In this section the proposed method is applied to study the stability of equilibria of single-port and multi-port DC networks with CPLs.
5.5.1 Single-port system
A schematic representation of a DC network with a single CPL is shown in Figure 5.1. Observe that the combination of the resistive load rp and the CPL, acts as a ZIP load connected to the capacitor C. In view of Remark 5.2, the current sink is omitted for brevity.
Define the state vector x = col(q, ϕ), where q is the capacitor’s electric charge and ϕis the inductor’s magnetic flux. Then the system can be modeled by
˙ which is well defined in the set
Ω+={(q, ϕ)∈R2|q >0}. Notice that the power balance equation takes the form
H˙ =− ∇HTR∇H
The equilibria of the system (5.18), (5.19) are computed in the following lemma.
Lemma 5.3. The system (5.18)-(5.19) admits two equilibria given by
The equilibrium points are real if and only if ∆≥0 or equivalently P ≤Pmaxe , Pmaxe := rpvg2
4r`(r`+rp) . (5.20) Through straightforward computations, it can be shown that the Jacobian of the vector field in the right hand side of (5.18) has a positive eigenvalue at the equilib-rium point (¯qu,ϕ¯u), and hence this equilibrium is unstable. Furthermore, it can be shown that for small values of the load power, the Jacobian is negative definite at the equilibrium point (¯qs,ϕ¯s).
Considering the results of Lemma 5.3, the equilibrium (¯qs,ϕ¯s) is proposed as the candidate for nonlinear stability analysis. To use the results of Corollary 5.1, compute Hence, according to Corollary 5.1, if the condition (5.22) is satisfied, then the equilibrium (¯qs,ϕ¯s) is asymptotically stable. Notice that if P < min{Pmaxe , Pmaxs }, then the existence of the asymptotically stable equilibrium point (¯qs,ϕ¯s) is guaran-teed.
Remark 5.4. Let ¯v := qC¯s denote the voltage of the CPL at the equilibrium (¯qs,ϕ¯s).
Then the physical equivalent of the term−Cq¯2sqP in (5.21), is the inverse of a nonlinear negative resistor with the resistance This resistor is in parallel with rp, i.e.,
R+Z(q, ϕ) =
is the equivalent resistor. Interestingly, the physical interpretation of the condition (5.22) is that the equivalent resistance is positive (and thus a passive element) at the equilibrium point, i.e., req(¯v)>0.
As the dissipation matrix R is diagonal, and the Hamiltonian of the states are decoupled, the largest k in (5.14)—and hence the largest Lk contained in Ωp—can be constructed explicitly.
To streamline the presentation of the result, consider the constants ηi :=− u¯i
RiiG¯ui, (5.23)
and the constant vectors
`i := col(¯x1,x¯2,· · · ,x¯i−1, ηi
Mii,x¯i+1,· · · ,x¯n). (5.24) Corollary 5.2. Consider the system (5.13) with the quadratic Hamiltonian (5.16) with diagonalR >0 and M>0. Assume that ¯x∈Ωp. Define
kd:= min
i=1,...,m
H(`i) ,
with (5.23) and (5.24). Then, an estimate of the ROA of the equilibrium ¯x is the sublevel setLkd of the shifted Hamiltonian function H(x) defined in (5.17).
Proof. The proof follows analogously to the proof of Theorem 5.2, and hence is
omitted.
Remark 5.5. Notice that the set Lk in the case considered in the corollary is the ellipsoid
Bearing in mind that the dissipation matrix R is diagonal, and using Lemma 5.2, Ωp is computed as Remark 5.6. The interpretation of (5.26) is that the closer the load power toPmaxs is, the smaller the ROA is. According to (5.22), this means that, with a fixed ¯qs, larger resistancesr` and rp result in a larger ROA. Hence, in the proposed method, small parasitic elements provide a small region of attraction in the absence of the load resistance and the voltage source output resistance.
Now assume that (5.22) holds. Using Corollary 5.2, the set Lkd with kd=H(qmin,ϕ¯s),
is an estimate of the ROA. Furthermore, using (5.25), this set can be written as the oval
Table 5.1: Simulation Parameters of the Single-Port CPL vg(V) r` (Ω) rp(Ω) L(µH) C(mF) P(kW)
24 0.04 0.1 78 2 1
Figure 5.2: Phase plane of the system (5.18)-(5.19) with the parameters given in Table 5.1.
This set guaranteesq > qmin for all solutions starting within the oval.
The results are evaluated by a numerical example of the network shown in Figure 5.1, with the parameters given in Table 5.1. The maximal power for existence of the equilibrium and its local stability is computed as Pmaxe = 2.57 kW and Pmaxs = 2.33 kW, respectively. Notice that the CPL satisfies the conditions (5.20) and (5.22), since
P < Pmaxs < Pmaxe .
Figure 5.2 shows the phase plane of the system (5.18)-(5.19). The estimate of the ROA (the oval (5.27)) is shown in blue, and all other converging solutions are shown in light gray. It is evident that the proposed method provides an appropriate estimate of the ROA, as the solutions just beneath this region (in dark gray), diverge from the equilibrium.
5.5.2 Multi-port networks
The stability of a complete multi-port DC network with CPLs can be investigated in a similar fashion. It is assumed that the capacitors and the inductors are not ideal, i.e., it is considered that all the inductors have a resistance in series and the capacitors posses a resistor in parallel. Moreover, it is assumed that the constant power loads are connected to a capacitor in parallel. This feature amounts to the capacitive effect of the input filters for this type of loads; see [10, 20, 22]. LetiM ∈Rl represent the currents through the inductors, and vC ∈ Rc denote the voltages across the capacitors, where l and c are the number of inductors and capacitors, respectively. With a little abuse of notation, define the state vector x= col(q, ϕ)∈ Rc+l and the control vector u=−P ∈Rc, whereq ∈Rc denotes the electric charge of the capacitors, andϕ∈Rl denotes the magnetic flux of the inductors. Then the network dynamics of a multi-port network admits a port-Hamiltonian representation
given by are positive definite matrices associated with the resistances, and Γ ∈ Rl×c is the matrix associated with the network topology. The vector uc∈Rc+l is constant and its components are linear combinations of the voltages and currents of the sources in the network. Similar to the case of the single-portRLC circuit with a CPL, and using Theorem 5.2, an ellipsoid can be computed here as an estimate of the ROA.