A  power  network  (nodes,  branches)  is  regulated  by  flow  equations  based  on  the  1^st  and  2^nd Kirchhoff Laws.

LAW 1: the net flow in a node of the network is zero:

The  network  topology  is  a  graph  that  may  be  described  by  a  branch-node  incidence  matrix  T (composed of elements with values -1, 1 or 0 only). Nodal injections are described by a vector L. The 1st Law may be translated into the matrix equation

LAW 2: the flow in a branch is proportional to the difference in potential P between its extreme nodes:

This may be globally translated into a matrix equation where B is a diagonal matrix:

F=B T^tP

The combination of the two Laws produces a well-known circuit equation

T B T^t P = L  or   Y P = L

where Y is sometimes called a nodal-admittance matrix and P is a vector of nodal potentials.

Question 1

Admit that in a network with n nodes and m branches, one has available k measurements, with k > n. These measurements may by on a mix of injections L, nodal potentials P and branch flows B.

Admit that these measurements are contaminated with noise. Therefore, the measurements do not form a set compatible with the circuit equation or the Kirchhoff Laws.

Admit that this noise is Gaussian, and independent for each measurement. Admit that the variance is any case is small.

One wishes therefore to find a set of Potentials Pˆ that would minimize some reasonable definition of an error between the measurement vector and the vector of values (F, L or P) that is compatible with the circuit equations.

Question 2

Admit that some of the measurement errors are gross errors (much larger than the errors admitted previously), and that it is unknown where such gross errors occur. These may severely contaminate the estimation of Pˆ.

Discover  which  measurements  contain  gross  errors  (instead  of  small  errors)  and  achieve  an estimation of Pˆ ignoring these gross errors.

Question 3

Admit now that there are switches scattered in the network branches. They can assume a state of open (S = 0) or closed (S = 1). An open switch interrupts the branch flow and eliminates this branch from the network (namely, from matrix T).

Admit that there are measuring devices that report each switch status.

Assume that, beside the k measurements of (F, L or P), some switch status signals are missing – so, the network topology becomes unknown.

The challenge is double: to guess correctly the network topology and thus to estimate Pˆ.

Problem 4: Modelling fiber flow in fiberboard manufacturing

Company: Sonae Industria - Produção e Comercialização de Derivados de Madeira, S.A
Company representative:
Abstract: Wooden chips and resin are the two main components in the manufacturing of medium-density fiberboard (MDF). One of the most important parts of the production process is the combination of these two materials into a fiber mat which is then pressed into boards. This process involves first applying high temperatures to the chips to refine them into fibers and then sending these fibers to flow into a blow-line together with water vapour, where they are combined with resin injected at different points (injectors). The company would like to understand better the behaviour of the fiber flow on the blow-line in what concerns speed, pressure and dead zones. This information about the process is then to be used for the optimization of the design of the blow-line to increase efficiency and reduce costs.

Problem 5: Modelling drying process in paper manufacturing

Company: Euroresinas - Indústrias Quimicas, S.A.
Company representative: Paulo Cruz
Abstract: Euroresinas produces different types of resin-impregnated paper at their factories. After the paper goes through the resin bath, it is lead through a drying process which is a fundamental step in the process. Although the temperature is controlled by a heating system, the actual values inside the chain of dryers are only measured at a few points away from where the paper is actually flowing. The company would like to be able to model the temperature profile inside the dryers to better understand problems such as dusting and sticking, as well as the optimization of different variables related to bathing and drying times and energy consumption.