Parameter Setting Strategy for the Controller of the DFIG Wind Turbine Considering the Small-Signal Stability of Power Grids

Due to the increasing penetration of the wind generation, the stability, especially the small-signal stability, of the power grid is much related to it. Currently, few studies considered the impact of the parameter settings of the wind turbine controller on the small-signal stability of the grid under the full range of wind conditions. In this paper, we propose a framework for deriving a set of controller parameters by interiorizing their impact on the power system stability, based on an analytic model of a 15th-order single DFIG-infinite grid connection under all wind speeds. The study results on a real wind turbine show that the controller parameters optimized for a specific wind speed may not feasible for other operational conditions yet the proposed framework can obtain a set of parameters guaranteeing the power system stability under all wind speeds.


I. INTRODUCTION
The energy transition and environmental concerns require the generation of electricity from clean and renewable sources, therefore, the wind has become the world's fastest-growing energy source. Among different wind harvest technologies, the doubly-fed induction generators (DFIGs) have become the mainstream of wind turbines (WT) in the past several decades [1].
The DFIG WT control system includes back-to-back converter controller, pitch angle controller and phase-locked loop controller [2], which contain several PI links, thus the stability of its grid-connected system is closely related to the control parameters and corresponding control strategy. Small-signal stability issue of the grid, caused by the large-scale connected DFIG WT, is increasingly serious [3], [4]. Therefore, how to set the parameters of the control system of the DFIG WT to ensure the small-signal stability of the power grid has become The associate editor coordinating the review of this manuscript and approving it for publication was Tariq Masood . an urgent problem to be solved. However, the traditional controller parameters design method determines the feasible range of each control parameter by the classical control theory without considering the small-signal stability of power grids [5].
In order to analyze the impact of the wind power integration on the small-signal stability of the power system, researchers have established a small-signal model of the WT DFIG system in [6]- [8], in which the converter controller strategy just adopts the vector-oriented control strategy, and the main focus is the influence of generator electrical parameters and WT mechanical parameters on system's smallsignal stability. In recent years, the direct power control (DPC) strategy of the WT with DFIG back-to-back converter has gradually become a research hotspot due to its simple structure and good dynamic performance [9]- [11], however, it has more controller parameters to be tuned to maintain the stability of power system. In [11], the participation factors are used to identify the dynamic mode which describes the association between the eigenvalues and state variables of linear dynamic systems. In [12], a mathematical model of the DFIG and the power system is established to simulate the stability of the power system with variable wind speeds, but the model cannot obtain parameters that can guarantee the system stability.
In general, for the existing studies, most of them focused on verifying the system stability with different wind speeds and electrical parameters without considering the controller parameters setting strategy to increase the stability of the power grid under the whole range of wind speed. Therefore, in this paper, the small-signal stability characteristics of the grid integrating the DFIG WT system controlled by the DPC are analyzed in-depth, and the controller parameter setting strategy is proposed to improve the stability of the power grid based on a derived 15 th -order analytic model of a common DFIG WT with an infinite bus system. By the Lyapunov small-signal stability analysis method, the correlation between the small-signal stability of the power grid and the control system parameters under all possible wind speed conditions is studied. Through the analysis of the small-signal stability under full wind speed, a universal set of parameters of the controller setting strategy is determined. Taking a DFIG WT with an infinite bus as an example, the effect of the WT control system parameters on the power grid smallsignal stability under different wind speeds are analyzed in detail, which verifies the effectiveness and practicability of the proposed strategy.
The rest of the paper is organized as follows. In Section II, three operation modes of the WT are discussed. In Section III, a small-signal stability analysis model of the DFIG WT is briefly introduced. In Section IV, the corresponding state equations are derived. In Section V, the theoretical basis and implementation steps of the parameters setting strategy of the controller used in the DFIG WT are proposed. A real example of a 2 MW wind turbine in the Laoqian Shan wind farm in Shanxi Province is studied in Section VI. Finally, some conclusions are drawn in Section VII.

II. THREE OPERATION MODES OF THE WT
A sketch of relationships among the output power P of a DFIG WT, WT rotor speed ω tur and pitch angle β versus wind speed is shown in Fig. 1. According to the variation of the wind speed, the operating characteristics of the WT can be divided into three different zones (AB, BC, CD). Point A corresponds to the cut-in wind speed, Point B corresponds to the constant rotating speed, point C corresponds to the rated wind speed, and point D corresponds to the cut-out wind speed.
• Zone AB -where the energy capture is optimized. The energy optimization is achieved by tracking the maximum wind power coefficient C max p through the rotating speed adjustment.
• Zone BC -where the WT rotor speed is limited to the rated value, i.e. ω tur = ω rated tur . • Zone CD -where the energy capture is limited. This mode corresponds to the situation when the wind speed is larger than the rated wind speed and less than the cutout wind speed, i.e. V rated ≤ V ≤ V out . The output power of the DFIG WT is then set to the rated value, i.e. P = P rated , by the operation of the pitch angle control. The mechanical power P m captured by the WT can be expressed by equation (1): where ω ref tur is the reference angle speed, λ is the blade tip speed ratio, λ opt is the optimum tip speed ratio, V is the wind speed, ω tur is the WT angle speed, R is the WT blade radius, ρ is the air density, and C p is the wind energy utilization factor.

III. MATHEMATICAL MODELS OF THE DFIG WT
The mathematical models of the DFIG WT include the WT shafting model, doubly-fed induction generators model, DC capacitors model, and control system model. Fig. 2 illustrates the structure of a double-fed turbine with an infinite bus system. The power and current reference directions for each branch are shown in Fig. 2.

A. MODEL OF THE DRIVE TRAIN
The drive train uses the two-mass model [4], given by (3)-(5).
2H gen dω r dt  where H tur and H gen are the inertia constants of the WT and the generator, respectively, θ s is the shaft twist angle, ω tur and ω r are the angle speeds of the WT and the generator rotor, respectively, ω 0 is the angle speed base of the system, D s is the damping coefficient, K s is the shaft stiffness, T e is the electromagnetic torque.

B. THE DFIG SYSTEM MODEL
The DFIG system model uses a second-order model that ignores the transient process of the stator: where E d and E q are the d and q axis voltages behind the transient reactance, respectively, L r is the rotor self-inductance, L m is the mutual inductance, R r is the rotor resistance, ω 1 is synchronous angle speed, s is the rotor slip, u rd , and u rq are the d and q axis rotor terminal voltage, respectively, i sd and i sq are the d and q axis stator-side current, respectively.

C. THE DYNAMIC MODEL OF THE DC CAPACITOR
cu dc du dc dt = P r + P g = u rd i rd + u rq i rq + (u gd i gd + u gq i gq ) (8) where i rd and i rq are the d and q axis current, respectively.

D. THE CONTROL SYSTEM MODEL 1) THE ROTOR-SIDE CONVERTER CONTROLLER MODEL
The rotor-side converter control strategy uses direct power control, and the control block diagram is shown in Fig. 4.  Two state variables x 1 , x 2 have been introduced, and control equations are given by (9)-(13).
where the superscript * indicates the reference value, P s and Q s are the stator side active and reactive power values, u rd and u rq are the d and q axis rotor voltage, respectively, ψ s is the stator flux linkage, U s is the amplitude of the terminal voltage. K Pi and K Ii (i = 1, 2) are proportional and integrating parameters, respectively, L s is the stator self-inductance.

2) THE GRID-SIDE CONVERTER CONTROLLER MODEL
The grid-side converter control strategy uses stator voltageoriented control, and the control block diagram is shown in Fig. 5. The stator voltage is oriented with the q axis and set i * gd to zero, which can limit reactive power exchanged between the grid-side converter and the system to zero. Three state variables x 3 , x 4 , x 5 have been introduced, and control equations are given by (14)-(19).
where i gd and u gd are the d-axis components of the gridside converter current and voltage. i gq and u gq are the q-axis components of the converter current and voltage, u dc is the capacitor terminal voltage. x Tg is the equivalent reactance of the transformer on the grid side of the back-to-back converter. K Pi and K Ii (i = 3, 4, 5) are the proportional and integration parameters of the PI block, respectively.

3) THE PITCH ANGLE CONTROL MODEL
The pitch angle control can improve the efficiency of wind energy conversion and limit the output power [2]. The mathematical model is given by (20)-(22).
where K Pβ and K Iβ are the proportional and integration parameter of the PI block, and state variable x β have been introduced. β is the pitch angle, T servo is the inertia time constant, P ref is the output power reference value.

4) THE PHASE-LOCKED LOOP CONTROLLER MODEL
The DFIG WT uses the phase-locked loop control to obtain the frequency, phase, and voltage amplitude of the power grid to achieve synchronization between the WT and grid [15]. The mathematical model is given by (23)-(24).
where K P6 and K I6 are the proportional and integration parameters of the PI block, and state variable x 6 has been introduced. u sq is the q axis stator voltage, θ p is the output phase of the phase-locked loop.

IV. STATE EQUATION OF THE SINGLE DFIG-INFINITE SYSTEM
According to the flow direction of each electrical quantity in Fig. 2, the voltage at the grid connection point is given by (25).
The voltages at the d-q axes are given by (26)-(27).
where u sd and u sq are the d and q axis stator voltage, respectively. The grid-side converter voltages are given by (28)-(29).
The power for the grid-side converters and the output powers of the stator of the DFIG are given by (30)-(33), respectively: State variables and input variables are given by (34).
For the state equations (3) Small-signal stability analysis model of the WT with DFIG can be obtained as follows where A = A − BD −1 C is the system state matrix. The detailed derivation is given in Appendix A. For the eigenvalue λ = σ ± jω of matrix A, the real part σ depicts the damping of the system, and the imaginary part ω indicates the oscillation frequency of the grid. Therefore, to increase the oscillation damping and reduce the oscillation period, we want a large |σ | and small ω.
Further, to quantitatively evaluate the contribution of the σ and ω to the small-signal stability, we define a stability margin Y i of λ i as Then an index of the small-signal stability margin of the grid can be defined as where N is the order of the system. It is obvious that the larger Y the more robust of the grid. Yet the equation (39) cannot guarantee that the real parts of all eigenvalues are negative. Therefore, the real parts of all eigenvalues should satisfy: where C is the stability threshold to guarantee enough stability margin.
To make Y as large as possible, the participation factor is commonly used to describe the sensitivity between the control parameters and the eigenvalues, i.e. though the sensitivity analysis, we can find the best way to set the control parameters to enlarge the stability margin. The participation factor p ki of the k-th variable x k of λ i can be defined as [15]: where u ki and v ki are the k-th row and the i-th column elements of the left and right characteristic vectors of A, respectively. The larger the module value of the p ki , the greater correlation between the state variable x k and the mode λ i , which can be used to describe the relativity between each eigenvalue and the control parameters K p1 ∼ K p6 , K pβ .

V. CONTROL SYSTEM PARAMETER SETTING STRATEGY
When the wind speed is higher than the rated wind speed, the output power of the WT and the related characteristic equation does not change with the wind speed (zone CD in Fig. 1). In other words, if a certain set of controller parameters can maintain the small-signal stability of the power grid at the rated wind speed, it can guarantee the stability of the system even above the rated wind speed. By contrast, when the WT runs in the zone AB and zone BC, the output power of the WT and the corresponding characteristic equations vary with the wind speed. The impact of the parameters of the control system on the small-signal stability of the system under different wind speeds must be analyzed in detail.
As the most frequent operational condition and the highest efficiency of the turbine lies in zone CD, we would like to set controller parameters in a way that the largest stability margin Y happens in this zone. However, to maintain the smallsignal stability of the grid under all possible wind speed, the selected parameters in zone CD may need to be altered as small as possible to regain the stability in other zones. Therefore, the following controller parameter setting strategy is proposed.
Firstly, a feasible region of the controller parameters is determined according to empirical experiences, and a set of initial parameters is randomly generated within the feasible region. Secondly, participation factors between the controller parameters and eigenvalues are calculated. Then controller parameters are adjusted in a descending order of the participation factors to ensure the small-signal stability of the power grid. Due to the nonlinear nature of the system, the eigenvalue position and the control parameters cannot always be changed synchronously. Therefore, the control parameters adjustment step d cannot be too large. In this paper, we set d = 0.01.
The controller parameters setting strategy of the DFIG WT can be further elaborated as follows: Step 1: Initialization.
Step 1.1: Set the maximum number of iterations to D, and a = 0.
Step 2: Start with the rated wind speed.
• Sort eigenvalues λ i (i = 1, 2, .., N ) in ascending order of Y i and for each λ i , sort K pj (j = 1, 2, . . . , M ) in descending order of modulus of p ki , where M is the number of non-zero p ki .
• For each λ i , successively adjust the control parameters K pj (j = 1, 2, . . . , M ) within the feasible range (one at a time) to make λ i move towards left while maximizing Y .
• If the system is stable, go to Step 3, else go to Step 1.3 and a = a + 1.
Step 3: Reduce the wind speed as a step of 0.1 m/s.
Step 3.2: If there are unstable eigenvalues, sort the unstable eigenvalues in ascending order of Y i . For each unstable eigenvalue, control parameters K pj (j = 1, 2, . . . , M ) are adjusted in the descending order of modulus of p ki (i = 1, 2, .., N ) to make the eigenvalue move into the stable region. If all eigenvalues are stable, go to Step 3.3, otherwise, go to Step 1.3 and a = a + 1.
Step 3.3: If the wind speed is greater than or equal to the cut-in wind speed, go to Step 3. Otherwise, go to Step 4.
Step 4: Test. From the rated to the cut-in wind speed, calculate eigenvalues λ i (i = 1, 2, .., N ) with a decreasing step of 0.1m/s for the wind speed. If all eigenvalues satisfy equation (42), go to step5 and set a = a + 1, otherwise, go to Step 1.3 and a = a + 1.

VI. CASE STUDY
Taking a DFIG WT with an infinite bus as an example (Fig. 2). The rated power of the WT is 2MW, the cut-in wind speed is 3m/s, the rated wind speed is 10.4m/s, and the cut-out wind speed is 25m/s. The feasible range and randomly initialized value of the control parameters K p1 ∼ K p6 , K pβ are reported in Tab. 1.
The eigenvalues in the ascending order of the Y i at the rated wind speed are shown in Tab. 2. Among them, λ 1,2 is located on the right side of the coordinate system, and Y is 3021.72.   The participation factors of the state variables x 1 ∼ x 6 , x β of each eigenvalue are shown in Tab. 3. It should be noted that the participation factors of the eigenvalues 6, 12, 14, 15 and the corresponding control parameters are 0 (omitted in Tab. 3), therefore λ 6 , λ 12 , λ 14 , λ 15 cannot be modified by simply changing controller parameters.
In order to clearly show the impacts of variable parameters on the eigenvalues, the trajectories of eigenvalues varied with K Pi (i = 1, 2, . . . , 5) are shown in Fig. 6. The trajectories of K P6 and K Pβ are ignored as they do not have an influence on the changes of the system stability.
It can be seen from Fig. 6 that the stability and the stability margin of the power system can be altered by changing K Pi , and the trend is similar to that in Tab. 3. It is manifest from Tab. 3 that to make λ 1,2 move to the left, we should adjust the controller parameters in the order of K P4 , K P5 , K P2 , K P1 , K Pβ as their values are decreasing correspondingly. The changes of the controller parameters in the given order and their relevant Y values are given in Tab. 4. Tab. 5 shows the eigenvalues after the adjustment of the controller parameters.   It can be easily seen that after the alteration of the controller parameters, all the eigenvalues for the rated wind speed    Starting from the controller parameters in Tab. 4, we further reduce the wind speed at a step of 0.1 m/s to check the stability of the grid. The controller parameters can maintain the stability of the power system until the wind speed reduced to 3.8m/s. The eigenvalues of the system at the wind speed of 3.8m/s are given in Tab. 6.
The participation factors of the unstable eigenvalue λ 1,2 are shown in Tab. 7.
The control parameters that can maintain the small-signal stability of the grid at 3.8m/s after adjusting K P1 and K P2 are given in Tab. 8 and Tab. 9 shows the corresponding eigenvalues.
Similarly, using the proposed method we obtain a new set of controller parameters to regain the small-signal stability of the grid. This new set of controller parameters can maintain system stability until its cut-in speed.
To test the validity of the controller parameters under all wind speeds, we recalculate the eigenvalues of the system from the rated wind speed to the cut-in speed. The trajectories of the eigenvalues are shown in Fig. 7.
It is obvious that with the latest set of controller parameters given in Tab. 8, the turbine does not impose any threat to the stability of the grid under all wind conditions.

VII. CONCLUSION
The settings of controller parameters to increase the stability of the power system are needed especially with the fast and vast penetration of wind generation in the power system. In this paper, an overall mathematical model of a DFIG WT of direct power control at the rotor-side with an infinite bus system is established for all wind conditions. Based on the in detail analysis of the correlation between the WT controller parameters and the eigenvalues, a WT control system parameters setting strategy considering the small-signal stability of the grid is proposed, which cannot only guarantee the stability of the system under all wind conditions but also increase the stability margins.
The simulation on a real case shows that there is an inflection point in the trajectory of some eigenvalues as the wind speed changes, and it is impossible to determine universal controller parameters according to the traditional analysis only at a certain wind speed. Control system parameters setting thus must consider the small-signal stability of the grid under all possible wind speed.
The proposed strategy in this paper can maximize the small-signal stability margin of the grid represented by the index Y at the rated wind speed, while at other wind speeds, it can ensure the small-signal stability of the grid by finetuning the control parameters. The framework proposed in this paper can provide a theoretical basis for controller parameters setting of wind farms and also have the advantage of simplicity and high efficiency in calculation.