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Fig. 13.4 Comparison of rotor speed and bus frequency measurements for the IEEE 14-bus system

Example 13.1 Bus Frequency Measurements for the IEEE 14-Bus System

Figure 13.4 shows the rotor machine speed as well as the bus 1 frequency computed as described above for the IEEE 14-bus system. The disturbance consists in line 2-4 outage for t = 1 s. In the simulation, Tf = 0.1 s is used. The low-pass filter of the bus measurement allows following the trend of the rotor speed but removes swing oscillations. As expected, the smaller Tω, the more accurate the measurement output.

Chapter 14

Loads

This chapter describes static and dynamic nonlinear loads. Since traditional loads used in power flow and transient analysis are constant PQ and shunt admittances, the loads described in this chapter are also called non-conforming loads. These are the voltage dependent load (Section 14.1), the ZIP load (Section 14.2), the frequency dependent load (Section 14.3), the exponential recovery load (Section 14.5), the thermostatically controlled load (Section 14.6), the Jimma’s load (Section 14.7), and the mixed load (Section 14.8).

Non-conforming loads are generally initialized after the power flow analysis. However, there is no particular di culty in including non-conforming loads in power flow equations. This possibility is taken into account in the following formulation of non-conforming loads. Moreover, according to the notation given in Chapter 9, load powers ph and qh are preceded by a minus because these powers are absorbed from the bus.

14.1Voltage Dependent Load

Voltage Dependent Loads (VDLs) are loads whose powers are monomial functions of the bus voltage magnitude, as follows:

where v0 is the initial voltage at the load bus as obtained by the power flow solution. Other parameters are defined in Table 14.1. Equations (14.1) can be directly included in the formulation of power flow equations. However, VDLs are generally initialized after the power flow analysis, and p0 and q0 are computed based on constant PQ load powers pL0 and qL0:

Example 14.1 Network PV Curves Considering Load

Characteristics

The standard CPF analysis provides an information about the existence of power flow solutions if considering constant PQ load models. Figure 14.2 shows the e ect of load characteristics for the determination of PV curves for the IEEE 14-bus system. The higher the exponents αp and αq , the higher the maximum loading level.

Fig. 14.2 PV curves using di erence load characteristics for the IEEE 14-bus system

14.2ZIP Load

Polynomial or ZIP loads are loads whose powers are a quadratic expression of the bus voltage:

where v0 is the initial voltage at the load bus as obtained by the power flow solution. Other parameters are defined in Table 14.2. If the ZIP load

is initialized after the power flow analysis, the parameters in (14.3) can be defined based on the PQ load powers pL0 and qL0:

In this case, a PQ load must be connected at the ZIP load bus.

Table 14.2 ZIP load parameters

14.3Frequency Dependent Load

A generalized exponential voltage frequency dependent load is modeled by the following set of DAE [130]:

where the frequency deviation Δω is approximated by filtering and di erentiating the bus voltage phase angle θ (see Figure 14.3). The parameters p0 and q0 are the initial active and reactive powers, respectively, computed after the power flow solution and based on the PQ load active and reactive powers pL0 and qL0 as defined in (14.2), and v0 and θ0 are the voltage magnitude and phase angle determined by the power flow analysis.

Table 14.3 defines the parameters of the frequency dependent load whereas Table 14.4 depicts some typical exponent values for characteristic loads [25].

14.4Voltage Dependent Load with Dynamic Tap Changer

Figure 14.4 depicts a simplified model of voltage dependent load with embedded dynamic tap changer.1 The transformer model consists of an ideal circuit with tap ratio m, hence the voltage on the secondary winding is vs = vh/m. The voltage control is obtained by means of a quasi-integral anti-windup regulator. All constant parameters are defined in Table 14.5.

1A more detailed model can be obtained using an ULTC (see Subsection 11.2.2 of Chapter 11) feeding a voltage dependent load (see Section 14.1).

Kd + Kis

−

vref

mmin

Fig. 14.4 Voltage dependent load with dynamic tap changer

Table 14.5 Load with dynamic tap changer parameters

The algebraic equations of the device are:

The reference voltage sign is negative due to the characteristic of the stable equilibrium point (see Example 11.3).

If voltage dependent loads with embedded dynamic tap changer are initialized after the power flow analysis, the powers p0 and q0 are computed based on the PQ load powers as in (14.2), and the state variable m and the voltage reference vref are initialized as follows:

where v0 is the bus voltage obtained by the power flow solution.

Example 14.2 E ect of Tap Changer Dynamics on Transient Analysis for the IEEE 14-Bus System

Figure 14.5 compares the results of the time domain integration for the IEEE 14-bus system using constant PQ loads, constant impedance loads and voltage dependent loads with embedded dynamic tap changer. Tap changer dependent loads are initialized after the power flow solution and have the following data: αp = αq = 2, Kd = 0 and Ki = 0.1 1/s/pu. When considering tapchanger embedded loads, the response of the system in the first seconds after the disturbance is similar to that of the system with constant impedance load models (see also Example 10.2 of Chapter 10). This is due to the slow response of tap changers. After some seconds, tap changers are able to regulate the voltage magnitude and the bus voltage trajectories approximate those of the system with constant PQ load models.

Fig. 14.5 E ect of tap changer dynamics in transient analysis for the IEEE 14-bus system

14.5Exponential Recovery Load

This section describes an exponential recovery load based on the model proposed in [128, 155]. Equations are:

−ph = xp/Tp + pt

where ps and pt are the static and transient real power absorptions, which depend on the load voltage:

The power flow solution and the PQ load data are used for determining the values of p0, q0 and v0. In particular, p0 and q0 are determined as in (14.2). A PQ load is required to initialize the exponential recovery load bus. All parameters are defined in Table 14.6.

Table 14.6 Exponential recovery load parameters