Dynamic characteristics of virtual inertial response provision by DFIG-based wind turbines

Power converter technology partially or fully electrically decouples the wind energy source from the grid which results in the decrease of system inertia. However, when those units participate in virtual inertial response their electromechanical dynamics become coupled to the grid electromechanical modes. To date, there were no comprehensive studies on how do different elements and parameters of a wind energy conversion system (WECS) impact its virtual inertial response provision. This is important from the standpoint of understanding the expected wind farm response during frequency containment process as well as from the standpoint of developing better inertial response controllers. In this paper we have investigated how do operating point, line-side and machine-side converter, phase-locked loop and pitch angle control impact the inertial response of the total power controlled type III WECS (DFIG) which is one of the most common wind turbine topologies used today. We show that the operating point, pitch angle control and outer loop of the machine-side converter have a visible impact on strength of the inertial response, while other elements do not and some can even be neglected in inertial response studies.


Introduction
A high rate of penetration of renewable energy sources (RES) in the last decade has brought along certain problems for power systems of today as well as for power systems of tomorrow. Variable and stochastic RES (of which wind energy and solar photo-voltaic (PV) energy are the most prolific representatives) are connected to the grid via power electronic interface which ensures power production at the rated grid frequency. Connection of this converter-connected generation and decommissioning of large synchronous units has a couple of consequences: 1. grid inertia is reduced since power electronics decouple the rotating mass from grid frequency (in the case of PVs there is no rotating mass); 2. traditionally, these sources operate at the maximum power point and do not ensure a certain amount of upward reserves 1 That is why a lot of attention has been given to developing auxiliary control algorithms for type-III and type-IV wind energy conversion systems (WECSs) which enable the utilization of their decoupled kinetic energy to respond to frequency disturbances, usually named virtual or synthetic inertia [1][2][3][4][5][6][7][8]. Moreover, WECSs can be operated according to some sub-optimal power curve which ensures a certain amount of power reserve during normal operation [9][10][11][12][13]6,14]. Then, droop control can be added to enable the participation of wind turbine generators (WTGs) in primary frequency control (PFC). This is a heavily researched topic on which we will spend no more time on and we refer the reader to [15] to read more about frequency support services from WPPs. Nevertheless, WECSs are complex electromechanical systems and there were no comprehensive studies on how exactly do different parameters and the operating point impact the inertial response and droop control capabilities of a type-III or type-IV WECS. Type-III wind turbines are usually called doubly-fed induction generator (DFIG) wind turbines and we will use this term onward.
of the converter and machine models does not significantly influence the transient response of the DFIG; DC voltage can be assumed constant.
In [17], the same authors analyzed the impacts of the following aspects on system frequency response: control strategy (power or torque), maximum-power-point-tracking (MPPT) characteristic, initial loading and auxiliary inertial controller parameters. They concluded that torque control is more stable than power control, MPPT curve provides a self-stabilizing mechanism, initial loading has a significant impact of inertial response provision due to the converter limits and that various power and frequency responses could be obtained depending on the design of the inertial controller, but no sensitivity analysis was done. In some of our recent work [18,19], we have analyzed the small-signal response of a total power controlled variable speed wind turbine (VSWT) and we have shown the following: impact of the operating point on the combined VIR and PFC depends on the control structure, more precisely on the power tracking and set-point curve. Weaker response is generally recorded for higher wind speeds. However, the design of the deloaded curve around the maximum rotor speed results in a stronger and more oscillatory response which can sometimes be unstable. We have also shown that the inertia of the wind turbine doesn't have a significant impact on the grid frequency following a disturbance except in the aforementioned region where bigger wind turbines result in deteriorated frequency response. The former is in line with the well-known decoupling effect of power converters, while the latter is a consequence of the power set-point algorithm. Pitch angle servomechanism time constant doesn't have an effect at above rated wind speeds, while in the region around maximum rotor speed slower pitch angle control results in a more oscillatory behaviour of the grid frequency. However, in [18,19], simplified wind turbine model was used (generator and electrical control were not taken into account). The grid was replaced by a simple low-order system frequency response model with turbine dynamics. Recently, research has shown [20,21] that there is a link between the virtual inertial response (VIR) and phase-locked loop (PLL) which can affect both the small-signal stability of the power system and the strength of the VIR. Arani and Mohamed [22] analyzed the impacts of droop control in DFIGs on microgrid and weak grid stability and they also concluded that torque control is more stable than power control. They have also shown that pitch angle controller does not have a significant impact on droop control, but no studies on inertial response were done. Quan and Pan [23] analyzed the impact of operating point of a total power controlled DFIG on simultaneous provision of inertial response and droop response using the model linearization and they have concluded the following: power injection is stronger under lower wind speeds in the medium wind speed region (where power is proportional to the cube of the generator speed). They did not analyze behaviour under low wind speeds nor under high wind speeds. On the other hand, Hu et al. [24] showed that the inertial response of a torque-controlled DFIG is stronger under higher wind speeds (in the MPPT region). They also showed that the higher PI gains of the speed controller weaken the inertial response due to the bigger restraining effect. However, they used a simplified DFIG model. Based on the literature survey, a comprehensive study is required to analyze the impact of various parameters on virtual inertial response from DFIGs. With that being said, this paper is based upon preliminary analysis from [25] and it contains much more exhaustive analysis.

Contribution
Based on the literature survey, one can see that there were no comprehensive studies on the sensitivity analysis of various DFIG elements on inertial response provision. These elements are: machine-side converter (MSC) and line-side converter (LSC) controller parameters, PLL parameters, pitch angle controller parameters and initial operating point). The active power response of a DFIG-based wind turbine and the impact of the aforementioned parameters heavily depend on the type of the control design of the WECS. Furthermore, no multidimensional analysis was done in any of the surveyed literature from Section 1.1. In this paper, we have focused on the total power control as one of the frequently used schemes in the literature. The main contribution of this paper is to understand how different elements of a DFIG WECS impact the provision of inertial response in order to facilitate further research regarding wind turbine control design as well as to shed light onto the fact that different responses from wind farms may be expected during frequency containment process. To the best of our knowledge, this is something that was not done before. Furthermore, we note that the response of the WECS to frequency disturbance differs between different types of models in some aspects. This is an important finding because using different models can result in arriving to contradictory conclusions while doing power systems research.
Therefore, the contributions of this paper are as follows: • analysis of the impact of the DFIG operating point on VIR for the whole operating regions from cut-in to cut-out; • multidimensional sensitivity analysis of the impact of the DFIG control system parameters on the dynamic characteristics of the VIR; • to clarify, illustrate and discuss in detail the characteristics of the total power control of DFIG WECS and how it affects the provision of virtual inertia; • finding that inertial response sensitivity to initial conditions depends on the type of MPPT control algorithm.
The rest of the paper is organized as follows: In Section 2, the test system used for simulations and methodology are presented. In Section 3, we analyze and discuss the impact of the aforementioned variables on the VIR provision by the total power controlled DFIG WECS. Section 4 concludes the paper.

Methodology
Fig . 1 shows the test system used in the simulations. It is a twomachine system consisting of a DFIG-based wind power plant (consisting of 15 aggregated 2 MVA turbines) and a 75 MVA synchronous generator interconnected through a series of lines and transformers. Wind power penetration is equal to 28.5% of total installed capacity. Loads are connected to bus 6. We understand that this is not the most realistic depiction of a power system, but it is the simplest form of a system for studying frequency dynamics. The synchronous machine represents the rest of the bulk power system while isolating the average frequency dynamics from other factors such as grid topology,

M. Krpan and I. Kuzle
Electric Power Systems Research 178 (2020) 106005 interaction between different controllers, different turbine types, etc. which are beyond the scope of this paper. The classical model of the synchronous machine is used and it is equipped with a TGOV1 turbinegovernor model and a IEEET1 automatic voltage regulator (AVR). In all test cases, a frequency disturbance is induced by connecting the 5 MW load at t = 1 s which is equal to 5% of the total generation capacity and 10% of pre-disturbance load. Parameters of the test system are given in the Appendix. Electromechanical transient simulation (RMS values) is conducted in DIgSILENT PowerFactory 2019 software package. Fig. 2 shows the overall control system of the DFIG model used in this paper. MPPT or deloaded operation is set by setting the mode flag to 0 or 1, respectively. Fig. 3 shows the generator power vs. rotor speed curve for both MPPT and deloaded operation. However, since we are focusing on VIR, the DFIG will be operating according to the MPPT curve. Parameters of the complete wind turbine system are given in the Appendix. Now, we briefly present the mathematical model of the DFIG wind turbine used in this paper. Wind turbine mechanical power is calculated as (1) [26]: where the aerodynamic power coefficient C p is a complex analytic expression and can be found in [27]. Electrical power set-point P ppt (power-point tracking) depends on the operating region of the turbine (Fig. 3) and is described by (2)-(4): In the region B − C/B′ − C′ generator power is proportional to the cube of the generator speed and a coefficient k g which can be calculated using an iterative method due to the nonlinearity of the aerodynamics [28]: In the region C′ − D′ in deloaded mode, the maximum speed is reached and the power set-point is obtained from the estimated wind speed [29,30]: Generator power and speed at points ABCD/A′B′C′D′ are constant coefficients calculated during the turbine design. At points C and D′ power set-point is equal to the nominal and maximum deloaded power, respectively-and the pitch angle controller restricts the turbine speed to the maximum speed. Shaft dynamics are described by the two-mass model (5): DFIG model is an existing element in DIgSILENT PowerFactory software modelled in rotor reference frame. It is controlled with the modulation factors P* mq and P* md from the MSC. P* mq and P* md are expressed in the rotor reference frame, however they are initially obtained in the stator-flux reference frame in which the MSC operates.
LSC as well as the DC capacitor are additionally considered as well. LSC operates in the stator-voltage reference frame. Both MSC and LSC are modelled with the fundamental frequency model (for RMS studies) of the voltage source converter element with sinusiodal PWM. The converter AC voltage V ac is related to the converter DC voltage V dc by (6): where P* mr and P* mr are real and imaginary parts of the modulation index depending on the reference frame used. All the MSC and LSC PI controllers are non-windup to prevent the windup effect of the integral controller. Pitch control is described by (7). Pitch controller is also a nonwindup controller. Pitch angle β is limited between 0 and 27°and the pitch rate is limited to ± 10°/s.
3. Dynamic characteristics of virtual inertial response

Impact of initial wind speed
The DFIG is initialized at point B (Fig. 3) and the wind speed is linearly increased from ∼7 m/s to ∼25 m/s. Pitch control becomes active around 12 m/s. Inertial response is not considered in zone AB because of the low speed and the increased possibility of stalling. Impact of the initial conditions (initial wind speed) is visible in Fig. 4a-d. Generally, when the inertial response is activated at point B (in this case

M. Krpan and I. Kuzle
Electric Power Systems Research 178 (2020) 106005 around 6.9 m/s), the generator speed will drop resulting in shifting the power set-point curve from BC to the line AB. This in turn results in slightly lower peak time, lower peak value and a larger undershoot of the power injection. When the wind speed is high enough, the generator speed will stay in the BC region. Here, the peak value of the inertial response falls linearly with the wind speed while there is no significant impact on the time when the peak value is reached. The explanation can be found in the small-signal stability model of the simplified one-mass wind turbine system which has been derived in [18,23]; transfer function G(s) which relates the power set-point P* to the virtual inertia power output δP is The gain of this transfer function falls with the increasing initial generator speed (i.e. wind speed) which has been illustrated in Fig. 4d. Once the pitch control becomes active and the power reference and generator speed are controlled to be constant, there is a step increase in the peak value compared to the previous wind speed step when pitch control was inactive. Then, the peak value of the inertial response falls non-linearly with the respect to the increasing wind speed reflecting the nonlinear nature of the pitch angle in the aerodynamic model. Slower pitch control action is reflected in the peak time compared to when only rotor speed control is utilized: peak time jumps from 1.9 s to around 2.05 s and falls of with increasing wind speed. This falloff can be attributed to the high nonlinearity of the aerodynamic part and the shortcomings of the analytical C p curve at higher wind speeds [16]. The peak value of the inertial response can vary between 0.045 p.u. and 0.040 p.u. depending on the wind speed.
On the other hand, Hu et al. [24] used speed-controlled DFIG model with inverse MPPT characteristic (rotor controller controls the speed rather than power) and they report stronger inertial response at higher wind speed which brings us to the first conclusion: inertial response sensitivity is not the same for total power controlled DFIG and for the speed-controlled DFIG. Value of the virtual inertia constant should be dynamically changed as the function of the generator speed in order to achieve better inertial response and to achieve a consistent power injection with respect to both the pre-disturbance power output and the rated power.

Impact of PLL parameters
Frequency signal which is used as an input to the virtual inertia controller is estimated using a PLL (Fig. 5) which measures the stator voltage at Bus 1. In the stator flux reference frame, the d-axis coincides with the stator flux vector and the q-axis coincides with stator voltage vector, and it is 90 ∘ ahead of the d-axis. The PLL controls the angle θ such that the d-axis component of the stator voltage is zeroed. In this case, θ is the angle of the stator voltage vector relative to the voltage angle at the reference bus (0 ∘ ). Inputs to the PLL are the real and imaginary components of the stator voltage in the global (positive-sequence, xy) reference frame. Estimated grid frequency is equal to f s = ω s /2π. The PLL model is described with (9): We vary K p PLL and K i PLL linearly between 1 and 35, and 20 and 35, respectively. Fig. 6 shows the dynamic characteristics of the inertial response for different PI parameters for a below rated and an above rated wind speed. The simulations have shown a few things: firstly, if K p PLL and K i PLL are large enough, they do not have a significant impact on the strength of the inertial response ( Fig. 6a and c). However, for certain combinations of K p PLL and K i PLL where one or both of those gains are small, the peak is significantly higher (Fig. 6b and d) than it is for larger gains, but the complete behaviour is more oscillatory and undesirable. Smaller PI gains resulted in worse tracking and stronger oscillations.
Secondly, For smaller PI gains, the DFIG model exhibits a behaviour similar to a non-minimal phase shift system which is visible through the initial undershoot in Fig. 6b and d: in the initial moments following a disturbance, the DFIG output power is momentarily reduced further aggravating the grid frequency dynamics which in turn results in a stronger response. Worse stator voltage angle tracking will indirectly influence the DFIG dynamics because this angle is used for transforming between rotor reference frame and stator flux reference frame in the rotor-side control system. Furthermore, with smaller PI gains the damped frequency of the PLL mode is close to that of the electromechanical modes of the system which means that the PLL with participate in the electromechanical oscillations of the system. However, PLL gains are typically large and the PLL mode is well damped [31].
To the contrary, Ma et al. [20] report weaker inertial response under smaller PLL PI gains and well-damped behaviour. In their paper, they have investigated the impact of PLL dynamics on inter-area oscillations between two systems connected with a weak tie-line. We did not notice any such behaviour in our test system, even with increasing both line lengths from 10 km to 110 km. There can be multiple reasons for this discrepancy between the two results: grid topology, types of excitation systems and power system stabilizers, controllers parameters, wind turbine generator model, etc. This is a complex issue which needs separate and in depth studies which are beyond the scope of this paper.
This behaviour is independent of whether or not the virtual inertial response is active or not and it mostly depends on the PLL itself. The participation of PLL in electromechanical oscillations is visible by plotting the trajectories of the PLL mode and the system electromechanical mode (Fig. 7). PLL mode is related to the PLL state variables x PLL and . Synchronous generator (SG) mode has a frequency of around 1.5 Hz and can be considered a local mode. Fig. 7a shows that the parameters of the PLL PI controller do not have a significant impact on the SG mode. On the other hand, PLL mode is close to the imaginary axis for small K p PLL values (weak damping) and this oscillatory behaviour will dominate the response as seen from Fig. 6b and d. By increasing the proportional gain of the PLL, damping of this mode is increased and at certain point PLL mode ceases to be oscillatory (it is completely damped). Fig. 7c shows that the participation factor of PLL state variables in the SG mode increases for larger K p PLL gain, but does not depend on the virtual inertia coefficient.

Impact of MSC control loops
Machine side converter control system usually consists of a slower outer loop which generates the q and d axis rotor current references and a faster inner loop which generates the q and d axis rotor voltage references or similar PWM control signals (in this case, they are q and d axis modulation factors which is one possible option in DIgSILENT PowerFactory). MSC is operated in a stator-flux reference frame where q axis corresponds to the active power control and d axis corresponds to the reactive power control. Since the reactive power control is not a topic of this research, only the q axis parameters will be studied.
We mentioned that the outer loop is slower and the inner loop is faster. Parameter tuning of a PID controller is not a straightforward process and depends on the desired performance as well as on the model of the system and the type of control (power, torque or speed control [32,24,20]). Nevertheless, Hansen et al. report that better performance can be achieved with stronger integral gain. Therefore, we will approach the analysis of the impact of MSC control loops parameters on virtual inertial response by considering both weaker and stronger PI action.

Outer loop
K p outer and K i outer are linearly varied between 1.5 and 5, and 0.5 and 8, respectively. Fig. 8 shows the impact of MSC outer PI controller on the strength of the inertial response. For smaller PI gains the virtual inertial response is stronger (higher apex) for both below rated and above rated wind speed (Fig. 8a and d). This is because the stronger action of the outer PI loop restrains the change of the generator power more [24]. This in turn results in weaker power injection from DFIG. However, if the outer control loop is fast enough (large PI gains) than there is no impact on the strength of the inertial response as shown by the blue shaded areas in Fig. 8a and d. Furthermore, weaker PI gains will also results in faster peak time as shown in Fig. 8b and e. In time domain, this is illustrated with two responses for some characteristic strong and weak combinations of PI gains (Fig. 8c and f). In summary, this means that the dynamics of the slower outer loop should be taken into account when studying inertial response dynamics. They can only be neglected if the PI gains are large such that the whole loop has faster set-point tracking.

Inner loop
K p inner and K i inner are linearly varied between 0.1 and 1.4, and 10 and 100, respectively. Impact of the much faster inner loop dynamics is negligible as shown in Fig. 9. Impact on the strength of the inertial response is in the order of 10 −4 (Fig. 9a) and the impact on the peak time is in the order of 10 −2 (Fig. 9b). Inertial response peak time surface plot has some random fluctuations that exist due to the interaction of certain combinations of PI parameters that may influence the behaviour of the model and the numerical integration (i.e. smaller proportional and/or integral gains). Compared to the below rated wind speed scenario when the pitch angle control is not active (Fig. 9a), strength of the inertial response is much more sensitive to > K 1 p inner (Fig. 9d) for about an order of the magnitude (10 −3 ). The only visible impact is in the initial fast transient behaviour as shown in Fig. 9c. Therefore, the dynamics of the inner loop can be neglected in DFIG inertial dynamics and they don't have an influence on the inertial response.

Impact of pitch control
Pitch control is active only at above rated wind speeds to keep the rotor from over-speeding. There are four main parameters which we have studied to see how they impact the DFIG inertial response: proportional and integral gain of the PI controller (K p pitch and K i pitch , respectively) that generates the pitch servo reference β*, time constant of the pitch servo mechanism T s and the pitch rate limit.  (Fig. 10a). Time at which the peak of the active power injection occurs is longer for bigger K p pitch while the K i pitch doesn't have a significant contribution (Fig. 10b). Fig. 10c shows two DFIG inertial responses for a couple of characteristic combinations of the pitch PI controller. We can conclude that the pitch PI controller parameters should be taken into account when studying virtual inertial dynamics during above rated wind speeds

Pitch servomechanism time constant and rate limiter
Generally, peak value of the inertial response decreases (Fig. 11a) and the peak time of the inertial response increases (Fig. 11b) for increasing the pitch servo time constant. However, this sensitivity is not significant: peak value changes are in the order of 10 −4 and peak time changes are in the order of 10 −2 . Therefore, pitch servo time constant for some characteristic values does not have a significant impact on the DFIG inertial response (Fig. 11c). Pitch rate limit has been changed from ± 3°/s to ± 13°/s and it also does not influence the dynamics of the inertial response (Fig. 11d).

Impact of LSC and DC capacitor
Line-side converter keeps the DC capacitor voltage constant and controls the power factor of the LSC. Inner current control loops will be disregarded in this section since they are fast and do not influence the inertial response as shown in section 3.3.2. Varying the GSC DC loop parameters does not influence the inertial response of DFIG, as can be seen in Fig. 12a-d. Furthermore, the capacitance of the DC link capacitor does not impact the inertial response dynamics (Fig. 12e). Generally, LSC and DC link dynamics can be neglected in inertial response studies.

Conclusion
In this paper we have thoroughly investigated the sensitivity of virtual inertial response to various parameters of the WECS using multidimensional analysis. These parameters are: initial operating point, machine-side and line-side converter controller parameters, pitch angle control parameters and PLL parameters. The conclusions are listed below.
Impact of the wind speed on the strength of the inertial response depends on the type of machine-side converter control. For total power controlled WECS, the response is weaker with increasing wind speed, while for the speed control with inverse MPPT characteristic the response is stronger with increasing wind speed. Once the pitch control becomes active, the peak is initially slightly higher than the instance before pitch angle control activation. Then, the response also becomes weaker with increasing wind speed. Near the cut-in speed, inertial response will reduce the generator speed which will result in shifting from the MPPT curve to the quasi-constant rotor speed line. This, in turn, results in a lower peak value and peak time. The gradient of the power vs. rotor speed curve in this region has an impact on the dynamics, but detailed analysis of this region was out of the scope of this paper.
Small values of PLL PI gains will result in in more oscillatory behaviour and weak damping of the local mode. Following a disturbance, the DFIG power is momentarily reduced further aggravating the grid frequency although the actual peak value is higher. On the other hand, large gains result in strong tracking, no oscillations and smaller peak value of the inertial response. Generally, PLL dynamics can be neglected if the PLL is fast and the modes are well-damped.
Outer control loop of MSC has an impact on virtual inertial response provision. If the outer loop ihas smaller PI gains, the inertial response is stronger and peak time is shorter. This is because it will take a longer time for the weaker controller to restrain the power changes towards the set-point.
Inner loop of the MSC control, DC voltage loop and the inner loop of the LSC and the DC capacitor dynamics can be neglected in the inertial response studies since they are very fast.
Between all the parameters of the pitch control subsystem, proportional gain of the PI controller has the most significant impact on the inertial response. Larger proportional gain will results in bigger power output. Inertial response is not significantly sensitive to integral gain nor to the pitch servo time constant.
Future research may include: comparative analysis between different DFIG-based wind turbine models and control structures, virtual inertia dynamics of full converter WECS with different control structures and droop dynamics of DFIG/full-converter wind turbines.  Overhead line parameters: rated voltage: 220 kV; rated current: 0.4 kA; resistance: 0.05 Ω/km; reactance: 0.488 Ω/km; length: 10 km.