5.5 Stability investigation using a frequency domain approach
5.5.5 Analysis of the VSC subsystem Capacitor connected at the PCC
PCC
In Section 4.7.4 the effects of a capacitor connected at the PCC has been studied with eigenvalue analysis. It has been shown in Section 4.7.4 that the dc-side-resonance-related eigenvalues (λ1,2) are better damped when the capacitor is connected at the PCC. However,
instability has been identified in the system, and it is related to the resonance of the ac side LCL circuit. This is further investigated using the approach developed in this section. In Figures 5.16 and 5.17 the VSC conductance and susceptance are plotted for power transfers of +1 and −1, respectively, and with the DVC parameters as defined for case 2. The VSC conductance and susceptance when there is no ac capacitor is also plotted for the sake of comparison. It can be seen that, when an ac capacitor is connected to the PCC, the VSC subsystem is non passive for some frequencies in both power directions. This is different from the case with no ac capacitor in which the VSC subsystem is passive for negative power transfers and non passive for positive power transfers. Nevertheless, the passivity of the VSC subsystem can be checked at each particular resonance frequency. Concerning the dc side resonance phenomenon, for the case when the power transfer is +1, the VSC conductance is more positive as compared with the case with no ac capacitor meaning that the resonance is better damped. When the power transfer is in the opposite direction, i.e. −1 the VSC conductance is still negative but it is higher, compared to the case when there is no ac capacitor.
From the figures, it can be seen that the VSC conductance presents fairly high resonance peaks. However, at those frequencies, the dc grid resistance is very small, since the dc grid impedance is capacitive. If at the ac-side resonance frequencies the dc grid resistance and the VSC conductance are neglected, the stability of the system can be approximately
100 100.2 100.4 100.6 100.8 101 0 50 100 0 0.1 0.2 Re[ F ][pu] Re[ e G][pu]0 100 100.2 100.4 100.6 100.8 101 −100 −50 0 −0.1 0 Frequency [pu] Im[ F ][pu] Im[ e G][pu]0
Figure 5.16: Unstable case. Frequency response of eG0and F with DVC paramters set as case 2 and
SCR as 5: Solid gray: G0(s). Solid black: F for a power transfer of +1 and no ac
capacitor, . Dashed: F for a power transfer of +0.5 and ac capacitor. Dotted: F for a power transfer of +1 and ac capacitor
assessed by claiming that the system is stable if
Fy(ωacres) eG0y(ωacres) < 1 (5.72)
where ωacres is the ac-side resonance frequency under examination. From Figure 5.16,
the VSC susceptance peak at the resonance frequency 2.8 pu is around −100 pu and the corresponding dc grid reactance is around -0.034 pu, which makes FyGy equal to 3.4 pu,
meaning that (5.72) is not fulfilled. At the second resonance frequency, 4.8 pu, the VSC susceptance peak is around −70 pu, while the magnitude of eG0is around −0.004 pu which means that (5.72) is fulfilled. On the other hand, from Figure 5.17, the VSC susceptance peak is around 80 pu and the dc grid reactance is around −0.032 pu for a resonance fre- quency of 2.95 pu, which fulfills (5.72) since FyGy is negative. The same is true for the
second resonance frequency, since the VSC conductance resonance peak is positive, and the dc grid reactance is negative.
Although the VSC admittance and the dc grid impedance has been analyzed considering their definition in this thesis, it is recommendable that the analysis is performed as carried out in [44]. In that way, the ac side resonance is captured without the influence of the VSC system, and then VSC admittance seen as shown in Figure 5.18 can give better information on whether or not the resonance is amplified or damped.
100 100.2 100.4 100.6 100.8 101 −100 −50 0 0 0.1 0.2 Re[ F ][pu] Re[ e G][pu]0 100 100.2 100.4 100.6 100.8 101 −50 0 50 −0.1 0 Frequency [pu] Im[ F ][pu] Im[ e G][pu]0
Figure 5.17: Sstable case. Frequency response of eG0and F with DVC paramters set as case 2 and
SCR of 5: Solid gray: G0(s). Solid black: F for a power transfer of −1 and no ac
capacitor, . Dashed: F for a power transfer of −0.5 and ac capacitor. Dotted: F for a power transfer of −1 and ac capacitor
5.6
Conclusions
In this chapter, the dc-side dynamics of the two-terminal VSC-HVDC system has been studied using a frequency domain approach. The VSC-HVDC system has been modelled as a SISO feedback system, in which two subsystems have been defined: the VSC and their dc grid subsystems. The corresponding transfer functions have been derived and the passivity properties have been studied. It has been shown that the dc grid subsystem is an unstable system which can be approximated to a marginally stable system, eG0. eG0 has
been found to be a passive subsystem, meaning that it is not the source of the instability. However, the dc grid subsystem presents a resonance which can interact with the VSC subsystem. The VSC subsystem has been found passive when the VSC, which controls the direct-voltage, absorbs power from the dc grid. This means that, when the VSC absorbs power from the dc side, the system is stable even for high DVC gains. The VSC subsystem is non passive when the VSC injects current into the dc grid which means that there is a risk that the resonance phenomenon developed in the dc side becomes amplified due to the non passive behaviour of the VSC subsystem.
The VSC admittance has been defined also in this chapter, and it is shown that in the un- stable cases, the VSC-subsystem presents a “negative conductance” characteristic at the frequencies of interest. When the dc-side resonance encounters a negative VSC conduc- tance, the resonance can be amplified depending on the size of the negative VSC conduc- tance. It has been shown that the following influence the magnitude of the negative VSC conductance:
1. The amount of active power injected by the VSC into the dc grid. The more power is injected into the dc grid, the more negative the VSC conductance.
2. The DVC proportional gain. The higher the DVC proportional gain, the more nega- tive the VSC conductance.
3. The SCR of the ac system to which the VSC it is connected. The weaker the system, the more negative the VSC conductance.
The analysis has been performed for a particular control system. However, the procedure is not restricted to the control system assumed in this chapter. If the interaction between the VSC which controls the direct voltage and the dc grid dynamics is to be investigated, the next procedure can be followed:
1. Identify the resonance frequency and the resonance peak of the dc-side. Commercial tools which calculates the harmonic impedance of electrical networks can be used for this purpose.
2. Determine if the converter conductance is negative at the resonance frequency. De- termine also if the Nyquist stability criterion is fulfilled.
3. If the system is unstable, investigate if the magnitude of the converter admittance can be decreased by modifying the controller structure.
Simulations in a multi-terminal
configuration
In this chapter, simulations are performed in order to further investigate the dc network dynamics in VSC-MTDC configurations. A four-terminal HVDC system is modelled in PSCADTM, and its performance when using two control strategies, the voltage-margin and
the voltage-droop control, is tested. Some events, such as converter disconnections and operating point changes, are tested for both control strategies and different controller para- meters. The effect of other control loops, not studied analytically in this thesis, are investi- gated in this section as well through simulations.
6.1
System description
The system under analysis is a radial four-terminal VSC-HVDC system, as depicted in Figure 6.1. The cables are modelled as Π sections and their lengths are as shown in the figure. The equivalent cable inductance, capacitance, and resistance per kilometer are as indicated in Table 3.2. The VSCs are two-level converters and they have their ratings as
Figure 6.1: System under analysis
indicated in Table 3.1 and Table 4.1, i.e. 600 MVA power rating, 300 kV rated line-to- line voltage in the ac side, and ±300 kV rated pole-to-pole voltage in the dc side. In
this example, VSC1 and VSC2 are set to control the direct-voltage, following a strategy
defined in the next sections, and VSC3 and VSC4 are set to control the power. Table 6.1
summarizes the controller parameters considered in the simulations. The controllers of VSC1and VSC2are selected two times faster than VSC3and VSC4in order to provide a fast
voltage regulation in the dc side of the system. In VSC1 and VSC2, the recommendation
that the DVC should be ten times slower than the VCC is adopted [44, 45].
Table 6.1: Controller parameters in per unit
Parameter VSC1 VSC2 VSC3 VSC4
VCC bandwidth (α) 8.0 8.0 4.0 4.0
DVC nat. res. frequency (ωn) 0.8 0.8 − −
DVC damping factor (ξ) 1.0 1.0 − −
VCC proportional gain (kp) 2.0 2.0 1.0 1.0
VCC integral gain (ki) 0.02 0.02 0.01 0.01
DVC proportional gain (kpe) 3.078 3.078 − −
DVC integral gain (kie) 1.231 1.231 − −
Furthermore, a current limiter is added in order to limit the output of the DVC, as illustrated in Figure 6.2. The limits are set ±1.3 pu. In some cases, the current limiters are deactivated in order to investigate the effect on the system dynamics.
Figure 6.2: Implementation of a current limiter