Improved Reduced-Order Model for PLL Instability Investigations

There is a rapid increase in the installed capacity of offshore wind power plants (WPPs) worldwide, and this trend will continue in the context of the recent EU energy crisis and the Ukraine war. As the share of the capacity of wind power plants increases in power systems, it is essential that WPPs can stay connected to the grid during different operational events and provide ancillary services. Recent research has identified the Phase Locked Loop (PLL) converter control as a contributing factor to instabilities driven by large-signal disturbances (i.e. severe grid faults). This type of instability is referred to as grid-synchronisation stability (GSS). In light of this, grid codes are being revised to secure the connection of WPPs and specify the services that need to be provided. To study GSS, it is essential to accurately model the wind turbine (WT) system and grid conditions. However, the actual WT models are often unavailable, black-boxed, or computationally too heavy to model in detail. Thus, simplified reduced-order models (ROM) resembling the actual system behaviour are the need of the hour. This paper presents an improved reduced-order model that considers a systematic approach to modelling the misalignment of the PLL angle in the converter states. Moreover, it considers the time-varying parameters and the step changes of initial states after fault triggering, which otherwise are neglected in the present literature. Our focus in this paper is to propose a novel WT ROM and demonstrate its effectiveness for offline studies. Based on transient stability studies, the improved reduced-order model accurately tracks the grid-angle post-grid disturbances, and its trajectory is a good match compared to a detailed simulation model in the time domain.


I. INTRODUCTION
In 2020, worldwide offshore wind generation experienced significant growth, reaching 25 TWh with capacity additions of 6 GW, representing a 12% increase compared to 2019. To achieve 8000 TWh by 2030, as outlined in the Net Zero Emissions by 2050 Scenario, annual offshore wind capacity growth needs to be increased to 80 GW [1]. With the rapid deployment of wind energy, offshore wind power plants (WPPs) are expected to play a crucial role in ensuring stability and control in future power grids.
Traditionally, transient/large-signal stability studies have focused on analysing the power system dynamics using RMS The associate editor coordinating the review of this manuscript and approving it for publication was Emilio Barocio. simulations, assuming that stability is primarily influenced by the electromechanical states of conventional power plants [2]. However, under high renewable penetration conditions, the grid-tied power-electronic renewable generation exhibit very high control dynamics during system transients such as fault ride-through (FRT) and fast-acting control actions. The time constants of such controls are smaller than those of a typical electromechanical phenomenon, resulting in possible interactions with the electrical states of the system [3]. Consequently, current industry practices have evolved to use EMT simulations to investigate the transient stability of grid-tied renewable generation.
Recent research from EMT studies has revealed that under weak grid conditions, grid-tied renewable generation with grid-following controls is prone to be small-signal unstable due to the interactions between the grid and its controls, in which the phase-locked-loop (PLL) plays a crucial role. Aside from this, another unstable phenomenon related to the PLL is emerging, driven by large-signal disturbances (i.e. grid faults). This type of instability is referred to as grid-synchronisation stability (GSS) [4], [5].
In a wind turbine, the PLL provides the instantaneous grid voltage, phase angle, and frequency. Despite the wide variety of PLL algorithms, the most widely used PLL is based on the synchronous reference frame (SRF) approach. In fact, some recent proposals entirely rely on the SRF-PLL dynamics, which are enhanced with a prefilter such as doubly decoupled SRF PLL, moving average SRF PLL, prefilter moving average SRF PLL [6]. In the absence of the modified PLL structure, considering only the basic version of the SRF-PLL structure provides a less complex analysis.
In order to investigate GSS, it is essential to model the WT converter system and the grid conditions accurately. However, the actual EMT models are often unavailable, black-boxed, or computationally too heavy to model in detail. Simulating a manufacturer-provided EMT model for an entire wind farm can take a long time, and stability can only be observed for the predefined events simulated. This could result in missing out on critically stable events. Hence, simplified reduced-order models resembling the original system behaviour have gained prominence in stability studies.
In [7], a methodology for a reduced-order model (ROM) based on power loss conservation in a solar farm is presented. A similar concept was extended to wind farms in [8]. Our present work only focuses on ROMs for wind turbines. Owing to several advantages, type-4 WT with back-to-back converters has gained prominence for offshore wind applications. In several works, including [9], [10], a type-4 WT is modelled just as a grid-side converter (GSC), where the turbine generator and the rotor-side converter (RSC) are not considered; instead, to observe their effects, the active power is simulated through a controlled current source. This significantly reduces the complexity of the WT model.
For a GSC, in [11], [12], [13], [14], [15], and [16], the fast system dynamics are neglected when analysing the slow PLL dynamics. As a result, the inner current loop is approximated as a unity gain. Consequently, the type-4 WT is simply modelled as a controlled current source with its phase angle regulated by an SRF-PLL. Furthermore, in [13], the SRF-PLL is decomposed into two feedback loops -''grid synchronisation loop'' and ''self-synchronisation loop'', and the authors show how the low-frequency PLL dynamics are affected when the impedances of grid changes. In [14], the SRF-PLL is approximated as a synchronous machine, where the active current reference is constrained based on the system's stable equilibrium points.
In general, the WT ROMs in [11], [12], [13], [14], [15], and [16] address some nonlinear characteristics, such as low-frequency PLL dynamics, interactions between the PLL and the system, and the impact of weak grids on the transient system behaviour. However, there is further scope for improvement. The existing ROMs lack a systematic approach to modelling the misalignment of the PLL angle. They also lack consideration of time-varying parameters and neglect the jumps in initial states at the instant of fault triggering. The ROMs must behave similarly to a detailed EMT model; otherwise, the application of such ROMs will be limited to certain cases.
This paper aims to improve the WT ROMs to investigate grid-synchronisation stability. Time-domain studies are also presented to support the improved performance. The main contribution of this paper is as follows, 1) A systematic approach is proposed to model the misalignment of the PLL and system reference frames. It is known that PLL misalignment causes incorrect measurement of real-world signals, which can lead to control instability. This work will present a clear modelling approach, including the voltages and currents of the system and control reference frames of the converter. 2) An improved PCC voltage equation is introduced, resulting in a time-varying WT ROM. The proposed ROM considers time-varying parameters such as ramp changes in grid frequency and fault current injections, which have been neglected in existing literature. 3) An analytical equation is provided to calculate the jump in initial states after grid disturbances. Different initial states can lead to different trajectories that may converge or diverge from the equilibrium point. A slightly different initial condition can result in an unstable system that was originally stable. The proposed set of equations for calculating phase/frequency jumps is shown to match the trajectories of a detailed EMT WT model.
The proposed WT ROM is computationally faster than detailed EMT models owing to its reduced-order nature. Since assumptions are considerably reduced, the proposed ROM is highly accurate over a wide range of frequencies.
The proposed ROM also has a higher degree of application since the time-varying terms are considered. Several grid disturbances are discussed and simulated in the paper. Section II presents the methodology for the improved reduced-order model of the WT converter system and the grid. Section III presents the reduced-order model's performance and is compared against a detailed WT simulation model in PSCAD. Finally, Section IV presents the conclusion. Fig. 1a and Fig. 1b are presented in order to better explain the grid-synchronisation instability event physically. During severe grid faults, the fault point voltage drops down close to zero. According to requirements in some grid codes, the wind turbines need to inject controlled active and reactive currents to an inductive impedance load with a very low voltage at the terminal. For weak systems, the amount of active and reactive current injections during a fault is important for stability, and VOLUME 11, 2023 if the fault is not cleared in time, the converter may lose synchronism. It must be noted that synchronisation instability also concerns other grid disturbances, such as grid voltage phase jumps, large frequency changes, and disconnection of transmission lines in the external grids, among others.

A. REDUCED ORDER MODEL OF WIND TURBINE
In this work, a type-4 WT with only the GSC is considered. Under grid faults, once the dc chopper is activated, the dc voltage is assumed to be steady during the faults (as seen in Fig. 1a). Further, the inner current controller is considered fast enough to regulate the DQ-domain currents. Consequently, the dynamics of the inner current loop are avoided. This assumption is valid since fast system dynamics can be neglected when analysing slow PLL dynamics.
Furthermore, the shunt capacitor of the filter is also not considered, where in the later section, it will be shown that the capacitor impact on GSS is negligible when the current is controlled on the grid-side inductor of the LCL filter in the detailed WT model. Subsequently, the reduced-order representation of the WT and grid in the DQ domain is illustrated in Fig. 1c.
A PLL with SRF approach is used since such a structure provides a less complex yet highly intuitive analysis. It is well known that the PLL misalignment causes a wrong measurement of the real-world signals, which can cause instability in the controls.  From Fig. 2b, the misalignment in converter currents and voltages can be written as (1) and (2).
where,  (1) and (2) introduces nonlinearities in the control system. It must be noted that the WT system is modelled in a DQ frame, rotating at a fixed frequency ω 0 . The grid frequency could be different, but that is addressed by assuming that the grid voltage in DQ is rotating with (ω g − ω 0 ). The parameters of the system are considered to be time-varying. This is used later in this paper to model different grid disturbances and fault current injection.
As per Fig. 1b, based on superposition principle, the phase voltage v pcc can be derived as, Furthermore, (4) represents the DQ voltages at PCC in the system reference frame. It must be noted that in (4), L g (= dL g dt ) represents the change in grid inductance during a grid disturbance; in steady state operationL g = 0. In (4), the symbol 's' is the Laplace operator, where, for example, L g si s d = L g (di s d /dt). A systematic approach to modelling the misalignment of the PLL angle is presented in (4), (5) and (7), where all the converter states (voltage and currents) are transformed to the control reference frame.
The RHS of (4) can be simplified as (5) In order to track θ df , a standard SRF-PLL tries to keep vpcc c q to zero, and alsoθ g = ω g − ω 0 . The control equations of an SRF PLL are given by (6).
where, k p and k i are the PLL controller gains, while from (5), Equation (7) is an improved PCC voltage equation, where the third term(L g i c q ) is a novelty of this paper. It should be noted that i c d , i c q , V g , and ω g are not necessarily time-invariant during the PLL synchronisation analysis, i.e. the grid frequency or a post fault active current recovery might change as a ramp function.
Furthermore, subtractingθ g =ω g from both sides of (8), and solving forδ results in a swing equation look alike, where, If a time-invariant model is considered, i.e. by removing the derivatives in (10), we obtain the same model discussed in [17]. The ROM can then be rewritten in ODE form as, where, x 1 = δ and x 2 =δ, and On the other hand, if a system parameter varies over time, such as post-fault active current ramp injection from the VOLUME 11, 2023 converter, the ROM can then be rewritten in ODE form as, where, x 1 = δ, x 2 =δ and x 3 = i c d , and Similarly, for system parameter variation such as ramp in the grid frequency, the ROM can be formulated by considering the grid-frequency derivatives defined in (10). It must be noted that the improved ROM (9) has been derived based on differential equations without any RMS assumptions.

B. INITIAL VALUE AFTER A DISTURBANCE
When the system parameters change continuously, the state variables also continuously change. However, if there is a step-change in the system, e.g. due to a fault, there will be an impulse in the state equations, as (10) contains derivatives of parameters. The easiest way to deal with this discontinuity is to solve the ODEs piece-wise. The ODE can be solved numerically in periods where there is no step change in the parameters, and it is enough to update the initial values [17] at the beginning of each period to address the step changes.

1) JUMP IN INITIAL STATES FOR δ
Jumps in θ df are dependent on the derivatives in (6a), where the change in θ df can be represented as, where, [X ] = X post−event − X pre−event . It must be noted that since in the second term of (15),δ is bounded, therefore, for a very short time, the term does not change much, and hence it can be neglected. Further, subtracting [θ g ] from both sides of (15) results in (16) that describes the jump in initial states for δ postdisturbance,

2) JUMP IN INITIAL STATES FORδ
The method described in Section II-B1 can be similarly applied to obtain the analytical equation for the change in the PLL frequency. For frequency we have, θ df = k p vpcc c q + k i vpcc c q dt (17) The second term on the right-hand side of (17) can be treated similarly to (15).
By replacing, and subtracting [θ g ] from both sides of (19) results in, (20) where, subscript pr and po is the is the pre-event and postevent units, respectively. To better understand (16) and (20), it is advised to go to Section III-A.

III. CASE STUDIES
In this section, the proposed ROM's performance is analysed and compared against an EMT switching simulation model of a WT, implemented digitally in PSCAD, where the system parameters are taken from the CIGRE benchmark model, C4.49 [9]. Table 1 presents the parameters describing the WT system considered in this study. The proposed ROM is tested against grid disturbances such as grid phase jumps, frequency change, grid voltage sags, fault ride-through and grid impedance change.

A. GRID VOLTAGE PHASE ANGLE JUMP
As discussed in Section II-B, a step-change in the system parameters can cause a discontinuity in the solution of (9). To better present this, a simulation is performed where the grid phase angle is stepped up from 0 • to 20 • at t=0.1 sec. Consequently, two time periods are defined, one before the step and one after. The analysis of the first period (before the step) can be neglected as it is assumed that the step is applied when the system is at the equilibrium point, as shown in (21).
As per (16), a 20 • phase jump in θ g causes a negative 20 • jump in δ. The PLL angle is constant during the step because there is no other parameter change, and the PLL needs time to synchronise to the new angle. However, according to (20), this angle shift causes a step-change in the PLL frequency: It can be seen from Fig. 3a that the jump in δ andδ is identical to the one calculated in (16) and (20). It is seen that the PSCADδ has a slight overshoot and delay; this can be accounted for by the finite current controller bandwidth in PSCAD simulations and the infinite current controller bandwidth considered in the ROM. However, the trajectories until the steady state is similar to the one obtained from PSCAD simulations, as shown in Fig. 3b.

B. GRID FREQUENCY VARIATION
The grid frequency is changed by a 10Hz/s ramp from 50Hz to 52Hz. Since no step-change occurs, the continuity in the trajectories is preserved before, during and after the ramp (see Fig. 4). A very low amplitude oscillation is observed in the PSCADδ trajectory; this is due to the PWM switching within the GSC. A good match in δ andδ trajectory is observed between the PSCAD simulations and proposed ROM.

C. GRID VOLTAGE SAG
The grid voltage magnitude is stepped down from 1.0 pu to 0.9 pu at t=0.1 sec. Similar to Section III-A, the jumps in δ andδ can be computed for a grid voltage sag. As per Fig. 5, a good match in δ andδ trajectory is observed between the PSCAD simulations and the proposed ROM.

D. FAULT RIDE THROUGH
In this case study, a complete Fault ride-through is simulated. When a symmetrical fault occurs (the grid voltage is stepped down to 5%), the active current is stepped down to 0.1 pu, and the reactive current is stepped up to -1 pu. When the fault is cleared after 100 ms (the voltage is stepped up back to  1pu), the reactive current is set to pre-fault value; however, the active current is ramped up (e.g. 2.84e4 A/s), not to stress the mechanical structure. The sequence of events can be observed in Fig. 6 (sub-figure three). Based on the step changes, the entire simulation can be split into three periods, whose initial states can be computed from (16) and (20). As seen in Fig. 6, there is a good match between the proposed ROM and the  Furthermore, when a fault with 0.1% grid voltage is applied and is cleared after 400 ms, it is observed in Fig. 7 that the system becomes unstable. Once again, there is a good match between the ROM and the detailed PSCAD model.

E. GRID IMPEDANCE VARIATION
The last test case is a step-change in the grid impedance. This case can be compared to opening a line in the transmission system. The system, in this case (see Fig. 8), is stable when the grid SCR is changed from 3.3 to 1.6. A good match in the trajectories is observed between the PSCAD simulations and the proposed ROM.

F. DISCUSSION
A comparison of the computational time for the proposed ROM and the PSCAD simulation model is presented in Table 2. All the study cases are listed. It is seen that the proposed WT ROM is computationally much faster than the detailed EMT model owing to its reduced-order nature.  Our proposed ROM is generalised, i.e. all the system parameters have derivatives which allow ramp changes in parameters, such as frequency ramp changes in grid frequency and ramp changes in post-fault active current, among others. It should be noted that previous RoMs cannot model ramps; therefore, they cannot be compared one-to-one; these characteristics illustrate the novelty of our work.
It is also seen that the dynamic performance of the proposed reduced order model is quite accurate over a wide range of frequencies owing to the consideration of time-varying terms. However, some deviations exist in the high-frequency range, mainly driven by component modelling in PSCAD. For example, a low-magnitude oscillation in the PLL frequency is seen, which can be attributed to the PWM's switching dynamics. Also, a slight delay with an overshoot in the PLL frequency is observed at the instant of a disturbance. This behaviour is introduced due to the current control loop (i.e. tuning a fast controller results in high overshoots, and the controller modelling inherently introduces a delay). Such high-frequency dynamics can be ignored when analysing the overall low-frequency synchronisation dynamics (i.e. the PLL acts as a low pass filter) due to PLL. A limitation of the proposed reduced-order model is that it holds as long as the current controller has a high bandwidth compared to the PLL bandwidth. Additionally, the dc bus voltage was considered constant during faults (considering the action of the dc chopper); however, this may not always be true. In light of this, modifying the voltage (7) to consider system dynamics/damping applicable due to dc bus voltage variation could be fruitful.
Our focus in this paper was to propose a novel WT ROM and demonstrate its effectiveness for offline studies. Regarding the availability of detailed system parameters, we understand that this may not always be possible, especially for complex systems. However, we believe that our proposed approach is still useful in cases where relevant system parameters may be obtained through system identification techniques. We will address this challenge in future research. Furthermore, the focus of this paper is on modelling and analysis and not on design. As long as the PLL dynamics are slower than the current controller, the proposed ROM is applicable. When the dynamics are comparable, new solutions must be developed; this paper targets a specific problem and provides solutions.
In general, the proposed ROM has a high confidence in application in transient stability investigations of WTs. The advantages are two-fold. Firstly, during the system modelling stage -since actual WT EMT models are often unavailable or black-boxed, the proposed model could be used or fitted as not much modelling data is required. Also, since it considers time-varying terms, testing all types of large symmetrical disturbances is possible. Additionally, using the proposed ROM reduces the computation burden considerably. Secondly, the trajectories obtained are trustworthy during transient stability assessment -since the proposed ROM is accurate over a wide range of frequencies. In the end, since the WT ROM is equivalent to a swing equation of a synchronous generator, analytical stability boundaries can be obtained without running time domain simulations.

IV. CONCLUSION
An improved reduced-order model of a type-4 wind turbine was developed to investigate large signal/ gridsynchronisation stability. The methodology presents a systematic way to model the misalignment of the PLL angle in the converter states. A novel PCC voltage equation is derived, resulting in a time-varying WT ROM. Further, the reduced-order model considers the jump in initial states post-disturbance, which has been neglected in the existing literature.
Based on the modelling and studies carried out in this paper, it is observed that post grid disturbances, the proposed reduced-order WT model correctly tracks the angle and frequency, and its trajectory is a good match when compared to a detailed simulation model in PSCAD. The time-domain simulations showed that the assumptions such as neglecting the current controller due to slow PLL bandwidth and ignoring the shunt capacitor filter, are valid considerations when analysing the large-signal/ grid-synchronisation stability of a type-4 WT.
A limitation of the proposed WT ROM is that it holds as long as the current controller has a high bandwidth compared to the PLL bandwidth. When the dynamics are comparable, new solutions must be developed. Additionally, as a future scope of this work, the dc bus voltage variation may be considered along with adapting the proposed ROM for unbalanced grid disturbances.
In general, the proposed WT ROM is computationally faster than detailed EMT models due to its reduced-order nature. Since assumptions are considerably reduced in the modelling, the proposed ROM is highly accurate over a wider range of frequencies. The proposed ROM also has a higher degree of application since the time-varying terms are considered.