Study on simple three-bus system

The intrinsic voltage-unstable nature of VSCs in current-limited mode has been elucidated above by two very simple examples. Here, the analysis is expanded to a larger system, namely that depicted in Figure 58. It comprises a grid, a VSC-HVDC converter, and a load in between. Such sketch could exemplify for example:

 A large load fed by two production areas, one being based on conventional generation (Vg) and the other being based on remote HVDC transmission (WPP or interconnector, VAC).

 A large load fed by two conventional generation areas, one of them (VAC) containing HVDC too and having lost its main production facilities and being left with HVDC alone.

Figure 58 - Electrical model of three-bus system.

As simplifying assumptions, let us consider ZLN1 = ZLN2 and VAC = Vg as complex numbers. In practice this means the load power is equally shared by AC grid and VSC-HVDC. These hypotheses, though quite restrictive, do not qualitatively affect the results, as will be shown later.

The generic approach that will be applied is derivation of an equivalent Thevenin circuit of the network as seen from the load terminals. Once the parameters Veq and Zeq of such equivalent circuit have been found, the voltage stability profile is immediately determined. Complex numbers will be used in the following.

(a) Normal operation

A further assumption for normal operation is that the VSC is able to control the voltage V̅_C so as to match Vg. If the load admittance is increased the equal power sharing between grid and VSC-HVDC is maintained. Under these assumptions, the converter can be modelled by a voltage source V̅_C and the equivalent circuit parameters will be:

V̅_eq= V̅_C ∙ Ż_LN2

Ż_LN1+ Ż_LN2+ V_g ∙ Ż_LN1

Ż_LN1+ Ż_LN2= V_g (23)

Ż_eq = Ż_LN1Ż_LN2

Ż_LN1+ Ż_LN2= Ż_LN2

2 (24)

Onshore AC voltage control

where complex numbers have been used. Changes in impedance distribution and/or active/reactive load sharing (differences in angle/magnitude of Vg and V̅_C) will reflect into the value of the above parameters and influence voltage stability accordingly. The results are therefore not affected qualitatively by the added assumption and the methodology can be applied regardless of the actual impedance distribution and converter voltage angle and magnitude.

(b) Current-limited operation

If current limitation is reached, the needed control freedom for maintaining constant VAC is lost by the converter. Following the approach suggested above, the VSC-HVDC station is modelled as a complex current source I̅_C (dashed current generator in Figure 58). The Thevenin equivalent parameters will be given, in complex numbers, by:

V̅_eq^' = V_g + Ż_LN2I̅_C (25)

Ż_eq^' = Ż_LN2 (26)

The effect on the voltage stability of the system is evidently dramatic in terms of equivalent impedance, since it doubles under the current assumptions, drastically reducing the theoretical transmittable power. The effect on the equivalent voltage V̅_eq^' , on the other hand, depends on the control strategy for I̅C, whose magnitude is limited, but whose angle can be changed by control means. Angle and magnitude of Ż_LN2 play a role too in determining the value of V̅_eq^' .

To understand more clearly how the angle control strategy of I̅_C affects the magnitude of V̅_eq^' , let us expand the expression of I̅_C as a complex number (real axis aligned with Vg):

I̅_C = I_Cr + jI_Ci | I_C = √I_Cr² + I_Ci² = I_N (27)

Writing the magnitude of Eq. (25) and rearranging (detailed calculation in Appendix 4):

V_eq^'2

Z_LN2² = (I_Cr+ V_g ZLN2

cos θ_LN2)

2

+ (I_Ci - V_g ZLN2

sin θ_LN2)

2

(28)

where θ_LN2= atan^XLN2

R_LN2 is the line impedance angle. The equation is that of a circle in the complex converter current plane with centre in C = (- ^Vg

Z_LN2cos θ_LN2; ^Vg

Z_LN2sin θ_LN2) and radius r = ^Veq'

Z_LN2. Parameterising the equation for different values of the equivalent voltage magnitude, plotting the corresponding circles and superimposing the circles with constant converter current magnitude IC, Figure 59 is obtained, which contains interesting information in terms of how to increase the equivalent voltage and therefore the transmittable power. In Figure 59, it is supposed that IC be limited to the nominal current IN, which is also used as a parameter. Furthermore, the same transmission line parameters as above have been used and Vg = 1 pu still holds.

Interesting properties can be noticed in Figure 59:

 Maximisation of transmittable power happens for maximum V^’eq. For a given value of IN, maximum voltage is achieved for strongly negative imaginary current. In the ideal case of lossless transmission line a pure negative imaginary current achieves maximisation of V^’eq. This can clearly be seen in Eq. (28) too.

 On the other hand, just imposing ICr = 0 in reality may not be straightforward, nor sensible, for the following reasons:

o The relation between real and imaginary parts of the converter current with converter active and reactive power depends on the actual operational scenario (power angle). Since the VSC is supposed to fulfil other control objectives (P, VDC, Q, VAC control) before entering current limitation, setting ICr = 0 and ICi = IN

in current-limited mode may be in contrast with other desired control features.

o Raising the equivalent voltage V^’eq may actually lead to an excessively high converter voltage, which would not be reachable within the VSC’s capabilities.

 In spite of the above, it is expected that increasing the voltage at the load bus is achieved if supplementary reactive power is provided in order to make up for the increasing consumption along the lines as the load increases. The grid voltage source Vg is uncontrolled, but the converter current I̅_C can be selected to be mainly reactive.

Intuitively, hence, it is expected that maximisation of V^’eq is obtained by feeding reactive power.

Figure 59 - Constant equivalent voltage circles and intersection with converter current locus.

To exemplify the effects of different converter current angles on the voltage stability of the system depicted in Figure 58, the PV curves are plotted for ideal case (normal operation, no current limitation, VAC = Vg) and two values of V^’eq in current-limited mode in Figure 60. The complex expression of IC for the two cases is reported too, highlighting the improvement a proper control of the current angle can bring about in the PV profile of the system: maintaining the same maximum current modulus (IN = 0.4 pu) 8° decrease in current angle can nearly provide an additional 0.1 pu of transmittable power, improving the long-term stability. The arrows illustrate the operational point path as load increases in the worst of the two cases (V^’eq = 1.05 pu).

In order to more thoroughly correlate the above results with the real control philosophy of an HVDC station, the converter behaviour during current limitation is of utmost importance. More

-1.5 -1 -0.5 0 0.5 1 1.5

0 0.5 1 1.5 2

ICr [pu]

I Ci [pu]

IN = 0.5 0.4 0.3

0.2 0.1

V´eq = 0.95 pu V´eq = 1.0 pu V´eq = 1.05 pu V´eq = 1.1 pu

Onshore AC voltage control

specifically, the current prioritisation philosophy needs be taken into consideration. Current prioritisation in SRF (dq-frame) is a known feature and detailed description is hence demanded to Appendix 1.

Figure 60 - PV curves for three-bus system in ideal normal operation and current-limited (IN = 0.4 pu) mode with two different current angles.

Here, dynamic simulations are used to understand how the current prioritisation approach relates to the above analysis. The model in Figure 58 was implemented in dynamic simulation software.

The VSC-HVDC station, being connected to a stiff voltage source on the DC side, is controlled in P-VAC fashion according to Figure 11 (Chapter 3), but the VAC droop block is substituted by a PI to guarantee zero voltage error in normal operation. The load admittance is progressively increased and the power reference of the VSC-HVDC is increased accordingly, so as to maintain equal load sharing.

The three following cases are simulated:

1. Ideal case: VSC-HVDC has sufficient current capability to maintain VAC = 1.0 pu regardless of the power injection – i.e. IN = ∞.

2. Limited case with Id priority: the VSC-HVDC has current limited below IN = 0.4 pu and the active power is given priority over the reactive power – i.e. ICd is controlled to its unlimited reference while ICq is limited according to the maximum current.

3. Limited case with vector priority: again, the converter current is limited below IN = 0.4 pu. However, no prioritisation of ICd (PC) or ICq (QC) is done – i.e. the reference current vector is only limited in its magnitude, without changing its angle.

The PV curves of the system for the three cases are plotted in Figure 61. It is apparent that, as hypothesised above, priority on P over Q is deleterious to the long-term stability (dashed line), while vector limitation, where Q can grow together with P, performs better and provides a 0.15 pu additional margin in the active power that can be delivered to the load.

The graphs in Figure 62 and Figure 63 shed further light on the operation of the system, respectively reporting the converter current in both complex plane (relative to Vg) and converter SRF and the converter active and reactive power (PC and QC respectively). It can be seen that by letting QC grow along with PC to support the voltage, more active power can eventually be

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

P [pu]

V [pu]

Ideal V'eq = 1.1 pu V'eq = 1.05 pu Critical loads

IC = 0.397 + j 0.069 pu I

C = 0.398 + j 0.014 pu

delivered to the load and produced by the VSC-HVDC. The current plot in Figure 62 is a further proof of the analytical observations derived with Eq. (28) and Figure 59.

Figure 61 - Load PV curves from dynamic simulation in ideal unlimited case, Id-priority and vector limitation. P is in pu of V g 2/ZLN2.

Figure 62 - Complex and SRF converter current locus for two current limited cases, in pu of converter ratings.

Several more investigations could be done on the issue, in particular pursuing the following objectives:

 Address the influence of the main assumptions behind the analysis, mainly the purely symmetrical nature of the electrical network and operational scenario.

 Investigate how the converter voltage and reactive power limits related to the available DC voltage (described above in Section 5.2) affect the operation under current limitation.

 Demonstrate a similar phenomenon can occur on larger, more realistic, heavily loaded power systems.

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

P [pu]

V [pu]

Ideal case Vector limit.

Priority on I

d

0 0.5 1

-1 -0.5 0 0.5

ICr [pu]

I Ci [pu]

0 0.5 1

-1 -0.5 0 0.5

ICd [pu]

I Cq [pu]

Vector limit.

Priority on I

d

Onshore AC voltage control

Nevertheless, the presented findings are deemed to be quite generic, and for example the graph in Figure 59 should be usable on a wider scale to explain the mechanisms leading to voltage collapse and determine how to improve VSC-HVDC control in current-limiting mode.

Figure 63 - Time plot of converter active and reactive power in pu of converter ratings.