4.1. Concept Description
The phase imbalance compensation (PIC) concept is shown in
Figure 1. In this concept, a phase wide series compensation is implemented in such a way that the impedance at the fundamental frequency remains the same under both compensation schemes, i.e.,
, where the prime symbol denotes quantities in the classical compensation scheme and the double prime symbol denotes quantities in the phase imbalance concept. In order to achieve this, consider phase
. The relation between capacitor
and inductor
is derived as shown in Formula (6), where
and
are additions due to the PIC,
is the inductance of the uncompensated line, and
is the capacitance of the series capacitor under the classical compensation scheme.
The same applies for phase
and solving Equation (6) for
ultimately leads to Equation (7), where
represents the degree of asymmetry. This equation sets the relation between the additional inductance and capacitance in each phase that applies the PIC. For each phase, a single degree of freedom exists.
As a result of the imbalanced compensation, the resonance frequency of each asymmetrically compensated phase is different. With asymmetrical compensation deployed in phases
and
(see
Figure 1),
. The resonance frequency is then defined as Equations (8)–(10) for phase
,
and
. Since no additional impedances are included in phase
, it should be noted that
.
Figure 4 shows the vector and phasor diagrams of the phase impedances at a sub-synchronous frequency
for the classical as well as PIC concepts. For the classical compensation concept, the phase impedances are identical. At
these impedances are equal to the damping (
Figure 4a) and result in a balanced phasor (
Figure 4c). In the PIC, the phase impedances at
are different (
Figure 4b), which lead to an unbalanced phasor (
Figure 4c). As mentioned in
Section 3, the vector and phasor diagrams at the fundamental frequency are the same for both compensation concepts.
When a disturbance occurs in the grid, the unbalanced phase impedances will create a non-uniformly distributed
. This
is defined as given in Reference (11).
With no series compensation, is zero. With the classical compensation, the in each phase is identical, as the maximum current in each phase is the same (due to the same in each phase). On the other hand, for each phase that implements an asymmetrical compensation, is different depending on , which, in turn, is a function of the degree of asymmetry in that respective phase. As such, the phase imbalance concept is able to alter the (and electrical torque) at sub-synchronous frequencies, and, therefore, potentially mitigate DFIG-SSR. The larger , the smaller and the smaller the . With decreasing , the resulting subsynchronous oscillations are smaller.
4.2. Developed Analytical Model for Phase Imbalace Compensation
In order to create an understanding of how the PIC alters the dynamic behaviour of the power system, an analytical model describing the relation between the inductance and capacitance of the imbalanced phase and the resonance frequency of that phase, is developed next. The starting point for this derivation is
, as
. Consider phase
. If
differs from
by
, then the relation between
and
is given by Equation (12).
Substituting Equation (7) into Formula (12) yields Equation (13) and, after rearranging, it yields Equation (14).
Solving Equation (14) for
results in Equation (15).
Substituting
in Equation (15) gives Formula (16).
Finally, solving Formula (16) results in Equation (17).
The relation between
,
, and
is derived by substituting Equation (5) in Equation (17), which ultimately results in Formula (18).
The degree of asymmetry
for phase
is then defined as given in Equation (19) and represents the analytical model of the PIC concept. The same relation holds for phase
.
Figure 5 is derived from Equation (19) and shows different compensation levels and different degrees of asymmetry. The resulting change in resonance frequency is
. Since any
requires an additional series capacitor, the resonance frequency
will always increase.
is always positive, as is shown in
Figure 5, and it is, therefore, not possible to reduce the resonance frequency below
using the series resonance PIC scheme. Furthermore, for any given
,
is inversely proportional to
.
From
Figure 5, it can be observed that an imposed
in the PIC can be achieved by modifying either
or
in the design phase. For fixed
, a modification of
(which is a quantity at
) results in an altered active power transfer limit and altered
. For fixed
, modification of
results only in altered sub-synchronous behaviour (i.e., altered
), as
should be respected for each value of
.
4.3. Development of the Power System Model and DFIG Model
To assess the effectiveness of PIC in mitigating DFIG-SSR, the development of the power system and DFIG model is briefly described next. Due to its highly local nature, the investigation of DFIG-SSR requires a detailed model of the DFIG. Modelling of the grid is required in such a way that it excites the oscillation mode, which, ultimately, will result in the interaction between the DFIG and the grid. Therefore, many investigations on DFIG-SSR reported in literature use the topology of the IEEE First Benchmark Model [
21]. The model in the current work is modified in two ways. First, the synchronous generator model is replaced by a detailed DFIG model. Second, the equivalent grid is modelled as a voltage source with a series impedance, so that the grid contains a sub-synchronous resonance point at 30 Hz, which corresponds to
at 36%. The equivalent grid has a short circuit ratio of 10. The single line diagram of the power system under consideration (including all relevant network data) is given in
Figure 6. Several practical studies have used similar models for investigating DFIG-SSR [
22,
23,
24,
25].
Detailed modelling of the DFIG is crucial, as the adverse interactions are influenced by the damping available at the DFIG terminal. A generic detailed EMT wind turbine model following the IEC standard 61400-27-1 [
26] was developed in PSCAD as part of the MIGRATE project [
27]. Since the controls presented in the IEC standard are meant for root mean square applications, several adaptations and additions were implemented in order to make the models applicable for EMT analysis. The development of the detailed model is reported in Reference [
28]. This model contains, among others, a carefully designed filter to address switching harmonics, where the filter design is based on Reference [
29]. Furthermore, the semiconductor switches are modelled in detail, including the thyristor’s turn on and turn off resistances, giving a more realistic damping behaviour of the DFIG. The mechanical dynamics of the wind turbine generator were modelled using a two mass model. The classical double loop control was used as a control structure of the grid as well as the rotor side converter of the DFIG. The implemented controls are the same as the ones used in the authors’ previous work (Figures 8 and 9 in Reference [
1]). The DFIG’s parameters are given in
Appendix A. The behaviour and fault response of the developed model was successfully validated against vendor models. Further discussion on the model development is out of the scope of the current work.
4.4. PIC Evaluation for DFIG-SSR Mitigation
Prior to evaluating the effectiveness of the PIC in mitigating DFIG-SSR, the influence of the PIC on the impedance characteristic of the transmission line in the modified IEEE First Benchmark Model is assessed. One of the contributions of this work is the derivation and validation of Formula (19) through repetitive EMT simulations of an actual system, considering different values for the degrees of asymmetry across one and two phases. To this end, low
values are chosen as these have a more pronounced effect on the impedance characteristic. The results are shown in
Figure 7. Let
and
be equal to 0.2 and 0.5, respectively. Substituting these values for the degree of asymmetry and
equal to 36% in Equation (19) results in
of 14 and 10 Hz for phase
and phase
. As is shown in
Figure 7a, the exact same results are achieved using EMT simulations, where the impedance characteristic of each phase is determined using the dynamic frequency scanning technique [
30].
Figure 7b shows the impedance characteristics of phase
for different levels of
. These results are also in line with Formula (19) and
Figure 5. Higher
values decrease
.
When the DFIG is added to the overall system, the equivalent impedance characteristic is changed by following the actual
will be different from 30 Hz. As shown in
Figure 8, the connection of the DFIG to the classical series compensated transmission line reduces
to 10 Hz. The converter controller parameters, inductances from the transformers and the DFIG machine, and the inductance and shunt capacitance from the filter at the GSC are additional impedances that alter the impedance characteristic of the DFIG. The additional impedance introduced by these components is one of the reasons for the reduced
.
To identify whether there exists an adequate degree of asymmetry capable of mitigating DFIG-SSR, optimisations were performed in PSCAD. is defined as given in Equation (7). An optimisation approach is selected, as it can quickly and efficiently search through wide parameter ranges. Two independent optimisations were performed. In the first optimisation, the goal was to find the adequate degree of asymmetry when only phase is compensated using PIC, i.e., single-phase imbalance compensation. The objective function of this optimisation is given by Formula (20), where performance index is developed to monitor the performance of the optimisation. The performance index is a measure of the energy associated with the oscillation. The threshold of below which DFIG-SSR can be considered mitigated was determined using extensive empirical analysis. These analyses consisted of thousands of EMT simulations in PSCAD in which each has slightly different converter controller parameters. In these simulations, the transmission line was compensated using the classical compensation scheme. For each simulation, the performance index was recorded. The empirical analysis showed that DFIG-SSR can be considered mitigated when is smaller than 0.07 per unit, as this represents a case with negligible oscillation energy. As a reference, under the classical series compensation scheme, is equal to approximately 0.215 per unit.
In the second optimisation, the goal was to find adequate degrees of asymmetry
and
when phase
and phase
are compensated using PIC, i.e., two-phase imbalance compensation. The objective function is given by Formula (21). In both optimisations, DFIG-SSR, when using phase imbalance compensation, is triggered by a switching action, which causes a permanent radial connection of the DFIG to the unbalanced compensation.
To illustrate a case where DFIG-SSR is successfully mitigated when the transmission line is equipped with classical series compensation,
Figure 9 shows the response of the DFIG to a fault before and after modification of the rotor side converter controller parameters. After 1.5 s, a switching action caused the DFIG to be radially connected to the series capacitor (classical compensation), which resulted in DFIG-SSR in the case of unmodified parameters. After modification of the RSC controller parameters, DFIG-SSR was mitigated.
The optimisation results are given in
Figure 10a for the single-phase imbalance and in
Figure 10b for the two phase imbalance compensation. Approximately 3000 iterations for the single-phase imbalance compensation and 4000 iterations for the two-phase imbalance compensation were taken as sufficient to judge the optimisation’s convergence, where each iteration includes an EMT simulation. The Genetic Algorithm [
31] was used as an optimisation algorithm, where the settings (e.g., initial population, mutation rate, pairing method, etc.) were chosen as recommended in Reference [
32]. The influence of the algorithm’s settings on the optimisation results is out of the scope of the current work and is left as a topic for further research.
In neither of the cases, converged to a value smaller than 0.07 per unit and, consequently, no adequate degree of asymmetry capable of mitigating DFIG-SSR was identified. On the contrary, for the investigated DFIG, the PIC concept consistently performed either the same as or worse than the classical series compensation scheme. This is concluded by observing the evolution of as the degree of asymmetrical change. It shows that decreases with increasing values of , but never reaches a value below 0.215 per unit, which is the value of for the classical series compensation scheme. For the sake of consistency, the optimisations were performed multiple times without affecting the findings.
One way to assess in more detail the influence of
on mitigating DFIG-SSR (and, thus, on
), is to observe how the
at
,
, changes with changes in the degree of asymmetry
. Using Equation (11),
can be observed by observing the peak current
. The higher
, the stronger
and the larger
, i.e., the poorer the capability for mitigating DFIG-SSR. Different levels of asymmetry in phase
were evaluated.
Table 1 provides the resonance frequency and maximum phase current
for five values of
. For the analysis, the operating point of the DFIG was fixed at a windspeed of 10 m/s (the nominal windspeed of the DFIG is 10.2 m/s). DFIG-SSR occurs mainly below nominal windspeeds [
1], which is the reason why the windspeed of 10 m/s was selected. However, lower windspeeds will not alter the main conclusions of the analysis. For comparison purposes, the results from the classical compensation scheme are included as well.
Two observations can be made. First, it is confirmed again that the resonance frequency increases by reducing the degrees of asymmetry. This is in line with Equation (19). Second, with decreasing
(i.e., increasing
),
increases, which implies that, with lowering degrees of asymmetry,
increases.
Figure 11 shows the measured instantaneous current in phase
for three cases: no series compensation, classical series compensation, and phase imbalance compensation. It shows a worse performance of the phase imbalance series compensation compared to the classical series compensation. This
-
-
relation explains why
is relatively large for small values of
and small for larger values of
.
Lastly, to explain why
is increasing as
decreases, it is necessary to investigate the damping characteristic of the DFIG. Using the perturbation-based dynamic frequency scan [
30], the damping of the DFIG was estimated at windspeeds of 6 m/s (i.e., below nominal windspeed), 10 m/s (i.e., around nominal windspeed), and 20 m/s (i.e., above nominal windspeed). The results are given in
Figure 12.
The following conclusions can be drawn from
Figure 12. First, Considering the
of 10 Hz, any
achieved by means of phase imbalance compensation, where
remains in the negative resistance region, will deteriorate the damping provided by the DFIG. To achieve an
outside the negative resistance region requires
-values, which are not economical for practical applications. For different windspeeds, the negative resistance region of the DFIG is given in
Table 2.
Second, for the same , the damping worsens with decreasing windspeeds. Therefore, the design of the PIC as a mitigation solution for DFIG-SSR needs to be validated across several operating conditions.
4.5. Mitigating Torsional SSR with PIC
The analytical model that was developed in
Section 4.2 proved that, for any degree of asymmetry in the phase imbalance compensation, the resulting resonance frequency is increased. Furthermore,
Section 4.4 showed that, in the negative resistance region of a DFIG, the negative damping increases with a growing resonance frequency. It was concluded that the PIC cannot mitigate DFIG-SSR, as the resulting shift in
deteriorates the electrical damping even further.
Why is it then that the PIC is theoretically able to mitigate classical SSR, as was shown in Reference [
8]? This is due to the fact that the negative resistance region for a conventional generator is rather limited compared to a DFIG. This is briefly discussed by using two examples. For a wind power plant in China, torsional interaction (SSR) was investigated. Using the complex torque coefficient method, electrical damping coefficients under different wind power output levels were calculated (
Figure 13 [
33]).
The possibility of a torsional interaction with
of 31 Hz was identified. At this frequency, the damping coefficients were negative. Compared to the DFIG damping (
Figure 12), the negative resistance region in
Figure 13 is limited. Increasing
with as little as 2 Hz results in positive damping and eliminates the adverse torsional SSR interaction. The PIC is capable of achieving this shift in
and, therefore, could be a potential mitigation solution.
In another practical example, SSR analysis was carried out to investigate the torsional characteristics of the Hanul nuclear power plant in Japan [
34]. Increased probability of torsional interaction between the generator and a fixed series capacitor was identified, where, depending on the operating condition,
ranged between 35 and 39 Hz. A 2–5 Hz shift in
would result in positive damping and, therefore, eliminate the adverse torsional interactions. In this case, the PIC could be one of the potential solutions to achieve this.