure allows for a simple and efficient implementation. The optimal transportation assignments can be calculated by a sweep through the graph, starting in the nodes without children, and iterating upwards. This implies that each gate in an irrigation network only needs to receive information from the gates downstream and send information to the gates upstream.

Even stronger results are derived for string graphs. Firstly, it is shown how to give optimal feed-forward for planned disturbances. These planned disturbances could for example be farmers taking water out of an irrigation network. This requires minor modifications to the aforementioned controller structure, where the information about the planned disturbances can be communicated by a sweep through the graph. Secondly, it is shown how to allow for production in every node. This requires two sweeps, with one going in the upstream direction and one going in the downstream direction. These sweeps can be done in parallel, and thus the implementation time is unaffected. The resulting controller is applied to a more realistic simulation model for irrigation networks, where it outperforms a simple P controller in response to both step changes and disturbance rejection. For disturbance rejection of low-pass filtered disturbances, the performance is close to the theoretical maximum attained using a centralized controller with a perfect model.

The optimal control problem is also studied from a localized perspective, where each node tries to maximize its own utility. To coordinate, each node is presented with a price for having a certain level at each time point. It is shown how to calculate prices so that that the nodes’ optimal levels align with the socially optimal levels. These prices can also be calculated by a sweep through the graph.