7.10 Rate feedback compensation

At higher corner frequency values, Figure 7.25 shows that PID groups 2, 3, and 4 provide additional phase bias in the 3-8 rad/s modal frequency range. The example of detailed tuning of AVRs in the power plant with three remote generators is described in section 7.11.

Figure 7.20 Case W: Unit on-line. Bode plot of .

Proportional plus Integral Compensation

Simple PI Compensation

The frequency response of a simple PI compensator modified with a series shift block is shown in Figure 7.27. The PI with series lag compensator shown above can be converted to PID form.

Rate feedback compensation

Method of analysis
Tuning of the Excitation System (ES)
Rate feedback compensation using Frequency Response Methods
Rate feedback compensation using the Root Locus Method

Note that the top corner frequency is independent of the velocity feedback parameters, KF and TF. The desired shape of the open/closed loop system is shown in Figure 7-38.

Figure 7.28 Generator and Excitation System with field-voltage feedback compensation.

Tuning of AVRs with Type 2B PID compensation in a three- generator system

The three-generator, 132 kV power system

The characteristics of this generator brushless generator and power system are: (i) the lines are long with a surge impedance load (SIL) of 45 MW; (ii) at the rated output of the station, the lines are heavily loaded (about 1.7xSIL); (iii) with the interruption of a line the loading on the second. To ensure that the tuning of the PID covers a range of operating conditions, the 75 generation/operating conditions shown in Table 7.9 are examined.

Table 7.9 Power system operating conditions

The frequency response characteristics of the brushless exciter and genera- tor

These two responses are shown in Figure 7.43 along with the phase response of the open-loop transfer function. Based on PID parameter Set 4 in Table 7.6, let's determine the terminal voltage response of the closed-loop system to a +1% step change in the generator reference voltage.

Figure 7.42 Envelopes of frequency responses between the generator terminal voltage transducer and the exciter field voltage, V trn /V exf , for the feasible range of operating

Summary, Chapter 7

An emphasis in the studies was to determine a basis for evaluating a suitable set of PID 2B parameters to meet the dynamic performance specifications over a wide range of operating conditions. The purpose of new technique called the 'phase matching', explained in Appendix 7–I.5, is to improve the robustness of the generator controls to variations in the gain of the voltage control loop.

The analysis is based on an operating condition chosen to be the 'best' representation of those frequency response characteristics over the range of operating conditions. The studies using this technique have shown the importance of obtaining good models and parameters for both exciter and generator, preferably validated through tests.

Appendix 7–I

Depending on the position of the gain crossover in the range 0.7 - 2.5 rad/s, the phase margin is the difference between the open-loop phase response ( ) and. 2 produces excessive phase conduction and therefore the phase margin of the OLTF is greater than the desired value PMdes.

Figure 7.49 The brushless AC exciter is based on the AC8B Excitation System Model [12]

Types of Power System Stabilizers

Introduction

This chapter discusses the design of velocity signal synthesizing prefilters and highlights some issues that may be detrimental to the performance of the resulting prefilter and velocity PSS. When designing the PSS, attention should be paid to reducing the effects of the torsional modes of the turbogenerator unit on its dynamic performance [5].

Dynamic characteristics of washout filters

Time-domain responses
Frequency-domain responses

The responses of two identical washout filters in series to a step input of 1 unit are shown in Figure 8.2 for values of the washout time constant of 4 and 8 s. Consequently, the responses of two such filters to a ramp of 1 unit/s are shown in Figure 8.3 for a range of values of the washout time constant from 1 to 10 s.

Table 8.1 Analytical expressions for responses of washout filters to step and ramp inputs

Performance of a PSS with electric power as the stabilizing sig- nal

Transfer function and parameters of the electric power pre-filter
Dynamic performance of a speed-PSS with an electric power pre-filter

Note the minus sign of the PSS output signal on the summing connection to the AVR. Care must be taken to apply negative feedback from the PSS output signal to the AVR summing connection.

Figure 8.5 Structure of the PSS with electric power as the stabilizing signal. Note the negative sign of the PSS output signal at the summing junction to the AVR.

Performance of a PSS with bus-frequency as the stabilizing sig- nal

Dynamic performance of a speed-PSS with a bus-frequency pre-filter The bus-frequency pre-filter delivers a synthesized speed signal to a PSS whose design is
Degradation in damping with the bus-frequency pre-filter

The real parts of the mode shifts for the bus-frequency PSS are significantly degraded (by a factor of 1/c) with respect to the speed-PSS. The ratio of the true rotor speed to the bus frequency at the modal frequency is 1.57; this ratio agrees well with the value c= 1.59 in the table.

Figure 8.6 (a) Open-loop system . (b) Open-loop frequency response for Case A: (i) pre-filter and PSS transfer function given by (8.10) (the unstable mode in the open-loop system is )

Performance of the “Integral-of-accelerating-power” PSS

Introduction
Torsional modes introduced by the speed stabilizing signal
The Ramp Tacking Filter (RTF) The RTF is a low-pass filter of the form,

It should be noted that the RTF tracking function is turned off if it deviates noticeably. The value of the time constant T9 more or less determines the angular frequency of the RTF.

Figure 8.9 Frequency responses of the Ramp Tracking Filter for N = 1, T 9 = 0.1, , M= 5 or M = 4.

Conceptual explanation of the action of the pre-filter in the IAP PSS

Action of the pre-filter, no washout filters
Effect of the washout filters and integrators on the performance of the pre- filter
Dynamic performance of the complete pre-filter
Potential causes of degradation in performance of the pre-filter of the IAP PSS

RTF, but also how closely the RTF's output tracks the input to the RTF. Degradation of prefilter performance can be attributed to a number of causes.

Figure 8.11 The action of the pre-filter. (i) The implementation of (8.13) and (8.16)

The Multi-Band Power System Stabilizer

It is interesting to investigate the nature of the frequency response for MB-PSS omitting washout filters, velocity transducers and torsion (notch) filters. The frequency responses for three bands and the output of the MB-PSS are shown in Figure 8.24; they closely match Figures 5 and 6 in [16].

Figure 8.22 Multi-Band PSS. SD: Speed Transducer. (See [16], [17] for details).

Concluding remarks

The use of the latter signal as the stabilization signal for a 'true' speed PSS has been found to reduce the effective damping gain of the PSS (by as much as 40% in the cases studied). However, the attenuation gain of the PSS can be increased to compensate for the gain reduction.

Pons, "Extensive PSS use in large systems: the Argentinian case," in Power Engineering Society Summer Meeting, 1999. Beaulieu, "Practical experience with the use of power system stabilizers," in Power Engineering Society Summer Meeting, 1999.

Appendix 8–I

8–I.2 Steady states at the input and output of the RTF and associated tracking errors for mechanical input power RTF and associated tracking errors for mechanical input power. Since the Laplace transform of is , the Laplace transform of each of the input functions is simple.

Figure 8.26 Differential filter G df The input-output transfer function is

Basic Concepts in the Tuning of

PSSs in Multi-Machine Applications

Introduction

Eigenvalues and Modes of the system

The -element column vector, , is the right eigenvector of the matrix corresponding to the eigenvalue. In the case of the oscillating modes, assuming the mode is unique, the right and left eigenvectors of the complex conjugate eigenvalues are also complex conjugates.

Mode Shape Analysis

Example 1: Two-mass spring system
Example 2: Four-mass spring system

The form of the answers is again consistent with the results in Figure 9.2 and/or Table 9.2. The second mode shows the behavior of masses relative to a frame of reference.

Figure 9.1 (a) A two-mass system free to move in the x-direction on a flat surface, (b) the general form of the parameters and variables for the j th mass.

Participation Factors

Example 4.3

Similarly, the kth column represents the participation factors of the modes in the kth state. It can be confirmed that the sum of the participation factors covering the eight eigenvalues is

Table 9.5 Participation factors for the speed states in the system modes.

Determination of the PSS parameters based on the P-Vr approach with speed perturbations as the stabilizing signal

The P-Vr transfer function in the multi-machine environment
Transfer function of the PSS of generator i in a multi-machine system The basic concepts for the determination of the parameters of a PSS in a single machine sys-

For example, for PSS tuning of a generator in a multi-machine power system, phase information on the P-Vr transfer function is determined from field tests [10] or based on SMIB models with machine inertia. constant set to a very large value in the generator of interest [11], [12]. The damping gain (at the machine's MVA rating) of the PSS determines the extent of the shift to the left.

Figure 9.9 (a) Model of a generator in a multi-machine power system; (b) conceptually, with shaft dynamics on all machines disabled.

Synchronising and damping torque coefficients induced by PSS i on generator i

In other words, the damping gain ki per unit of the PSS is the realized one. This observation is illustrated in the SMIB cases by a comparison of the inherent coupling coefficients in Figures 5.5(a) and 5.18.

Figure 9.11 Model of generator i in a multi-machine system with (i) shaft dynamics on all machines disabled and (ii) switches S del and S PSS in rotor angle and PSS paths, respectively.

Zywno, “Using Power System Stabilizers to Improve Overall System Stability,” Power Systems, IEEE Transactions on, vol. Gibbard, “Robust design of power system stabilizers with fixed parameters over a wide range of operating conditions”, Power Systems, IEEE Transactions on, vol.

Application of the PSS Tuning Concepts to a Multi-Machine Power System

Introduction

The PSSs in a new power station must be set to meet the damping and other performance criteria of the system operators over the range of system operating conditions and contingencies. Normally, the main emphasis is placed on the dynamic performance of the multi-machine power system after large-signal disturbances.

A fourteen-generator model of a longitudinal power system

Power flow analysis
Dynamic performance criterion The dynamic performance criterion requires

Data for the power flow analysis of the six normal operating conditions given in Table 10.1 are provided in Appendix 10–I.2. Included in Appendix 10–I.2 are relevant results of the analysis, such as reactive outputs of generators and SVCs, together with tap positions on generator and network transformers.

Table 10.2 Generation conditions for six power flow cases.

Eigen-analysis, mode shapes and participation factors of the 14- generator system, no PSSs in service

Eigenvalues of the system with no PSSs in service
Application of Participation Factor and Mode Shape Analyses to Case 1 Consider in Figure 10.2 the unstable, oscillatory mode (designated ‘Mode L’)

Let's look at graphs of not only the magnitudes of its participation factors (PF), but also its mode shape (MS); the plots are shown in figure 10.3. In the participation factor plot 'W' and 'DEL' are the rotor speed and angle perturbations respectively.).

Figure 10.2 reveals that there are five unstable oscillatory modes, one stable oscillatory mode that does not satisfy the dynamic performance criterion and seven other lightly-damped os-cillatory modes with damping ratios less than 0.1.

The P-Vr characteristics of the generators and the associated synthesized characteristics

As detailed in Section 5.10.6.1, the 'best fit' characteristic for these studies is considered to be that which lies in the middle of the magnitude and phase bands formed by the P-Vr characteristics1. The more or less invariant nature of the phase responses of the P-Vr characteristics is also explained in Section 9.4.1.

The synthesized P-Vr and PSS transfer functions

Blocks that can accommodate complex poles and zeros are desirable in the PSS structure, as will be seen from the form of PSS transfer functions in (10.4) and (10.6) below. For illustrative purposes, the very short time constants (6.7 ms) of the low-pass filter are used here to minimize its influence in the range of modal frequencies.

Figure 10.22 Structure of the PSS for analysis and design purposes

Synchronising and damping torque coefficients induced by PSS i on generator i

For PSS-induced synchronizing and damping torque coefficients, the rotor angle path is open and the PSS path is closed (see Figure 10.23). The responses are shown in Figure 10.25 for two generators for the operating condition Case 1; coefficients are in units of generator rating.

Figure 10.23 Model of generator i fitted with a PSS in a multi-machine power system;

Dynamic performance of the system with PSSs in service

Assessment of dynamic performance based on eigen-analysis
Assessment of dynamic performance based on participation and mode- shape analysis

In Chapter 13, it will be shown that the smaller increments in the inter-area mode shift mode with PSS amplification are due to (i) smaller values of the participation factor of the generators participating in the mode, and (ii) the influence of the interaction between their PSSs [4]. A plot of the rotor modes for Case 1 with increasing gain is shown in Fig. 10.26(a).

Figure 10.26 Tracking of rotor modes for values of PSS damping gain 0 to 150% (30 pu)

Intra-station modes of rotor oscillation [6], [7]

Note that the PSS design procedure based on the P-Vr characteristic does not explicitly attempt to shift the intra-station modes directly to the left in the complex s-plane. The design of an auxiliary controller specifically for damping the intra-station modes is proposed in [8].

Table 10.12 Case 1: Heavy load. Intra-station modes for four units at SPA_4 and PPS_5.

Correlation between small-signal dynamic performance and that following a major disturbance

A transient stability study based on the fourteen-generator system
The analysis of modal interactions [10], [11], [12]

The peak amplitudes of the velocity perturbations in Figure 10.32(a) are 1.5 to 2%, which is not small. The second-order terms complement the information provided by the first-order linear approximation of the power system equations.

Table 10.14 Behaviour and type of the rotor modes, Case 1;

Summary: Tuning of PSSs based on the P-Vr approach

This is caused by the production of a positive or negative damping torque induced on generator i by the action of the PSS fitted to machine j [4]. Furthermore, we observe in Figure 10.26 that the mode shifts associated with the inter-area modes are smaller than those of the local modes.

The following Tables 10.15 and 10.16, together with Table 10.11, show the values of the modes of rotor oscillation for Cases 1 to 6 with the PSSs off and in service. The transformer tap ratios listed in Table 10.21 are based on the convention shown in Figure 10.35.

Table 10.15 Rotor modes and modes shifts for medium-heavy and peak loads, Cases 2 & 3

Tuning of FACTS Device Stabilizers

Introduction
A ‘simplistic’ tuning procedure for a SVC
Theoretical basis for the tuning of FACTS Device Stabilizers
Tuning SVC stabilizers using bus frequency as a stabilizing sig- nal

Use of bus frequency as a stabilizing signal for the SVC, BSVC_4

The residuals of the transfer function for the inter-area modes K, L and M are then calculated for the operating conditions 1 to 6. The residuals are from the SVC transfer function for the range of operating conditions, Cases 1 to 6.