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Application to Some Canonical Power Systems

Dans le document The DART-Europe E-theses Portal (Page 59-63)

In this section, the results of Section 4.2 are applied to three different problems of power systems containing constant power loads. These comprise the standard volt-age stability (in a static sense) of AC power systems with “light” active power load and the study of existence of equilibria of MT-HVDC networks and DC microgrids

4.4.1 Voltage stability (in a static sense) of AC power sys-tems

The standard static analysis of voltage stability in AC power systems assumes the dynamics is in steady state, and concentrates its attention on the algebraic equations relating the active and reactive power, with the voltages and the phase angles—the well-known power flow equations. In [121] it was first suggested to investigate the sign of the real parts of the eigenvalues of the power flow Jacobian matrix as an indicator of voltage stability. This sensitivity analysis of the voltage magnitudes with respect to changes in the active and reactive power flows is the prevailing approach to analyze voltage stability in AC networks as explained in power systems textbooks,i.e., [56, Chapter 14] and [119, Chapter 7]. In this subsection it is shown how the analysis framework developed in Section 4.2 can be applied to carry out this analysis, for the particular case of power systems with “light” active power load.

Consider a high-voltage AC power network with n ≥ 1 PQ nodes. Denote by Vi > 0, δi and Qi the voltage magnitude, phase angle and the reactive power load demand at nodei, respectively. The analysis is restricted to scenarios in which the following standard “light” active power load (also called “decoupling”) assumption is satisfied [56, Chapter 14.3.3], [99, Assumption 1].

Assumption 4.3. δi−δj ≈0 for all i, j ∈ {1, . . . , n}.

With Assumption 4.3, the reactive power flow at nodeiis given by [56, 108, 99]8 QZIP,i=Vi

The reactive power demand QZIP,i at thei-th node is described by a, so-called, ZIP model, namely

QZIP,i := YiVi2+kiVi+Qi .

The term ZIP load refers to a parallel connection of a constant susceptanceYi ∈R, a constant reactive current ki ∈ R, and a constant power Qi ∈ R load. Then, the (algebraic) reactive power balance equation is obtained as

YiVi2+kiVi+Qi

8The reader is referred here to Chapter 2 for a detailed derivation of the power flow equations and particularly to the developments leading to equation (2.22) which appears by the end said chapter.

can be rewritten as (4.2). If the reasonable assumption thatYi <0 for at least one node is considered, then the matrix A satisfies Assumption 4.1.

Consider the following notion of voltage stability, called voltage regularity—also referred as static (or long-term) voltage stabilty—for the system (4.2) with the pa-rameters (4.14), which relates the analysis of Section 4.2 to standard power system practice, see [56, Chapter 14], [119, Chapter 7] and the more recent work [108].

Definition 4.3(cf. [65, 45]). A positive root ¯xof the system (4.2) is voltage-regular if the Jacobian ∇f(x)

x=¯x, with f(x) given in (4.3) with the parameters (4.14), is Hurwitz.

The following remarks concerning the application of the results of Section 4.2 to this particular problem are in order.

R1 The coefficients −bi are the constant reactive powers extracted or injected into the network, being positive (capacitive) in the former case, and negative (inductive) in the latter. As indicated in Section 4.2, sharper results—i.e., that ¯xmax is the only stable equilibrium, and a simpler structure of the set E of initial conditions for the distinguished solutions—are available if the signs of the coefficients bi are known. Hence, the proposed conditions have a direct interpretation in terms of reactive power demand.

R2 The solution ¯xmax of the system (4.13) represents the physically admissible steady state with the highest values of voltage magnitudes at each node, which is the usually desired high-voltage operating point.

R3 Lemma 4.5 in the Appendix implies that the Jacobian of the dynamics (4.3) evaluated at any stable equilibrium point is Hurwitz. Hence, if case V of Proposition 4.2 applies then the dominant equilibrium is voltage-regular in the sense of Definition 4.3. Consequently, Proposition 4.2 provides a constructive procedure to evaluate the existence of a unique dominant and voltage-regular solution in power systems with constant power loads.

4.4.2 Multi-terminal HVDC transmission networks with con-stant power devices

An MT-HVDC network with n power-controlled nodes (P-nodes) and s voltage-controlled nodes (V-nodes), interconnected by m resistive-inductive (RL) transmis-sion lines, can be modeled by [92]

τI˙t=−It−h(V),

LI˙=−RI +BP>V +B>VVV, CV˙ =It− BPI−GV,

(4.15)

where It ∈ Rn, V ∈ Kn+, I ∈ Rm and VV ∈ Rs. The matrices R, L, G, C, and τ are diagonal, positive definite of appropriate sizes. The physical meaning of each state variable and of every matrix of parameters is given in Table 4.1. Furthermore, B= stack (BV,BP)∈R(s+n)×m denotes the, appropriately split, node-edge incidence

Table 4.1: Nomenclature for the model (4.15).

State variables It P-nodes injected currents V P-nodes voltages

I Line currents

Parameters L Line inductances C P-nodes shunt capacitances R Line resistances G P-nodes shunt conductances

τ Converter time constants VV V-nodes voltages

matrix of the network.9 The open-loop current injection at the power terminals is described by

h(V) = stack Pi

Vi

,

where Pi ∈ R denotes the power setpoint.10 See also [132] for a systematic model procedure of HVDC systems using the port-Hamiltonian framework.

By simple algebraic computations, it can be shown that (4.15) admits an equi-librium if and only if the algebraic equations

0n =−h( ¯V)− BPR1BP>+GV¯ − BPR1B>VVV, (4.16) have real solutions for ¯V ∈ K+n. Notice that (4.16) is equivalent to the right hand side of (4.3) ifx:= ¯V and

A :=BPR1B>P +G, bi :=Pi, w :=−BPR1B>VVV.

Given thatBP is an incidence matrix,R and Gare diagonal positive definite matri-ces, then, the termBPR−1B>P is a Laplacian matrix and thus it is positive semidef-inite. Consequently, A = A> is positive definite and Assumption 4.1 is satisfied and the results of Section 4.2 can be used to analyze the existence of equilibria of the dynamical system (4.15). This, through the computation of the solutions of

˙

x=f(x), takingf as the right hand side of (4.16).

As in Remark R1 of the previous example, the coefficients −bi are the powers extracted or injected into the network, being negative in the former case and positive in the latter; the observation of Remark R2 is also applicable in this example.

4.4.3 DC microgrids with constant power loads

A model of a DC microgrid, with n ≥ 1 converter-based distributed generation units, interconnected by m ≥ 1 resistive-inductive (RL) transmission lines, can be

9Recall the preliminaries on graph theory and on power systems reported in Chapter 2.

10The first equation in (4.15) represents the simplified converter dynamics, see [92, Section II, equation (18)] and [92, Figure 4]. The converter usually has a PI current control, see the equations (27) and (28) of [92]. For simplicity, the analysis of this chapter is restricted to to study equilibria of the network without the PI. Nonetheless, the presented methodology applies also to the closed-loop scenario.

Table 4.2: Nomenclature for the model (4.17).

State variables

It Generated currents

V Load and bus voltages

I Line currents

Parameters

Lt Filter inductances

L Line inductances

C Shunt capacitances

Rt Filter resistances

R Line resistances

External variables

u Control input (converter voltage)

IZIP Yi: Constant impedance ki: Constant current Pi: Constant power

written as [26]

Ltt =−RtIt−V +u, CtV˙ =It+BI−IZIP(V),

LI˙ =−B>V −RI,

(4.17) where It ∈ Rn, V ∈ Kn+, u ∈ K+n and I ∈ Rm as well as Rt, R, Lt, L and Ct are diagonal, positive definite matrices of appropriate size; the physical meaning of each term appears in Table 4.2. Denote byB ∈Rn×m, the node-edge incidence matrix of the network. The load demand is described by a ZIP model,i.e.,

IZIP(V) = YV +k+ stack

Pi

Vi

,

where Y ∈ Rn×n is a diagonal positive semi-definite matrix, k ∈ Rn is a constant vector, andPi ∈R.

Some simple calculations show that, for a given u= ¯u constant, the dynamical system (4.17) admits a real steady state if and only if, the algebraic equations

0n=Rt1 u¯−V¯

− BR1B>V¯ −IZIP( ¯V), (4.18) have real solutions for ¯V ∈ Kn+. Definingx:= ¯V and

A:=R−1t +Y +BR−1B>, bi :=Pi, w :=R−1t u¯−k.

the system (4.18) can be written in the form (4.2). Similarly as for the MT-HVDC model, it can be shown thatA is a positive definite matrix and, hence, satisfies the conditions in Assumption 4.1. Therefore, the results of Section 4.2 can be applied to study the solutions of the steady state equation (4.18).

Once again, Remarks R1 and R2 of Subsection 4.4.1 are also applicable here.

Dans le document The DART-Europe E-theses Portal (Page 59-63)