Учебники / 0841558_16EA1_federico_milano_power_system_modelling_and_scripting

Fig. 15.5 Piece-wise saturation model

Polynomial Interpolation

This model consists in computing the coe cients of the polynomial that best interpolates three points of the saturation curve [272]. The three points are associated with ψ0.8 = 0.8, ψ1.0 = 1.0 and ψ1.2 = 1.2 pu, as shown in Figure 15.6. The saturation curve is assumed to be linear for ψ < 0.8. For ψ ≥ 0.8 one has:

The saturation factors s1 and s2 are computed as:

s2 = 1 − ia1.2 ib1.2

and, along with the slope for ψ < 0.8, completely define the saturation curve. The main issues of this model are (i) the point at which the polynomial is

Fig. 15.7 Generator rotor angles using a constant synchronous speed reference

Example 15.4 E ect of Using the Center of Inertia for the IEEE 14-Bus System

Figures 15.7 and 15.8 show the di erence between rotor angles referred to a constant synchronous speed and the same angles but using a COI speed reference. These figures represent a line 2-4 outage at t = 1 for the IEEE 14-bus system. At a first glance, the angle trajectories of Figure 15.7 could lead to think that the system is losing synchronism. Actually, the relative di erences among rotor angles remain bounded, thus the system is stable. This conclusion is straightforward if using the COI speed reference.

15.1.10Dynamic Shaft

A dynamic mass-spring model is used for defining the shaft of the synchronous machine. Figure 15.9 depicts the shaft scheme (springs are in solid gray). The complete set of di erential equations that describe the dynamic shaft is as follows:

Fig. 15.8 Generator rotor angles using a COI speed reference

ω˙ IP = (−DIP (ωIP − ωs) − D12 (ωIP − ωHP ) − D23 (ωIP − ωLP )

ω˙ EX = (−DEX (ωEX − ωs) − D45 (ωEX − ω)

+KEX (δ − δEX ))/2HEX

Example 15.5 Transient Behavior of Dynamics Shafts

Figure 15.10 shows the transient behavior of typical shaft rotor speed dynamics. The plot is obtained considering a dynamic shaft for the synchronous machine 1 of the IEEE 14-bus system. All shaft data are given in Appendix D.

15.1.11Sub-synchronous Resonance

Figure 15.11 depicts a generator with shaft dynamics and compensated line, which represents a simple model for studying the sub-synchronous resonance (SSR) problem. Shaft dynamics are the same as those described in previous Section 15.1.10 and are modeled as high, intermediate and low pressure turbine masses, exciter mass and machine rotor. The scheme of Figure 15.11 is one of the simplest models that may show the SSR phenomenon. SSR is a well-known problem of undamped oscillations that may occur when the transmission line to which the machine is connected is compensated by a series capacitor [140, 141, 142, 355].

The dynamics of the RLC circuit cannot be neglected since the line presents two modes whose frequency can be roughly estimated as Ωb(1 ±

xC /xL). For typical values of the inductive and capacitive reactances, the

Fig. 15.10 Dynamic shaft rotor speed dynamics

Fig. 15.11 Generator with dynamic shaft and compensated line

lower of these two frequencies can be close to one of the mechanical oscillations of the generator shaft. Thus, beyond a certain value of the compensation level, the machine may experiment a negative damping of one of the mechanical modes that results in dangerous stresses on the shaft. This phenomenon can be also described in terms of Hopf bifurcation [45, 205, 206].

A simple model used for studying SSR is presented in [361]. The machine is modelled through (15.9), (15.51) and

15.2Induction Machine

Induction machine models can be formally formulated using the Park’s approach. However, since induction machine rotors have no salient poles and since the rotor angular position is generally irrelevant, the Park’s two-reaction approach is not strictly necessary. This section describes three models of increasing complexity. These are pure mechanical model, single-cage rotor model, and double-cage rotor model. Each machine model includes a mechanical torque, which can thus be modeled separately and then included as a common ancestor class.

Table 15.5 defines the parameters of all induction machine models described in this section. Since a typical study related to induction machines is the start-up transient, the parameter list includes start-up parameters that control if and when the machine is started (i.e., sup and tup in Table 15.5). If the machine is marked for start-up, the slip is σ = 1 (e.g., rotor speed ω = 0) and the machine status is u = 0 for t ≤ tup.

Table 15.5 Induction machine parameters

15.2.1Initialization

As discussed in Chapter 4, the standard power flow problem is formulated describing loads as constant power consumptions. However, induction motors do not behave as constant power consumptions. Thus, if the machine dynamic equations are initialized after the power flow analysis, there will be certainly a data inconsistency. In fact if one uses the bus voltage v¯0 and the active power p0 to compute the machine slip and mechanical torque, then the reactive power consumed by the machine is assigned. Generally, this reactive power

is not equal to the one obtained as the solution of the power flow analysis. To solve this inconsistency, there are two possibilities:

1.Taking into account that induction motors are generally compensated, a shunt capacitor bank can be included to fix the reactive power mismatch at the machine bus.

2.The dynamic machine model can be directly included into the power flow problem. In this case the data to be imposed in the power flow analysis is the mechanical torque at the machine shaft. This method is certainly the most precise. Furthermore, if there is a capacitor bank, this can be modelled using its real capacity.

A detailed discussion on this topic can be found in [262].

15.2.2Torque Model

A typical model that can be used for the mechanical torque is a quadratic function of the rotor speed:

and given the relationship between the slip σ and the speed ω in pu, e.g. σ = 1 − ω, the torque/slip characteristic becomes:

In some cases, the previous model is not adequate, for example, it does not allow taking into account the machine duty-cycle. In this case, it is necessary to provide a function of time τm(t). Since accounting for any possible behavior is not possible, the better solution is likely to provide a series of (τm, t)-value pairs.

15.2.3Electromechanical Model

The electrical circuit for the first order induction motor is depicted in Figure 15.13. Only the mechanical state variable is considered, being the circuit in steady-state condition. The di erential equation is:

Fig. 15.13 Electrical circuit of the first-order induction machine model

whereas the power injections are:

The negative sign of the active and reactive power indicates that the machine is working as a motor. If the machine cannot work as a brake, then the di erential equation undergoes an anti-windup limiter that activates if σ ≤ 0 and σ˙ < 0 (see Appendix C).

15.2.4Detailed Single-Cage Model

The simplified electrical circuit for the single-cage induction motor is depicted in Figure 15.14. Equations are formulated in terms of the real (d-) and imaginary (q-) axes, with respect to the network reference angle. In a synchronously rotating reference frame, the link between the network and the stator machine voltages is as follows:

The di erential equations in terms of the voltage behind the the stator resistance rS are: