4 MODELLING OF VSC-BASED EQUIPMENT FOR DYNAMIC SIMULATIONS
4.1 STATCOM model for dynamic simulations
Figure 4.1 shows a schematic representation of the STATCOM and its control variables. The voltage source converter, whose AC terminal is connected to the low-voltage side of the OLTC transformer, is charged with the control of the AC voltage magnitude Vv. The VSC station uses a DC capacitor, one that plays an important role in the dynamics of the converter. In the steady-state regime, its voltage is kept constant at the nominal value Edcnom and the current in the capacitor becomes zero, ic0 , something that can be inferred from the expression of the capacitor current (4.1) in which the derivative of the voltage with respect to time is zero. During the dynamic regime, however, failures on the AC network are reflected in the DC link voltage, pushing the capacitor to a charging/discharging stage.
51 Figure 4.1 Schematic diagram of the STATCOM and its control variables
The VSC dynamics are accurately captured if the dynamics in the DC capacitor are suitably represented when it is exposed to changes in the current flowing through it. The computation of the voltage oscillations at the DC bus requires the determination of both the current injected at the DC bus IdcR(4.2) and the capacitor currenticIdcRIdc.
dc
c dc
i C dE
dt (4.1)
0v dcR
dc
I P
E (4.2)
where P0v is the nodal power injected at the DC bus (3.11), previously derived in Section 3.1.
Therefore, the expression (4.3) serves the purpose of calculating the DC v oltage dynamics of the STATCOM (Castro et al., 2013).
dc dcR dc
dc
dE I I
dt C
(4.3)
where Idcis the selected control variable acting upon the DC voltage. This last mathematical expression permits the calculation of the DC voltage dynamics in the voltage source converter. At this point, paying attention to the value of the capacitor Cdc is necessary as it plays a c rucial role when attaining the DC voltage dynamics. Its value is estimated according to the energy stored in the capacitor: Wc12C Edc dc2 . Arguably, the electrostatic energy stored in the capacitor bears a resemblance with the kinetic energy of the rotating electrical machinery, where this energy can be further related to the inertia constant that impacts the motion equation of a certain rotating machine.
Following the same reasoning, the electrostatic energy stored in the DC capacitor can be associated with an equivalent inertia constant Hc [s] as W H Sc c nom, where Snom would correspond to the rated apparent power of the VSC. This time constant may be taken to be approximately
52
1
c s
H
by drawing a parallel between the converter and a bank of thyristor-switched capacitors of the same rating; however, it is recommended that Hc5ms so as to avoid numerical instabilities (de Oliveira, 2000). Hence, the value of the capacitor is computed as in (4.4).2
2 nom c
dc
dc
S H
C E (4.4)
The stability of the DC voltage ensures a smooth operation of the STATCOM; the voltage control at the DC side of the VSC is carried out through the regulation of the DC current entering or leaving the converter, Idc. The implementation of the DC voltage controller is shown in Figure 4.2 (Castro et al., 2013), in which the error between the actual voltage Edc and Edcnom is processed through a PI controller to obtain new values of the DC c urrent Idc at every time-step. Hence, the DC voltage dynamics is basically determined by the gains Kpe and Kie.
Figure 4.2 DC voltage controller of the VSC
The equations arising from the block diagram corresponding to the DC voltage controller are shown in (4.5)-(4.6).
dcaux
ie dc dcnom
dI K E E
dt (4.5)
dc pe dc dcnom dcaux
I K E E I (4.6)
The converter keeps the capacitor charged to the required voltage level by making its output voltage lag the AC system voltage by a small angle (Hingorani and Gyugyi, 1999). This angular difference is computed as v , where is the angle of the phase-shifting transformer and v represents the angle of the AC terminal voltage of the VSC. During the dynamic regime, the power balance on the DC side of the converter is achieved by suitable control of the angle . The PI controller des igned to meet such a purpose is shown in Figure 4.3 (Castro et al., 2013).
53 Figure 4.3 DC power controller for the DC side of the VSC
where the 0 entry of the sum block of the PI regulator signifies that there is no external power injection at the DC bus. The differential equations that represent the dynamic behaviour of this power controller are given by (4.7)-(4.8).
0 aux
ip v
d K P
dt
(4.7)
0
pp v aux
K P (4.8)
The modulation index is responsible for keeping the AC terminal voltage magnitude at the desired value; to such an end, the AC-bus voltage controller depicted in Figure 4.4 is employed (Castro et al., 2013). This is a first-order controller which yields small changes in the modulation index dma by comparing the actual voltage Vv and the scheduled AC voltage Vvspec. The new value of the modulation index increases or decreases according to the operating requirements conditions.
The differential equation for the AC voltage controller is shown in (4.9).
Figure 4.4 AC-bus voltage controller of the VSC
a
ma
vspec v
a maK V V dm
d dm
dt T
(4.9)
To guarantee that the VSC operates within its operating limits, a limit checking of the terminal current, previously derived in (3.13), must take place IvIvnom. Hence, the overall dynamics of the VSC are well captured by equations (4.3)-(4.9).
54
The STATCOM differential equations representing the DC voltage dynamics, the DC current controller and the AC voltage controller are discretised as shown in (4.10)-(4.13).
( ) 0.5 ( ) ( ) 0.5 ( )
Edc dc t t dc t t dc t dc t
F E tE E t E (4.10)
( ) 0.5 ( ) ( ) 0.5 ( )
dcaux
I dcaux t t dcaux t t dcaux t dcaux t
F I tI I tI (4.11)
( ) 0.5 ( ) ( ) 0.5 ( )
aux aux t t aux t t aux t aux t
F
t
t
(4.12)
( ) 0.5 ( ) ( ) 0.5 ( )
dma a t t a t t a t a t
F dm tdm dm tdm (4.13) where
1
( ) ( ) ( )
dc t dc dcR t dc t
E C I I (4.14)
1
( ) ( ) ( )
dc t t dc dcR t t dc t t
E C I I (4.15)
( ) ( )
dcaux t ie dc t dcnom
I K E E (4.16)
( ) ( )
dcaux t t ie dc t t dcnom
I K E E (4.17)
( ) 0 ( )
aux t K Pip v t
(4.18)
( ) 0 ( )
aux t t K Pip v t t
(4.19)
1
( ) ( ) ( )
spec
a t ma ma v v t a t
dm T K V V dm (4.20)
1
( ) ( ) ( )
spec
a t t ma ma v v t t a t t
dm T K V V dm (4.21)
The discretised differential equations of the VSC are appended to those corresponding to the active and reactive power balances of the power network at the AC terminal of the VSC and the high-voltage side of the OLTC transformer. These equations were derived in Section 3.3 but for the sake of completeness they are reproduced in equations (4.22)-(4.25).
cal
k kltc dk k
P P P P
(4.22)
cal
k kltc dk k
Q Q Q Q
(4.23)
v v dv vltc
P P P P
(4.24)v v dv vltc
Q Q Q Q
(4.25) Equations (4.10)-(4.13) and (4.22)-(4.25) make up the necessary set of equations that must be solved together with the equations of the whole network including synchronous generators and their controls to carry out dynamic simulations of power systems containing STATCOM devices. To solve such non-linea r set of equations, the Newton-Raphson method is employed for reliable55 dynamic simulations. Hence, in connection with Figure 4.1 where the STATCOM is assumed to be connected at bus k of the power system, the linearised matrix equation (4.26), around a base operating point, provides the computing framework with which the time-domain solutions are performed (Castro et al., 2013).
11 12
21 22
dc
dcaux
aux
a
i i
k k
k k
v v
i
v v
E dc
I dcaux
aux dm a
P
Q V
P
Q V
F E
F I
F F dm
J J
J J (4.26)
where J11 comprises the first-order partial derivatives of the nodal active and reactive power mis matches with respect to the voltage angle and voltage magnitude of the AC-side voltages, i.e., both the AC terminal of the VSC and the high-voltage side of the OLTC transformer. Likewise, J12 contains the partial derivatives arising from the nodal active and reactive powers with respect to the state variables of the converter. The matrix J21 consists of partial derivatives of the VSC discretised differential equations with respect to AC voltages. Lastly, J22 is a matrix that accommodates the first-order partial derivatives of the VSC discretised differential equations with respect to its own control variables. These matrices are shown in (4.27)-(4.28).
11
k k k k
k k v v
k k k k
k k v v
v v v v
k k v v
v v v v
k k v v
P P P P
V V
Q Q Q Q
V V
P P P P
V V
Q Q Q Q
V V
J , 12
0 0 0 0
0 0 0 0
0 0 0 0
v v
dc a
v v
dc a
P P
E dm
Q Q
E dm
J (4.27)
21
0 0
0 0 0 0
0 0 0 0
dc dc
aux aux
a a
E E
v v
v v
dm dm
v v
F F
V
F F
V
F F
V
J , 22
0 0
0 0
0
0 0 0
dc dc
dcaux dcaux
aux aux aux
a
E E
dc a
I I
dc dcaux
dc aux a
dm a
F F
E dm
F F
E I
F F F
E dm
F dm
J (4.28)
56
It is worth recalling that a dynamic simulation requires as its main input the steady-state initial conditions of the dynamic system to be solved. This s teady-state equilibrium point is c omputed in advance using the power flow model that has been addressed in Section 3.3, thus ensuring a reliable dynamic solution.
Considerations for the OLTC transformer during the dynamic operating regime
The adopted modelling approach calculates together the state variables of the VSC station and that of the OLTC transformer through the power-flow solution, as presented in Chapter 3. However, during the dynamic operating regime the time constants involved in the tap controllers of the OLTC transformer are large in comparison with the very rapid time responses afforded by a VSC station.
Therefore, from now on in this chapter and with no loss of applicability, the OLTC transformer dynamics are omitted, keeping the tap values furnished by the power-flow solution as constant parameters during the dynamic state. Indeed, the long-term impact that the operation of the OLTC transformer might have on the dynamic performance of the newly developed VSC-based equipment models could also be assessed (if the control loop of the OLTC taps is considered), however, this lies outside the main focus of the research carried out in this thesis.