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**^{t}P

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.