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14.6Thermostatically Controlled Load

This section describes a dynamic load with temperature control based on the model given in [130]. This device is initialized after the power flow solution and needs a PQ load connected at the same bus to properly initialize the state variables. The control diagram is depicted in Figure 14.6 that represents the following equations:

where the state variable x undergoes an anti-windup limiter and the algebraic variable G undergoes a windup limiter.

Fig. 14.6 Thermostatically controlled load

The power flow solution provides the initial voltage v0 and active power p0 that are used for determining the gain K1 and the maximum conductance gmax, as follows:

gmax = KLg0

322 14 Loads

where g0 = p0/v02 and KL (KL > 1) is the ceiling conductance output ratio. Finally, the initial load temperature is Θ0 = Θref and Table 14.7 defines all constant parameters required by this device.

Table 14.7 Thermostatically controlled load parameters

14.7Jimma’s Load

This section describes a load similar to a ZIP model except for the dependence of the reactive power on the time derivative of the bus voltage [152, 341]. This device is not included in the power flow analysis and thus requires a PQ load connected at the same bus to be properly initialized. Since in transient stability analysis bus voltages are not state variables, the time derivative is defined using an auxiliary state variable xv and a high-pass filter similar to the bus frequency measurement device described in Section 13.5 of Chapter 13 (see Figure 14.7). The di erential equation is:

and the power injections are defined as:

where the load parameters pz0, pi0, pp0, qz0, qi0 and qp0 are computed as in (14.4). The power flow analysis provides the initial voltage v0 that is needed for computing the Jimma’s load power injections. Table 14.8 defines the parameters of this device.

where p0 and q0 are computed based on the PQ load active and reactive powers pL0 and qL0 as defined in (14.2). The power flow solution provides the initial voltage v0 that is needed for computing the power injections. Table 14.9 defines all constant parameters of this devices.

Table 14.9 Mixed load parameters

Fig. 15.1 Synchronous machine scheme

reducing machine equations to static phasors if the machine rotor is also rotating at the synchronous speed. Less evident is the advantage of the Park’s transformation in case the rotor is not rotating at the synchronous speed. Nevertheless, the Park’s model is a standard de facto and is so widely used that machine data are generally available according to the axes of the Park’s transformation.

Following subsections are organized as follows. The complete set of machine parameters are defined in Subsection 15.1.1. Then the machine variable initialization procedure is outlined in Subsection 15.1.2. The remainder presents the machine equations. The material is organized taking into account implementation issues. Since some equations are common to all models, it is convenient to create a base class that includes such common equations and then to import the base class into specific machine models. No homopolar equations are given since the system and the machine are considered perfectly balanced.

15.1.1Synchronous Machine Parameters

Table 15.1 defines the complete set of synchronous machine parameters. Factors αp and αq are used in case of multiple generators connected to the same bus and indicate the fraction of active and reactive powers that each machine provides with respect to the total power produced by the static generator

defined in power flow analysis. The sum of these factors for the machines connected to the same bus has to be 1.

Table 15.1 Synchronous machine parameters

15.1.2Initialization

Dynamic models of synchronous machines are not included in standard power flow analysis. Thus a PQ, PV or a slack generator are required to impose the desired voltage and active power at the synchronous machine bus. Once the power flow solution is determined, v0, θ0, p0 and q0 at the generator bus are used for initializing the machine state variables, the field voltage vf and the mechanical torque τm. Example 9.2 of Chapter 7 describes the initialization of the two-axis synchronous machine model. The initialization procedure for other machine models is basically the same. The only di erences are:

1.Higher order machine models require the initialization of state variables associated with magnetic fluxes and or sub-transient emfs. These can be obtained directly by the algebraic equations (15.11) and any set of magnetic equations (assumed steady-state) provided in Subsection 15.1.5.

2.If more than one machine is connected to the same bus, the total injected powers p0 and q0 obtained by the power flow analysis have to be multiplied by the factors αp and αq , respectively.

3.In case of the classical machine model (i.e., emf behind the transient reactance), equation (9.11) has to be substituted for:

328 15 Alternate-Current Machines

In fact, for the classical model, xq is not defined.

15.1.3Common Equations

The equations common to all machine models are the interface with the network and mechanical di erential equations. The power injection ph and qh at bus h are:

whereas the link between the network quasi-static voltage phasor vh θh and machine voltages vd and vq are:

where δ is the machine rotor angle. Mechanical di erential equations are:

where Ωb is the base synchronous frequency in rad/s (314.16 rad/s at 50 Hz) and ωs is the reference frequency in pu. If the reference frequency is the synchronous one, then ωs = 1 pu. Some references define the machine starting time as M = 2H (or TM = 2H) and use this quantity in (15.5) instead of the inertia constant H.

Finally, one can define auxiliary equations for the input mechanical torque and field voltage. For the mechanical torque:

where τm0 and vf 0 are the initial values of the mechanical torque and the field voltage, respectively. As discussed in Example 9.1 of Chapter 9, τm and vf are auxiliary algebraic variables that allows easily interfacing the machine model with other devices such as turbine governors and automatic voltage regulators.

15.1.4Stator Electrical Equations

Stator electrical equations link the voltages to currents and magnetic fluxes, as follows:

˙q = Ωb(raiq − ωψd + vq )

ψ

While required for electro-magnetic transients, flux dynamics are relatively fast for transient stability studies. In fact, the inverse of the base frequency

Furthermore, considering that rotor speed deviations are small, one can assume ω ≈ 1 in (15.10). Hence:

The three models above, namely (15.9), (15.10) and (15.11), can be used indi erently. Thus, it is convenient to create a class for each electrical equation model and then to form the complete machine model by importing the required electrical equation class. Clearly, using model (15.9) makes sense only in very detailed analyses that require a precise formulation of electromagnetic dynamics. The most common choice adopted by most power system books is (15.11), which also allows removing the variables ψd and ψq from the machine model.

15.1.5Magnetic Equations

As discussed in the introduction of this section, the transfer functions that link stator fluxes with stator currents and the field voltage provide a certain degree of arbitrariness in the synchronous machine model. Most complete models introduce one state variable per rotor winding, real or equivalent. Thus for the machine depicted in Figure 15.1, four state variables and their associated di erential equations are required for most detailed models. Finally, two algebraic equations allow defining stator fluxes as functions of the stator currents, the field voltage and the rotor state variables. Simplified models consist in downgrading one or more rotor state variables to algebraic ones.

The dynamic response of damper windings is faster than that of the dc field winding and of the rotor-core induced currents. The standard notation

defines sub-transient (indicated by a double superscript ) the fast dynamics of damper windings and transient (indicated by a single superscript ) the dynamics of the dc field winding and of the rotor-core induced currents. Finally, steady-state quantities are said synchronous and are indicated without superscripts.

Sauer-Pai’s Model

The Sauer-Pai’s model is as follows [269]:

Finally, the following algebraic equations complete the model:

Marconato’s Model

The complete d- and q-axis diagrams of the Marconato’s model are depicted in Figure 15.2. The di erential equations are [184]: