In this chapter, we will develop two case studies of problems that involve the solution of linear simultaneous equations:

1. the solution of Kirchhoff's laws in a simple electrical circuit (Section 4.1), and

2. the search for a set of inputs to a “discrete-time linear system” that will bring the system to a desired state (Section 4.2).

The first case study will be developed in some detail, while the second will be much more briefly described. As we proceed, we will introduce notation to help us express ideas concisely and precisely. We will try to describe the choices that are made in formulating a model. You may already be very familiar with these models; however, the reasoning we present here may help you to pose and answer questions that arise in formulating your own models. We emphasize that the formulation of a new problem may involve many stages and refinements and that our presentation hides some of this process.

Analysis of a direct current linear circuit

Motivation

In designing a large integrated circuit to be fabricated or even a large circuit consisting of discrete components, it is very important to be able to calculate the behavior of the circuit without going to the expense of actually building a prototype. Furthermore, because of manufacturing tolerances, manufactured component values may differ from nominal. The values may also drift over the lifetime of the circuit. Both manufacturing tolerances and drift of component values can be interpreted as changes in the values of components from their nominal values in the base-case circuit.

In this chapter we will introduce six case studies:

1. production, at least-cost, of a commodity from machines that have minimum and maximum machine capacity constraints (Section 15.1),

2. optimal routing in a data communications network (Section 15.2),

3. least absolute value estimation (Section 15.3),

4. optimal margin pattern classification (Section 15.4),

5. choosing the widths of interconnects between latches and gates in integrated circuits (Section 15.5), and

6. the optimal power flow problem in electric power systems (Section 15.6).

The first and third case studies will draw from the previous formulations in Sections 12.1 and 9.1, respectively. The sixth case study combines the formulations from Sections 15.1 and 6.2. These three case studies will be introduced briefly, concentrating on the extensions from the previous formulations. They further illustrate the idea of incremental model development. The second, fourth, and fifth case studies introduce new material and will be developed in more detail. All six of these case studies will turn out to be optimization problems with both equality and inequality constraints. The first three have linear constraints, while the last three have non-linear constraints. Transformations will be applied to the fourth and fifth to deal with the non-linear constraints.

Least-cost production with capacity constraints

This case study generalizes the least-cost production case study from Section 12.1.

Motivation

Recall the least-cost production case study discussed in Section 12.1.

In this chapter we will introduce two case studies:

1. multi-variate linear regression (Section 9.1), and

2. state estimation in an electric power system (Section 9.2).

Both problems will turn out to be unconstrained optimization problems of the special class of least-squares data fitting problems [84, chapter 13].

Multi-variate linear regression

Some of this section is based on [103] and further details can be found there. The development assumes a background in probability. See, for example, [31, 103].

Motivation

In many applications, we have a hypothesized functional relationship between variables. That is, we believe that there are some dependent variables that vary according to some function of some independent variables. The simplest relationship that we can imagine is a linear or affine relationship between the variables.

For example, we may be trying to estimate the circuit parameters of a black-box circuit by measuring the relationship between currents and voltages at the terminals of the circuit. We will have to try several values of current and voltage to characterize the circuit parameters. As in the circuit case study of Section 4.1, we could either:

• apply vectors of current injections and measure voltages, interpreting the currents as the independent variables and the voltages as the dependent variables, or

• apply vectors of voltages and measure currents, interpreting the voltages as the independent variables and the currents as the dependent variables.

In this chapter, we will develop two case studies of problems that involve the solution of non-linear simultaneous equations. Both case studies will involve circuits, but the non-linearities will arise in different ways. The case studies are:

1. the solution of Kirchhoff's laws in a non-linear direct current (DC) circuit (Section 6.1), and

2. the solution of Kirchhoff's laws in a linear alternating current (AC) circuit where the variables of interest are not currents and voltages but instead are power (and “reactive power”) injections (Section 6.2).

The first case study will draw on the development from the linear circuit study described in Section 4.1 and we will not repeat in detail those issues that were already presented in Section 4.1. The second case study will be discussed in considerable detail.

The progression from the case study of Section 4.1 to the case studies of Sections 6.1 and 6.2 are examples of model extension and development. By developing the model incrementally, we can treat a few issues at a time without being overwhelmed by trying to analyze all the issues at once. In developing your own models, you might also build them step-wise, rather than all-at-once. Even an unrealistically simple initial model can provide valuable insights into more realistic cases. For example, most DC circuits in practice include at least some non-linear components. Nevertheless, formulation of the linear DC circuit case study in Section 4.1 has provided most of what we need to formulate a non-linear DC circuit case study.