SVC Control Components and Models
4.2 MEASUREMENT SYSTEMS
4.2.2 The Demodulation Effect of SVC Voltage-Measurement System The SVC voltage-measurement systems have an inherent demodulation effect
98 SVC CONTROL COMPONENTS AND MODELS
independent phase control is employed using a phase-locked oscillator in either a 6-pulse or 12-pulse scheme and when the SVC capacitors are resonant with the ac system near the 3rd harmonic. The measurement system in this case comprises a multiplier circuit instead of a rectifier circuit. For a normal bus voltage off2V1sin 2pf0t, the output of the multiplier is 2V21sin2 2pf0t, which is equivalent to
2V21sin2 2pf0tcV21(1−cos 4pf0t) (4.6) If a phase-locked loop (PLL) voltage regulator (see Section 4.3.2) is used, then the PLL will ignore the 2nd harmonic components and the controller will respond toV21. Should there be a 3rd harmonic component in the bus voltage, the dc output of the measurement system (multiplier) will be proportional to (V21+V23) in each phase. Thus, this feedback signal is unaffected by the phases of 3rd harmonic components relative to the fundamental-frequency signals. The three different TCR phases respond symmetrically to the total rms voltage and do not augment the preexisting 3rd harmonic currents in the system, therefore obviating 3rd harmonic instability.
With squared voltage, or (voltage)2, feedback, the reference voltage input could also be squared to achieve proportionality between the bus voltage and reference voltage.
4.2.2 The Demodulation Effect of SVC Voltage-Measurement System
MEASUREMENT SYSTEMS 99 Rs Ls Cs
Rp
Lp
I0 sin(q0t ) e (t )
Figure 4.5 Fault clearing in a series-compensated network with a shunt reactor.
fs c 1
2pfLsCs (4.7)
The network impedance also shows a pole (peak impedance) at frequencyf1
f1 c 1
2pf(Ls+Lp)Cs (4.8)
As the shunt inductanceLpis usually very large compared to the series induc- tance Ls, the frequencyf1 can be approximated by
f1 ≈ 1
2pfLpCs (4.9)
If the applied fault is subsequently cleared the bus voltage, e(t) is given by e(t)cE0cos(2pf0t) +E1e−(t/t1)cos(2pf1t) (4.10) where
E0 cX0I0 冢1− ff0s 冣2
冢1− ff10 冣2 cthe magnitude of the fundamental-voltage component
100 SVC CONTROL COMPONENTS AND MODELS
E1 cX0I0 冢ffs1冣2 −1
冢ff01冣2 −1 cthe magnitude of subsynchronous-voltage component
X0 c LsLp
Ls+Lp (2pf0)cthe equivalent parallel reactance ofLs andLp at fundamental frequency
t1 c2 Ls+Lp
Rs+Rp cthe decay-time constant of the subsynchronous component An estimate of the magnitude of different quantities can be obtained from the following realistic example [17]: Let the system have a short-circuit level of 10,000 MVA at 735 kV when it is series-compensated by 30%, and further- more, let theX
/
Rratio of the system reactance be 30 and that of the 330-MVAR shunt inductor be 400 at 60 Hz. Using the preceding equations, it can be com- puted that a subsynchronous-frequency component magnitudeE1 c0.4 pu and frequencyf1 c7Hz, with a decay-time constantt1 c1.37 s added to the fun- damental component. The current and voltage waveforms corresponding to this case are depicted in Fig. 4.6. It is seen that the short-circuit current is composed of a 32.8-Hz component corresponding to the series-resonance mode frequency, fs, that is given by Eq. (4.7). Similarly, voltages and currents of different fre- quencies that correspond to various other excited modes also get added to the fundamental.−150
−100−50 0 50 100 150
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−2
−1 0 1 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
(pu/100 MVA)
Short-Circuit Current
(pu) Voltage e(t )
Figure 4.6 Current and voltage waveforms at fault application and clearing on the circuit.
MEASUREMENT SYSTEMS 101
0 0.1 0.2 0.3
−2
−1 0 1 2
00 0.2 0.4 0.6 1 0.8
0
54 60 60
6 66
0.1 0.2 0.3
−2
−1 0 1 2
00 0.2 0.4 0.6 1 0.8 ADDITION
Time (s)
(a) (b)
Time (s) MODULATION
Frequency (Hz) Frequency (Hz)
Figure 4.7 Time- and frequency-domain representation of a voltage: (a) 0.5 pu, 6 Hz added to a fundamental of 1 pu, 60 Hz and (b) 1-pu fundamental modulated by a 0.5-pu, 6-Hz voltage.
4.2.2.2 Modulation The phenomenon of modulation occurs under differ- ent circumstances, such as during a power swing when the magnitude of the fundamental voltage varies with the frequency of the modulating signal. A fundamental-frequency voltage undergoing modulation can be expressed as
v(t)c[V0+V1sin(2pf1t+J1)] sin 2pf0t (4.11) Equation (4.11) can be rewritten as
v(t)cV0sin 2pf0t+ V1
2 cos[2p(f0 −f1)t−J1]
− V1
2 cos[2p(f0+f1)t+J1] (4.12) It is seen that modulation results in two sideband frequencies,f0−f1andf0+f1 besides the fundamental frequency f0.
A comparative study of the addition and modulation phenomenon—together with their corresponding frequency spectra, as obtained from a fast Fourier transform analysis—is presented in Fig. 4.7.
4.2.2.3 The Fourier Analysis–Based Measurement System Let the voltage to be measured be given by
v(t)cV0sin(2pf0t+J0) +V1sin(2pf1t+J1) (4.13)
102 SVC CONTROL COMPONENTS AND MODELS
A single-phase Fourier analysis–based measurement system will generate an output
Vmeas(t)cV0+Vdcos(2pfdt+a−J0) +Vscos(−2pfst −a−J0) (4.14) where
Vdc
V1sin冢pff01冣
p冢1− ff10冣 (4.15)
Vs c
V1sin冢pff01冣
p冢1+ff10冣 (4.16)
acJ1+p冢1− ff10 冣 (4.17)
fdcf0 −f1 (4.18)
fs cf0+f1 (4.19)
It is seen from these equations that when a frequency componentf1is added to the fundamental frequencyf0, the output comprises the following components:
1. A constant dc voltageV0 representing the magnitude of the fundamental component.
2. A sinuosidal time-varying voltageVd of frequency fdcf0 −f1. 3. A sinusoidal time-varying voltageVs of frequencyfs cf0+f1.
To exemplify the preceding list, if a 0.25-pu, 6-Hz voltage is added to a 1-pu, 60-Hz fundamental, the output is composed of the following components:
1. A dc voltageV0 c1pu.
2. An ac voltageVd c0.027 pu of frequencyfd c60 − 6c54Hz.
3. An ac voltageVs c0.022 pu of frequencyfs c60+ 6c66Hz.
The input and output signals of the single-phase measurement system, along with their frequency spectra, are shown in Fig. 4.8.
In a 3-phase Fourier measurement system, if the 3-phase voltages are bal- anced the output is obtained as a vectorial addition of the three input voltages, as follows:
MEASUREMENT SYSTEMS 103
Network Control system
Frequency Spectra 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−2
−1
0 0.5 1 1.5 0 1 2
10
0 20 30 40 50 60 70 80 90 100 0
0.5 1
0 10 20 30 40 50 60 70 80 90 100 0
0.01 0.02 0.03 Measurement
System
u (t ) = 1.0 sin(2p60t ) + 0.25 sin(2p6t )
Figure 4.8 The measurement technique and frequency spectrum obtained by using Fourier analysis.
Vmeasc 13(Vameas+Vbmeas+Vcmeas) (4.20) The frequency components inVmeasare determined by the phase sequence of the added frequency component f1 Eq. (4.13), as given in the following table:
Sequence of Added
Frequency,f1 Generated Frequency Output,Vmeas Positive fdcf0 −f1 V0+Vdsin(qdt+Jd) Negative fscf0+f1 V0+Vssin(−qst+Js)
Zero — V0
where Vd andVs are given by Eqs. (4.15) and (4.16), and the phase shiftsJd
andJs are, respectively, obtained as
JdcJ1−J0+p冢32 − ff10冣 (4.21)
Js c−J1−J0−p冢12 − ff10 冣 (4.22)
which shows that if the frequency f1 in Eq. (4.13) has a positive-phase sequence, only the sideband f0 −f1 is produced. On the other hand, iff1 has a negative-phase sequence, then only the sideband f0+f1 will only be gener-
104 SVC CONTROL COMPONENTS AND MODELS
ated. Usually both positive- and negative-sequence components of frequency f1 are present at the input; therefore, a linear combination of both sideband frequencies f0−f1 andf0+f1 will be produced at the output.
4.2.2.4 Coordinate Transformation–Based Measurement Systems Measurement systems such as those based on coordinate transformation, which compute the rms value by squaring the input voltage, also produce additional frequencies, as described here. Let the input voltage be expressed as
v(t)cV0sin 2pf0t+V1sin 2pf1t (4.23) Then
v2(t)cV20sin2 2pf0t+V21sin2 2pf1t+ 2V0V1sin 2pf0tsin 2pf1t c V20
2 (1−cos 4pf0t) + V21
2 (1−cos 4pf1t)
+V0V1[cos 2p(f0−f1)t −cos 2p(f0+f1)t] (4.24) Iff0 andf1 are 60 Hz and 6 Hz, respectively, then the output of the measure- ment system will contain, in addition to dc, the frequency components 12 Hz (2f1), 54 Hz (f0−f1), 66 Hz (f0+f1), and 120 Hz (2f0). These measurement systems therefore produce not only the sideband frequenciesf0−f1 andf0+f1 but also the frequencies 2f1 and 2f2.
4.2.2.5 ac
/
dc Rectification–Based Measurement Systems Measure- ment systems involving ac/
dc rectification are nonlinear in nature and hence produce the frequencies 2f1 in addition to the sideband frequenciesf0+f1 and f0 − f1. A comparison of the Fourier analysis–based and rectification-based measurement systems is presented in Fig. 4.9. The rectification system is also shown to introduce an error in the computation of the mean value of the input voltage.4.2.2.6 Filtering Requirement Any disturbance in the power system excites all the natural modes of oscillation in the network. Corresponding to each single resonant frequencyf1, the measurement systems will generally pro- duce sideband frequencies f0+f1 andf0 − f1 on the dc side. However, mea- surement systems based on rms voltage computation or rectification will gen- erate two additional frequency components: 2f1 and 2f0. The frequencies that lie within or close to the bandwidth of the voltage regulator (typically 30 Hz) appear as superimposed oscillations in the susceptance-order signal Bref and modulate the TCR currents. If these frequency components are not filtered, they tend to reinforce the natural oscillation modes, resulting in different forms of resonance and instabilities (see Chapter 5).
MEASUREMENT SYSTEMS 105
0.1
0 10 20 30 40 50 60 70 80 0.15 0.2 0.25 0.3 0.35 0.4 0.95
1
0 0.01 0.02 0.03 1.033 1.05
0.1
0 10 20 30 40 50 60 70 80 0.15 0.2 0.25 0.3 0.35 0.4 0.95
1
0 0.01 0.02 0.03 1.05 Measurement
System:
Rectifier Type
Measurement System:
Fourier Type Measurement Systems’ Input
Fundamental: 60 Hz, 1 pu positive sequence +
Component: 6 Hz, 0.25 pu positive sequence and 0.25 pu negative sequence
(pu)
(pu)
Time (s) Time (s)
Frequency (Hz) Frequency (Hz)
(pu)
(pu)
Figure 4.9 The comparison between two types of measurement systems: rectifier and Fourier.
For instance, in systems without series compensation, a network-resonant pole, typically at 85 Hz, excites an 85-Hz oscillation that is superimposed on the fundamental 60-Hz voltage wave. The measurement system then generates the frequencies 25 Hz (f1 − f0), 120 Hz (2f0), 145 Hz (f0 +f1), and 170 Hz (2f1) in addition to dc at its output. The frequencies of 145 Hz and 170 Hz lie well outside the bandwidth of the voltage regulator and hence are not significant, but the 25-Hz frequency component may lead to harmonic instabil- ity. This instability can be avoided by installing band-reject (i.e., notch) filters on the ac side of the voltage transducer. These filters are tuned to the critical network-resonant modes, which typically lie within 80–100 Hz.
The choice of installing the filter on the ac side or dc side of the measurement system is based on its effect on the phase margin of the system. It is found that a band-reject filter on the ac side reduces the phase margin to a much lesser extent than a corresponding filter on the dc side and is therefore used widely.
The filter is designed to cause a minimum phase lag at 60 Hz to not affect the normal SVC response. For example, in the Hydro-Quebec James Bay system, a combination of notch filters (80 Hz and 96 Hz) are provided [18]. If in any ac system the first network-resonant frequency lies beyond 2f0 (120 Hz), these ac notch filters are not required.
106 SVC CONTROL COMPONENTS AND MODELS
Because of the saturation of the SVC coupling transformer, sometimes 2nd harmonic currents are injected into the system, causing the inception of 2nd harmonic voltage oscillations. This 120-Hz (f1) voltage component gets trans- lated to 60 Hz (f1 − f0), 120 Hz (2f0), 180 Hz (f0+f1), and 240 Hz (2f1) on the dc side of the voltage transducer. The 60-Hz frequency component in the control loop tends to augment the original 120-Hz oscillation in the system, leading to 2nd harmonic instability (see Section 5.4). To mitigate this effect, a notch filter tuned to 60 Hz is connected to the dc side.
Provided on the dc side of the voltage transducer are two additional notch filters: one tuned to 120 Hz, the other to 360 Hz. It may be recalled that the 360-Hz ripple is caused by the 6-pulse rectification process. There is a need to filter the 120-Hz frequency component; otherwise, it may generate a 3rd harmonic sideband-frequency component in the SVC bus voltage. The other high-frequency components, arising on the dc side and caused by the demodu- lation effect of the voltage transducers, are eliminated through a low-pass filter having a time constant in the range of 1–8 ms.
Furthermore, in series-compensated ac networks, the interaction between the series capacitors and shunt reactors results in the excitation of the shunt-reactor mode having a frequency in the range of several to 20 Hz. This mode is deci- sive in determining the controller stability. A much-improved SVC response is obtained by filtering this mode and is achieved by placing a high-pass filter with a break frequency of about 25 Hz on the ac side.
The total arrangement of filters in the voltage measurement system and the typical characteristics of the filters within the context of the Hydro-Quebec James Bay system SVC [18] are depicted in Fig. 4.10.