사각형입니다.

https://doi.org/10.6113/JPE.2019.19.2.569

ISSN(Print): 1598-2092 / ISSN(Online): 2093-4718



SSCI Mitigation of Series-compensated DFIG Wind Power Plants with Robust Sliding Mode Controller using Feedback Linearization


Penghan Li*, Linyun Xiong*, Jie Wang, Meiling Ma*, and Muhammad Waseem Khan*


†,*School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, Shanghai, China



Abstract

A robust controller is designed based on feedback linearization and sliding mode control to damp sub-synchronous control interaction (SSCI) in doubly fed induction generator (DFIG) wind power plants (WPPs) interfaced with the grid. A feedback-linearized sliding mode controller (FLSMC) is developed for the rotor-side converter (RSC) through feedback linearization, design of the sliding mode controller, and parameter tuning with the use of particle swarm optimization. A series-compensated 100-MW DFIG WPP is adopted in simulation to evaluate the effectiveness of the designed FLSMC at different compensation degrees and wind speeds. The performance of the designed controller in damping SSCI is compared with proportional-integral controller and conventional sub-synchronous resonance damping controller. Besides the better damping capability, the proposed FLSMC enhances robustness of the system under parameter variations.


Key words: Feedback linearization, PSO, Sliding mode control, SSCI, Wind power plant


Manuscript received Jun. 22, 2018; accepted Dec. 21, 2018

Recommended for publication by Associate Editor Zheng Wang.

Corresponding Author: jiewangxh@sjtu.edu.cn Tel: +86-13524666398, Shanghai Jiao Tong University

*Sch. Electron. Informat. Electr. Eng., Shanghai Jiao Tong Univ., China



Ⅰ. INTRODUCTION

Sub-synchronous resonance (SSR) of wind power plants (WPPs) is a condition where wind turbines exchange energy with the grid at one or more natural frequencies below the fundamental frequency of the system [1], [2]. A new type of SSR called sub-synchronous control interaction (SSCI) has been identified in recent years [3], [4]. Studies have demonstrated that doubly fed induction generator (DFIG) WPPs are susceptible to SSCI due to fast-acting converter control [5]-[10].

Several control methods using flexible AC transmission systems have been proposed to mitigate SSR (or SSCI) in power systems, such as unified power flow controllers [11], static synchronous compensators [12], and thyristor-controlled series compensators [13]. Modification of existing controller is another option for damping SSCI. To stabilize the SSR of practical power systems, a combined control strategy composed of an excitation damping control and a generator terminal damping control was proposed [14]. With the derivative of the rotor speed deviation as the control input, a fuzzy logic control strategy was developed to damp SSR [15]. An adaptive controller was used for supplementary control to alleviate SSCI in DFIG WPPs [16]. A fuzzy based adaptive control strategy was proposed for the grid-side converters (GSC) and rotor-side converters (RSC) of DFIG to alleviate SSCI [17]. A parallel damping control strategy was proposed using the rotor torque analysis method [18]. The issue of SSCI was addressed by embedding suppression filters into the converter controller of a DFIG [19]. A filter-less damping control method was proposed for DFIG WPPs to alleviate SSCI [20]. However, the nonlinear dynamics of DFIGs was not considered in previous works’ design procedures. Therefore, a nonlinear controller is developed in the current study to transcend the limitation of linear control schemes.

The feedback linearization (FL) method retains the nonlinearity through nonlinear feedback and coordinate transformation. Transformed systems are independent of predefined operations [21]. This characteristic is suitable for highly nonlinear DFIGs. However, a system under FL control could be sensitive to parameter variations. Sliding mode control (SMC) has been widely used to improve robust properties [22]. A high-order SMC was designed for grid synchronization and power optimization of wind farms [23]. For DFIG-based WPPs, a fractional-order SMC was developed for maximum power point tracking [24]. An SMC was developed for GSC of DFIG to damp SSCI [25]. However, the power rating of GSC is only 25% of gerarator rating, which limits the effectiveness of the control method. Moreover, the impact of the number of online wind turbines (WTs) was neglected in [25], and the configuration of realistic WPPs was not considered in electromagnetic transient (EMT) simulation in [25]. Therefore, in the present work, a sliding mode controller is developed for RSC based on feedback linearization to damp SSCI at different compensation degrees, wind speeds, and number of online WTs. The design procedures of the proposed controller include feedback linearization, design of the sliding mode controller, and parameter tuning with use of particle swarm optimization (PSO). Eigenvalue analysis and EMT simulation based on a realistic WPP are performed to evalute the effectiveness of the proposed control method.

The remainder of this paper is organized as follows. The model of the DFIG WPP is presented in Section II. In Section III, the system is feedback linearized. In Section IV, the sliding mode controller for RSC is developed, and PSO is utilized for parameter tuning. In Section V, the effectiveness of the designed FLSMC is tested and compared with that of the PI controller and SSRDC at different compensation degrees and wind speeds. Concluding remarks are presented in Section VI.



Ⅱ. SYSTEM MODEL

As shown in Fig. 1, a 100 MW DFIG WPP is interfaced with series-compensated transmissions [10]. The WPP is an aggregation of 50 wind turbines, each with 2 MW power rating [25]. System data are presented in the Appendix.


그림입니다.
원본 그림의 이름: image2.png
원본 그림의 크기: 가로 3110pixel, 세로 1554pixel

Fig. 1. DFIG-based WPP interfaced with the grid.


The studied system includes the induction generator, shaft system, and BtB converter [10]. Shaft dynamics is expressed as follows:

그림입니다.
원본 그림의 이름: image1.png
원본 그림의 크기: 가로 2328pixel, 세로 1784pixel        (1)

where ωr and ωm are rotor angular frequency and rotor shaft angular frequency, respectively; Tae and Te are aerodynamic torque and electromagnetic torque, respectively; γ denotes the torsion angle; D denotes the torsion damping coefficient; H represents the inertia constant; Ks represents the torsion stiffness coefficient; Ng represents the gear ratio.

The electrical dynamics of the system is obtained by using Kirchhoff’s current law and Kirchhoff’s voltage law, which are expressed as follows:

그림입니다.
원본 그림의 이름: image3.png
원본 그림의 크기: 가로 2703pixel, 세로 1118pixel        (2)

그림입니다.
원본 그림의 이름: image4.png
원본 그림의 크기: 가로 2703pixel, 세로 501pixel       (3)

그림입니다.
원본 그림의 이름: image5.png
원본 그림의 크기: 가로 2025pixel, 세로 1051pixel             (4)

그림입니다.
원본 그림의 이름: image6.png
원본 그림의 크기: 가로 2172pixel, 세로 1118pixel           (5)

그림입니다.
원본 그림의 이름: image7.png
원본 그림의 크기: 가로 2860pixel, 세로 1017pixel     (6)

그림입니다.
원본 그림의 이름: image8.png
원본 그림의 크기: 가로 1692pixel, 세로 1051pixel              (7)

where subscripts q and d denote quadrature and direct axes, respectively; Rrf and Lrf are the resistance and inductance of the RSC filter, respectively; ω1 is system angular line frequency; Srd and Srq denote RSC input switching signals; udc denotes the voltage across the DC-link capacitor (Cdc); us, is, Ls, and Rs are stator voltage, current, inductance, and resistance, respectively; ug, ig, Lgf, and Rgf are GSC filter voltage, current, inductance, and resistance, respectively; usc denotes the voltage across the compensation capacitor (Csc); u denotes the voltage at the PCC; i denotes the current of the transmissions; e is the grid voltage; LL and RL are transmission inductance and resistance, respectively.

The system is expressed as follows:

그림입니다.
원본 그림의 이름: image9.png
원본 그림의 크기: 가로 1608pixel, 세로 751pixel             (8)

where 그림입니다.
원본 그림의 이름: image10.png
원본 그림의 크기: 가로 1160pixel, 세로 368pixel; 그림입니다.
원본 그림의 이름: image11.png
원본 그림의 크기: 가로 1056pixel, 세로 368pixel;그림입니다.
원본 그림의 이름: image12.png
원본 그림의 크기: 가로 4101pixel, 세로 292pixel. The expressions of 그림입니다.
원본 그림의 이름: image13.png
원본 그림의 크기: 가로 440pixel, 세로 284pixel and 그림입니다.
원본 그림의 이름: image14.png
원본 그림의 크기: 가로 409pixel, 세로 284pixel are given in the Appendix.



Ⅲ. FEEDBACK LINEARIZATION OF DFIG-BASED WPP

System feedback linearizability is determined by relative degree, which depends on system outputs. The following condition should be fulfilled:

그림입니다.
원본 그림의 이름: image15.png
원본 그림의 크기: 가로 1139pixel, 세로 335pixel               (9)

where hi(x) is the system I th output; ri denotes relative degree; L denotes a Lie derivative. Condition (9) should be fulfilled for both outputs. Thus, we can calculate the following:

그림입니다.
원본 그림의 이름: image16.png
원본 그림의 크기: 가로 2328pixel, 세로 884pixel       (10)

그림입니다.
원본 그림의 이름: image17.png
원본 그림의 크기: 가로 2505pixel, 세로 867pixel      (11)

The calculations expressed by Eqs. (10) and (11) demonstrate that the matrix is nonsingular.

그림입니다.
원본 그림의 이름: image18.png
원본 그림의 크기: 가로 2109pixel, 세로 684pixel

The relative degree is 그림입니다.
원본 그림의 이름: image19.png
원본 그림의 크기: 가로 1254pixel, 세로 251pixel, indicating that the system can be partially linearized [26]. The system is devided into two subsystems as follows:

그림입니다.
원본 그림의 이름: image20.png
원본 그림의 크기: 가로 1045pixel, 세로 468pixel               (12)

그림입니다.
원본 그림의 이름: image21.png
원본 그림의 크기: 가로 690pixel, 세로 468pixel                 (13)

where 그림입니다.
원본 그림의 이름: image22.png
원본 그림의 크기: 가로 137pixel, 세로 201pixel represents new states; 그림입니다.
원본 그림의 이름: image23.png
원본 그림의 크기: 가로 137pixel, 세로 218pixel represents the remaining states. The system states are as follows:

그림입니다.
원본 그림의 이름: image24.png
원본 그림의 크기: 가로 805pixel, 세로 335pixel                (14)

We choose coordinate transformation as follows

그림입니다.
원본 그림의 이름: image25.png
원본 그림의 크기: 가로 638pixel, 세로 551pixel                  (15)

Then, we obtain the following:

그림입니다.
원본 그림의 이름: image26.png
원본 그림의 크기: 가로 2839pixel, 세로 894pixel  (16)

The linearized system is obtained with the use of the linear control method as 그림입니다.
원본 그림의 이름: image27.png
원본 그림의 크기: 가로 795pixel, 세로 251pixel;

그림입니다.
원본 그림의 이름: image28.png
원본 그림의 크기: 가로 2390pixel, 세로 1118pixel       (17)

where v is the input for linear control. Physical control laws are computed from Eq. (17) as follows:

그림입니다.
원본 그림의 이름: image29.png
원본 그림의 크기: 가로 2505pixel, 세로 1217pixel     (18)

The stability of the untransformed subsystem is ensured using zero-dynamic theory [9].



Ⅳ. SMC DESIGN BASED ON FL MODEL


A. Damping Controller Design

Sliding mode control is used based on the linearized system to improve robustness. The sliding mode surfaces are defined as follows:

그림입니다.
원본 그림의 이름: image30.png
원본 그림의 크기: 가로 1692pixel, 세로 784pixel             (19)

where 그림입니다.
원본 그림의 이름: image31.png
원본 그림의 크기: 가로 805pixel, 세로 318pixel and 그림입니다.
원본 그림의 이름: image32.png
원본 그림의 크기: 가로 909pixel, 세로 284pixel; c1 and c2 are positive constants. Reference values 그림입니다.
원본 그림의 이름: image33.png
원본 그림의 크기: 가로 210pixel, 세로 284pixel and 그림입니다.
원본 그림의 이름: image34.png
원본 그림의 크기: 가로 210pixel, 세로 318pixel are computed from reference rotor reactive power 그림입니다.
원본 그림의 이름: image35.png
원본 그림의 크기: 가로 221pixel, 세로 284pixel and active power 그림입니다.
원본 그림의 이름: image36.png
원본 그림의 크기: 가로 221pixel, 세로 284pixel as follows:

그림입니다.
원본 그림의 이름: image37.png
원본 그림의 크기: 가로 1421pixel, 세로 751pixel              (20)

The equivalent control is derived in the solution of 그림입니다.
원본 그림의 이름: image38.png
원본 그림의 크기: 가로 627pixel, 세로 301pixel as follows:

그림입니다.
원본 그림의 이름: image39.png
원본 그림의 크기: 가로 962pixel, 세로 584pixel                 (21)

The reaching control is added to the equivalent control as follows:

그림입니다.
원본 그림의 이름: image40.png
원본 그림의 크기: 가로 2057pixel, 세로 518pixel        (22)

where 그림입니다.
원본 그림의 이름: image41.png
원본 그림의 크기: 가로 179pixel, 세로 251pixel, 그림입니다.
원본 그림의 이름: image42.png
원본 그림의 크기: 가로 189pixel, 세로 251pixel, 그림입니다.
원본 그림의 이름: image43.png
원본 그림의 크기: 가로 179pixel, 세로 251pixel, and 그림입니다.
원본 그림의 이름: image44.png
원본 그림의 크기: 가로 189pixel, 세로 251pixel are positive parameters. A Lyapunov function is selected as follows:

그림입니다.
원본 그림의 이름: image45.png
원본 그림의 크기: 가로 795pixel, 세로 451pixel               (23)

The time derivative of Eq. (23) is calculated as follows:

그림입니다.
원본 그림의 이름: image46.png
원본 그림의 크기: 가로 2693pixel, 세로 1084pixel     (24)

Hence, the designed control method is stable. With the substitution of Eq. (22) into Eq. (18), the control law is derived as follows:

그림입니다.
원본 그림의 이름: image47.png
원본 그림의 크기: 가로 3162pixel, 세로 978pixel   (25)


B. Parameter Tuning with PSO

Trial-and-error method is commonly used to determine SMC parameters [27]. For superior transient and steady-state performance, evolutionary algorithms, such as PSO and genetic algorithm (GA), are applied for SMC parameter tuning [28]. The advantages of PSO over other evolutionary techniques, such as GA, are as follows.

• Its implementation is easier.

• It requires less parameter tuning.

• It has higher memory ability than GA, and it is cheap in terms of computation time.

Therefore, PSO is used for FLSMC parameter tuning through the following procedure.

1) Initialization: The proposed damping controller has six control gains (c1, c2, ε1, ε2, k1, and k2); therefore, six particle positions are generated randomly as follows:

그림입니다.
원본 그림의 이름: image48.png
원본 그림의 크기: 가로 2557pixel, 세로 717pixel       (26)

where xi,j is the position of the jth particle in the ith group, and Xi0 is the original value.

The velocity for position updating is expressed as follows:

그림입니다.
원본 그림의 이름: image49.png
원본 그림의 크기: 가로 1660pixel, 세로 784pixel            (27)

where vj,min and vj,max are minimum and maximum velocities, respectively, of member j of the particles; xj,min and xj,max are the lower and upper bounds, respectively; N represents the dimension interval. Initial velocity is generated between vj,min and vj,max. Other parameters are determined as follows:

그림입니다.
원본 그림의 이름: image50.png
원본 그림의 크기: 가로 1379pixel, 세로 851pixel             (28)

where wi and wf denote initial and final weights, respectively; α is the decay constant; a denotes the accelerating constant.

2) Evaluating Best Solution and Updating: The fitness value, which is expressed as follows, is assessed:

그림입니다.
원본 그림의 이름: image51.png
원본 그림의 크기: 가로 2286pixel, 세로 318pixel        (29)

where F denotes the fitness value, and λl denotes the system eigenvalues. The optimization objective is to drive all eigenvalues to the left half plane. The individual best solution is updated after each iteration.

3) Velocity and Position Updating: Velocity and position updating is expressed as follows:

그림입니다.
원본 그림의 이름: image52.png
원본 그림의 크기: 가로 2975pixel, 세로 684pixel   (30)

그림입니다.
원본 그림의 이름: image53.png
원본 그림의 크기: 가로 2057pixel, 세로 301pixel          (31)

where r1 and r2 are uniformly distributed random numbers within [0, 1]; x* and x** represent individual and global best solutions, respectively.

4) Stopping Criteria: The search will stop if one of the following criteria is fulfilled. (i) The fitness value of the global best solution becomes smaller than a specified number, or (ii) the time counter is greater than the specified maximum number. Otherwise, the system will continue iterating until Step 2. The controller gains are presented in the Appendix.



Ⅴ. SIMULATION RESULTS AND ANALYSIS

A diagram of the proposed control method is presented in Fig. 2. SSRDC [8] and a PI controller [10], whose diagrams are presented in Figs. 3 and 4, respectively, are used for comparison. The parameters are presented in the Appendix.


그림입니다.
원본 그림의 이름: image54.png
원본 그림의 크기: 가로 3131pixel, 세로 1387pixel

Fig. 2. Diagram of the FLSMC.


Fig. 3. PI controller: (a) RSC. (b) GSC.

그림입니다.
원본 그림의 이름: image55.png
원본 그림의 크기: 가로 2735pixel, 세로 1044pixel

(a)

그림입니다.
원본 그림의 이름: image56.png
원본 그림의 크기: 가로 2735pixel, 세로 993pixel

(b)


Fig. 4. SSRDC. (a) RSC. (b) GSC.

그림입니다.
원본 그림의 이름: image57.png
원본 그림의 크기: 가로 2839pixel, 세로 1571pixel

(a)

그림입니다.
원본 그림의 이름: image58.png
원본 그림의 크기: 가로 2839pixel, 세로 1071pixel

(b)


A. Eigenvalue Analysis

The system eigenvalues are computed using the MATLAB function “linmod” [7]. System eigenvalues at different compensation levels, wind speeds, and number of online WTs are shown in Fig. 5. The figure shows that the FLSMC and SSRDC improve the capability of DFIGs in damping SSCI, and the capability of the proposed controller is better. Moreover, the proposed FLSMC stabilizes the studied system under varied operating conditions.

As presented in Fig. 5, a negative correlation exists between system damping and compensation level, whereas system damping has a positive correlation with wind speed. The number of online WTs has a nonlinear impact on SSCI, especially using PI controller and SSRDC. At a given wind speed and compensation level, a certain number of online WTs, which can cause severe damping, exists. However, system damping generally improves with increased online WTs when FLSMC is used. This characteristic guarantees that the system stability will not be undermined with the incorporation of additional WTs in the future, once the system stability is ensured in the early-stage construction of WPP. In terms of WPP planning and practical power systems, this property of the FLSMC has major significance.


Fig. 5. System eigenvalues. (a) At varied compensation levels. (b) At varied wind speeds. (c) At varied number of online WTs.

그림입니다.
원본 그림의 이름: image59.png
원본 그림의 크기: 가로 2703pixel, 세로 1739pixel

(a)

그림입니다.
원본 그림의 이름: image60.png
원본 그림의 크기: 가로 2703pixel, 세로 1843pixel

(b)

그림입니다.
원본 그림의 이름: image61.png
원본 그림의 크기: 가로 3319pixel, 세로 2260pixel

(c)


B. Electromagnetic Transient Simulation

In this section, the performance of the proposed FLSMC is evaluated using EMT simulation. Series compensation of 75% is provided to the transmissions at 그림입니다.
원본 그림의 이름: image62.png
원본 그림의 크기: 가로 325pixel, 세로 201pixel s with 7 m/s wind speed. Responses from the DFIG-based WPP are presented, including PCC voltage, active power of transmissions, and DC-link voltage.

In Figs. 6(a) and 6(b), the presented waveforms show the effectiveness of the FLSMC and SSRDC in mitigating SSCI. Furthermore, the designed FLSMC exhibits superior damping performance; it consumes less time than does SSRDC to damp SSCI. In addition, Fig. 6(c) demonstrates the capability of the proposed FLSMC in damping fluctuations of the DC-link voltage. Given the proposed control laws, which are two nonlinear equations calculated online, the simulations require slightly higher CPU time to complete for FLSMC compared with the PI controller and SSRDC.


Fig. 6. DFIG responses for wind speed of 7 m/s and compensation level of 75%. (a) PCC voltage. (b) Transmission line active power; (c) DC-link voltage.

그림입니다.
원본 그림의 이름: image63.png
원본 그림의 크기: 가로 2703pixel, 세로 2015pixel

(a)

그림입니다.
원본 그림의 이름: image64.png
원본 그림의 크기: 가로 2724pixel, 세로 1886pixel

(b)

그림입니다.
원본 그림의 이름: image65.png
원본 그림의 크기: 가로 2682pixel, 세로 1834pixel

(c)


As presented in Fig. 7, the simulation is performed at 45% series compensation, whereas Fig. 8 demonstrates system responses at 9 m/s wind speed. In addition to the better damping capability of the proposed FLSMC, transient SSCI problems are observed to be less severe at lower compensation levels or higher wind speeds.


Fig. 7. DFIG responses for wind speed of 7 m/s and compensation level of 45%. (a) PCC voltage. (b) Transmission line active power. (c) DC-link voltage.

그림입니다.
원본 그림의 이름: image66.png
원본 그림의 크기: 가로 2745pixel, 세로 2025pixel

(a)

그림입니다.
원본 그림의 이름: image67.png
원본 그림의 크기: 가로 2765pixel, 세로 1848pixel

(b)

그림입니다.
원본 그림의 이름: image68.png
원본 그림의 크기: 가로 2745pixel, 세로 1823pixel

(c)


Fig. 8. DFIG responses for wind speed of 9 m/s and compensation level of 75%. (a) PCC voltage. (b) Transmission line active power. (c) DC-link voltage.

그림입니다.
원본 그림의 이름: image69.png
원본 그림의 크기: 가로 2682pixel, 세로 1977pixel

(a)

그림입니다.
원본 그림의 이름: image70.png
원본 그림의 크기: 가로 2682pixel, 세로 1800pixel

(b)

그림입니다.
원본 그림의 이름: image71.png
원본 그림의 크기: 가로 2682pixel, 세로 1826pixel

(c)


In practice, DFIG parameters vary to a certain extent. The variations of the rotor resistance (Rr), stator resistance (Rs), stator leakage inductance (Lls), rotor leakage inductance (Llr), and mutual inductance (Lm) could influence the characteristics of the SSCI [3]. However, at the frequencies in this study, the impedances of the inductance  (Lls, Llr, and Lm) are much larger than those of the resistances (Rr, Rs), i.e., jωL >> R. Thus, the analysis mainly focuses on the impact from the variations of the inductance (Lls, Llr, and Lm). When SSCI occurs, the increasing saturation degree of the leakage flux and main flux leads to decreases in the values of Lls, Llr, and Lm [8], [29]. Fig. 9 presents the simulation results at 75% series compensation and 7 m/s wind speed with Lls, Llr, and Lm reduced to 80% of the nominal value. As shown in Fig. 9, the SSRDC can still damp oscillations. However, more than 4 s is required for the SSRDC to eliminate the SSCI, much longer compared with that shown in Fig. 6. Moreover, due to parameter variations, the fluctuation range of DC-link voltage is larger with SSRDC. On the contrary, the performance of the FLSMC is not much affected by DFIG parameter deviation. Although the fluctuations in system responses increase slightly, FLSMC is still capable of mitigating SSCI within 3 s. The simulation results demonstrate the strong robustness of the proposed controller against parameter variations.


Fig. 9. DFIG responses under parameter variations for wind speed of 7 m/s and compensation level of 75%. (a) PCC voltage. (b) Transmission line active power. (c) DC-link voltage.

그림입니다.
원본 그림의 이름: image72.png
원본 그림의 크기: 가로 2609pixel, 세로 2009pixel

(a)

그림입니다.
원본 그림의 이름: image73.png
원본 그림의 크기: 가로 2609pixel, 세로 1806pixel

(b)

그림입니다.
원본 그림의 이름: image74.png
원본 그림의 크기: 가로 2609pixel, 세로 1806pixel

(c)


We present the potential of SSCI in a realistic DFIG WPP divided into different groups connected by cables, as illustrated in Fig. 10. The WPP is divided into four groups (G-1 to G-4). Each group is composed of 20 wind turbine generators (WTGs), and all groups are linked to PCC via collector cables. G1 and G2 consist of WTG-1, each rated 2 MW; G3 and G4 consist of WTG-2, each rated 2.3 MW. WTG-1 is used in this analysis, and the parameters of WTG-2 are provided in the Appendix. The cable length varies between 0.5 and 2 km, as shown in Fig. 10 [30]. The layout of a realistic wind farm is presented in detail in [31]. EMT simulation is performed under different wind power output conditions. In this case, G-1, G-2, G-3, and G-4 produce 100%, 75%, 50%, and 25% of their rated capacity, respectively. Series compensation of 75% is given to the transmission line at 그림입니다.
원본 그림의 이름: image62.png
원본 그림의 크기: 가로 325pixel, 세로 201pixel s. Responses from the DFIG-based WPP are presented in Fig. 11. As shown in Fig. 11, the WPP with the PI controller has SSCI. On the contrary, oscillations are damped with SSRDC and FLSMC. Moreover, less time is required to completely eliminate SSCI under the proposed FLSMC. The simulation is repeated at the series compensation of 45%, as illustrated in Fig. 12. The damping performance of FLSMC is superior to that of the PI controller and SSRDC.


그림입니다.
원본 그림의 이름: image75.png
원본 그림의 크기: 가로 3308pixel, 세로 1995pixel

Fig. 10. Series-compensated WPP with collector cable.


Fig. 11. DFIG responses based on realistic WPP for wind speed of 7 m/s and compensation level of 75%. (a) PCC voltage. (b) Transmission line active power. (c) DC-link voltage.

그림입니다.
원본 그림의 이름: image76.png
원본 그림의 크기: 가로 2536pixel, 세로 1936pixel

(a)

그림입니다.
원본 그림의 이름: image77.png
원본 그림의 크기: 가로 2536pixel, 세로 1732pixel

(b)

그림입니다.
원본 그림의 이름: image78.png
원본 그림의 크기: 가로 2536pixel, 세로 1758pixel

(c)


Fig. 12. DFIG responses based on realistic WPP for wind speed of 7 m/s and compensation level of 45%. (a) PCC voltage. (b) Transmission line active power. (c) DC-link voltage.

그림입니다.
원본 그림의 이름: image79.png
원본 그림의 크기: 가로 2599pixel, 세로 1925pixel

(a)

그림입니다.
원본 그림의 이름: image80.png
원본 그림의 크기: 가로 2599pixel, 세로 1697pixel

(b)

그림입니다.
원본 그림의 이름: image81.png
원본 그림의 크기: 가로 2599pixel, 세로 1773pixel

(c)



Ⅵ. CONCLUSION

SSCI is caused by the interaction of DFIG-based WPPs with series compensation. Thus, a robust nonlinear controller is designed in this work using feedback linearization and sliding mode control to mitigate SSCI in consideration of the parameter variations and inherent nonlinearities of DFIGs. FLSMC is developed for RSC through feedback linearization, design of the sliding mode controller, and parameter tuning with use of PSO. The performance of the FLSMC is assessed and compared with that of the PI controller and SSRDC. Eigenvalue analysis is performed under varied operating conditions. With regard to the superior capability of FLSMC in alleviating SSCI, (i) system damping deteriorates with decreased wind speed or increased compensation levels, and (ii) the number of online WTs has a nonlinear impact on SSCI damping. EMT simulation is subsequently carried out to validate eigenvalue analysis. EMT simulation results show that the designed FLSMC enhances SSCI damping and robustness properties of the studied system.



APPENDIX

TABLE I SYSTEM PARAMETERS

Parameter

Value

Stator resistance (Rs)

0.0084 p.u.

Stator inductance (Ls)

0.1670 p.u.

RSC filter resistance (Rrf)

0.0083 p.u.

RSC filter inductance (Lrf)

0.1323 p.u.

GSC filter resistance (Rgf)

0.0015 p.u.

GSC filter inductance (Lgf)

0.1500 p.u.

DC-link capacitance (Cdc)

10 mF

Generator damping coefficient (Dg)

0.5 p.u.

Turbine damping coefficient (Dt)

2.5 p.u.

Generator inertia constant (Hg)

0.5 p.u.

Turbine inertia constant (Ht)

2.5 p.u.

Transmission line resistance (RL)

0.02 p.u.

Transmission line inductance (LL)

0.0016 p.u.


TABLE II PARAMETERS OF WTG-2

Parameter

Value

Stator resistance

0.0051 p.u.

Stator inductance

0.1317 p.u.

RSC filter resistance (Rrf)

0.0112 p.u.

RSC filter inductance (Lrf)

0 p.u.

GSC filter resistance (Rgf)

0.1924 p.u.

GSC filter inductance (Lgf)

0.2117 p.u.

Generator damping coefficient (Dg)

0 p.u.

Turbine damping coefficient (Dt)

0.3 p.u.

Generator inertia constant (Hg)

0.5 p.u.

Turbine inertia constant (Ht)

4 p.u.


TABLE III CONTROLLER GAINS

Parameter

Value

Parameter

Value

K1

0.0001

T1

0.0500

K2

0.0000

T2

0.0050

K3

0.0001

T3

0.0250

K4 

0.0001

T4

0.0025

K5

0.1000

T5

0.0500

K6

1

T6

100

K7 

0.1000

T7

0.0500

K8

1

T8

100

G

30

T9

0.0500

k1

100

T10

100

k2

200

c1

200

ε1

1

c2

500

ε2

0.5

 

 


Expressions of 그림입니다.
원본 그림의 이름: image14.png
원본 그림의 크기: 가로 409pixel, 세로 284pixel and 그림입니다.
원본 그림의 이름: image13.png
원본 그림의 크기: 가로 440pixel, 세로 284pixel

그림입니다.
원본 그림의 이름: image82.png
원본 그림의 크기: 가로 2004pixel, 세로 4253pixel

그림입니다.
원본 그림의 이름: image83.png
원본 그림의 크기: 가로 2954pixel, 세로 7384pixel



REFERENCES

[1] A. Moharana and R. K. Varma, “Subsynchronous resonance in single cage self-excited induction generator based wind farm connected to series-compensated lines,” IET Gener. Transm. Distrib., Vol. 5, No. 12, pp. 1221-1232, Dec. 2011.

[2] H. Cho, S. Oh, S. Nam, and B. Lee, “Non-linear dynamics based sub-synchronous resonance index by using power system measurement data,” IET Gener. Transm. Distrib., Vol. 12, No. 17, pp. 4026-4033, Sep. 2018.

[3] X. Xie, X. Zhang, H. Liu, H. Liu, Y. Li, and C. Zhang, “Characteristic analysis of subsynchronous resonance in practical wind farms connected to series-compensated transmissions,” IEEE Trans. Energy Convers., Vol. 32, No. 3, pp. 1117-1126, Sep. 2017.

[4] L. Xiong, P. Li, F. Wu, and J. Wang, “Stability enhancement of power systems with high DFIG-wind turbine penetration via virtual inertia planning,” IEEE Trans. Power Syst., to be published.

[5] V. B. Virulkar and G. V. Gotmare, “Sub-synchronous resonance in series compensated wind farm: A review,” Renew. Sustain. Energy Rev., Vol. 55, pp. 1010-1029, Mar. 2016.

[6] P. Li, J. Wang, L. Xiong, and F. Wu, “Nonlinear controllers based on exact feedback linearization for series-compensated DFIG-based wind parks to mitigate sub-synchronous control interaction,” Energies, Vol. 10, No. 8, pp. 1182, Aug. 2017.

[7] P. Li, L. Xiong, F. Wu, M. Ma, and J. Wang, “Sliding mode controller based on feedback linearization for damping of sub-synchronous control interaction in DFIG-based wind power plants,” Int. J. Electr. Power Energy Syst., Vol. 107, pp. 239-250, May 2019.

[8] G. Irwin, A. Jindal, and A. Isaacs, “Sub-synchronous control interactions between type 3 wind turbines and series compensated ac transmission systems,” in Proc. IEEE Power Energy Gen. Meet., pp. 1-6, Jul. 2011.

[9] P. Li, J. Wang, F. Wu, and H. Li. “Nonlinear controller based on state feedback linearization for series-compensated DFIG-based wind power plants to mitigate sub-synchronous control interaction,” Int. Trans. Electr. Energ. Syst., e2682, 2018.

[10] H. A. Mohammadpour and E. Santi, “Modeling and control of gate-controlled series capacitor interfaced with a DFIG-based wind farm,” IEEE Trans. Ind. Electron., Vol. 62, No. 2, pp. 1022-1033, Feb. 2015.

[11] X. Zhu, M. Jin, X. Kong, J. Zhao, J. Liu, and Q. Zhou, “Subsynchronous resonance and its mitigation for power system with unified power flow controller,” J. Mod. Power Syst. Clean Energy, Vol. 6, No. 1, pp. 181-189, Jan. 2018.

[12] S. Wang, S. Lu, H. Xiao, Z. Xu, X. Cui, and X. Wu, “SSO suppression method and effectiveness of STATCOM in an identical multi-machine system,” J. Engineering, Vol. 2017, No. 13, pp. 1483-1487, Oct. 2017.

[13] R. Zheng, T. Joseph, S. Wang, and J. Liang, “A control strategy for TCSC to mitigate SSR with local measurements,” IET International Conference on AD and DC Power Transmission, pp. 1-6, 2017.

[14] X. Xie, H. Liu, and Y. Han, “Coordinated design of supplementary excitation damping controller and voltage-sourced converter based generator terminal subsynchronous damping controller for subsynchronous resonance suppression: a case study,” Electr. Power Components Syst., Vol. 44, No. 5, pp. 565-577, Feb. 2016.

[15] M. R. Qader, “Fuzzy control strategy for camping sub-synchronous resonance,” J. Electr. Eng. Technol., Vol. 13, No. 5, pp. 1791-1797, Sep. 2018.

[16] J. Taherahmadi, M. Jafarian, and M. N. Asefi, “Using adaptive control in DFIG-based wind turbines to improve the subsynchronous oscillations of nearby synchronous generators,” IET Renew. Power Gener., Vol. 11, No. 2, pp. 362-369, Apr. 2017.

[17] R. Mahalakshmi and K. C. S. Thampatty, “A fuzzy based adaptive controller for grid connected DFIG based windfarm for damping SSR oscillations,” TENCON 2017 IEEE Reg. Conf., pp. 2138-2143, 2017.

[18] F. Zhao F. Zhao, L. Wu, J. Zhang, X. Gao, H. Liu, H. Wang, and Y. Yang, “Suppression method of parallel-damping controller for DFIG,” J. Engineering, Vol. 2017, No. 13, pp. 880-884, Oct. 2017.

[19] H. Liu, X. Xie, J. He, H. Liu, and Y. Li, “Damping DFIG-associated SSR by adding subsynchronous suppression filters to DFIG converter controllers,” IEEE Power Energy Soc. Gen. Meet., pp. 1-5, 2016.

[20] H. K. Nia, F. Salehi, M. Sahni, N. Karnik, and H. Yin, “A filter-less robust controller for damping SSCI oscillations in wind power plants,” IEEE Power Energy Soc. Gen. Meet., pp. 1-5, Jan. 2018.

[21] S. M. Mohiuddin, M. A. Mahmud, A. M. O. Haruni, and H. R. Pota, “Design and implementation of partial feedback linearizing controller for grid-connected fuel cell systems,” Int. J. Electr. Power Energy Syst., Vol. 93, pp. 414-425, Dec. 2017.

[22] L. Xiong, P. Li, H. Li, and J. Wang, “Sliding mode control of DFIG wind turbines with a fast exponential reaching law,” Energies, Vol. 10, No. 11, 1788, Nov. 2017.

[23] L. Xiong, P. Li, F. Wu, M. Ma, M. W. Khan, and J. Wang, “A coordinated high-order sliding mode control of DFIG wind turbine for power optimization and grid synchronization,” Int. J. Electr. Power Energy Syst., Vol. 105, pp. 679-689, Feb. 2019.

[24] L. Xiong, J. Wang, X. Mi, and W. Khan, “Fractional order sliding mode based direct power control of grid-connected DFIG,” IEEE Trans. Power Syst., Vol. 33, No. 3, pp. 3087-3096, May. 2018.

[25] P. Li, J. Wang, L. Xiong, and M. Ma, “Robust nonlinear controller design for damping of sub-synchronous control interaction in DFIG-based wind farms,” IEEE Access, to be published.

[26] M. A. Mahmud, H. R. Pota, and M. J. Hossain, “Dynamic stability of three-phase grid-connected photovoltaic system using zero dynamic design approach,” IEEE J. Photovoltaics, Vol. 2, No. 4, pp. 564-571, Oct. 2012.

[27] M. Han, J. Fan, and J. Wang, “A dynamic feedforward neural network based on gaussian particle swarm optimization and its application for predictive control,” IEEE Trans. Neural Networks, Vol. 22, No. 9, pp. 1457-1468, Sep. 2011.

[28] P. Siano and G. Mokryani, “Assessing wind turbines placement in a distribution market environment by using particle swarm optimization,” IEEE Trans. Power Syst., Vol. 28, No. 4, pp. 3852-3864, Nov. 2013.

[29] H. M. Jabr and N. C. Kar, “Effects of main and leakage flux saturation on the transient performances of doubly-fed wind driven induction generator,” Electr. Power Syst. Res., Vol. 77, No. 8, pp. 1019-1027, Jun. 2007.

[30] R. K. Varma and A. Moharana, “SSR in double-cage induction generator-based wind farm connected to series-compensated transmission line,” IEEE Trans. Power Syst., Vol. 28, No. 3, pp. 2573-2583, Aug. 2013.

[31] S. Liang, Q. Hu, W. Lee, “A survey of harmonic emissions of a commercially operated wind farm,” IEEE Trans. Ind. Appl., Vol. 48, No. 3, pp. 1115-1123, May 2015.



그림입니다.
원본 그림의 이름: image84.jpeg
원본 그림의 크기: 가로 150pixel, 세로 175pixel

Penghan Li was born in Shanxi, China, in 1993. He received his B.S. degree in Electrical Engineering from Harbin Institute of Technology, Harbin, China, in 2016. He is working toward his Ph.D. degree in electrical engineering in the School of Electronic Information and Electrical Engineering at Shanghai Jiao Tong University, Shanghai, China. His research interests include power system nonlinear control, sub-synchronous interaction, and DFIG control.


그림입니다.
원본 그림의 이름: image85.jpeg
원본 그림의 크기: 가로 135pixel, 세로 160pixel

Linyun Xiong was born in Chongqing, China, in 1993. He received his B.S. degree in Electrical Engineering from Sichuan University, Chengdu, China, in 2015. He is working toward his Ph.D. degree in the School of Electronic Information and Electrical Engineering at Shanghai Jiao Tong University, Shanghai, China. His research interests include power system nonlinear control and Hamiltonian systems.


그림입니다.
원본 그림의 이름: image86.png
원본 그림의 크기: 가로 143pixel, 세로 169pixel

Jie Wang was born in Jiangsu, China, in 1960. He received his M.S. degree from the North China Electric Power University, Baoding, China, in 1991, and his Ph.D. degree from Northeastern University, Shenyang, China, in 1998. From 1999 to 2001, he researched nonlinear control and Hamiltonian systems in Shanghai Jiao Tong University, Shanghai, China. From 1982 to 2002, he worked on industrial automation and power system automation. He is presently working in the field of power system stability and control.


그림입니다.
원본 그림의 이름: image87.jpeg
원본 그림의 크기: 가로 166pixel, 세로 190pixel

Meiling Ma was born in Jiangsu, China, in 1991. She received her B.S. and M.S. degrees from Nanjing Normal University, Nanjing, China, in 2012 and 2015, respectively. From 2015 to 2017, she worked in the field of nonlinear dynamics of power systems in Nanjing Normal University Taizhou College, Taizhou, China. She is currently working toward her Ph.D. degree in the School of Electronic Information and Electrical Engineering at Shanghai Jiao Tong University, Shanghai, China. Her research interests include power system transient stability and Hamilton realization of power systems.


그림입니다.
원본 그림의 이름: image88.jpeg
원본 그림의 크기: 가로 136pixel, 세로 161pixel

Muhammad Waseem Khan received his B.E. degree from the Department of Electrical Engineering, Sarhad University of Science and Information Technology, Peshawar, Pakistan, in 2012, and his M.S. degree from the Department of Electrical Engineering, Shanghai Jiao Tong University, Shanghai, China, in 2017. He is currently a Ph.D. research scholar in the Department of Electrical Engineering, Shanghai Jiao Tong University, Shanghai, China. His research interests include microgrid control, optimization, and management using multi-agent systems for the integration of distributed renewable energy generation.