Setpoint Feasibility and Stability of Wind Park Control Interactions at Weak Grids

Argüello Andrés

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Abstract

The controllers of wind parks that are connected to weak grids can induce unstable oscillations near the fundamental frequency. Such phenomenon can be studied with equivalent impedances of the wind generators, which depend on their operational setpoints at the fundamental frequency. The feasibility of such setpoint, i.e., solution to the power flow, does not depend on the control parameters. However, oscillations from unfeasible setpoints and control instabilities both occur at weak grids and near the fundamental frequency. This letter presents an exploratory study to map the conditions where both phenomena occur, using sensitivity studies comprising multiple setpoints and control parameter tunings. The results are visually presented as the regions which inform the configurations leading to unfeasible setpoints, and unstable control interaction scenarios. Amongst the results, it can be observed that the trajectory of the eigenvalues in the Nyquist plots towards an unfeasible setpoint approaches a fundamental frequency instability.

Keywords

Control interaction; impedance; power quality; resonance; stability; wind generator

I. Introduction

TRADITIONAL bulk synchronous power plants such as thermal and hydro plants are being replaced gradually by dispersed inverter-based generators such as wind and solar generators [

1]. This change carries new challenges to the power quality of power grid such as unstable oscillations due to control interactions of the power-electronics controllers with power grid [2].

In response, [

3] updates the stability definitions to encompass power-electronics interfaced technologies. One of the key aspects highlighted in the update is the stability of wind parks (WPs) connected to weak grids, which is a rather common scenario as WPs can be located at remote sites away from the main transmission system. This characteristic particularly affects the stability of oscillations near the fundamental frequency due to the loss of terminal voltage stiffness, as highlighted in [4].

When connected to weak grids, i.e., with a low short-circuit ratio (SCR) ( $S C R < 2$ , which is relative to rated generator capacity) [

3], Type-III and Type-IV WPs can become unstable mainly due to the phase-locked loop (PLL) not being able to synchronize with the terminal voltage. They are also influenced by the other control such as DC bus voltage control, power control, and AC voltage control, and the DC bus voltage dynamics [5]-[7]. The instability manifests as symmetrical oscillations centered near the fundamental frequency (sustained or undamped), and has been reported to occur in a wide range of frequencies [8]. These oscillations are more problematic when they excite torsional modes of neighboring generation sites [9].

As mentioned in [

3], there is a relationship between the control setpoint of the generators (defined by their active power injection, reactive power injection, and terminal voltage) and the control stability at weak grids. This is noted when the generator impedance models for stability studies are used, where the setpoint affects the impedance profile, especially near the fundamental frequency [6].

So, before analyzing the stability, it is first necessary to ensure that the operational setpoint is feasible. A setpoint is classified as feasible if the system reaches a constrained numerical solution at the steady state, i.e., if the power flow at fundamental frequency can be solved within preset operational limits. Setpoints with high active power injection and low terminal voltage, which are more likely to occur at weak grids, can be unfeasible. Thus, the setpoint feasibility and stability of control interactions with power grid can be easily mixed up as they both occur near the fundamental frequency at weak grids, but are conceptually different.

This letter presents a series of charts illustrating the results of a sensitivity study, which show separate regions where unfeasibility and instability occur. Several active power and terminal voltage setpoints are presented with different grid strengths for a range of control parameters associated with unstable oscillations near the fundamental frequency [

3], [10].

The insights derived from this letter can serve as a basis for system operators during planning phase to map the two phenomena when performing power transfer capability studies to determine the stability margins in case of contingencies leading to weaker grids. It can also be used in post-event analysis to diagnose the true cause of oscillations.

Finally, it is important to clarify that this letter studies “weak grid oscillations” defined in [

4], as those caused by unstable controls interacting with a relatively high-impedance (low SCR) grid. The phenomenon of “subsynchronous oscillations due to series capacitors, i.e., subsynchronous resonance” [4] is out of the scope of this letter.

II. Regions of Unfeasibility and Instability

Figure 1 shows an example of the unfeasible and unstable regions of PLL integral gain, where $V_{t}$ is the terminal voltage; and P_t is the active power. These are groups of points represented as a surface, since the number of scenarios studied is limited. All points within them are either unfeasible (black) or unstable (colored). The regions present multiple setpoints and grid strength using the grid SCR at WP terminals. Each color indicates a different value of control gain, for example, the yellow one indicates that the value of the PLL integral gain $K_{i P L L}$ is 10 times its base value.

Fig. 1 Example of unfeasible and unstable regions of PLL integral gain. (a) V_t = 0.8 p.u.. (b) V_t = 1.0 p.u..

To build the regions, a power flow at fundamental frequency is run first at every setpoint [

11]. If feasible, the result is used to calculate the impedance models of the generators, and later on, the stability of control interactions with power grid is checked with Nyquist impedance criterion [12].

It is not the intention of this letter to draw the regions analytically. The regions are used as a support tool to show the conditions under which each phenomenon occurs, in a qualitative fashion. However, the regions can be built numerically using power flow software for the unfeasible regions and impedance-based equivalents for the unstable regions.

A. Sensitivity Study Setup

1)　Test System

The test system used in this letter is shown in Fig. 2, which models a 64 MVA WP with a single-machine equivalent circuit.

Fig. 2 Simplified WP circuit for sensitivity study.

In Fig. 2, $V_{s c}$ is the grid voltage; $Z_{s c}$ , $Z_{t 1}$ , and $Z_{t 2}$ are the RL branches representing the grid equivalence ( $V_{s c} = 220$ kV, $X / R = 10$ ), the main WP transformer (220 kV/35 kV, 100 MVA, $| Z_{t 1} | = 0.1$ p.u., $X / R = 50$ ), and the step-up transformer (32×2 MVA, 35 kV/0.69 kV, $| Z_{t 2} | = 0.08$ p.u., $X / R = 5$ ), respectively; $C_{p}$ models a shunt capacitor bank; $Z_{g e n}$ and $Z_{W P}$ are the matrices of the positive- and negative-sequence impedances of the wind generator and WP, respectively; $I_{t}$ is the current flowing to generator terminal; $Q_{t}$ is the reactive power absorbed at generator terminals; and PCC stands for point of common coupling.

2)　Power Flow Feasibility

The power flow at the fundamental frequency is run for each SCR and $P_{t}$ to calculate the reactive power setpoint $Q_{t}$ for a given terminal voltage $V_{t}$ . Feasibility is constraint to the limits of apparent power $S = \sqrt[]{P_{t}^{2} + Q_{t}^{2}} \leq 1.2$ p.u. and the power flow reaching a solution within 100 iterations. The tested setpoints for the SCR, $P_{t}$ , and $V_{t}$ range from 0.5 to 4 in steps of 0.5, from 0.1 p.u. to 1 p.u. in steps of 0.1 p.u., and from 0.8 p.u. to 1 p.u. in steps of 0.1 p.u., respectively.

3)　Model of Wind Generators

The suitable Type-III and Type-IV generator models to study weak grid oscillations are given in [

13] and [9], respectively. The common topologies of the converter controllers are shown in Figs. 3-7, where

F_{v}

is the voltage filter;

v_{d t}

and

v_{q t}

are the d- and q-axis components of terminal voltage

V_{t}

, respectively;

ω_{r}

and θ_r are the rotor speed and angle, respectively;

ω_{P L L}

and

θ_{P L L}

are the PLL frequency and angle, respectively;

ω_{0}

is the fundamental frequency;

i_{d g}

and

i_{q g}

are the d- and q-axis currents of grid-side converter (GSC), respectively;

i_{d r}

and

i_{q r}

are the d- and q-axis currents of rotor-side converter (RSC), respectively;

v_{d g}

and

v_{q g}

are the d- and q-axis voltages of GSC, respectively;

L_{G S C}

is the feedforward inductance gain of GSC;

v_{d c}

and

v_{a c}

are the DC and AC bus voltages, respectively;

P

and

Q

are the active and reactive power at terminals, respectively; PI_PLL, PI_GSC, PI_P, PI_Q, PI_ac, and PI_dc are the proportional-integral (PI) gains of different controllers; and the superscript ref represents the reference value.

Fig. 3 PLL topology for Type-III and Type-IV generators.

Fig. 4 Internal current control of converters.

Fig. 5 Current reference control of GSC for Type-III generator.

Fig. 6 Current reference control of RSC for Type-III generator.

Fig. 7 Current reference control of GSC for Type-IV generator.

It is important to highlight that this letter aims for a qualitative analysis of the relationship between the setpoint and the control instability, and does not focus on specific control topologies nor tunings, which vary from vendor to vendor.

If the operational setpoint is feasible, the positive- and negative-sequence impedance matrices of the generator can be calculated from the frequency response of its terminal current $I_{t} (f)$ to the terminal voltage $V_{t} (f)$ disturbances at frequency $f$ . Figure 8 presents an example of sequence impedance profiles near fundamental frequency of the Type-III and Type-IV generators (0.69 kV, 2 MVA), where subscripts pp, pn, np, and nn represent the positive-positive, positive-negative, negative-positive, and negative-negative sequence impedance terms, respectively. Notice that the profile is more complex for the Type-III generator due to the slip of the induction machine. Moreover, such coupling terms have a comparable magnitude to the diagonal terms.

Fig. 8 Sequence impedance profiles near fundamental frequency. (a) Type-III generator. (b) Type-IV generator.

The base values for parameters of the Type-III and Type-IV generators tested in the sensitivity study are shown in Table I. These are the PI control gains of PLL ( $K_{p P L L}$ , $K_{i P L L}$ ), the current control loop of GSC ( $K_{p G S C}$ , $K_{i G S C}$ ), the active power control loop ( $K_{p P}$ , $K_{i P}$ ), the reactive power control loop ( $K_{p Q}$ , $K_{i Q}$ ), the AC bus voltage control loop ( $K_{p v a c}$ , $K_{i v a c}$ ), and the DC bus voltage control loop ( $K_{p v d c}$ , $K_{i v d c}$ ). The DC bus capacitance value ( $C_{d c} = 0.01$ F) is also tested. Both generator controllers are tuned equally for comparison.

TABLE I Base Values for Parameters of Type-IIII and Type-IV Generators

Parameter	Value	Parameter	Value
K_pPLL, K_iPLL	15, 45	K_pvac, K_ivac	2, 20
K_pP, K_iP	0.1, 3	K_pvdc, K_ivdc	8, 400
K_pQ, K_iQ	0.1, 1	K_pGSC, K_iGSC	0.83, 5

These control gains are chosen for this letter as they interact with the fundamental frequency signals as well as the operational setpoint (active/reactive power and terminal voltage) [

7]. The DC bus voltage control gains and the DC bus capacitance are also evaluated as this subsystem, which has been reported to interact with the PLL at weak grids [10].

4)　Range of Generator Parameters for Sensitivity Study

The multiplier array for the PLL gains is set to be [0.1, 1, 10, 100, 200] in order to match the gains evaluated in [

8] for the weak grid stability studies. The multiplier array for all other control gains is set to be [0.1, 1, 10].

The intention of using large variation intervals of the gains is to explore the possibility of control instabilities at weak grids with poorly tuned controllers, given that the original control tuning in Table I is stable at almost all tested setpoints.

5)　Stability Criterion

The stability of converter-based systems can be studied with the Nyquist criterion [

4], [12], and the impedance matrices of WP and grid in the positive-sequence and negative-sequence domain are shown as:

\{\begin{array}{l} Z_{s c} (f) = [\begin{matrix} z_{s c, p p} (f) \\ z_{s c, n p} (f) \end{matrix} \begin{matrix} z_{s c, p n} (f) \\ z_{s c, n n} (f) \end{matrix}] \\ Z_{W P} (f) = [\begin{matrix} z_{W P, p p} (f) \\ z_{W P, n p} (f) \end{matrix} \begin{matrix} z_{W P, p n} (f) \\ z_{W P, n n} (f) \end{matrix}] \end{array}

(1)

where the subscripts WP and sc represent WP and power grid, respectively.

The sequence impedance matrices $Z_{W P}$ and $Z_{s c}$ are calculated at the PCCs of the WP and the power grid, respectively. The generalized Nyquist criterion dictates that if the eigenvalues of the matrix product $Z_{s c} Z_{W P}^{- 1}$ encircle $(- 1,0)$ in the complex plane, the system is unstable. An example shown in Fig. 9(a) is considered for a Type-III WP with $S C R = 1$ , $P_{t} = 0.5$ p.u., $V_{t} = 1$ p.u., a PLL bandwidth of 16.73 Hz, and no capacitor bank.

Fig. 9 Unstable oscillation of Type-III WP after change in active power. (a) Nyquist plot (eigenvalues). (b) EMT simulation results (current at PCC).

The power flow is feasible, and when the output power is reduced to $P_{t} = 0.1$ p.u., the impedance-based Nyquist criterion dictates that unstable oscillations are expected at $60 \pm 8.5$ Hz. The electromagnetic transient (EMT) simulation results in Fig. 9(b) validate the unstable oscillations of the WP.

B. Sensitivity Study Results

The most relevant results of this letter are summarized as follows. The stability of generators near the fundamental frequency is most sensitive to PLL gains, active power control, and current control, because they lead to the largest regions.

The unfeasible and unstable regions from the sensitivity study of the PLL gains are shown in Fig. 10.

Fig. 10 Unfeasible and unstable regions from sensitivity study of PLL gains. (a) K_pPLL (Type-III WP). (b) K_pPLL (Type-IV WP). (c) K_iPLL (Type-III WP). (d) K_iPLL (Type-IV WP).

The larger unstable regions are reached for the lower proportional gains and the larger integral gains, which are associated to a large bandwidth (BW) with low damping, as confirmed by the PLL tuning scenarios in Fig. 11 (Scenario 1: $K_{p P L L} = 15$ , $K_{i P L L} = 45$ ; Scenario 2: $K_{p P L L} = 0.15$ , $K_{i P L L} = 45$ ; Scenario 3: $K_{p P L L} = 1500$ , $K_{i P L L} = 45$ ; Scenario 4: $K_{p P L L} = 15$ , $K_{i P L L} = 0.45$ ; and Scenario 5: $K_{p P L L} = 15$ , $K_{i P L L} = 4500$ ), where Scenarios 3 and 5 lead to the largest oscillations.

Fig. 11 PLL performance for different tunings. (a) Frequency response. (b) Response to step $Δ v_{q} = 0.05$ p.u..

The PLL is more susceptible to instability if the generator is operated at low-voltage setpoints. However, if properly tuned, it provides a large stability margin even under large variations of the tunings of other controllers.

The unfeasible and unstable regions from the sensitivity study of the active power control loop are shown in Fig. 12 for the Type-III WP. The Type-IV WP does not produce unstable regions. Notice that as for the PLL, smaller proportional gains and larger integral gains lead to instability, especially at low-voltage setpoints. The Type-IV WP does not present any regions because the power controllers are located in the machine-side converter, which is decoupled from the grid dynamics by the DC bus.

Fig. 12 Unfeasible and unstable regions from sensitivity study of active power control loop. (a) K_pP. (b) K_iP.

The unfeasible and unstable regions from the sensitivity study of the DC bus voltage control loop are shown in Fig. 13 for the Type-III WP. Again, the Type-IV WP does not lead to instability regions. In this case, notice that this system is not very relevant for the stability analysis unless operating at low terminal voltages and low gains.

Fig. 13 Unfeasible and unstable regions from sensitivity study of DC bus voltage control loop. (a) K_pdc. (b) K_idc.

The unfeasible and unstable regions from the sensitivity study of the current control loop of GSC are shown in Fig. 14. Notice that for both WPs, lower proportional gains severely increase the risk of instability at weak grids. Additionally, too high proportional gains can also lead to instability with low power injections and at weaker grids. This means the controller must be properly tuned specially in WPs connected to weak grids. The integral gains do not lead to relevant unstable regions.

Fig. 14 Unfeasible and unstable regions from sensitivity study of current control loop of GSC. (a) K_pGSC (Type-III WP). (b) K_pGSC (Type-IV WP). (c) K_iGSC (Type-III WP). (d) K_iGSC (Type-IV WP).

C. Discussion and Qualitative Takeaways

1) Influence of Setpoint Variables

Generator impedances and the system stability are functions of the setpoint and grid strength. Studies in the literature tend to isolate one variable at the time [

7], but to study the effect of the setpoint with better accuracy, the combined influence of V_t(Q_t), P_t, and SCR is recommended. These variables of the setpoint have a coupled effect, as the grid strength, the terminal voltage, and active power injection define the reactive power injection, which in turn changes the generator impedance profile.

This can be visualized through the irregular shape of the colored regions along the axes for fixed control gain values, and how these regions change for different values of terminal voltage.

2) Higher Risk Control Gains

Higher risk control gains are more likely to result in instability, as observed by the growing tendency of the regions.

However, inadequate control tunings (see the case of $K_{p P L L} = 0.1$ ) at weak grid can also result in unstable oscillations. Most works in the literature leading to instabilities near the fundamental frequency use PLLs with very large gains, thus, resulting in large bandwidths [

8]. Properly tuning the PLL with low gains reduces the risk of the unstable control interactions with power grid.

The active and reactive power control gains do not lead to unstable regions in the Type-IV WP. This occurs because the Type-IV WP power control is located in the machine-side converter, which is decoupled by the action of the DC bus voltage from the power grid dynamics.

The gains of the controller for DC bus voltage do not lead to significant regions in either of the WPs, and different values of the DC capacitor do not lead to instability regions. This is because the original PLL gains have a good stability margin. If they are poorly tuned with large BW, such gains may have a greater effect on the stability of weak grid oscillations.

3) Higher Risk Setpoints

As expected, most problems appear at low SCR and high P_t, i.e., unfeasible setpoints occur for $S C R \leq 1$ and $| P_{t} | \geq 0.5$ p.u.. The generators at lower-voltage setpoints are more susceptible to instability as the controller has greater difficulty in remaining synchronous with the grid while providing the required active power of the setpoint. Such condition is analog to a weaker grid. Moreover, the risk for unstable oscillations is higher when the generators absorb reactive power due to the change in the impedance profile, which depends on the setpoint.

4) Distinction Between Regions

In order to test for instability, the setpoint feasibility has to be confirmed first. As expected, the region of unfeasible setpoints is located with the largest power injections and at weaker grids. This occurs for both the Type-III and Type-IV WPs. Additionally, it increases in size for the lower values of terminal voltage. Moreover, the regions of instability may appear at any place of the chart, but also tend to increase their size with the lower terminal voltages.

5) Type-III WP v.s. Type-IV WP

Type-III WPs appear to be more susceptible to the instability than Type-IV WPs, when evaluated under the same control and setpoint conditions. This can be explained by the better decoupling of the Type-IV WPs via the DC bus, which isolates the active power control stage and rotating machine from the power grid.

6) Risk Associated to Wind Conditions

As expected, the higher power injections are more likely to lead to setpoint unfeasibility at weak grids. The high wind condition has been described as a key factor for instability in [

4]. However, as shown by the results, low power injections can also lead to instability with poor control tunings.

D. Numerical Quantification of Risk

From the previous charts, a numerical risk index can be derived as the ratio of the area corresponding to the phenomenon to the total area of the chart. For example, Fig. 10(c) and (d) presents the regions from the sensitivity study of the PLL integral gain. Two risk indices can be derived from the figures: ① unfeasibility risk index $U_{R}$ , which is the percentage of all scenarios leading to unfeasible setpoints; and ② instability risk index $I_{R}$ for each value of gain, which is the percentage of all scenarios leading to unstable weak grid oscillations.

The risk indices corresponding to Fig. 10(c) and (d) are shown in Table II. Notice that Table II allows to compare the risk among different scenarios numerically in order to show which operational setpoints lead to higher risk of unfeasibility or instability, and which control tunings are more susceptible to instability. For example, Type-III WP has a higher risk of unstable weak grid oscillations, and the risk increases for both generators at the lower terminal voltages.

TABLE II Risk Indices Calculation from Chart Results

$V_{t}$ (p.u.)	$U_{R}$ (%)	$I_{R}$ of Type-III WP (%)					$I_{R}$ of Type-IV WP (%)
$V_{t}$ (p.u.)	$U_{R}$ (%)	0.1 $K_{i P L L}$	$K_{i P L L}$	10 $K_{i P L L}$	100 $K_{i P L L}$	200 $K_{i P L L}$	0.1 $K_{i P L L}$	$K_{i P L L}$	10 $K_{i P L L}$	100 $K_{i P L L}$	200 $K_{i P L L}$
0.8	9	1	3	2	8	61	0	0	0	15	58
0.9	8	0	0	2	5	50	0	0	0	15	59
1.0	6	0	0	2	4	43	0	0	0	14	56
1.1	6	0	0	2	4	36	0	0	0	11	50

E. Relationship Between Setpoint Unfeasibility and Fundamental Frequency Instability

An interesting outtake can be observed is that by using the Nyquist plot, unfeasible setpoints can be appreciated as instabilities at the fundamental frequency. The Nyquist plots of Fig. 15 show the change in eigenvalues after transitioning to a high-power setpoint close to the limit of feasibility.

Fig. 15 Relationship between unfeasible setpoint and instability at fundamental frequency for Type-III WP ( $S C R = 1$ , $V_{t} = 1$ p.u.). (a) $P_{t} = 0.5$ p.u. (stable). (b) $P_{t} = 0.95$ p.u. (unstable).

This result is validated with EMT simulation by increasing the active power output of the generator in sequential steps of $Δ P_{t} = 0.05$ p.u., starting at $P_{t} = 0.5$ p.u. until reaching the unfeasible setpoint at $P_{t} = 0.95$ p.u.. The current magnitude at the PCC of WP is shown in Fig. 16. Notice that the signal only has fundamental frequency, as this instability is not related to the controllers but to the unfeasibility of the setpoint.

Fig. 16 Unstable oscillation at fundamental frequency for Type-III WP after entering unfeasible setpoint of P_t=0.95 p.u..

III. Conclusion

Oscillations resulting from unfeasible setpoints and control instability of WP controllers both occur near the fundamental frequency and at weak grid. However, these are different phenomena. The control instability can be studied with impedance models, but they depend on the operational setpoint. To map the conditions for each phenomenon, i.e., the circuit configurations leading to instability or unfeasibility, this letter presents a sensitivity analysis comparing Type-III and Type-IV generators at multiple setpoints and control tunings, and the results are presented as charts separating unfeasible and unstable regions of setpoint.

The results demonstrate that weak grids, low terminal voltages, high active power injections, and poorly-tuned PLLs increase the risk of instability. In addition, Type-III generators are just as vulnerable as or even more vulnerable than Type-IV generators, as the latter have a better electric decoupling from the the controllers and the rotating machine through the DC bus of the converter, which reduces their influence on the impedance profile of the generator.

As an interesting result, it is also noted that the trajectory of the eigenvalues in the Nyquist plot that lead to unfeasible setpoints approaches a fundamental frequency instability.

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