1. Introduction
The generation of renewable electrical energy has undergone an important development to face the global climatic crisis [
1,
2]. The remarkable growth of wind power (WP) initiates a transition phase based on large-scale energy efficiency [
3]. During 2021, new WP installations reached 94 GW worldwide, reaching a cumulative installed capacity of 837 GW, which represents a growth of 12% compared to 2020 [
4]. Most wind turbines (WT) installed are variable-speed wind turbines (VSWT) based on either the doubly fed induction generator (DFIG) or the synchronous generator via full converter (SGFC), due to their versatility and efficiency [
5]. However, the intermittent nature of the wind resource is an important factor in the technological development of WTs [
6]. For example, the conventional response of a VSWT to possible disturbances in the power system, such as a frequency deviation caused by an imbalance between electric generation and demand, does not include the natural contribution of its inertia to provide support, as a conventional synchronous generator would [
7]. In the research for solutions to these drawbacks, many authors have chosen to simulate the DFIG-WT in the MATLAB/Simulink
® environment, whose models are highly robust. However, studies that involve many WTs and their interactions with the electric power system lead to complex problems with many different components characterized by very diverse time scale dynamics. Particularly, the detailed modeling of the power electronic converter (PEC) of each WT in a larger system leads to a multiscale problem that requires so high a computational effort that it is almost intractable with conventional modeling approaches. Therefore, studying efficient methods that manage to reduce the computational burden without compromising the accuracy of the dynamic model of the VSWT and PEC is necessary.
In the literature, various studies have shown the possibility of acting on the sophisticated control system of a WT to provide a wide range of ancillary services [
8]. In this context, detailed VSWT models have been developed. In ref. [
9], the authors model the DFIG to investigate the contribution of a WT to the power system frequency control. Mainly, the impact of the different regulator settings and the system inertia are investigated and the results are evaluated from a computational point of view. In [
10], the authors model a variable-speed wind system based on a DFIG with a linear PI controller, with the stator connected directly to the grid and the rotor connected through a back-to-back converter. The results are evaluated in a simulation environment where a refinement depends on the computational robustness implemented. The MATLAB software presents a sophisticated DFIG modeling, as mentioned in ref. [
11], where the WT is simulated with the Wind Energy Conversion System (WECS), using the maximum power point tracking (MPPT) method to extract the optimal power and applying the Backstepping controller to control reactive power and electromagnetic torque in order to test the performance and robustness of the system. The results show a great precision in the behavior of the WT with respect to the datasheets. However, MATLAB simulates the entire WT system and is not segmented, which is inefficient for certain specific VSWT applications. In this sense, the authors have proposed new WT-DFIG modeling methods focusing on the parameters under study. For example, in [
12], the derivation of the mathematical model of a WT is based on formulas that calculate the mechanical and electrical power of the WT, the results showing that the proposed model is simple with low computational burden. However, the model can only be applied in voltage control mode—though the model of several DFIGs connected to the grid (e.g., with hundreds of buses) is a significant computational challenge [
13]. Similarly, ref. [
5] presents a detailed simulation model in MATLAB/Simulink
® for WT failure analysis. Among other relevant studies, ref. [
14] presents a robust mathematical model of VSWT linked to the rotor in an experimental and simulation way for a WECS in different wind speed conditions.
The importance of using detailed WT modeling for ancillary services applications, especially in isolated systems, is evident [
15]. In this context, supplementary control strategies to provide WTs with effective inertial response to supply the developed grid requirements is indispensable [
7]. In this sense, the studies presented in [
8,
16,
17] present a model of the short-term dynamics of the VSWT, using the simplified electromechanical model, where the dynamics of the PEC and the electrical generator are represented through a first order transfer function. The benefits of the model have been tested in a simulation environment considering the real operating conditions based on measurement data recorded on an insular power system. In addition to the simplifications allowing the increase of computational performance in simulations, the technique is suitable for the integration of VSWT in large-scale power systems where the research interest is to predict the dynamic response of a WT in terms of its mechanical variables, active power, and grid frequency (by measuring the rotor speed of synchronous generators) within a Load Frequency Control scheme. Despite its contrasted benefits, the model has a significant limitation: the impossibility of extending the scope of such studies to represent a power system in a three-phase form. This is crucial to assess the dynamics of the voltage at the point of common coupling (PCC), to predict the variability of current injected in the grid, and to serve as a test bench to propose and validate voltage control strategies through controlled injection/absorption of reactive power, among others. Hence, further research is needed to evaluate PEC modeling in conjunction with grid-connected VSWT.
The study of PECs is as important as the rest of the VSWT’s components. Thus, when dynamic analysis is applied to large-scale power systems with their disaggregated components, the enormous computational burden required to simulate detailed models of PECs, within a timeframe comprising from a few seconds to a couple of minutes (time in which the physical phenomena related to the stability of frequency/voltage in the grid occurs), might greatly limit the performance of this type of study. To overcome this issue, various alternatives have been investigated to model the PECs in a less complex way without harming the numerical accuracy, with respect to the results that a traditional detailed model would provide [
18]: e.g., references [
19,
20,
21] present techniques based on predictive control of infinite states to simplify the control logic of PECs, allowing one to shorten the execution time of the control loops with respect to classical linear controllers. However, to meet the control objectives, a greater number of computational calculations are required. This situation can be solved by reducing the number of sectors necessary for the vectorial decomposition of the three-phase voltage at the PCC by means of the use of lookup tables [
20,
21]. This is achieved by keeping a detailed representation of the three-phase inverter (6 or 9 power transistor bridge). On the other hand, [
22] presents a simplification focused on a three-phase inverter represented by a bridge of six controlled current sources, modulated by Sinusoidal Pulse Width Modulation (SPWM), demonstrating the reduction of computational effort. The study presented in [
18] improves the modeling by reducing the order to just three controlled current sources that are governed by a pair of linear controllers in coordinates
d−
q. The results have been evaluated at the simulation level.
Hence, given the need to have a computer tool that allows short-term dynamics power system analysis and traditional frequency stability studies to be done with the least computational effort, this paper presents a simplified model of a VSWT rather than a detailed representation of the WT components that would be intractable when simulating frequency events in large-scale power systems, keeping a compromise between simplicity, flexibility, and accuracy based on [
8]. Additionally, the proposed model includes a representation of the grid–connection interface based on the PEC presented in [
18] that has been evaluated in a separate way. Therefore, the main contribution of this work is the proposal of a novel simplified electromechanical model of a VSWT for grid-integration studies that is based on the fusion of the models presented in [
8] and [
18]. In this way, the VSWT representation is extended by controlling variables with the help of the PEC with minimal computational effort. Finally, this paper goes further by presenting comparisons between simulations and experimental studies under controlled conditions.
The remainder of this paper is organized as follows:
Section 2 presents the detailed modeling of WT and PEC, where the simplified models of these components are explained through mathematical equations.
Section 3 shows the results of the study, where
Section 3.1 presents the validation of the proposed simplified model with respect to the detailed model available in the MATLAB/Simulink
® library, with separate and joint validations of VSWT and PEC. In
Section 3.2, the test-bench for emulating the time-domain behavior of some variables of interest provided by the proposal in an actual microgrid laboratory is presented. In this section, a critical discussion of the results is offered.
Section 4 summarizes the conclusions of the paper.
2. Modelling of Variable-Speed Wind Turbine
Representing the short-time dynamics of the VSWT, the simplified electromechanical model proposed in [
8] and summarized in
Figure 1a has been used for the purposes of this paper. This model, designed initially for a DFIG-WT (Type III), can be applied to represent a wind turbine with a synchronous generator via full converter (SGFC-WT, Type IV,
Figure 1b) due to the similarities between their mechanical topologies and because, within the time frame considered in the load-frequency control studies, the electromagnetic time constants are negligible compared to the mechanical ones. This fact allowed us to represent the dynamics of the power electronic converter and the electrical generator by a first order transfer function with time constant
. Nevertheless, in this work, we intend to improve the representation of this component by introducing an approximate representation of the power electronic converter, its controllers, and the electrical generator to allow this simplified VSWT model to be used in three-phase power systems studies in a computer simulation environment. In this paper, we will refer to this approach as a grid-side dynamics VSWT model.
Figure 1 shows the main components of a DFIG-WT (Type III), and a SGFC-WT (Type IV). In both cases, the constructive similarity of these two models can be appreciated: a wind rotor; a back-to-back PEC composed of a rotor side converter (RSC), in the case of DFIG-WT, and machine side converter (MSC), in SGFC-WT, and a grid side converter (GSC); an electrical generator controlled by an RSC or MSC (under MPPT efficiency criteria); a pitch controller; and a GSC that governs the flow of active and reactive power injected into the grid at the point of common coupling (PCC).
These similarities allow us to propose a generic model of VSWT, which we have outlined in
Figure 2. An attempt has been made to maintain a modular structure, such that it allows the addition and/or improvement of some components, in addition to making it possible to implement additional control strategies without this implying greater difficulties and conflicts between the variables involved [
7,
16]. In the diagram,
Tt and
Tem are the mechanical and the electromagnetic torque, respectively.
Pg is the total output active power,
t and
g are the angular speed of the turbine and the electric generator, and
v and
β denote the wind speed and the blade pitch angle. At this point, it is important to mention that the short-term operation of the blocks: wind rotor, pitch angle controller, mechanical system, and active power controller (by MPPT) were successfully validated by simulation in [
8]. This modular design incorporates two new and essential components: an approximate representation of the power electronic converter and the generator, and a pair of closed-loop controllers that regulate the active and reactive power injected into the grid.
Figure 3 shows the block diagram of the simplified electromechanical model of a VSWT, which takes part of the simplified WT model proposed by the authors in [
8] and the simplified PEC model presented by the authors in [
18]. Both are carefully articulated to achieve adequate performance and in adherence to the theoretical foundations of each of them.
The following lines present the formulation and a description of each of the components that make up the proposed model.
The set of Equation (1) shows the conversion of the input and output variables of the mechanical system block to per unit (p.u.) magnitudes.
where:
,
, and
represent torque, power, and angular speed; subscripts
and
represent the variables referring to the generator and the turbine, respectively;
,
, and
are the grid frequency, the number of pole pairs of the electrical generator and the gearbox ratio, respectively.
Equations (2)–(6) represent the dynamics of the wind rotor [
8]:
where:
,
, and
represent rotor radius, air density, and power coefficient, respectively;
is the tip speed ratio;
and
are the power and speed constants, respectively, depending on the constructive characteristics of the wind rotor; and
–
are constants for approximation of the power coefficient for three-blade wind turbines and that are related to their construction characteristics; these constants are described in detail in refs. [
23,
24].
The VSWT speed controller uses an MPPT algorithm to extract power optimally. In
Figure 4, this characteristic is represented by Equation (7), where
is the optimization constant, whose value depends on the type of turbine. Segment A–B in
Figure 4 is the start zone of the WT. Then, the optimization zone modifies the rotor speed by adjusting the points of maximum power in segment B–C. During operation of segment C–D, the mechanical speed of the WT is approximately constant until reaching the nominal electrical power. Finally, if the rotor speed exceeds point D, the pitch angle controller is enabled.
The PEC subsystem in
Figure 3 is implemented according to the simplified solution provided by the authors in [
18]. The PEC has a three-phase representation modeled by means of a controlled current source. This source must inject three-phase currents
,
and
whose amplitude and phase will be defined according to the control criteria implemented and the applications assigned to the converter. For the generation of the current command signals, the theory of the
d−
q coordinate system is applied, as explained below:
In order to control the active power,
, a PI closed-loop controller is used, generating the output a signal
, represented in Equation (8). This signal is applied to a first-order delay function, introduced to represent the controller time to reach the control variable,
.
The reactive power control
, is similar to the previously described scheme; in this case, the control variable is
, Equation (9).
The generated signals (
and
), are transformed in real time by means of a phase closed loop (PLL); then, the reference signals are evaluated by the Park transform by means of Equation (10). Finally, to provide feedback to the PI controllers of each of the control loops, the instantaneous active and reactive powers injected by the controlled current sources are measured. The output signals are
,
and
, in amperes, as shown in
Figure 3. For further details, ref. [
18] presents the detailed modeling of the PEC.
where:
,
, and
are the direct axis, quadrature, and homopolar components, respectively;
,
, and
are three-phase currents, and
is the angular frequency of the grid voltage at PCC.
4. Conclusions
This paper presents a simplified model for a variable-speed wind turbine by introducing an approximate representation of the power electronic converter. The objective of this model is to reduce the computational effort and to allow one to analyze the dynamics of instantaneous electrical variables, enabling the extension of the scope of network studies related to the behavior of the grid frequency (primary frequency control) and power quality studies in high-penetrated wind power systems. All of these are at a lower cost and computational time. Validation of the proposed simplified model is compared with a Wind Turbine Doubly-Fed Induction Generator block (Phasor Type) from MATLAB/Simulink® and with laboratory experiments under controlled conditions. The main novelty of this paper is the development of a simplified electromechanical model of a variable speed wind turbine considering an electronic power controller that optimizes the computational effort, with the aim of having a computer tool that allows the performance of dynamic analyses that would be intractable when simulating frequency events in large-scale power systems with complete models.
The results show a minimal delay between the proposed model and the DFIG-WT MATLAB/Simulink®, representing approximately 3% of the WT inertia constant. In general, the transient and steady-state response have accurate approximations.
Regarding the proposed model of the PEC that constitutes the extended VSWT, it is obvious that the results are not perfect due to external factors, i.e., there is a small voltage fluctuation in the PCC due to the actual prototype being connected to the mains, whose Thevenin impedance at that point is greater than that considered in the simulation models. Even so, the experimental results show that the electrical variables obtained in practice are similar to the reference models implemented in simulation within the time frame analyzed.
The computational effort in both models has been compared; the results show that the proposed model reduces the computational time by 80% with respect to the detailed model. This computational benefit is achieved even though the proposed model uses the “continuous” mode for its numerical solver in MATLAB, while the reference model uses the “phasor” approximation to reduce its computational burden.
The results show a high correlation of these variables between the fluctuations of the proposed and the detailed model.
Finally, the reduction of the simulation time achieved with the proposal also makes possible the emulation in real time of certain electrical variables of the VSWT in the laboratory, as has been demonstrated in this work. Furthermore, having a real test bench to assess the integration of wind generation in a utility grid or its interaction with other agents into an islanded microgrid constitutes a valuable means of verifying the effectiveness of different techniques aimed at improving the dynamic characteristics of such systems.