Problem Formulation

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

+

is1

v_s1

+ vsn

isn

...

CP L₁ +

v₁ i₁

CP Lm

+ vm

i_m ...

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 ∈C^m and I ∈C^m are the vectors of generalized Fourier transforms of the port voltages and injected currents, respectively, and K ∈ C^m 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∈C^m×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) +G^H(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

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

V_iconj(I_i) +S_i = 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 argumentjω0is 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

f_i(I) = 0, (3.5)

where the complex mappingsf_i :C^m →C are defined as f_i(I) :=I^H e_ie^>_i G

I+I^HK_ie_i+S_i, (3.6) where e_i ∈ R^m is the i-th Euclidean basis vector. The fact that the mappings f_i(·) 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 ϑ_i(I) = 0 have a real solution, with the mappings ϑ_i :R^m →R given by

ϑ_i(I) :=I^>[e_ie^>_i G(0)]I+I^>s_ie_i+P_i, (3.7) where I ∈ R^m are the currents, G(0) ∈ R^m×m is the DC gain of the impedance matrix, and s_i ∈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 P_i, Q_i, G and K, is a nonlinear analysis daunting task. In practical scenarios, how-ever, the values of P_i and-or Q_i 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 powersP_i and Q_i 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, . . . , P_m)∈R^m

Q := col(Q₁, . . . , Q_m)∈R^m S := col(S₁, . . . , S_m)∈C^m,

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

• PFAQ^I ⊂ R^m denotes the set of P that are inadmissible for all Q. That is, if P ∈ PFAQ^I there is no steady state no matter what Q is.

• Q^IFAP ⊂R^m is the set ofQthat are inadmissible for all P. That is, ifQ∈ Q^IFAP

there is no steady state no matter what P is.

• S^I ⊂R^m ×R^m represents the set of (P, Q) that are inadmissible. That is, if (P, Q)∈ S^I 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.

• PFSQ^A ⊂R^m set of P that are admissible for someQ. That is, if P ∈ PFSQ^A there is a Q (that can be computed) for which there is a steady state.

• Q^AFSP ⊂R^m set ofQthat are admissible for some P. That is, ifQ∈ PFSP^A there is a P (that can be computed) for which there is a steady state.

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

By full characterization of the sets it is meant that

PFSQ^A ∪ PFAQ^I =R^m (form≤2) Q^AFSP∪ Q^IFAP =R^m (form≤2) S^A∪ S^I =R^m×R^m (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 PFAQ^I and Q^IFAP allows to rule out “bad”P sandQs, respectively. The setS^Iis useful in the scenario when the devices at the ports transfer constant power with a specified power factor PF_i= ^P_Sⁱ

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

Finally, for m ≤ 2, the sets PFSQ^A and Q^AFSP 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 setS^A is provided. See Section 3.5 for an illustration of these scenarios in two numerical examples.