Optimum design of proportional-integral controllers in

INTERNATIONAL TRANSACTIONS ON ELECTRICAL ENERGY SYSTEMS Int. Trans. Electr. Energ. Syst. (2015) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/etep.2120

Optimum design of proportional-integral controllers in gridintegrated PMSG-based wind energy conversion system Saurabh M. Tripathi2*,†, Amar Nath Tiwari1 and Deependra Singh2 1

Department of Electrical Engineering, Madan Mohan Malaviya University of Technology, Gorakhpur, (U.P.), India 2 Department of Electrical Engineering, Kamla Nehru Institute of Technology, Sultanpur, (U.P.), India

SUMMARY In this paper, a direct-driven small permanent magnet synchronous generator (PMSG) wind energy conversion system (WECS) with a back-to-back power converter working in grid-connected mode is discussed. Control structures based on field-oriented control and voltage-oriented control mechanisms are proposed for machine-side converter and grid-side converter, respectively. Optimization approaches namely ‘modulus optimum’ and ‘symmetric optimum’ are used to obtain analytical expressions for the selection of the parameters of involved proportional-integral (PI) controllers in different control loops. A transient system simulation using SimPowerSystem is built to evaluate the performance of the PMSG-based WECS by employing selected values of PI controller parameters both under varying wind conditions and under symmetrical and asymmetrical grid-fault conditions. Copyright © 2015 John Wiley & Sons, Ltd. key words:

Fault ride-through (FRT) field-oriented control (FOC); grid-side converter (GSC); machineside converter (MSC); modulus optimum (MO); permanent magnet synchronous generator (PMSG); symmetric optimum (SO); voltage-oriented control (VOC); wind energy conversion system (WECS)

1. INTRODUCTION With the recent advances in power converter technology and availability of permanent magnet materials, direct-drive permanent magnet synchronous generators (PMSGs) have increasingly drawn the attention of wind turbine manufacturers [1]. PMSG offers high efficiency, low maintenance, no excitation system, and improved power factor and elimination of the gearbox [2–6]. Moreover, a full-scale power electronic converter between the PMSG and the utility grid offers not only additional technical performances such as complete decoupling from the grid, full controllability of the system for maximum wind power extraction, high performance, high precision, high reliability, wide operating range and improved capability of fault-ride-through (FRT) but also paybacks in the losses occurring in the power conversion stage [1,7]. Pulse width modulation (PWM) voltage source converter (VSC) is the state-of-the-art technology nowadays increasingly favored by all wind turbine manufactures. The possibility of high switching frequencies accompanied by an appropriate control strategy makes PWM converters fit for grid interface of wind generation system [7]. Most of the industrial controllers still in operation are based on PI control law, which offers the simplest, feasible, and most efficient solution to control problems [8–10]. Parametric tuning and optimization of the PI controller can be viewed as one of the most crucial engineering task during the commissioning of control system so as to obtain the desired control responses [11]. Parametric tuning of the PI controller is a compromise between speed of response and stability for small-signal disturbances as well as robustness to tolerate large-signal disturbances [12]. The relative stability of a control *Correspondence to: Saurabh M. Tripathi, Department of Electrical Engineering, Kamla Nehru Institute of Technology, Sultanpur—228 118 (U.P.), India. † E-mail: [email protected] Copyright © 2015 John Wiley & Sons, Ltd.

S. M. TRIPATHI ET AL.

system can be analyzed using two quantitative measures defined as ‘gain margin (GM)’ and ‘phase margin (PM)’ determined from the Bode plot of open-loop transfer function. Further, the locations of the closed-loop poles decide the transient performance and stability of a control system. A large amount of research work has already been carried out in the past to determine the PI controllers’ parameters [13]. Numerous tuning techniques are summarized in [11,14] and the references therein. Each tuning technique has its own features and constraints. Brief descriptions of a few PI tuning and optimization techniques are presented in the following paragraphs along with their key features and limitations so as to stimulate the readers to compare and judge each of these techniques individually for their superiority. One of the most popular Zeigler-Nichols (Z-N) method offers a rough estimate of the PI controller parameters, which need to be further adjusted heuristically by the designers so as to achieve the desired closed-loop response and is particularly suitable for the system with monotonic step response (S-shape response) [13,15]. The PI controller tuned according to Z-N method may cause high overshoot, large oscillation, and longer settling time in closed-loop response of a higher-order system [13]. The approach based on internal model control is another most commonly used PI tuning technique, which essentially requires the plant model to be reduced to first or second order before controller design rules could be applied [16]. However, the choice of the reduction method greatly influences the resulting controller design [16]. In order to address different aspects of the controller design problem, several methods employ optimization techniques. For instance, [17] copes with controller optimization with respect to the criteria ISE (integral-square error), IAE (integral-absolute error), and ITAE (integral time absolute error). By focusing on the square of the error function, the ISE criterion penalizes the positive as well as negative values of the errors, whereas by focusing on the magnitude of the error function, the IAE criterion penalizes either the positive or negative values of the errors [18]. Because both ISE and IAE indices weight all errors equally independent of time, their minimization may result in step response with relatively small overshoot but a longer settling time [19,20]. The long duration transients in step response can be penalized by proposing the criteria ITSE (integral time square error) and ITAE [18,19]. Although the ITSE and ITAE criteria may overcome the disadvantages associated with the ISE and IAE criteria, respectively, the process of deriving the analytical formula becomes complex and time-consuming [20]. Mostly, PI tuning techniques are model based, and therefore, the PI controller might not lead to specified control performance if the controlled plant is uncertain [10,21]. This is why the designers always look for robust control design by synthesizing a controller for which the closed-loop system is stable and the specified control performance could be achieved despite plant uncertainty [21,22]. In fact, the system’s robustness (i.e., the system’s ability to withstand changes in its parameters before becoming unstable) is highly reliant on the greatness of GM and PM [21]. Intensive researches have been published by many authors on designing PI controllers so as to meet GM and PM specifications. Besides robustness, a good closed-loop control performance is also vital, and therefore, a trade-off between the robustness and the control performance is usually taken into consideration while synthesizing the PI/PID controller parameters [21]. For instance, a PID controller has been designed in [21] based on non-linear optimization wherein the closed-loop bandwidth is maximized for specified GM and PM with constraint on overshoot ratio so as to satisfy the criteria of both robustness and closedloop control performance in the design. Genetic algorithm (GA)-based methods have recently received great interest in searching global optimal solution in PI controller design problem [20]. However, the natural genetic operations of GAbased methods often lead to enormous computational efforts compared with other traditional methods to give an optimal solution [13,20]. Moreover, the premature convergence of GA algorithm not only degrades its performance but also reduces its search capability [13,20]. Further, some deficiencies in GA performance are also apparent in applications where the parameters being optimized are highly correlated [20]. The particle swarm optimization (PSO) is one of the modern heuristic optimization algorithms, which can circumvent the premature convergence and can generate high-quality solution to a PI controller design problem with no complicated evolutionary operations (i.e., selection, crossover, and mutation), and therefore, it reduces the computational efforts to some extent compared with the GA method [10,20]. Nevertheless, the PSO is usually employed as a minor compensatory tuner [10]. Copyright © 2015 John Wiley & Sons, Ltd.

Int. Trans. Electr. Energ. Syst. (2015) DOI: 10.1002/etep

OPTIMUM DESIGN OF PI CONTROLLERS

Loop-shaping criteria viz. modulus optimum (MO) and symmetric optimum (SO) are two fairly simple and generic PI controller tuning techniques for designing ‘optimal linear-control system’ in the frequency domain when certain parameters of the system are known [23–25]. These criteria are introduced in [9,11,12,25,26] and have conventionally been used for the evaluation of involved PI controller parameters in vector control structures of electric motor drives [12,23,27]. An experimental performance evaluation of PI tuning techniques based on MO and SO criteria applied to the field-oriented control structure of a permanent magnet synchronous motor drive has been presented in [28]. A discrete-time equivalent to MO and SO criteria has been investigated in [23] and applied to the vector control of grid-connected VSC. MO and SO prove to be a feasible controller design technique with a simple extension to apply it even on higher-order plants using the active damping approach proposed in [24]. To design PID controller for ‘type-p control loops’, which is characterized by the presence of ‘p’ integrators in the open-loop transfer function, the SO criterion has been extended in [9]. The MO and SO criteria present at least two major advantages in the design of control loops—(i) complete plant model is not a requisite and (ii) set-point response of the closed-loop system is satisfactory [9]. Among different techniques of PI controller parameters’ tuning, which differ in complexity, flexibility, and requirement of plant knowledge, there is always a need for a simple, straightforward, and intuitive technique requiring least plant knowledge and offering the best possible control performances [11,21]. Despite the widespread usage of PI controllers in the control structures of WECSs employing different sorts of generator technology, there are only a very few literatures that address the optimum design of controller parameters in wind power applications. For instance, the authors of [29] presented an optimum coordinated controller design for PMSG-based WECS where controller parameters are determined by optimizing the considered performance indices such as peak overshoot and settling time. It has been realized that the optimum design of PI controllers in the control structures of a WECS so as to obtain the best possible trade-off between closed-loop control performance and robustness is still a topic that is regarded to be poorly reported, and this paper addresses this need by presenting a systematic design procedure for a grid-connected WECS employing small PMSG along with the sizing of components and evaluation of involved PI controllers in different control loops of the same using easily understandable MO and SO criteria. The problem is formulated as ‘Tune the involved PI controller parameters in the control structures of a PMSG-based WECS such that the best possible trade-off between closed-loop control performance and system’s robustness margin could be realized’.

2. SYSTEM DESCRIPTION AND CONTROL PHILOSOPHY The proposed WECS mainly consists of a wind turbine, a PMSG with surface-mounted permanent magnets, a frequency converter built by two current regulated PWM VSCs namely (i) machine-side converter (MSC) and (ii) grid-side converter (GSC), and a common dc-link capacitor in between as shown in Figure 1. 2.1. Wind turbine model The output power of the wind turbine [30] is expressed as Pm ¼ 0:5ρC p AV 3ω

(1)

where Pm, ρ, Cp, A, R, and Vω are the turbine output power, air density, power coefficient, swept area (= π R2), radius of the turbine blades and wind speed, respectively. The power coefficient Cp is a function of the tip-speed ratio λ and pitch angle θ [31], expressed as 151 18:4 (2) 0:58θ 0:002θ2:14 13:2 exp C p ðλ; θÞ ¼ 0:73 λi λi 1 1 0:035 ¼ λi λ þ 0:08θ θ3 þ 1

(3)

λ ¼ ωr R=V ω

(4)

where ωr is the turbine rotational speed, and for lower to medium wind speeds, θ can be set to zero [32]. Copyright © 2015 John Wiley & Sons, Ltd.



Figure 1. Proposed PMSG-based wind energy conversion system.

2.2. Generator model The stator voltage equations of PMSG in d-q reference frame [33,34] are expressed as vds ¼ Rs ids þ Ld

d ids ωs Lq iqs dt

(5)

d iqs þ ωs Ld ids þ ωs λm (6) dt where ids and iqs are the d - q axes stator currents, Rs is the stator resistance, Ld and Lq are the d - q axes stator inductances, λm is the rotor flux, and ωs is the electrical speed. The electromagnetic torque [34] is given as 3 P (7) T e ¼ λm iqs þ Ld Lq ids iqs 2 2 where P is the number of PMSG poles. It is noticeable that for low-speed PMSG with surface-mounted permanent magnets, d and q axes inductances are the same; i.e., Ld ≈ Lq [32]. The parameters of PMSG [35] are listed in the appendix. vqs ¼ Rs iqs þ Lq

Copyright © 2015 John Wiley & Sons, Ltd.



2.3. DC-link capacitor A dc-link capacitor is essential to filter out dc voltage ripples across it [36]. For satisfactory PWM control, the dc-link voltage VDC is selected [31] as follows pffiffiffi pffiffiffi V DC > 2 2V LL = ma 3 (8) where VLL is the line-line root-mean-square (r.m.s.) voltage on the ac side of the PWM converter and ma is the modulation index considered equal to maximum value of one. It is to be noted that the constraint (8) on VDC is from the r.m.s. voltages on the ac sides of both MSC and GSC. The dc-link capacitor value is estimated [31] as pffiffiffi C ¼ 0:9I peak = 4 2πf V DCripple (9) where Ipeak is the permissible peak ac line current, f is the grid frequency, and V DCripple is the permissible ripple dc-link voltage (2%). 2.4. Coupling AC inductor The value of coupling ac inductor in between GSC and the grid is calculated [31] as pffiffiffi Lg ¼ 3ma V DC = 12αs f gsw ip

(10)

where αs is safety factor (120%), fgsw is the switching frequency of the GSC (10 kHz selected), and ip is the peak-to-peak permissible ripple current (5%). 2.5. Maximum power point tracking In view of the fact that any amount of power generated by the wind generation system can be injected into the grid, a grid-integrated WECS is always operated at maximum power point (MPP) to maximize the generation and utilization of power [1,37]. For maximum power point tracking (MPPT), the PMSG is operated in variable–speed–variable–frequency mode, where the rotor speed is allowed to vary in sympathy with the wind speed by maintaining λ to the optimum value [38]. The wind speed is measured using an anemometer, and a reference rotor speed corresponding to the MPP is generated using Equation (11). ωr ¼ λopt V ω =R

(11)

2.6. Controller blocks Controller blocks are chosen to be PI controllers for different control loops. Small overshoot, good damping of oscillations, and fast response are the three fundamental goals of the designers for the synthesis of involved PI controller in a control loop. The evaluation of PI controller parameters is one of the key issues in the design of a cascaded control structure where inner loops are designed to achieve fast response and outer loop is designed to achieve optimum regulation and stability [12]. As this paper deals with the synthesis of involved PI controllers in different control loops of a PMSG-based WECS using MO/SO criteria, brief ideas of these criteria clearly stating the objectives of their formulations are presented in the following subsections. 2.6.1. Modulus optimum. The MO is an optimization criterion generally dealing with the PI controller design for second-order control systems having one dominant time constant and other minor time constant [24,39]. The MO is formulated with the design objective to obtain a controller that gives the magnitude of the frequency response of the closed-loop transfer function as flat and as close to unity for a large bandwidth as possible [11,25,40]. If G(p) is the closed-loop transfer function

from the set point to d n j GðjωÞ j ¼ 0 for as many n the output, the controller is determined in such a way that G(0) = 1 and dωn ω¼0

as possible [11]. Copyright © 2015 John Wiley & Sons, Ltd.



This method is used because of its simplicity and fast response. The standard form of the control system’s transfer function for the MO is achieved by canceling the dominant time constant, and therefore, the system’s performance in response to any disturbance applied anywhere other than the set point is not optimal [25]. The MO results in a fast and non-oscillatory closed-loop time response for a large class of plants [25]. When it is possible to approximate the plant model with the benchmark transfer function of the form K1 ; T 1 >> T 2 (12) GP1 ðpÞ ¼ ð1 þ pT 1 Þð1 þ pT 2 Þ where K1 is the plant gain, T1 is the plant’s dominant time constant, and T2 is the plant’s parasitic time constant or minor time constant; the MO criterion leads to the proportional gain Kp1,MO and integral time constant Ti1,MO of the PI controller expressed as K p1; MO ¼

T1 and T i1; MO ¼ T 2 2 T 2K1

(13)

2.6.2. Symmetric optimum. When one pole of the open-loop transfer function is near to the origin or at the origin itself, the SO criterion is used for the evaluation of the PI controller parameters [12]. Suggested as an extension of the MO, the SO criterion is formulated with the design objective to obtain a controller that compels the magnitude of the closed-loop frequency response as close to unity in the widest possible frequency range as possible and maximizes the phase margin for given frequency so that the system can tolerate delays and, in addition, optimizes the behavior of the control system with respect to the disturbances affecting the control plant [9,12,24,26]. The system’s nonlinearities and time-varying parameters are also well-handled by the PI controller designed using the SO criterion. When it is possible to approximate the plant model with the benchmark transfer function of the form given by Equation (12), the SO criterion leads to the proportional gain Kp1,SO and integral time constant Ti1,SO of the PI controller expressed as T1 K p1; SO ¼ and T i1; SO ¼ a2 T 2 (14) a T 2K 1 Instead, when it is possible to approximate the plant model with the benchmark transfer function of the form given by K2 (15) GP2 ðpÞ ¼ p ð1 þ pT 2 Þ where K2 is the plant gain and T2 is the plant’s parasitic time constant; the SO criterion leads to the proportional gain Kp2,SO, and integral time constant Ti2,SO of the PI controller expressed as K p2; SO ¼

1 and T i2; SO ¼ a2 T 2 a T 2K2

(16)

The parameter ‘a’, which constitutes a trade-off between damping of the poles in the closed-loop transfer function and dynamic responses, can be chosen so as to achieve required damping ξ and desired performance [12,24] as a ¼ 2ξ þ 1

(17)

However, in literature, the parameter ‘a’ is constrained between 2 and 4 for optimization according to SO criterion [23]. The name of the SO criterion comes from the symmetry exhibited by the openloop frequency response [25,26].

3. FIELD-ORIENTED CONTROL OF MSC The main task of MSC is to extract the maximum power from the input source [7]. The field-oriented control structure for MSC consists of two control loops—(i) inner hysteresis-based current control loop and (ii) outer PMSG speed control loop. The actual rotor speed ωr of the PMSG is compared with the Copyright © 2015 John Wiley & Sons, Ltd.



reference rotor speed ωr estimated by the MPPT algorithm, and the speed error is processed through the outer speed PI controller so as to estimate the reference electromagnetic torque T e as follows 1 t (18) ∫ ωr ωr dτ T e ¼ K ps ωr ωr þ T is 0 where Kps is the proportional gain and Tis is the integral time constant of the outer speed PI controller. Transforming Equation (18) in Laplace domain, we have 1 þ pT is (19) T e ¼ K ps ω r ωr pT is The reference torque T e so generated is then used to estimate the reference q-axis stator current iqs using Equation (7). In order to achieve high torque to current ratio as well as to avoid demagnetization for the surface type PMSG, the reference d-axis stator current ids is typically set to zero [32,33]. Using inverse Park’s transformation of the reference d-q axes stator currents, the reference three-phase stator currents I sabc are estimated. The actual and reference three-phase stator currents are compared, and the resulting errors are processed into an inner hysteresis current controller (HCC) to generate switching signals for the MSC. The switching frequency for the MSC is limited within its specified range by proper selection of the hysteresis band [41,42]. 3.1. Design of speed PI controller Because hysteresis current control has negligible inertia and delay [30], and the actual current is limited within the tolerance band of the reference current, the HCC-based current control block can be viewed simply as a ‘unitary gain’ in the block diagram of outer speed control loop shown in Figure 2. The open-loop transfer function of the outer speed control loop is modeled as 1 þ pT is 1 3 P λm P (20) GOMS ðpÞ ¼ K ps pT is 1 þ pT s 4 p ð2J Þ where Ts is the sample time for speed control loop and J is the turbine-generator mechanical system inertia. The speed PI controller parameters addressed by SO are calculated as K ps ¼

8J and T is ¼ a2 T s 3 a P2 λm T s

(21)

On substitution of Equation (21) into Equation (20), GOMS(p) is simplified as GOMS ðpÞ ¼

1 þ p ða2 T s Þ p2 a3 T 2s þ p3 a3 T 3s

(22)

The closed-loop transfer function of the outer speed control loop is given as GCMS ðpÞ ¼

1 þ p ða 2 T s Þ 1 þ p ða2 T s Þ þ p2 a3 T 2s þ p3 a3 T 3s

(23)

The response of the outer speed control loop due to a change in the disturbance input is obtained from the transfer function derived as p a3 T 2s P þ p2 a3 T 3s P (24) GDMS ðpÞ ¼ 2J þ p ð2 a2 T s J Þ þ p2 2 a3 T 2s J þ p3 2 a3 T 3s J

Figure 2. Outer PMSG speed control loop. Copyright © 2015 John Wiley & Sons, Ltd.



The respective open-loop Bode plots, locations of closed-loop poles, step and disturbance responses of PMSG speed control loop with SO tuning of involved PI controller given by Equation (21) for the chosen values of a = 2, a = 2.4142, a = 3, and a = 4 are shown in Figure 3, and the outcomes are listed in Table I. It can be observed that the increment in parameter ‘a’ increases the PM, which, in turn, results in decreased sensitivity toward changes in the system’s parameter and, consequently, the improved system’s robustness. Moreover, more favorable closed-loop pole-zero placements can be obtained through variation in the parameter ‘a’. A lower value of ‘a’ (< 2.4142) results in poor damping of poles while a higher value of ‘a’ (≥2.4142) leads to improved damping but slower system response. Thus, the PM should be at least 45∘ so as to obtain system’s dynamic response without oscillations. It is also analyzed that the SO tuning of PI speed controller gives tracking performance with large overshoot caused by the forcing element in the numerator of GCMS(p). So, as another choice to decrease the overshoot and to enhance the performance of the controller, a first-order pre-filter GFMS(p) on the reference signal is employed [12]. GFMS ðpÞ ¼

1 1 þ p ða 2 T s Þ

(25)

Thus, the closed-loop transfer function GCMS(p) reduces to GCFMS ðpÞ ¼

1þp

ða2 T

sÞ

þ

p2

1 a3 T 2s þ p3 a3 T 3s

(26)

It is worth noticeable that the disturbance rejection capability is not affected because the pre-filter acts only at the reference signal outside of the control loop. Usually, a trade-off between the robustness and the control performance is taken into consideration while synthesizing the PI controller [21]. Consequently, the optimum selection of the speed PI controller parameters would be a compromise among

Figure 3. (a) Bode plot of open-loop transfer function and (b) root-locus of closed-loop transfer function for PMSG speed PI controller design addressed by—SO with a = 2; SO with a = 2.4142; SO with a = 3; and SO with a = 4. Performance of outer speed control loop—(c) step responses and (d) disturbance rejection capability. Copyright © 2015 John Wiley & Sons, Ltd.



Kps

7.12 5.90 4.75 3.56

Tuning criterion

SO (a = 2) SO (a = 2.4142) SO (a = 3) SO (a = 4)

0.0020 0.0029 0.0045 0.0080

Tis

PM (deg) 36.9 45.0 53.1 61.9

GM (dB) –Inf –Inf –Inf –Inf 1000 828 667 500

Crossover frequency (rad/s) Yes Yes Yes Yes

Whether closed loop stable? 0.5 0.707 1 1

Damping ratio 43.4 33.6 24.9 17.3

Overshoot (%)

8.3 7.5 11.8 20.5

Settling time (ms)

Without pre-filter

8.15 1.40 — —

Overshoot (%)

6.6 5.9 11.3 23.8

Settling time (ms)

With pre-filter

Step response

Table I. Outcomes of outer speed PI controller design.

0.03 0.03 0.04 0.05

Undershoot (%)

9.4 8.7 12.7 26.5

Settling time (ms)

Disturbance rejection capability




the performances as reviewed in Table I. Hence, the selected transfer function of speed PI controller is given by 1 þ 0:0029p (27) Gcs ðpÞ ¼ 5:90 0:0029p 4. VOLTAGE-ORIENTED CONTROL OF GSC The main task of GSC is to synchronize the WECS with the grid ensuring high quality of power delivered to the grid and to control the dc-link voltage [7]. The voltage-oriented control structure for GSC consists of two cascaded control loops—(i) inner current control loop and (ii) outer dc-link voltage control loop. The actual dc-link voltage VDC is compared with the reference value V DC , and the voltage error is processed through the outer dc-link voltage PI controller so as to generate the reference d-axis grid current idg as 1 þ pT iv V DC V DC (28) idg ¼ K pv pT iv where Kpv is the proportional gain and Tiv is the integral time constant of the outer dc-link voltage PI controller. In order to achieve unity-power factor operation of the WECS, the reference q-axis grid current iqg is typically set to zero [32,37]. By using Park’s transformation of the three-phase grid currents Igabc, the actual d-q axes grid currents idg and iqg are estimated. The actual and reference d-q axes grid currents are compared, and the resulting errors are processed into the inner current PI controllers so as to generate reference d-q axis GSC voltages. For grid synchronization of WECS, a phase-locked-loop technique [43] is used. 4.1. Design of inner current PI controllers The grid voltage equations in d-q reference frame, keeping grid voltage vector aligned to the d-axis [23], are didg ωg Lg iqg þ vdg (29) dt diqg þ ωg Lg idg þ vqg (30) eqg ¼ Rg iqg þ Lg dt where ωg is the angular grid frequency, vdg and vqg are the d-q axes GSC voltages, Rg and Lg are the coupling resistance and inductance, respectively. To facilitate independent control of d-q axes components of grid currents, the cross-couplings due to the coupling inductor are decoupled [12,37] by defining equivalent control signals as edg ¼ Rg idg þ Lg

udg ¼ vdg þ ωg Lg iqg þ edg

(31)

uqg

(32)

¼ vqg ωg Lg idg þ eqg

The signals udg and uqg are derived from the inner current control loop as 1 þ pT ic idg idg udg ¼ K pc pT ic 1 þ pT ic iqg iqg uqg ¼ K pc pT ic

(33) (34)

where Kpc is the proportional gain and Tic is the integral time constant of the inner current PI controllers. The decoupled d-q axes reference GSC voltages are then expressed as follows vdg ¼ udg þ ωg Lg iqg þ edg

(35)

vqg ¼ uqg ωg Lg idg þ eqg

(36)




By using inverse Park’s transformation, Equations (35), (36) are transformed into the reference a-bc voltages and are then applied to the PWM controller to generate switching signals for the GSC [32]. By using Equations (29)–(32), the process transfer function of d-q axes current control loops is obtained as I qg ðpÞ I dg ðpÞ K gc ¼ ¼ (37) 1 þ pT gc U dg ðpÞ U qg ðpÞ where Kgc = 1/Rg and Tgc = Lg/Rg. The open-loop transfer function of the inner current control loops as shown in Figure 4 is modeled as 1 þ pT ic 1 K gc (38) GOGC ðpÞ ¼ K pc pT ic 1 þ pT gc 1 þ pT sc where Tsc = Tc + 0.5 Tc = 1.5 Tc is the sum of minor time constants and Tc is the sample time for inner current control loop. The current PI controller parameters addressed by MO are calculated as T gc and T ic; MO ¼ T gc (39) K pc; MO ¼ 2 T sc K gc On substitution of Equation (39) into Equation (38), GOGC(p) with MO tuning of PI controller is simplified as GMO OGC ðpÞ ¼

1 p ð2 T sc Þ þ p2 2 T 2sc

(40)

The closed-loop transfer function of inner current control loop with MO tuning of PI controller is then expressed as GMO CGC ðpÞ ¼

1 1 þ p ð2 T sc Þ þ p2 2 T 2sc

(41)

With MO tuning of PI controller, the response of the inner current control loop due to a change in the disturbance input is obtained from the transfer function derived as p 2 T sc K gc þ p2 2 T 2sc K gc MO GDGC ðpÞ ¼ (42) 1 þ p 2 T sc þ T gc þ p2 2 T 2sc þ 2 T sc T gc þ p3 2 T 2sc T gc On the other hand, the current PI controller parameters addressed by SO are calculated as T gc and T ic; SO ¼ a2 T sc K pc; SO ¼ aT sc K gc

(43)

On substitution of Equation (43) into Equation (38), GOGC(p) with SO tuning of PI controller is simplified as 1 þ p ða2 T sc Þ GSO (44) OGC ðpÞ ¼ 2 3 2 p a T sc þ p3 a3 T 3sc The closed-loop transfer function of inner current control loop with SO tuning of PI controller is then expressed as 1 þ p ða2 T sc Þ GSO (45) CGC ðpÞ ¼ 1 þ p ða2 T sc Þ þ p2 a3 T 2sc þ p3 a3 T 3sc

Figure 4. Inner current control loop. Copyright © 2015 John Wiley & Sons, Ltd.



With SO tuning of PI controller, the response of inner current control loop due to a change in the disturbance input is obtained from the transfer function derived as GSO DGC ðpÞ

p2 a3 T 2sc K gc þ p3 a3 T 3sc K gc ¼ 1 þ p a2 T sc þ T gc þ p2 a3 T 2sc þ a2 T sc T gc þ p3 a3 T 3sc þ a3 T 2sc T gc þ p4 a3 T 3sc T gc (46)

The respective open-loop Bode plots, locations of closed-loop poles, step and disturbance responses of the inner current control loop with MO-based and SO-based tuning of involved PI controller parameters given by Equations (39) and (43), respectively, are shown in Figure 5, and the outcomes are listed in Table II. It is observed that the MO tuning of PI controller results in the highest PM (=65.5∘), whereas the SO tuning with a = 2 results in the lowest PM (=36.9∘). However, the increment in parameter ‘a’ increases the PM which, in turn, results in decreased sensitivity toward changes in the system’s parameter and, consequently, the improved system’s robustness. On the other hand, a lower value of ‘a’ (< 2.4142) results in poor damping of poles while a higher value of ‘a’ (≥2.4142) leads to improved damping but slower system response. It is also evident that MO tuning of current PI controller results in good response with small overshoot to a step-change in reference signal, but the disturbance rejection capability is the worst. On the other hand, the SO tuning of current PI controller leads to much better disturbance rejection capability. To compensate for the forcing element in the numerator of GSO CGC ðpÞ, a first-order pre-filter GFGC(p) on the reference signal is also employed [12]. GFGC ðpÞ ¼

1 1 þ p ða2 T sc Þ

(47)

Figure 5. (a) Bode plot of open-loop transfer function and (b) root-locus of closed-loop transfer function for inner current PI controller design addressed by—MO; SO with a = 2; SO with a = 2.4142; SO with a = 3; and SO with a = 4. Performance of inner current control loop—(c) step responses and (d) disturbance rejection capability. Copyright © 2015 John Wiley & Sons, Ltd.



Kpc

85.33 85.33 70.69 56.89 42.67

Tuning criterion

MO SO (a = 2) SO (a = 2.4142) SO (a = 3) SO (a = 4)

0.00690 0.00030 0.00044 0.00068 0.00120

Tic Inf –Inf –Inf –Inf –Inf

GM (dB) 65.5 36.9 45.0 53.1 61.9

PM (deg)

Whether closed loop stable? Yes Yes Yes Yes Yes

Crossover frequency (rad/s) 6.07 × 103 6.67 × 103 5.52 × 103 4.44 × 103 3.33 × 103 0.707 0.5 0.707 1 1

Damping ratio 4.32 43.4 33.6 24.9 17.3

Overshoot (%)

0.63 1.24 1.12 1.78 3.07

Settling time (ms)

Without pre-filter

8.15 1.40 — —

0.99 0.88 1.69 3.57

Settling time (ms) N/A

Overshoot (%)

With pre-filter

Step response

Table II. Outcomes of inner current PI controller design.

0.06 0.05 0.06 0.07 0.09

Undershoot (%)

26.9 6.6 9.5 13.0 17.8

Settling time (ms)





Thus, the closed-loop transfer function GSO CGC ðpÞ reduces to GCFGC ðpÞ ¼

1þp

ða2 T

sc Þ

þ

p2

1 a3 T 2sc þ p3 a3 T 3sc

(48)

The disturbance rejection capability is not affected from the employment of pre-filter on the reference signal. The highest crossover frequency is approximately 10 times lower than GSC switching frequency, which is vital to avoid switching noise interference. To satisfy the criteria of both robustness and closed-loop performance in the control design, the optimum selection of the current PI controller parameters would be a compromise among the performances as reviewed in Table II. Hence, the selected transfer function of current PI controller is given by 1 þ 0:00044p (49) Gcc ðpÞ ¼ 70:69 0:00044p

4.2. Design of the outer DC-link voltage PI controller Because Tsc is very small, hence, for the analysis of the outer dc-link voltage control loop, T 2sc and T 3sc can be neglected so as to approximate the inner current closed-loop transfer functions expressed as Equations (41) and (48) by an equivalent first-order transfer functions as GMO CGC ðpÞ≈

1 1 þ p ð2 T sc Þ

(50)

GCFGC ðpÞ≈

1 1 þ p ða2 T sc Þ

(51)

Combining Equations (50) and (51) and representing the approximated inner current closed-loop transfer function as GGCL(p), we obtain GGCL ðpÞ ¼

1 1 þ pT GCL

(52)

where TGCL = 2 Tsc for current PI controller addressed by MO criterion or TGCL = a2Tsc for current PI controllers addressed by SO criterion. Thus, the open-loop transfer function GOGV(p) of the dc-link voltage control loop as shown in Figure 6 is modeled as 1 þ pT iv 1 3 edg 1 (53) GOGV ðpÞ ¼ K pv pT iv 2 V DC 1 þ pT sv pC where Tsv = Tv + TGCL is the sum of minor time constants and Tv is the sample time for the outer dc-link voltage control loop. The dc-link voltage PI controller parameters addressed by SO are calculated as K pv ¼

2 CV DC and T iv ¼ a2 T sv 3 a edg T sv

(54)

On substitution of Equation (54) into Equation (53), GOGV(p) is simplified as GOGV ðpÞ ¼

1 þ p ða2 T sv Þ p2 a3 T 2sv þ p3 a3 T 3sv

(55)

Figure 6. Outer dc-link voltage control loop Copyright © 2015 John Wiley & Sons, Ltd.



The closed-loop transfer function of dc-link voltage control loop is then expressed as GCGV ðpÞ ¼

1 þ p ða2 T sv Þ 1 þ p ða2 T sv Þ þ p2 a3 T 2sv þ p3 a3 T 3sv

(56)

The response of the outer dc-link voltage control loop due to a change in the disturbance input is obtained from the transfer function derived as p a3 T 2sv þ p2 a3 T 3sv (57) GDGV ðpÞ ¼ C þ p ða2 T sv C Þ þ p2 a3 T 2sv C þ p3 a3 T 3sv C The respective open-loop Bode plots, locations of closed-loop poles, and step and disturbance responses of outer dc-link voltage control loop with SO tuning of involved PI controller parameters given by Equation (54) for the selected values of a = 2, a = 2.4142, a = 3, and a = 4 are shown in Figure 7, and the outcomes are listed in Table III. With these selections, an identical trend in the performance characteristics can easily be seen as was discussed for PMSG speed PI controller design. Further, to compensate for the forcing element in the numerator of GCGV(p), a first-order pre-filter GFGV(p) on the reference signal is also employed [12]. GFGV ðpÞ ¼

1 1 þ p ða2 T sv Þ

(58)

Thus, the closed-loop transfer function GCGV(p) reduces to GCFGV ðpÞ ¼

1þp

ða2 T

sv Þ

þ

p2

1 a3 T 2sv þ p3 a3 T 3sv

(59)

Figure 7. (a) Bode plot of open-loop transfer function and (b) root-locus of closed-loop transfer function for dclink voltage PI controller design addressed by—SO with a = 2; SO with a = 2.4142; SO with a = 3; and SO with a = 4. Performance of outer dc-link voltage control loop—(c) step responses and (d) disturbance rejection capability. Copyright © 2015 John Wiley & Sons, Ltd.



0.84 0.69 0.56 0.42

SO SO SO SO

(a = 2) (a = 2.4142) (a = 3) (a = 4)

Kpv

Tuning criterion

0.0037 0.0055 0.0084 0.0150

Tiv

PM (deg) 36.9 45.0 53.1 61.9

GM (dB) –Inf –Inf –Inf –Inf 534 442 356 267

Crossover frequency (rad/s) Yes Yes Yes Yes

Whether closed loop stable? 0.5 0.707 1 1

Damping ratio 43.4 33.6 24.9 17.3

Overshoot (%)

15.5 14.0 22.2 38.3

Settling time (ms)

Without pre-filter

8.15 1.40 — —

Overshoot (%)

12.4 11.0 21.1 44.7

Settling Time (ms)

With Pre-filter

Step response

Table III. Outcomes of outer dc-link voltage PI controller design.

8.29 9.74 11.8 15.4

Undershoot (%)

17.6 16.4 23.8 49.7

Settling time (ms)





From design point of view, the optimum selection of the dc-link voltage PI controller parameters would be a compromise among the performances as reviewed in Table III. Hence, the selected transfer function of dc-link voltage PI controller is given by 1 þ 0:0055p (60) Gcv ðpÞ ¼ 0:69 0:0055p

5. PERFORMANCE TEST RESULTS AND DISCUSSION Having designed the parameters of involved PI controllers, the effectiveness of the same is evaluated analytically through a detailed simulation model built in SimPowerSystem of MATLAB/Simulink by

Figure 8. Performance of the PMSG-based wind energy conversion system under varying wind condition (a) rotor speed, (b) PMSG stator voltages, (c) PMSG stator currents, (d) d-q axes PMSG stator currents, (e) generated active and reactive powers, (f) dc-link voltage, (g) grid voltages, (h) grid currents, (i) d-q axes grid currents, and (j) grid active and reactive powers. Copyright © 2015 John Wiley & Sons, Ltd.



Figure 9. Performance under symmetrical and asymmetrical-fault conditions when no specific MSC control is activated to enhance fault ride-through (FRT) capability of the PMSG-based wind energy conversion system (a) rotor speed, (b) PMSG stator voltages, (c) PMSG stator currents, (d) d-q axes PMSG stator currents, (e) generated active and reactive powers, (f) dc-link voltage, (g) grid voltages, (h) grid currents, (i) d-q axes grid currents, and (j) grid active and reactive powers.

implementing the optimally designed values of PI controller parameters and analyzing the performance of the PMSG-based WECS both under varying wind conditions and under grid-fault conditions. Various performance test results are shown in Figures 8–10. 5.1. Performance under varying wind conditions The simulation results are shown in Figure 8. An initial wind speed is considered as 8.5 m/s, which drives the wind turbine to rotate at a speed of 18.5 rad/s. The average active and reactive powers generated at wind speed of 8.5 m/s are about 3450 W and 0 VAr, respectively. After 0.75 s, as the wind speed starts increasing gradually from 8.5 to 11 m/s, the PMSG speed also starts increasing and smoothly tracks the reference PMSG speed, which is recursively calculated using the MPPT algorithm. The speed is settled at about 24 rad/s, which corresponds to the MPP at wind speed of 11 m/s. The Copyright © 2015 John Wiley & Sons, Ltd.



Figure 10. Performance under symmetrical and asymmetrical-fault conditions when specific MSC control is activated to store the active power surplus in the turbine-generator mechanical system inertia so as to enhance fault ride-through (FRT) capability of the PMSG-based wind energy conversion system (a) rotor speed, (b) PMSG stator voltages, (c) PMSG stator currents, (d) d-q axes PMSG stator currents, (e) generated active and reactive powers, (f) dc-link voltage, (g) grid voltages, (h) grid currents, (i) d-q axes grid currents, and (j) grid active and reactive powers.

corresponding increase in PMSG voltages and currents can be noticed, and the average active power generated increases to about 7250 W, but the average reactive power generated remains zero. After 1.75 s, the wind speed starts decreasing gradually from 11 to 8.5 m/s. The corresponding decrease in the PMSG speed, voltage, current, and average active power generated can also easily be observed. The actual d-q axes PMSG stator currents accurately track the reference values and results in balanced three-phase stator current, demonstrating good dynamics of the MSC controllers. On the other hand, irrespective of any change in wind speed, the dc-link voltage is quickly stabilized at its nominal value of 800 V, which confirms the effective control of the GSC. The grid voltage is almost constant at 415 V (r.m.s.), and the variation in wind speed directly affects the amount of grid currents and hence the average active power delivered to the grid by the GSC. It is noticeable that the average active power delivered to the grid is very close to the maximum captured turbine power at different wind Copyright © 2015 John Wiley & Sons, Ltd.



speeds, but the average reactive power delivered to/absorbed from the grid remains zero. Moreover, the total harmonic distortions (THDs) of grid voltages and currents are found within IEEE-519 standard limit. 5.2. Performance under symmetrical and asymmetrical grid-fault conditions The symmetrical and asymmetrical short-circuit faults at the grid side give rise to the symmetrical and asymmetrical grid-voltage dips, respectively [44]. The WECSs are not only required to supply power but also be able to survive under short-duration system faults and voltage unbalances by delivering active and reactive powers to the grid with a specific profile depending on the depth of the grid-voltage dip [45,46]. Such ability is often referred to as the FRT capability of WECSs. Under a grid-voltage dip, both the active and reactive current references for the GSC are given by the FRT requirement demanded by the utility operator [45]. According to the E.ON Netz fault response code [47], which is taken as the reference in this study, a 50% reduction in the grid voltage demands the reactive current equal to 100% of the rated system current. The FRT requirement is achieved by properly altering the grid active current reference to zero and reactive current reference in consistent with E.ON Netz fault response code [47]. 5.2.1. FRT capability of WECS without specific MSC control action. Figure 9 shows the simulation results when no specific MSC control action is taken for enhancement in FRT capability of WECS; i.e., only GSC is controlled to meet the FRT requirement. At time t = 2.53 s, a symmetrical grid-voltage dip of 50% of its nominal value is considered for 60 ms. During this symmetrical grid-voltage dip, average values of d and q axes grid currents are observed as 0 A and about 16 A, respectively. The average active and reactive powers delivered to the grid are noted as 0 W and about 4000 VAr, respectively, which demonstrates the fulfillment of the FRT requirement during symmetrical grid-voltage dip condition. At time t = 2.84 s, a grid-voltage dip of 50% in ‘phase-a’ of its nominal value is considered for 60 ms. During this asymmetrical grid-voltage dip, average values of d and q axes grid currents are observed as 0 A and about 5.2 A, respectively. The average active and reactive powers delivered to the grid are noted as 0 W and about 2100 VAr, respectively, which demonstrates the fulfillment of the FRT requirement during asymmetrical grid-voltage dip condition also. Because of the unbalanced grid phase voltages, the oscillations of the instantaneous active and reactive powers around their average values can easily be observed. Because the MSC and GSC are decoupled and no specific MSC control action is performed for enhancement in FRT capability of WECS, the PMSG speed, voltages, and currents remain unaffected during both symmetrical and asymmetrical grid-voltage dips. Further, the PMSG continues to generate the active power resulting in an imbalance between the active power generated and the same transferred to the grid, which then causes the dc-link voltage to increase uncontrollably. During the recovery of the dc-link voltage after the fault clearance, the GSC transfers more active power to the grid than the generated. Accordingly, the current idg reaches the limit (16 A) imposed by the PI controller during the dc-link voltage recovery. 5.2.2. FRT capability of WECS with specific MSC control action. The performance analysis presented earlier considers both the symmetrical and asymmetrical grid-voltage dip conditions to test the performance of the GSC for FRT when no specific MSC control action is taken for FRT enhancement. It is worth noticeable that the increased dc-link voltage may lead to system failure or even damage of both MSC and GSC [46] and, therefore, is intolerable particularly if the duration of grid-voltage dip is long. For this reason, many methods are proposed in the literature for the FRT enhancement of PMSG-based WECSs. A cheap solution with a simple control proposes the dissipation of the active power surplus in a braking resistor during grid-voltage dips [45,46,48,49]. Another method for FRT enhancement has been proposed in [45,48,50] using an energy storage system to absorb the active power surplus during the grid-voltage dip. FRT capability can also be enhanced by storing the active power surplus in the turbine-generator mechanical system inertia by increasing the rotor speed during the grid-voltage dip so as to maintain the dc-link voltage constant [46,51]. The analysis on FRT capability of the proposed PMSG-based WECS with specific MSC control action for its enhancement is considered in this subsection where the active power surplus is stored in the Copyright © 2015 John Wiley & Sons, Ltd.



turbine-generator mechanical system inertia by increasing the rotor speed during the grid-voltage dips so as to maintain the dc-link voltage constant. For this, the reference electromagnetic torque command during grid-voltage dip condition is changed to T e FRT ¼

E ga I ga þ E gb I gb þ E gc I gc ωr

(61)

In this way, whenever a grid-fault condition is detected, the active power generated by the PMSG is controlled at once to track the grid-side active power profile, thus retaining the dc-link voltage almost constant. The simulation results are shown in Figure 10 where the differences can primarily be noticed in MSC control action. No active power is delivered to the grid during the symmetrical and asymmetrical grid-voltage dip conditions considered as before achieving the FRT requirement; meanwhile, the specific MSC control action described earlier forces the generated average active power to zero, which avoids the uncontrollable increase in dc-link voltage. Thus, the q-axis PMSG current and, accordingly, the electromagnetic torque reduce to zero. As a result, there subsists a torque mismatch in the turbinegenerator mechanical system, which causes the rotor speed to increase. During recovery of the rotor speed after the fault clearance, the energy stored in the turbine-generator mechanical system inertia is delivered to the grid. For that reason, the currents iqs and idg reach the limit (16 A) imposed by the respective PI controllers during the rotor speed recovery.

6. CONCLUSION The designs of PI controllers employed in field-oriented and voltage-oriented control structures for machine-side and grid-side converters, respectively, of a grid-connected small PMSG-based wind energy conversion system were presented. Analytical expressions, closed-loop control transfer functions, and tuning criteria (MO and SO) for the involved PI controllers were also presented. Optimum selections of controller parameters were based on the preliminary analyses of the system stability and dynamic performance, and it was found that the SO tuning with a = 2.4142 of involved PI controllers in various control loops offered not only the satisfactory dynamic responses without oscillation but also good robustness margins. The selected PI controller parameters were applied in the simulation model, and the results were presented for varying wind conditions and discussed as well. Results were also presented to test the FRT capability of the WECS first, without specific MSC control action for FRT enhancement and subsequently, with specific MSC control action in which the active power surplus is stored in the turbine-generator mechanical system inertia by increasing the rotor speed during the grid-voltage dips so as to retain the dc-link voltage almost constant. With the proposed control structure, the PMSG-based WECS works satisfactorily both under varying wind conditions and under symmetrical and asymmetrical grid-fault conditions, confirming the effectiveness of the design of PI controllers.

7. LIST OF SYMBOLS AND ABBREVIATIONS 7.1. Symbols θ θg θs ξ αs λ λopt λm ρ ωr

Pitch angle Grid voltage phase angle PMSG rotor position angle Damping ratio Safety factor Tip-speed ratio Optimal tip-speed-ratio PMSG rotor flux Air density Turbine rotational speed / actual rotor speed




ωr ωs ωg a A C Cp edg,eqg Egabc f fgsw Gcc(p) Gcs(p) Gcv(p) GP1(p),GP2(p) GCFGC(p) GCFGV(p) GCFMS(p) GMO CGC ðpÞ GSO CGC ðpÞ GCGV(p) GCMS(p) GMO DGC ðpÞ GSO DGC ðpÞ GDGV(p) GDMS(p) GFGC(p) GFGV(p) GFMS(p) GGCL(p) GOGC(p) GMO OGC ðpÞ GSO OGC ðpÞ GOGV(p) GOMS(p) idg,iqg ids,iqs idg ,iqg ids ,iqs ip Ipeak

Reference rotor speed Electrical rotor speed Angular grid frequency Parameter constituting a tradeoff between damping of the poles in the closed-loop transfer function and dynamic response with SO tuning of PI controller Swept area DC-link capacitance Power coefficient d - q axes grid voltages Three-phase grid voltages Grid frequency Switching frequency of the GSC Transfer function of inner current PI controller Transfer function of outer speed PI controller Transfer function of outer dc-link voltage PI controller Benchmark transfer functions Closed-loop transfer function of inner current control loop with pre-filter on the reference signal Closed-loop transfer function of outer dc-link voltage control loop with pre-filter on the reference signal Closed-loop transfer function of outer speed control loop with pre-filter on the reference signal Closed-loop transfer function of inner current control loop with MO tuning of PI controller Closed-loop transfer function of inner current control loop with SO tuning of PI controller Closed-loop transfer function of outer dc-link voltage control loop Closed-loop transfer function of outer speed control loop Disturbance response function of inner current control loop with MO tuning of PI controller Disturbance response function of inner current control loop with SO tuning of PI controller Disturbance response function of outer dc-link voltage control loop Disturbance response function of outer speed control loop Transfer function of first order pre-filter on the reference signal of inner current control loop Transfer function of first order pre-filter on the reference signal of outer dc-link voltage control loop Transfer function of first order pre-filter on the reference signal of outer speed control loop First order approximation of inner current closed-loop transfer function Open-loop transfer function of inner current control loop Open-loop transfer function of inner current control loop with MO tuning of PI controller Open-loop transfer function of inner current control loop with SO tuning of PI controller Open-loop transfer function of outer dc-link voltage control loop Open-loop transfer function of outer speed control loop d - q axes actual grid currents d - q axes actual stator currents d - q axes reference grid currents d - q axes reference stator currents Peak-peak permissible ripple current Permissible peak ac line current




Igabc Isabc I sabc J K1,K2 Kp1,MO,Ti1,MO Kp1,SO,Ti1,SO Kp2,SO,Ti2,SO Kpc, MO,Tic, MO Kpc, SO,Tic, SO Kps,Tis Kpv,Tiv Ld,Lq Lg ma p Pm P R Rs Rg T e T e FRT T1 T2 Tc Tv Ts Tsc Tsv vdg,vqg vdg ,vqg Vω VDC V DC V DCripple VLL

Actual three-phase grid currents Actual three-phase stator currents Reference three-phase stator currents Turbine-generator mechanical system inertia Plant gains Proportional gain and integral time-constant of the PI controller addressed by MO tuning criterion with benchmark plant transfer function of the form GP1(p) Proportional gain and integral time-constant of the PI controller addressed by SO tuning criterion with benchmark plant transfer function of the form GP1(p) Proportional gain and integral time-constant of the PI controller addressed by SO tuning criterion with benchmark plant transfer function of the form GP2(p) Proportional gain and integral time-constant of inner current PI controller parameters addressed by MO tuning criterion Proportional gain and integral time-constant of inner current PI controller parameters addressed by SO tuning criterion Proportional gain and integral time-constant of outer speed PI controller Proportional gain and integral time-constant of outer dc-link voltage PI controller d - q axes stator inductances Coupling ac inductance Modulation index Complex frequency Turbine output power Number of PMSG poles Radius of the turbine blades PMSG stator resistance Resistance of the coupling ac inductor Reference electromagnetic torque Reference electromagnetic torque during grid-voltage dip Dominant time-constant in the benchmark plant transfer function GP1(p) Parasitic time-constant or minor-time constant in the benchmark plant transfer function GP1(p) or GP2(p) Sample-time for inner current control loop Sample time for outer dc-link voltage control loop Sample-time for outer speed control loop Sum of minor time-constants in open loop transfer function of inner current control loop Sum of minor time-constants in open loop transfer function of outer dc-link voltage control loop d - q axes GSC voltages d - q axes reference GSC voltages Wind speed Actual dc-link voltage Reference dc-link voltage Permissible ripple dc-link voltage Line-line r.m.s. voltage on the ac side of the PWM converter

7.2. Abbreviations r.m.s. FOC FRT GA GM GSC

Root-mean-square Field-oriented control Fault-ride-through Genetic algorithm Gain margin Grid-side converter




HCC IAE ISE ITAE ITSE MO MPP MPPT MSC PI PID PM PMSG PSO PWM SO THD VOC VSC WECS Z-N Method

Hysteresis current controller Integral-absolute error Integral-square error Integral-time absolute error Integral-time square error Modulus optimum Maximum power point Maximum power point tracking Machine-side converter Proportional-integral Proportional-integral-derivative Phase margin Permanent magnet synchronous generator Particle swarm optimization Pulse width modulation Symmetric optimum Total harmonic distortion Voltage oriented control Voltage source converter Wind energy conversion system Zeigler-Nichols Method

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APPENDIX : SPECIFICATIONS FOR PMSG-BASED WECS Air density: ρ = 1.229 kg/m2; rated wind speed: Vw = 11 m/s; rated turbine power: Pm = 7.68 kW; power coefficient: Cp max = 0.4412; tip-speed ratio: λopt = 5.66; blade radius: R = 2.6 m; stator resistance: Rs = 1.4 Ω; d-q axes stator inductances: Ld ≈ Lq = 5.8 mH; rotor flux: λm = 2.6 Wb; PMSG poles: P = 12; turbine-generator mechanical system inertia: J = 1 kg - m2; reference dc-link voltage: V DC ¼ 800 V ; dc-link capacitance: C = 1000 μF; coupling resistance: Rg = 1.85 Ω; coupling inductance: Lg = 12.8 mH; grid voltage: Eg = 415 V (r. m. s.); grid frequency: f = 50 Hz.

