On the Regulation of Wind Energy Conversion Systems Working as Conventional Generation Plants

Abstract

In this work, we obtain bounds for the wind speed interval in which a wind energy conversion system can be regulated in a similar manner to a conventional power generation plant. In particular, we conducted a steady-state analysis of a wind turbine coupled to a doubly fed induction generator (DFIG) that delivers power according to the electric grid requirements, and in a safe manner. In this sense, our main contribution is twofold. On the one hand, it involves expanding the secure operation window by adjusting the gearbox ratio, thus improving the reliability of the utility of grid integration. On the other hand, the WECS is controlled within new, safe wind speed intervals through a passivity-based controller and a proportional–integral controller, showing adequate performances in both cases.

1. Introduction

Historically, the main sources utilized in electric power generation have been hydroelectric power, nuclear energy, and fossil fuels [1]. The latter represents the global primary source. Nonetheless, it has been argued that renewable energy sources are of utmost importance for environmental and economic reasons [2,3,4]. In particular, wind power has proven to be an ecologically viable alternative for electric generation, with a continuously increasing installed capacity year after year [5]. The year 2020 was a notoriously positive period in the global wind industry, with a year-on-year growth of 53%, and more than 93 GW of wind power installed [6].

Among the wind energy conversion systems (WECSs) available in the market, those based on a double-fed induction generator (DFIG) have proven to be the most suitable options to serve as generating units. Based on a cost–benefit analysis, the main advantage of choosing a DFIG over other possible generators used in a variable-speed WECS lies in the dimensions of the rotor frequency converter. This size is directly related to the power delivery required from the DFIG, which is only a fraction of its nominal power (∼30%). In contrast, WECSs based on synchronous generators are more expensive since the size of the corresponding converter is fixed by its nominal power [2,6,7,8,9].

The problems that emerge when these generators are interconnected to the electrical grid have been addressed by multiple authors [10,11,12,13,14,15,16]. In most of these works, the authors centered their attention on the maximum power point tracking (MPPT) mode of operation while considering a unit power factor. Such an approach, however, neglects the reactive power dispatch capacity of the DFIG [9]. To understand the relevance of dispatching reactive power under the MPPT mode of operation in these energy conversion systems, see [17], where it is stated that the reliability of the utility grid integration of a DFIG-based WECS depends on the particular mode of operation, along with the chosen control strategy.

The maximum power tracking mode of operation is the most energetically efficient, i.e., it is aimed at extracting the maximum power available from the wind at all times; however, it may not be the most desirable option from a stability and control point of view. In particular, this has been relevant during contingencies, such as voltage dips and instabilities, since the electric power exchanged with the grid inherits the random nature of the wind. Such a problem was reported in connection with an existing power grid in 2006 [18,19], where the recommendation was to operate the WECS in a power regulation mode, as is done in conventional generation plants. However, unlike in a conventional plant, where the energy source is assumed to be constantly available and well-regulated [1], such an assumption ceases to be valid when the source undergoes random and spontaneous variations. For this reason, operating the DFIG WECS, similar to any other CGP, is notably challenging.

Recently, the authors have shown that the regulation problem can be posed and solved within a wind speed interval, regardless of the variations in the wind velocity between the so-called safe operating zone [20]. In this work, our main objective is to control the WECS in a secure manner, i.e., by satisfying the power demand from the electric network without exceeding the nominal values of the angular velocity and the currents in the windings of a DFIG, while considering the random nature of the wind. In this sense, we are interested in determining the wind speed regions where this can be achieved. We note that such regions can be extended by modifying construction parameters, such as the gearbox ratio.

Throughout the manuscript, we assume a given power demand together with a wind profile generated through a Weibull probability distribution. Additionally, we consider that all of the WECS parameters are known and constant and that the blade pitch angle is fixed at its maximum value.

Here, we begin by determining the safe operation zones where the WECS satisfies the power demand of the electric network. Then, we consider various gearbox ratios and find the corresponding limits where the WECS operates as a conventional plant for a series of active and reactive power demands. We obtain the safe electric and mechanical references that any controller must follow to regulate a WECS in a secure way, which is a commonly ignored challenge [5]. Thus, we address the control problem by implementing two independent control strategies, i.e., a passivity-based controller and a proportional–integral controller, respectively. Finally, we provide critical comments about the optimal operations of these types of systems.

Our contribution lies in exhibiting the conditions under which a WECS based on a DFIG can be safely and effectively controlled as a conventional plant. We also assess the probability of achieving such a goal at any given time due to the stochastic nature of the wind. We conclude our work with an example of a controlled WECS through a passivity-based controller and a proportional–integral controller, respectively, for the same control problem posed in [21].

The manuscript is structured as follows. In Section 2, we present the steady state of the system. In Section 3, we present the wind-safe zones for the WECS under the steady-state analysis developed in Section 2. In Section 4, the control of the WECS in the new, safe operation zones is shown through the implementation of two different control strategies. Section 5 provides a critical note on the impact of the random nature of the wind on the regulation problem. Finally, in Section 6, we make some final remarks and highlight further research directions.

2. Steady-State Analysis of the WECS

The steady-state analysis for the WECS electromechanical is obtained from the DFIG qd0 mathematical model. It is assumed that all inputs are known and adequately characterized [22]. The dynamics of the WECS are modeled by a system of first-order coupled ordinary differential equations. The ones corresponding to the stator voltages are as follows:

$$\begin{array}{c}\hfill L\frac{d}{dt}{i}_{qs}=-\left(\frac{L_M^2}{L}{r}_{gb}{\omega }_t-L{\omega }_s\right)i_{ds}-\frac{L_M{L}_r^{\prime }}{L}\frac{r_{gb}{\omega }_t}{a}{i}_{dr}-\frac{L_r^{\prime }}{L}{R}_s{i}_{qs}+\frac{L_M}{L}a{R}_r{i}_{qr}\\ \hfill +\frac{L_r^{\prime }}{L}{u}_{qs}-\frac{L_M}{L}a{u}_{qr},\end{array}$$

(1)

$$\begin{array}{c}\hfill L\frac{d}{dt}{i}_{ds}=\left(\frac{L_M^2}{L}{r}_{gb}{\omega }_t-L{\omega }_s\right)i_{qs}+\frac{L_M{L}_r^{\prime }}{L}\frac{r_{gb}{\omega }_t}{a}{i}_{qr}-\frac{L_r^{\prime }}{L}{R}_s{i}_{ds}+\frac{L_M}{L}a{R}_r{i}_{dr}\\ \hfill +\frac{L_r^{\prime }}{L}{u}_{ds}-\frac{L_M}{L}a{u}_{dr},\end{array}$$

(2)

while those for the rotor are

$$\begin{array}{c}\hfill L\frac{d}{dt}{i}_{qr}=\left(\frac{{L_M^{\prime }}^2}{L}{r}_{gb}{\omega }_t-L{\omega }_s\right)i_{dr}+\frac{L_M{L}_s}{L}a{r}_{gb}{\omega }_t{i}_{ds}-\frac{L_s}{L}{a}^2{R}_r{i}_{qr}+\frac{L_M}{L}a{R}_s{i}_{qs}\\ \hfill +\frac{L_s}{L}{a}^2{u}_{qr}-\frac{L_M}{L}a{u}_{qs},\end{array}$$

(3)

$$\begin{array}{c}\hfill L\frac{d}{dt}{i}_{dr}=-\left(\frac{{L_M^{\prime }}^2}{L}{r}_{gb}{\omega }_t-L{\omega }_s\right)i_{qr}-\frac{L_M{L}_s}{L}a{r}_{gb}{\omega }_t{i}_{qs}-\frac{L_s}{L}{a}^2{R}_r{i}_{dr}+\frac{L_M}{L}a{R}_s{i}_{ds}\\ \hfill +\frac{L_s}{L}{a}^2{u}_{dr}-\frac{L_M}{L}a{u}_{ds},\end{array}$$

(4)

which are coupled to the torque equation

$$\frac{J}{n_P}\frac{d}{dt}{r}_{gb}{\omega }_t=T_m({\omega }_t,v)-T_e(i_{qr},i_{dr},i_{qs},i_{ds})-B{r}_{gb}{\omega }_t,$$

(5)

where the electromagnetic torque is given by

$$T_e(i_{qr},i_{dr},i_{qs},i_{ds})=\frac{3}{2}\frac{L_M{n}_P}{a}\left(i_{dr}{i}_{qs}-i_{ds}{i}_{qr}\right).$$

(6)

Here, $L_{ls}$ and $L_{lr}$ are the stator and rotor leakage inductances, respectively, while $L_M$ is the magnetizing inductance. We will introduce the following shortcuts: $L_s=L_{ls}+L_M$, $L_r^{\prime }=L_{lr}{a}^2+L_M$, together with the auxiliary-squared inductances $L^2=L_{lr}{L}_s{a}^2+L_M{L}_{ls}$, ${L_M^{\prime }}^2=L_{lr}{L}_s{a}^2+L_M{L}_s$ that allow us to exhibit the structure of the system, allowing for a clearer dimensional analysis. The angular velocity of the wind turbine is ${\omega }_t$, which is related to the angular speed of the DFIG through the gearbox ratio ${\omega }_g=r_{gb}{\omega }_t$; ${\omega }_s$ denotes the synchronous speed of the generator. The mechanical torque is expressed by $T_m({\omega }_t,v)$, where we have exhibited its functional dependence on the wind speed v and where B is the damping coefficient of the wind turbine together with the rotor of the DFIG. The transformation ratio is given by $a=N_s/N_r$, where $N_s$ and $N_r$ represent the numbers of turns of the stator and rotor windings, respectively. The stator and rotor resistances are $R_s$ and $R_r$, $n_P$ is the number of pairs of poles, and J is the inertia coefficient.

The steady state of the WECS corresponds to the fixed points of the Equations (1)–(5), which can be reduced to the system of algebraic equations given by

$$\begin{array}{c}\hfill \left(u_{qs}-R_s{i}_{qs}^{*}\right)-{\omega }_s\left[\left(i_{ds}^{*}+\frac{i_{dr}^{*}}{a}\right)L_M+L_{ls}{i}_{ds}^{*}\right]=0,\end{array}$$

(7)

$$\begin{array}{c}\hfill \left(u_{ds}-R_s{i}_{ds}^{*}\right)+{\omega }_s\left[\left(i_{qs}^{*}+\frac{i_{qr}^{*}}{a}\right)L_M+L_{ls}{i}_{qs}^{*}\right]=0,\end{array}$$

(8)

$$\begin{array}{c}\hfill \left(u_{qr}-R_r{i}_{qr}^{*}\right)-\left(\frac{{\omega }_s-r_{gb}{\omega }_t^{*}}{a}\right)\left[\left(i_{ds}^{*}+\frac{i_{dr}^{*}}{a}\right)L_M+L_{lr}a{i}_{dr}^{*}\right]=0,\end{array}$$

(9)

$$\begin{array}{c}\hfill \left(u_{dr}-R_r{i}_{dr}^{*}\right)+\left(\frac{{\omega }_s-r_{gb}{\omega }_t^{*}}{a}\right)\left[\left(i_{qs}^{*}+\frac{i_{qr}^{*}}{a}\right)L_M+L_{lr}a{i}_{qr}^{*}\right]=0\end{array}$$

(10)

and

$$T_m\left({\omega }_t,v\right)-T_e\left(i_{qr}^{*},i_{dr}^{*},i_{qs}^{*},i_{ds}^{*}\right)-B{\omega }_g=0,$$

(11)

where we use the ∗ symbol to denote the equilibrium value of the corresponding variables.

The electrical network or grid is modeled as an infinite bus, whose voltage magnitude is equal to the nominal voltage in the stator windings aligned with the q-axis, i.e., it is given by the value of $U=u_{qs}$ with $u_{ds}=0$. In such a case, the equilibrium currents in the stator windings are decoupled and can be expressed in terms of the active and reactive power, $P_s$ and $Q_s$, respectively, as [21]

$$i_{qs}^{*}=\frac{2P_s^{*}}{3U}\mathrm{and}{i}_{ds}^{*}=\frac{2Q_s^{*}}{3U}.$$

(12)

Substituting (12) into (7) and (8) yields the equilibrium currents in the rotor windings

$$i_{qr}^{*}=\frac{2}{3U}\frac{a}{L_M}\left(\frac{R_s{Q}_s^{*}}{{\omega }_s}-L_s{P}_s^{*}\right)$$

(13)

and

$$i_{dr}^{*}=-\frac{2}{3U}\frac{a}{L_M}\left(\frac{R_s{P}_s^{*}}{{\omega }_s}+L_s{Q}_s^{*}\right)+\frac{aU}{L_M{\omega }_s}$$

(14)

Finally, using the equilibrium currents (12)–(14), we can solve (9) and (10) in favor of the voltages in the rotor windings to obtain their equilibrium form, namely

$$\begin{array}{c}\hfill u_{dr}^{*}=\frac{2}{3U}\left[L^2{\omega }_s^2\left(1-\frac{{\omega }_g^{*}}{{\omega }_s}\right)-a^2{R}_r{R}_s\right]\frac{P_s^{*}}{a{L}_M{\omega }_s}\\ \hfill +\frac{2}{3U}\left[R_s{L}_r^{\prime }\left(1-\frac{{\omega }_g^{*}}{{\omega }_s}\right)+a^2{R}_r{L}_s\right]\frac{Q_s^{*}}{a{L}_M}\\ \hfill +a\frac{R_r}{L_M{\omega }_s}U\end{array}$$

(15)

and

$$\begin{array}{c}\hfill u_{qr}^{*}=-\frac{2}{3U}\left[R_s{L}_r^{\prime }\left(1-\frac{{\omega }_g^{*}}{{\omega }_s}\right)+a^2{R}_r{L}_s\right]\frac{P_s^{*}}{a{L}_M}\\ \hfill -\frac{2}{3U}\left[L^2{\omega }_s^2\left(1-\frac{{\omega }_g^{*}}{{\omega }_s}\right)-a^2{R}_r{R}_s\right]\frac{Q_s^{*}}{a{L}_M{\omega }_s}\\ \hfill +\frac{1}{a}\frac{L_r^{\prime }}{L_M}\left(1-\frac{{\omega }_g^{*}}{{\omega }_s}\right)U,\end{array}$$

(16)

which are parameterized by ${\omega }_g^{*}=r_{gb}{\omega }_t$. This situation is illustrated in Figure 1, where we indicate that, to obtain the required value for ${\omega }_g^{*}$, one needs to solve the stationary torque balance Equation (11). Notice that the active and reactive powers of the DFIG are related to those demanded by the network through the common connection point (CCP in Figure 1) as

$$P_s^{*}+P_r^{*}=P\mathrm{and}{Q}_s^{*}=Q.$$

(17)

Additionally, the rotor’s active power is related to that of the stator through the slip relation

$$P_r^{*}=\left(1-\frac{{\omega }_g^{*}}{{\omega }_s}\right)P_s^{*}.$$

(18)

Therefore, the required active power from the stator is expressed in terms of the equilibrium angular velocity and the power demand as

$$P_s^{*}=\frac{{\omega }_g^{*}}{{\omega }_s}P.$$

(19)

The mechanical torque is expressed in terms of the wind velocity through the expression

$$T_m({\omega }_g^{*},v)=\frac{1}{2}\frac{r_{gb}}{{\omega }_g^{*}}\left(\rho v^3{C}_p\right)\left(\pi r_b^2\right),$$

(20)

where $C_p$ is the power coefficient of the wind turbine given by

$$C_p\left({\omega }_g^{*}\right)=\left[\frac{a_{1}v}{r_b}\frac{r_{gb}}{{\omega }_g^{*}}-\left(a_1{a}_2+a_3\right)\right]exp\left(a_2{a}_4-\frac{a_{4}v}{r_b}\frac{r_{gb}}{{\omega }_g^{*}}\right)+\frac{a_5{r}_b}{v}\frac{{\omega }_g^{*}}{r_{gb}}$$

(21)

where $r_b$ is the turbine radius, $\rho$ is the wind density, and $a_i^{\prime }s$ are the characteristic constants of the power coefficient [23]. Therefore, the torque balance Equation (11) yields a transcendental equation for the rotor angular velocity, which is expressed in terms of the wind velocity and the active and reactive power demands of the network, i.e.,

$$\pi \rho r_b^2{v}^3{C}_p({\omega }_g^{*},v)=2B\frac{{{\omega }_g^{*}}^2}{r_{gb}}+\frac{2n_P}{r_{gb}}P-\frac{4}{3}\frac{n_P{R}_s}{r_{gb}{U}^2}\left(\frac{{\omega }_s}{{\omega }_g^{*}}P+\frac{{\omega }_g^{*}}{{\omega }_s}{Q}^2\right).$$

(22)

In order to solve (22), we use the composed Newton–Raphson technique proposed in [24], whose convergence rate is sufficiently quick (on average, three iterations) to be considered as a viable online method to obtain the equilibrium references for the DFIG.

3. Secure Operating Zones for the WECS

The process of obtaining the largest safe zones for the WECS begins with knowing the construction data of a WECS. In this work, we consider a 2.4 MW WECS, which is equipped with a 2 MW DFIG. The wind turbine and the DFIG data are shown in

Table 1. Wind turbine parameters [25].

Parameter	Value
Rated mechanical power	2.4 MW
Turbine diameter	84 m
Wind speed range	3.5–12.1 m/s
Gearbox ratio ($r_{gb}$)	100
Damping coefficient (B)	0.071
$a_1$, $a_2$, $a_3$, $a_4$	0.73, 151, −0.58, −0.002
$a_5$, $a_6$, $a_7$, $b_1$, $b_2$	2.14, −13.2, −18.4, −0.02, −0.003
Maximum power coefficient ($C_{pM})$	0.44
Maximum tip-speed ratio (${\lambda }_M$)	7.2

and

Table 2. Generator parameters [25].

Parameter	Value
Rated stator power	2 MV
Rated stator phase voltage	398.4 V (rms)
Rated stator current	1760 A (rms)
Rated rotor current	1823 A (rms)
Rated rotor phase voltage	448 V (rms)
Rated stator frequency	50 Hz
Rated rotor speed	1800 rpm
Nominal rotor speed range	900–1800 rpm
Rated slip, turn ratio	−0.2, 2.94
Number of pole pairs	2
Stator winding resistance	2.6 m$\Omega$
Rotor winding resistance	2.9 m$\Omega$
Stator leakage inductance	87 $\mathsf{\mu }$H
Rotor leakage inductance	87 $\mathsf{\mu }$H
Magnetizing inductance	2.5 mH

, respectively, as reported in [25]. According to the steady-state analysis described in the previous section, we will consider

• $P=2$ MW: The maximum active power that can be delivered by the DFIG.
• $P=1.2$ MW: 60 % of the maximum active power that can be delivered by the DFIG.
• $P=0.0639$ MW: The minimum active power that can be delivered by the DFIG (corresponding to the minimum wind speed required [22]).

These are based on the network code of the United Mexican States for the year 2021, Chapter 3, Figure 3.5.1 in reference [26]. For CGP operating at this voltage level, the dispatch of reactive power must be carried out to achieve a $PF\ge 0.95$. In this sense, we consider such limitations for lagging and leading $PF$ conditions. The unitary $PF$ was considered as well.

Here, we will examine three case studies where we consider the critical operating conditions in terms of active and reactive power delivery, in accordance with the Mexican network code, using a randomly distributed wind velocity profile.

In addition, we evaluate the impact of adjusting the gearbox ratio on the width of the safe operating zones. The data obtained are presented in

Table 3. Safe zone while varying $r_{gb}$ for $P=2$ MW and $Q=\pm 0.55$ MVAr. The light-gray shading, which corresponds to an $r_{gb}$ below 60%, is no larger than the one corresponding to 100% $r_{gb}$. The up and down pointing arrows represent power factor leading and lagging, respectively. The darker gray area indicates that it is not possible to reduce the grarbox further than 50%.

Operation Condition				Generator Limits [m/s]			Final Limits [m/s]
P [MW]	Q [MVAr]	PF	${\mathit{r}}_{\mathit{gb}}$	${\mathit{v}}_{\mathit{min}}$ (Nom)	${\mathit{v}}_{\mathit{min}}$ (Curr)	${\mathit{v}}_{\mathit{max}}$ (Nom)	Lower	Upper	Size
$2.00$	0.00	1.00	100	3.97	5.85	7.27	5.85	7.27	1.42
$2.00$	$0.55$	0.96 ↓	100	3.97	6.04	7.27	6.04	7.27	1.23
$2.00$	$-0.55$	0.96 ↑	100	3.97	6.73	7.27	6.73	7.27	0.54
$2.00$	0.00	1.00	80	4.80	7.23	9.04	7.23	9.04	1.81
$2.00$	$0.55$	0.96 ↓	80	4.80	7.47	9.04	7.47	9.04	1.57
$2.00$	$-0.55$	0.96 ↑	80	4.80	7.85	9.04	7.85	9.04	1.19
$2.00$	0.00	1.00	60	6.20	9.56	12.00	9.56	12.00	2.44
$2.00$	$0.55$	0.96 ↓	60	6.20	9.89	12.00	9.89	12.00	2.11
$2.00$	$-0.55$	0.96 ↑	60	6.20	10.41	12.00	10.41	12.00	1.59
$2.00$	0.00	1.00	50	7.37	11.43	14.37	11.43	12.10	0.67
$2.00$	$0.55$	0.96 ↓	50	7.37	11.83	14.37	11.83	12.10	0.27
$2.00$	$-0.55$	0.96 ↑	50	7.37	12.45	14.37	12.45	12.10	−0.35
$2.00$	0.00	1.00	40	9.12	14.25	17.94	14.25	12.10	−2.15
$2.00$	$0.55$	0.96 ↓	40	9.12	14.75	17.94	14.75	12.10	−2.65
$2.00$	$-0.55$	0.96 ↑	40	9.12	15.53	17.94	15.53	12.10	−3.43

,

Table 4. Safe zone while varying $r_{gb}$ for $P=1.2$ MW and $Q=\pm 0.25$ MVAr. Larger size light-gray shading. The up and down pointing arrows represent power factor leading and lagging, respectively.

Operation Condition				Generator Limits [m/s]			Final Limits [m/s]
P [MW]	Q [MVAr]	PF	${\mathit{r}}_{\mathit{gb}}$	${\mathit{v}}_{\mathit{min}}$ (Nom)	${\mathit{v}}_{\mathit{min}}$ (Curr)	${\mathit{v}}_{\mathit{max}}$ (Nom)	Lower	Upper	Size
$1.20$	0.00	1.00	100	3.83	3.68	7.23	3.83	7.23	3.40
$1.20$	$0.25$	0.98 ↓	100	3.83	3.70	7.23	3.83	7.23	3.40
$1.20$	$-0.25$	0.98 ↑	100	3.83	3.70	7.23	3.83	7.23	3.40
$1.20$	0.00	1.00	80	4.68	4.48	9.00	4.68	9.00	4.32
$1.20$	$0.25$	0.98 ↓	80	4.68	4.57	9.00	4.68	9.00	4.32
$1.20$	$-0.25$	0.98 ↑	80	4.68	4.51	9.00	4.68	9.00	4.32
$1.20$	0.00	1.00	60	6.12	5.84	11.97	6.12	11.97	5.85
$1.20$	$0.25$	0.98 ↓	60	6.12	5.88	11.97	6.12	11.97	5.85
$1.20$	$-0.25$	0.98 ↑	60	6.12	5.88	11.97	6.12	11.97	5.85
$1.20$	0.00	1.00	50	7.29	6.95	14.35	7.29	12.10	4.81
$1.20$	$0.25$	0.98 ↓	50	7.29	7.00	14.35	7.29	12.10	4.81
$1.20$	$-0.25$	0.98 ↑	50	7.29	7.00	14.35	7.29	12.10	4.81
$1.20$	0.00	1.00	40	9.05	8.62	17.92	9.05	12.10	3.05
$1.20$	$0.25$	0.98 ↓	40	9.05	8.68	17.92	9.05	12.10	3.05
$1.20$	$-0.25$	0.98 ↑	40	9.05	8.68	17.92	9.05	12.10	3.05

and

Table 5. Safe zone while varying $r_{gb}$ for $P=0.0639$ MW and $Q=\pm 0.0159$ MVAr. Larger size light gray shading.The up and down pointing arrows represent power factor leading and lagging, respectively.

Operation Condition				Generator Limits [m/s]			Final Limits [m/s]
P [MW]	Q [MVAr]	PF	${\mathit{r}}_{\mathit{gb}}$	${\mathit{v}}_{\mathit{min}}$ (Nom)	${\mathit{v}}_{\mathit{min}}$ (Curr)	${\mathit{v}}_{\mathit{max}}$ (Nom)	Lower	Upper	Size
63,900	0.00	1.00	100	3.59	Does not exist	7.16	3.59	7.16	3.57
63,900	15,970	0.97 ↓	100	3.59		7.16	3.59	7.16	3.57
63,900	−15,970	0.97 ↑	100	3.59		7.16	3.59	7.16	3.57
63,900	0.00	1.00	80	4.48		8.95	4.48	8.95	4.47
63,900	15,970	0.97 ↓	80	4.48		8.95	4.48	8.95	4.47
63,900	−15,970	0.97 ↑	80	4.48		8.95	4.48	8.95	4.47
63,900	0.00	1.00	60	5.97		11.93	5.97	11.93	5.96
63,900	15,970	0.97 ↓	60	5.97		11.93	5.97	11.93	5.96
63,900	−15,970	0.97 ↑	60	5.97		11.93	5.97	11.93	5.96
63,900	0.00	1.00	50	7.16		14.32	7.16	12.10	4.94
63,900	15,970	0.97 ↓	50	7.16		14.32	7.16	12.10	4.94
63,900	−15,970	0.97 ↑	50	7.16		14.32	7.16	12.10	4.94
63,900	0.00	1.00	40	8.95		17.90	8.95	12.10	3.15
63,900	15,970	0.97 ↓	40	8.95		17.90	8.95	12.10	3.15
63,900	−15,970	0.97 ↑	40	8.95		17.90	8.95	12.10	3.15

. These tables are related to the operating conditions described in Section 2, providing a fixed active power and a reactive power that varies with respect to the power factor (leading and lagging conditions). Subsequently, these dispatch conditions are evaluated for different gearbox ratios $r_{gb}$.

The minimum and maximum values of wind speed are established by the design of the wind turbine. Similarly, the maximum predefined speed at which the DFIG can operate is taken as the upper limit. For the lower limit of the speed of the DFIG, two conditions are taken into account: (1) the minimum speed predefined by the design and (2) the minimum speed determined by the currents that circulate through its windings. Finally, between the restrictive operating limits of the wind turbine and the DFIG, the minimum and maximum wind speed ranges that guarantee the safe operation of the WECS are, thus, determined.

3.1. Case 1: P = 2 MW, Q = 0 MVAR and Q = ±0.55 MVAR

In this first case study, the operating conditions considered meet the requirements of the 2016 Mexican network code [26]; the results are shown in Table 3, where the gearbox ratio varies from 100% (design value) to 50%. The WECS safe operating zones correspond to the limits imposed by the minimum wind speed defined by the nominal currents in the DFIG stator and rotor windings, and the wind speed corresponding to the maximum nominal operating speed of the DFIG (800 rpm). It can be seen that if the size of the safe operating zone is used as the basis for the condition being 100% of the gearbox ratio (see the first three values in the last column of Table 3: 1.42 m/s, 1.23 m/s, and 0.54 m/s) for the three operating conditions, when reducing the gearbox ratio to 80% and 60%, the size of the safe zone increases. The largest increase in the size of the safe operating zone occurred for the operating condition with a gearbox ratio of 60%, and the largest corresponded to the unity power factor with a size of 2.44 m/s (cf. light-gray shading in Table 3). In the analysis process for this case study, it was observed that the DFIG is required to deliver its nominal active power, meaning that high wind speeds are required, to such a degree. In the case of a 60% gearbox ratio, the maximum wind limit (11.998 m/s) is almost equal to the maximum speed limit of the wind turbine (12.1 m/s).

It is also possible to observe that, in this case study, the gearbox ratio cannot be 50%. This is because the size of the safe zone decreases instead of increases. When operating in advance, the lower limits of the wind imposed by the currents in the stator and rotor windings exceed the maximum limit imposed by the wind turbine (cf. gray shading in Table 3). Thus, it can be said that the reduction of the gearbox ratio has a limit for the increase in the safe zone of operation, and that in this case study, it is 60%. The light-gray shading, which corresponds to an $r_{gb}$ below 60%, is no larger than the one corresponding to 100% of the gearbox ratio.

3.2. Case 2: P = 1.2 MW, Q = 0 MVAR and Q = ±0.25 MVAr

For this second case study, in order to show the operational flexibility of the WECS, a lower reactive power dispatch and 60% of the nominal active power were required. This condition allows us to see the impact that the reduction of the gearbox ratio has on the sizes of the safe zones for the WECS in different operating conditions. The results obtained are given in Table 4. Similar to case 1, the operating condition for a unit $PF$ is also included.

For this second case study, it can be observed that the reduction of the gearbox ratio increases the size of the safe zone of operation when $r_{gb}=100\%$. As in case 1, the gearbox ratio reduction to 60% is the best case because it has a greater increase with a size of 5.85 m/s for the three operating conditions of the WECS (cf. light-gray shading in Table 4). It was also observed that the reduction of the gearbox ratio below 60% causes the size of the safe operating zone to decrease. However, unlike the results in case 1, it can be seen that there is no condition where the lower limit of the wind speed is greater than the upper limit of the wind speed. However, it can be seen that for $r_{gb}=40\%$, the size of the safe zone is smaller than the safe zone for $r_{gb}=100\%$. Based on these results, it can be concluded that there seems to be a limit to how much the gearbox ratio can be reduced in order to increase the safe operational zones of the WECS.

3.3. Case 3: P = 0.063 MW, Q = 0 MVAr and Q = ±0.015 MVAr

In this last case study, the minimum active power that the DFIG can deliver to the electrical network was considered, considering the minimum wind speed at which the wind turbine could operate by design (3.5 m/s). In the case involving the dispatch of reactive power to the electrical network or grid, a condition that complies with the dispatch regulation was established, which is $\pm 0.01597$ MVAR. The operation of the WECS at unity power factor was also considered, as in the previous cases. The results are given in Table 5 and the most significant finding in the data obtained for this particular case is that the currents in the DFIG do not influence any of the safe operating zones found, so in Table 5, no data are reported for the lower limit of the wind speed. This is due to the minimum active power demand and low power factor that were imposed on the WECS for this case study. As in the previous cases, the size of the safe zone continues to increase until the $r_{gb}=60\%$ with a size of 5.96 m/s (cf. light-gray shading in Table 5), and then decreases. For $r_{gb}=40\%$, the size of the safe zone is less than that for $r_{gb}=100\%$, as in the second case study.

In general, the drawback of analyzing $r_{gb}$ is that it is not a variable parameter in the WECS. However, the analysis carried out shows its impact on the increase in the size of the safe operating zone of the WECS for the required power.

The determination and analysis of the safe areas of the WECS are not only important for controlling the delivery of power to the electrical network, they also provide very important information for the design and coordination of the electrical protections that act in the wind turbine. The data provided will serve as the foundation for effectively coordinating protections that will be in charge of preventing the WECS from working outside of the determined safe zones and damaging the system.

4. Control of WECS

Once we have determined the safe operating zones for the WECS, in this section, we show two control strategies for the regulation of the WECS operating as a conventional power plant. This is done by taking into account the power demanded by the network and the variability of the wind velocity. In both cases, we verify that, indeed, the currents in the DFIG windings never exceed their nominal values and, thus, the WECS is effectively controlled in a secure manner.

Here, we revisit the control problem formulated in [21], to control the WECS in the new, safe operating zones. All details regarding the implementation and tuning of the two control strategies can be consulted in Subsections 4.2 and 4.3 of [27]. In our control problem, we will assume that the dynamics of the electrical network and the back-to-back converter can be considered as inputs for the DFIG mathematical model presented in Section 2. The wind profile is given by a piecewise constant function whose mean and variance are obtained through sampling from a particular Weibull distribution that is consistent with the observed data (see Figure 2). Note that this wind profile is consistent with the safe wind operating region for the operation of the WECS (cf. Table 3, Table 4 and Table 5). Under these considerations, we guarantee the dispatch of the active and reactive powers demanded by the electrical network through adequate voltages in the DFIG rotor windings, ensuring the internal stability of the WECS.

4.1. The Passivity-Based Controller

In this section, we implement the passivity-based controller presented in [21]. The analysis was conducted for the control of a small WECS operating as a conventional generation plant; however, we will show that it works within the safe limits of the WECS considered in this manuscript, dispatching the demanded power as needed.

The voltages fed to the rotor windings are prescribed by the control law

$$\begin{array}{c}\hfill u_{qr}=R_r{i}_{qr}-\frac{{\omega }_s}{a}\left(1-\frac{{\omega }_g}{{\omega }_s}\right)\left[\frac{L_r^{\prime }}{a}{i}_{dr}+L_M{i}_{ds}\right]\\ \hfill -k_r\left[a^2{L}_r\left(i_{qr}-i_{qr}^{*}\right)+L_M\left(i_{qs}-i_{qs}^{*}\right)\right]\\ \hfill +k_s\left[L_s\left({}_{qs}-i_{qs}^{*}\right)+L_M\left(i_{qr}-i_{qr}^{*}\right)\right]\\ \hfill -k_m{\lambda }_{ds}\left({\omega }_g-{\omega }_g^{*}\right)\end{array}$$

(23)

and

$$\begin{array}{c}\hfill u_{dr}=R_r{i}_{dr}+\frac{{\omega }_s}{a}\left(1-\frac{{\omega }_g}{{\omega }_s}\right)\left[\frac{L_r^{\prime }}{a}{i}_{qr}+L_M{i}_{qs}\right]\\ \hfill -k_r\left[a^2{L}_r\left(i_{dr}-i_{dr}^{*}\right)+L_M\left(i_{ds}-i_{ds}^{*}\right)\right]\\ \hfill +k_s\left[L_s\left({}_{ds}-i_{ds}^{*}\right)+L_M\left(i_{dr}-i_{dr}^{*}\right)\right]\\ \hfill +k_m{\lambda }_{qs}\left({\omega }_g-{\omega }_g^{*}\right),\end{array}$$

(24)

where the gains must satisfy the conditions $k_r>0$, $k_m>0$, and

$$k_s>\frac{L_M^2}{B\left(L_s{a}^2{L}_r-L_M^2\right)L_r}{\parallel {\lambda }_r^{*}\parallel }^2{k}_m\mathrm{with}{\lambda }_r^{*}=L_r\left(\begin{array}{c}{i}_{qr}^{*}\\ i_{dr}^{*}\end{array}\right).$$

(25)

The controller is implemented directly in the dynamics of the WECS presented in Section 2.

In Figure 3 and Figure 4, we observe that the reference currents and angular velocity of the DFIG are satisfactorily followed in the dq0 frame. Therefore, with such a response for all of the variables of the DFIG, one can expect that the regulation of the active and reactive power delivered to the network is effectively fulfilled, as shown in Figure 5. We also note that the currents circulating through the windings of the DFIG do not exceed their nominal values during the regulation process, as shown in Figure 6. Nevertheless, we can also verify that there might be operating conditions for P and Q, such that the nominal angular velocity (188.49 [rad/s]) is surpassed, corresponding to the minimum active power demand considered (0.0639 [MW]) and the maximum wind speed available.

It is worth noting that a salient feature of the PBC control is that there are minor oscillations and peaks in the controller response when there are changes in the demanded powers for both active and reactive power. Such performance conditions are expected, yet they do not represent major problems when implementing the controller.

Finally, for this controller, one should be able to measure the entire state of the WECS; this is a situation that does not represent a technical problem due to the nature of the DFIG and the arrangement of the anemometers for the measurement of wind speed.

4.2. The Proportional–Integral Controller

Now we turn our attention to a PI controller. As in the previous case, the control inputs are the voltages in the rotor windings; in this case, the control law takes the usual form

$$u_{qr}=-k_p(i_{qr}-i_{qr}^{*})-k_i\int (i_{qr}-i_{qr}^{*})dt$$

(26)

and

$$u_{dr}=-k_p(i_{dr}-i_{dr}^{*})-k_i\int (i_{dr}-i_{dr}^{*})dt,$$

(27)

where $k_p$ and $k_i$ are the corresponding gains. The results for this controller performance are shown in Figure 7, Figure 8 and Figure 9, where we can see that the WECS is properly controlled.

In Figure 7, we show the regulation for the currents in the qd0 frame, while Figure 8 shows the regulation in the angular velocity of the generator. These results provide assurance of the adequate regulation for the power dispatch, as shown in Figure 9. Similar to the PBC case, we observe in Figure 10 that the nominal values for the currents are never exceeded and that the same operating conditions yield the same excess of angular velocity in the DFIG (see Figure 8). Notice as well that there are interesting stability properties for the closed-loop dynamics of the system, where only the currents in the DFIG rotor converge to their desired values (cf. Section 4, Equations (26) and (27)).

These results represent a significant advantage of the PI controller over the passive controller as the PI controller does not necessitate knowledge of the entire state of the WECS for its implementation. Additionally, its performance is better since it does not present oscillations when there are changes in the delivered powers. However, we recognize that formal proof of stability is still lacking for this type of controller, as opposed to the passivity-based control. Such an analysis is beyond the scope of this work.

5. On the Likelihood of Satisfying a Particular Power Demand

Finally, let us make a critical point regarding the random nature of wind variability. Thus far, we have determined the wind speed intervals in which it is safe to operate the WECS in the power regulation mode. That is, we provided an answer to the following problem: given a specific power demand from the electric grid at any given time, what is the minimum wind velocity required to satisfy the power demand? However, due to the unpredictability of the wind behavior, we also need to determine how likely it is that the WECS is capable of satisfying such a demand at the particular time that it is needed. To this end, we assume that the WECS is installed in a region where the wind velocity measurements form a sample of the Weibull probability distribution,

$$p\left(v\right)=\frac{k}{c}{\left(\frac{v}{c}\right)}^{k-1}exp\left[-{\left(\frac{v}{c}\right)}^k\right]\mathrm{for}v\ge 0,$$

(28)

whose variance intersects with the safe operating region. Here, the values for the shape and scale parameters are $k=4.5$ and c = 9.5 m/s, respectively. In this sense, the area under the curve of the probability distribution in the overlapping region yields the likelihood of dispatching the required power at any particular time.

Figure 11 shows that, based on the minimum and maximum operating wind speeds for the turbine (cf. Table 1), there is a 93% chance that the WECS can operate at any given time. For cases 2 and 3 of Section 3, there is a probability of up to 81% to meet the power demand, while the dispatch at nominal power (case 1) can only be expected to be met with a 25% chance at any particular time. These results are summarized in Figure 11.

Most of the time, the odds seem to be against extracting the nominal power; however, our study shows that whenever we have a lower power demand, one can operate the WECS in regulation mode. This might be especially relevant during contingencies or other low-demand conditions. In this sense, to the best of our knowledge, this is the first study where the regulation problem for this type of system is successfully addressed.

6. Conclusions

In this work, we addressed the regulation problem for a WECS operating as a conventional power generation plant. The results obtained allowed us to produce a twofold conclusion. On the one hand, we can categorically claim that it is indeed possible to achieve such a goal. On the other hand, such an operation heavily depends on the particular wind conditions at any given time. Since this is a random phenomenon, it is unlikely to deliver power requirements closer to the nominal capacity of the WECS. Nevertheless, this does not represent a completely undesired result since it opens up the possibility to operate this type of WECS in a regulation mode restricted to specific network conditions, e.g., during contingencies. In this sense, our recommendation is to adopt a dual operation, which involves operating the WECS in both MPPT and regulation modes, depending on the particular conditions of the grid. The MPPT mode is used to extract the maximum energy from the wind, while the regulation mode is used to stabilize the conditions of the electric network.

In addition, we have also shown that the gap in wind velocities defining the safe operating zones of a WECS based on a DFIG can be increased by decreasing the gearbox ratio. These results also show that there is a reduction limit for this constructive parameter corresponding to 60% of its nominal value. It remains to perform the analysis for variations of other parameters such as the blade pitch angle, which lies beyond the scope of our present discussion. Moreover, the bounds in the wind speed were obtained from a steady-state analysis of the WECS. The next step is to perform the dynamical analysis and control of this system to verify that these bounds are consistent with the full dynamics.

In sum, we have shown that WECS can be controlled in an adequate and safe manner for power demands below their nominal capacity, highlighting the importance of our result in the resilience of electrical networks where this type of generation system is important.