Problem Formulation

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3.2.1 Mathematical model of AC networks with CPLs

This chapter deals with LTI AC electrical networks with CPLs working in sinusoidal steady state at a frequency ω0 ≥ 0; see Fig. 3.1. The description of this regime in




+ vsn



CP L1 +

v1 i1


+ vm

im ...

Figure 3.1: Schematic representation of multi-port AC LTI electrical networks with n external voltage and current sources feeding m CPLs.

the frequency domain is1

V(jω0) = G(jω0)I(jω0) +K(jω0), (3.1) where2 V ∈Cm and I ∈Cm are the vectors of generalized Fourier transforms of the port voltages and injected currents, respectively, and K ∈ Cm captures the effect of the AC sources, all evaluated at the frequency ω0. Equation (3.1) can also be seen as the Thevenin equivalent model of the m-port AC linear network including the voltage and current sources and, then, G∈Cm×m should be interpreted as the frequency domain impedance matrix of the network at the frequencyω0.

In virtue of Remark 2.4 in the previous chapter, the following expression holds.

G(jω) +GH(jω)>0, ∀ω ∈R; (3.2) the reader is reminded that (·)H denotes the complex conjugate transpose.

The complex power of the i-th CPL is defined as the complex number

Si :=Pi+jQi, i∈ {1, . . . m}, (3.3) where Pi ∈ R and Qi ∈ R are the active and reactive power at the i−th port, respectively. Then, the CPLs constraint the network through the nonlinear relation

Viconj(Ii) +Si = 0, (3.4)

where conj(·) is the complex conjugate. In equation (3.4), and throughout the rest of the chapter, the qualifier i∈ {1, . . . m} is omitted.

It is established then that a necessary and sufficient condition for the existence of a sinusoidal steady–state (at a given frequencyω0) is that the complex equations (3.1), (3.3) and (3.4) have a solution, which is the question addressed in this chapter.

1A comprehensive development of this description is followed in the chapter of preliminaries;

see, particularly, Assumption 2.1 and Remark 2.4.

2To simplify the notation the argument0is omitted in the sequel.

3.2.2 Compact representation and comparison with DC net-works

Notice that, eliminating the voltage vector V, the system of equations (3.1), (3.3) and (3.4) can be compactly represented by the set ofcomplex quadratic equations

fi(I) = 0, (3.5)

where the complex mappingsfi :Cm →C are defined as fi(I) :=IH eie>i G

I+IHKiei+Si, (3.6) where ei ∈ Rm is the i-th Euclidean basis vector. The fact that the mappings fi(·) have complex domain and co-domain represents a major technical difficulty to establish conditions for existence of solutions of equations (3.5), (3.6).

This situation should be contrasted with the case of DC networks where the mappings have real domain and co-domain. Indeed, there exists a constant steady state if and only if thereal quadratic equations ϑi(I) = 0 have a real solution, with the mappings ϑi :Rm →R given by

ϑi(I) :=I>[eie>i G(0)]I+I>siei+Pi, (3.7) where I ∈ Rm are the currents, G(0) ∈ Rm×m is the DC gain of the impedance matrix, and si ∈R are the external (voltage or current) DC sources; see [9].

3.2.3 Considered scenario and characterization of the loads

Finding conditions for the solvability of equations (3.5) and (3.6), for arbitrary Pi, Qi, G and K, is a nonlinear analysis daunting task. In practical scenarios, how-ever, the values of Pi and-or Qi are fixed and conditions on G and K, such that an equilibrium exists, are looked for.3 Interestingly, it is shown next that this makes the problem mathematically tractable. The powersPi and Qi for which an equilibrium exists are said to be admissible, otherwise, they are called inadmissible.

To state the problem formulation in a compact manner, define the vectors P := col(P1, . . . , Pm)∈Rm

Q := col(Q1, . . . , Qm)∈Rm S := col(S1, . . . , Sm)∈Cm,

which are assumedgiven. Then, the conditions under which P and-or Q belong to either one of the following sets are identified.

• PFAQI ⊂ Rm denotes the set of P that are inadmissible for all Q. That is, if P ∈ PFAQI there is no steady state no matter what Q is.

• QIFAP ⊂Rm is the set ofQthat are inadmissible for all P. That is, ifQ∈ QIFAP

there is no steady state no matter what P is.

• SI ⊂Rm ×Rm represents the set of (P, Q) that are inadmissible. That is, if (P, Q)∈ SI there is no steady state.

3Nominal values ofGandK are usually available too.

The first contribution is the definition of LMIs—parameterised in P and Q—

whose feasibilityimpliesthatP and-orQbelong to either one of the aforementioned sets.

A second contribution is that, for the case of one- or two-port networks, i.e., m≤2, a full characterization of the following sets is provided.

• PFSQA ⊂Rm set of P that are admissible for someQ. That is, if P ∈ PFSQA there is a Q (that can be computed) for which there is a steady state.

• QAFSP ⊂Rm set ofQthat are admissible for some P. That is, ifQ∈ PFSPA there is a P (that can be computed) for which there is a steady state.

• (For m = 1) SA ⊂ Rm ×Rm set of (P, Q) that are admissible. That is, if (P, Q)∈ SA there is a steady state.

By full characterization of the sets it is meant that

PFSQA ∪ PFAQI =Rm (form≤2) QAFSP∪ QIFAP =Rm (form≤2) SA∪ SI =Rm×Rm (form= 1).

In other words, that the conditions of admissibility for P or Q are necessary and sufficient.

From the practical viewpoint, the inadmissibility sets PFAQI and QIFAP allows to rule out “bad”P sandQs, respectively. The setSIis useful in the scenario when the devices at the ports transfer constant power with a specified power factor PFi= PSi

i, in which case S is fixed. Another scenario of practical interest where the set SI is instrumental is whenP is fixed (possibly some elements zero) and someQi are fixed and the others are free. The main question in this case is if some specific values of the freeQi can enlarge the set of admissible P.

Finally, for m ≤ 2, the sets PFSQA and QAFSP provide a complete answer to the admissibility question when P is fixed and Q is free and, vice versa, when Q is fixed and P is free, respectively. Additionally, for the special case m = 1, a full characterization of the setSA is provided. See Section 3.5 for an illustration of these scenarios in two numerical examples.

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