One step ahead adaptive control technique featured by fuzzy-based least square estimator for wind systems

Abstract

The present paper proposes the application to a wind system of the One Step Ahead control scheme featured by a Fuzzy-based Least Square Estimator. The considered wind system power generation supplies an electrical load disconnected from the power supply grid. It is composed of a three bladed horizontal-axis wind turbine which drives a synchronous generator by means of gearbox: the mathematical model for both the horizontal-axis wind-turbine and the synchronous generator will be briefly outlined. The adaptive nature of the One-Step-Ahead control algorithm relies on providing consistent estimation of the controlled system. This is achieved by means of a Least Square Algorithm that is able to provide a good estimation of a linear discrete time model of the controlled system, nevertheless its convergence rate reduces very quickly. For this reason, Least Square Algorithm needs a resetting strategy, which allows the achievement of a compromise between estimation accuracy and convergence rate. This not only represents a very problem-dependent issue but also introduces weaknesses in term of control tracking errors, which in turns needs an extra control contribution (integral correction). The proposed Least Square Algorithm enhancement overcomes these issues managing differently the estimation accuracy and the convergence rate. In the Results section, the achievements of the application of the One-Step-Ahead algorithm to the wind system will prove the reliability of the suggested enhanced control technique.

Keywords

Wind system HAWT adaptive control parameter estimator

Introduction

Wind energy is one of the most cost-competitive renewable energy source and it can be converted into electrical energy by means of a wind system. Wind power generation plants can be installed practically everywhere, provided a sufficient wind intensity is available, and they do not produce pollutants during operation according to their sustainable nature. These aspects have contributed, in the last few years, to promote the diffusion of wind energy generation and, consequently, the wind turbine technology experienced a significant enhancement.

For electrical energy generation applications, the wind systems are generally equipped with an AC-DC-AC converter in order to provide the appropriate AC voltage and frequency (no active control). On the other hand, this solution presents energy losses associated with the power electronics (Lalor et al., 2005; Xu et al., 2018). For this reason, employing an active control system for the output voltage and frequency will reduce such energy losses. In this paper, the One Step Ahead controller, equipped with a Fuzzy-based Least Square Estimator, is applied to a wind generation system considered as an isolate source of power and composed of a horizontal-axis wind-turbine connected to a synchronous generator. The static and dynamic behavior of the wind generation system has been simulated by using appropriate mathematical models of the wind-turbine and synchronous generator.

Mathematical models simulating wind turbines dynamics can be divided into momentum, vortex, and finite difference models. The momentum models provide the wind turbine performance by using actuator surfaces and subdivide the flow domain in many stream-tubes (Cooperman and Martinez, 2014; Golnary and Moradi, 2018; Islam et al., 2006; Kishinami et al., 2005; Torresi et al., 2013, 2016). These models represent a good compromise between accuracy and computational costs and therefore they appear to be more suitable to describe the wind turbine dynamics for control applications. The vortex models approximate the turbine blade considering distributed surface vortices (Bermudez et al., 2000; Schweigler, 2012; Schmitz and Chattot, 2006; Sezer Uzol and Uzol, 2009). Finally, the finite difference models solve the fluid-dynamic equations using either finite volume or finite difference methods considering the rotor effects by means of an actuator surface (Hsu et al., 2014; Rajagopalan and Fanucci, 1985; Sayed et al., 2012).

Many authors describe appropriate mathematical models for the synchronous electrical generator (Nunes et al., 2004; Umans, 2014; Yin et al., 2007). In the present paper, a complete dynamical model of both the rotor and the stator winding will be presented, where only the voltage of rotor winding will be considered as a control variable. Finally, the dynamic equilibrium of the rotating parts of both the wind turbine and the synchronous machine will be considered.

Many papers deal with classical controls for wind generation systems using both open loop schemes (Hur et al., 2017; Martinez-Rodrigo et al., 2014; Tan and Islam, 2004), and a closed loop ones (Badihi et al., 2018; Boulamatsis et al., 2018; Gao and Gao, 2016; Hur, 2018; Menezes et al., 2018; Mullane et al., 2006; Pereira et al., 2018). Usually, the controlled system model is known only in terms of maps and/or numerical/experimental data. Consequently, the controller parameters need empirical/heuristic tuning to perform properly. On the other hand, adaptive controllers estimate, at each sample time, an appropriate model of the controlled system (Barambones et al., 2018; Narayana et al., 2017). Therefore, these controllers determine, on the basis of the estimated model, the control law. Such a feature makes the adaptive controllers suitable for non-linear and time varying systems, retuning themselves whenever the system conditions change because of disturbance actions. The One-Step-Ahead adaptive control technique, employed in this study, linearly estimates the dynamical behavior of the controlled system (Goodwin and Sin, 1986). This adaptive controller offers several attractive features over classical PID controllers, such as promptness, reliability and adaptive disturbances rejection, as can be found in Dambrosio et al. (2003), where a comparison the One-Step-Ahead adaptive controller and PID one has been carried out. In particular, the two features which play a key role for the One-Step-Ahead adaptive capability are represented by the model estimation accuracy and the model estimation convergence rate to controlled system. Nevertheless, as the iterations proceed, its adaptive capability reduces very quickly. Such a negative outcome is usually settled by resetting the covariance matrix (main responsible of the estimation convergence rate) to the initial values. Such a procedure is quite abrupt: it cancels the estimation knowledge stored in the covariance matrix, and it can possibly generate local instabilities.

The present paper proposes an enhancement in the model parameter estimation in order to keep a fairly good level of adaptivity. In the Results section a comparison between the performances obtained by means of the One-Step-Ahead controller featured with an ordinary parameter estimator, reported in Dambrosio (2018), and those produced by the same controller scheme but equipped with the proposed parameter estimator will be presented. The case study considers a wind system power generation which supplies an isolate electrical load disconnected from the power supply grid (Figure 1). It is composed of a three bladed horizontal-axis wind turbine which drives a synchronous generator by means of gearbox: the controlled variables are represented by the output voltage and frequency whereas the control ones are the wind turbine blade pitch and voltage of synchronous generator rotor winding. The parameter estimator proposed in this paper, unlike the ordinary ones, does not update the model parameters with a constant rate, but it considers a variable gain which allows to modulate the estimated model parameters updates according to the dynamic system condition. As it will be described in the following, such a gain will be determined by means of a fuzzy logic mapping system.

Figure 1.

Wind system power generation scheme.

Mathematical model of the horizontal axis wind turbine

The mathematical model of the horizontal axis wind turbine here considered relies on momentum conservation in both the axial and tangential directions by using Blade Element Momentum approach (Hansen, 2015; Tahani et al., 2014). For instance, the airfoil aerodynamic polars i.e. the lift ( $C_{L}$ ) and drag ( $C_{D}$ ) coefficients vs. angle of attack ( $α$ ), can be found in Islam et al. (2006). For each streamtube, the thrust and torque acting on the blade elements, at a distance $r$ from the center of the rotor, are given by

\begin{matrix} dT = ρ π r σ (1 - a)^{2} {V_{0}}^{2} C_{L} \frac{\cos ϕ}{si n^{2} ϕ} \\ (1 + \frac{C_{D}}{C_{L}} \tan ϕ) dr \end{matrix}

(1)

\begin{matrix} dQ = ρ π r^{2} σ (1 + a')^{2} (Ω r)^{2} C_{L} \frac{\sin ϕ}{co s^{2} ϕ} \\ (1 - \frac{C_{D}}{C_{L}} \frac{1}{\tan ϕ}) dr \end{matrix}

(2)

where $σ_{r} = B c / 2 π r$ is the local solidity while $B$ is the number of blades $c$ is the chord length at $r$ , respectively. Moreover, $ρ$ , $a$ , $a'$ , and $ϕ$ represent the air density, the axial, and tangential induction factors and the angle of relative wind. Finally, $Ω$ is the rotor angular speed and $V_{0}$ is the free stream velocity. From equations (1) and (2), the wind turbine torque, $Q_{T}$ , and power, $P$ , are given by

Q_{T} = \int_{Rmin}^{Rmax} dQ,

(3)

P = Ω Q_{T}

(4)

being $R_{\min}$ , $R_{\max}$ the internal and external rotor radii. Equation (4) allows one to evaluate the wind turbine power coefficient, $C_{p}$ :

C_{p} = \frac{P}{\frac{1}{2} ρ {AV}_{0}^{3}}

(5)

In the present paper, a three-bladed horizontal-axis wind-turbine, $B = 3$ , has been considered. Its external $R_{\max}$ and internal $R_{\min}$ rotor radii are equal to $5 m$ and $0.5 m$ , respectively. The nominal conditions are: $V_{0} = 10 m / s$ ; $P = 22 kW$ ; tip speed ratio $λ = Ω R_{\max} / V_{0} = 7$ . The blade geometry (chord and twisting angle) has been designed according the maximum rotor extracted power and maximum airfoil (NACA 0012) efficiency (max $C_{L} / C_{D}$ ) criteria for the outer blade section ( $2 / 3$ blade length) and constant chord criterion for the remaining inner part (Hansen, 2015; Tahani et al., 2014). The power coefficient $C_{p}$ versus the tip speed ratio $λ$ , obtained from the presented wind turbine model, is shown in Figure 2. Each curve has been obtained from equations (1)–(5) by using a fixed value of the collective pitch $β$ which, of course, affects the inflow angle $ϕ$ . Therefore each curve in Figure 2 refers to a different value $β$

Figure 2.

Wind turbine power coefficient.

Mathematical model of the synchronous generator

The wind turbine drives a three phase synchronous generator. Considering a balanced generator windings together with balanced external load, the governing equations (Umans, 2014) in a fixed axis reference system, $d - q$ , reads:

\begin{matrix} v_{f} = r_{f} i_{f} + L_{ff} \frac{d i_{f}}{dt} - \frac{3}{2} L_{af} \frac{d i_{d}}{dt}, \\ v_{d} = - r_{a} i_{d} + L_{af} \frac{d i_{f}}{dt} - L_{d} \frac{d i_{d}}{dt} + L_{q} i_{q} ω, \\ v_{q} = - r_{a} i_{q} - L_{q} \frac{d i_{q}}{dt} + (L_{af} i_{f} - L_{d} i_{d}) ω . \end{matrix}

(6)

where $r_{a}$ and $r_{f}$ are the stator and rotor internal resistances, respectively, while $L_{d}$ and $L_{q}$ are the synchronous inductances with respect to the $d$ and $q$ axes. Moreover, $L_{af}$ represents the mutual inductance between stator and rotor windings, $L_{ff}$ is the rotor self-inductance, and $ω$ is the angular frequency; $v_{f}$ and $i_{f}$ are the voltage and the current of the rotor windings which represent the generator excitation energy source. The variables $v_{d}$ , $v_{q}$ and $i_{d}$ and $i_{q}$ do not have a direct physical meaning, although they are related to the phase voltages and currents.

Considering the equilibrium of stator equivalent circuit, being the external load an ohmic-inductive impedance, with an inductance, $L$ , and a resistance, $R_{L}$ , the corresponding governing equations in the $d - q$ reference system are:

\begin{matrix} v_{d} = L \frac{d i_{d}}{dt}, \\ v_{q} = L \frac{d i_{q}}{dt}, \\ v_{d} = R_{L} (- i_{d} - i_{Ld}), \\ v_{q} = R_{L} (- i_{q} - i_{Lq}), \end{matrix}

(7)

where $i_{Ld}$ and $i_{Lq}$ are the inductance external load currents in the $d - q$ system. The dynamic equilibrium of the rotating parts of both the wind turbine and the synchronous machine can be written as:

\begin{matrix} \frac{d ω_{g}}{dt} = \frac{1}{J} (\frac{Q_{T}}{τ} - Q_{E}) = \\ \frac{1}{J} [\frac{Q_{T}}{τ} - \frac{3}{2} \frac{p}{2} [(L_{af} i_{f} - L_{d} i_{d}) - L_{q} i_{q} i_{q}]] \end{matrix}

(8)

where $J$ , $τ$ , and $t$ represent the moment of inertia of all the rotating parts, referred to the electrical generator shaft, the transmission ratio of the gear box and the time, respectively; $Q_{E}$ is the rotor electromagnetic torque, $ω_{g}$ is the rotor generator rotational speed and $p$ is the polar pairs number. Equations (6) (8) represent a set of eight equations in the eight unknowns $i_{d}$ , $i_{q}$ , $i_{f}$ , $v_{d}$ , $v_{q}$ , $i_{Ld}$ , $i_{Lq}$ , and $ω_{g}$ , which can be solved using the wind turbine torque computed by equstion (3). Finally, the phase voltages $v_{a}$ , $v_{b}$ , and $v_{c}$ can be obtained from the unphysical voltages $v_{d}$ , $v_{q}$ , by means of the following relations:

\begin{matrix} v_{a} = v_{d} \cos (p ω_{g} t) - v_{q} \sin (p ω_{g} t), \\ v_{b} = v_{d} \cos (p ω_{g} t - \frac{2}{3} π) - v_{q} \sin (p ω_{g} t - \frac{2}{3} π), \\ v_{c} = v_{d} \cos (p ω_{g} t + \frac{2}{3} π) - v_{q} \sin (p ω_{g} t + \frac{2}{3} π) . \end{matrix}

(9)

One-step-ahead adaptive control technique

The One-Step-Ahead (OSA) algorithm employes the Least Squares Algorithm (LSA) as parameter estimator; as illustrated in the scheme reported in Figure 3, relying on an appropriate time sequence of input and output data, it estimates, at each sample time, the parameters of an Auto-Regressive Moving Average (ARMA) model in state-space form, which approximates the controlled system (Brockwell and Davis, 2016; Goodwin and Sin, 1986; Paolella, 2018). From the estimated ARMA model, the control law can be designed in order to drive to zero the estimated control tracking error. The general n-order ARMA scheme that models a generic non-linear system, in predictor form with d-step ahead, can be written as:

Figure 3.

One-Step-Ahead algorithm scheme.

X_{i + d} = A (z^{- 1}) X_{i} + B (z^{- 1}) U_{i},

(10)

where $X = [x (1), x (2), \dots, x (n)]^{T}$ and $U = [u (1), u (2), \dots, u (m)]^{T}$ represent the state vector and input vector of the controlled system. The operators $A (z^{- 1})$ and $B (z^{- 1})$ are defined as follows:

\begin{matrix} A (z^{- 1}) = A^{0} + A^{1} z^{- 1} + \dots + A^{n - 1} z^{- (n - 1)}, \\ B (z^{- 1}) = B^{0} + B^{1} z^{- 1} + \dots + B^{m - 1} z^{- (m - 1)}, \end{matrix}

(11)

where $z^{- k}$ represents the $k$ -step time delay operator. Subscript $i$ refers to the vector evaluated at time $t = i Δ t$ . The LSA, in recursive form, provides an on-line parameter estimation of $A (z^{- 1})$ and $B (z^{- 1})$ . Equation (10), for one-step-ahead scheme, can be written as:

X_{i} = Θ Φ_{i - 1},

(12)

where:

\begin{matrix} Θ = [A^{0}, A^{1}, \dots, A^{n - 1}, B^{0}, B^{1}, \dots, B^{m - 1}], \\ Φ_{i - 1} = [X_{i - 1}, \dots, X_{i - n}, U_{i - 1}, \dots, U_{i - m}], \end{matrix}

(13)

In equation (12), $Θ$ represents the matrix of the ARMA model coefficients that, as far as a non-linear system is concerned, are generally unknown and time-varying. The LSA is able to estimate $Θ$ exploiting an appropriate system input and output time series, according the following recursive formulas:

\begin{matrix} Θ_{i}^{j} = Θ_{i - 1}^{j} + \frac{P_{i - 2}^{j} Φ_{i - 1}}{1 + Φ_{i - 1}^{T} P_{i - 2}^{j} Φ_{i - 1}} \\ [x_{i - 1} (j) - Θ_{i - 1}^{j} Φ_{i - 1}], \\ P_{i - 1}^{j} = P_{i - 2}^{j} - \frac{P_{i - 2}^{j} Φ_{i - 1} Φ_{i - 1}^{T} P_{i - 2}^{j}}{1 + Φ_{i - 1}^{T} P_{i - 2}^{j} Φ_{i - 1}} . \end{matrix}

(14)

In equation (14), $Θ_{i}^{j}$ indicates the j-th raw of the matrix $Θ_{i}$ , while $P_{i - 1}^{j}$ indicates the covariance matrix of $Θ_{i}^{j}$ and $x_{i - 1} (j)$ is the j-th state variable. The OSA control scheme uses the estimated ARMA model of the system at the i-th time step:

X_{i} = Θ_{i - 1} Φ_{i - 1},

(15)

in order to estimate the value of the control tracking error, which reads:

ξ_{i} = X_{i} - X_{i}^{*},

(16)

where $X_{i}^{*}$ represents the desired output vector at the i-th time step. Since the OSA controller cannot drive to zero the control tracking error $ξ_{i}$ at the same time step as the ARMA model estimation occurs, the estimated control tracking error expressed by equation (16) will be transposed one step in the future which, by using equation (15), reads:

ξ_{i + 1} = Θ_{i} Φ_{i} - X_{i + 1}^{*} .

(17)

Setting control tracking error $ξ_{i + 1}$ at zero and assuming that $Θ_{i} \approx Θ_{i - 1}$ , it is possible to determine the control vector $U_{i}$ :

\begin{matrix} U_{i} = (X_{i + 1}^{*} - A_{i - 1}^{0} X_{i} + \dots - A_{i - 1}^{n - 1} X_{i - n + 1} + \\ - B_{i - 1}^{1} U_{i - 1} + \dots - B_{i - 1}^{m - 1} U_{i - m + 1}) {(B_{i - 1}^{0})}^{- 1} . \end{matrix}

(18)

Equation (18) provides the control laws, which aim to converge to zero the control tracking error $ξ_{i + 1}$ . For the present non-linear system, the previous condition is not completly fulfilled. As a consequence, the matrix $Θ_{i - 1}$ must be continuously updated at each time step by using equation (14). These formulas are presented in recursive form and therefore need some initial values for the parameter estimation matrix $Θ_{i - 1}$ and for the covariance matrices $P_{i - 1}^{j}$ . When no better guess is available, the starting values will be:

\begin{matrix} Θ_{0} = 0, \\ P_{- 1}^{j} = k I, \end{matrix}

(19)

where the value of $k$ should be chosen according to the sensitivity of the controlled system with respect to the control variables.

Like for any other parameter estimator, the most relevant features for the LSA are represented by the parameter estimation accuracy and the model estimation convergence rate to the controlled system. The first one deals with the estimation error or, in other words, it measures how close (or how far) the estimated model output is to (from) the system output. On the other hand, the model estimation convergence rate, which is the first time derivative of the estimation error, denotes the estimator ability to keep track of the system variations (non linearities and/or time variances). Although these two features are not independent, since they refer to two different estimator skills, they might not be necessarily required at same time. In fact, when system experiences a significant transient period (e.g. system starting up, disturbance effects) from an estimation effectiveness point of view, the parameter estimator algorithm should present high estimation convergence rate and only fair estimation accuracy. On the contrary, when the system settles in the neighborhood of the equilibrium condition (e.g. system steady state and/or after disturbance effects have been sufficiently damped) the estimation accuracy plays a key role, whereas the estimation convergence rate has only a minor relevance. The ordinary LSA, as described by equation (14), is characterized by a very high initial convergence rate and a poor estimation accuracy. As the iterations proceed, the first rapidly decreases, since the $P^{j}$ matrix becomes small after a few iterations, whereas the estimation accuracy starts to improve. Such an issue has been taken into account considering a scheme in which the covariance matrix $P_{i - 1}^{j}$ is reset after a fixed number of iterations (Goodwin and Sin, 1986). This approach presents a few drawbacks: the first one concerns the reset value $P_{0}^{j}$ of the covariance matrix $P^{j}$ that results highly control problem dependent and, therefore, it represents a further parameter that must be chosen, weakening the generality of the OSA control algorithm. Secondly, this resetting strategy yields abrupt change in the covariance matrix $P^{j}$ , which in turn has an impact on the performance of the controlled system. Finally, adopting the reset procedure described by equation (19), LSA estimation accuracy and convergence rate are not improved when genuinely necessary; in fact, they present opposite behavior: right after matrix $P^{j}$ resetting, the estimation accuracy is poor and convergence rate is high; on the contrary, after few iterations, LSA provides an accurate estimation of $Θ_{i}$ but its convergence rate becomes unacceptable. Therefore, when the $P^{j}$ reset occurs the two main features of the LSA estimator algorithm suddenly change with no relation to the real controlled system condition. From equation (14), during the estimation process, the $P^{j}$ terms experience a decrease: at the beginning they assume high values and therefore the first of equation (14) allows large variation of $Θ_{i}$ , but its accuracy is poor. On the other hand, as the iterations go on, the second of equation (14) provides the most appropriate matrix $P^{j}$ for the LSA estimation, but its small values allow only small correction of the coefficients matrix $Θ_{i}$ , that is, low convergence rate. These considerations allow one to conclude that the LSA resetting strategy of the $P_{0}^{j}$ matrix tries to reach a compromise between the parameter estimation accuracy and the model estimation convergence rate but, in this way, it is not able to improve neither the first nor the latter according the system conditions. A better resetting strategy for the LSA parameter estimator has been proposed in Dambrosio (2018), where monitoring the $P^{j}$ decay, it is possible to determine a good $P^{j}$ value to be stored and then the most appropriate step iteration to retrieve such stored value. The new resetting strategy, on one hand, allows one to eliminate the local instabilities due to the original $P^{j}$ reset; nevertheless, once again it does not increase the estimation accuracy or the model estimation convergence rate according to the system conditions.

In the present paper a different approach is proposed for the LSA estimator algorithm: as shown in Figure 4, in order to avoid the $P^{j}$ resetting at prescribed value specified by equation (19), the second of equation (14) is replace by:

Figure 4.

One-Step-Ahead algorithm proposed scheme.

P_{i - 1}^{j} = P_{i - 2}^{j} - G^{j} \frac{P_{i - 2}^{j} Φ_{i - 1} Φ_{i - 1}^{T} P_{i - 2}^{j}}{1 + Φ_{i - 1}^{T} P_{i - 2}^{j} Φ_{i - 1}},

(20)

where $G^{j}$ represents a gain which enables to modulate the $P^{j}$ update. In this way, if the system experiences a significant transient period the $P^{j}$ updates have to be large enough to provide a good model estimation convergence rate and therefore the $G^{j}$ values should be chosen large. On the contrary, when the system settles in the neighborhood of an equilibrium condition the $P^{j}$ will decay after few iterations delivering a good estimation accuracy; in this condition there is no need further update $P^{j}$ and therefore the $G^{j}$ should assumes a quite small value. Of course large and small represent labels for values of the gain $G^{j}$ . Nevertheless, two considerations are in order: firstly, the gain $G^{j}$ varies, at least for the present application, in a narrow interval ( $[0, 1]$ ); then, as it will be shown in the Results section, though $G^{j}$ values does derive from an optimization process, the LSA algorithm, by replacing the second of equation (14) with equation (20), not only does not need any reset strategy, but it also presents good level of estimation convergence rate and estimation accuracy only when genuinely necessary. To confirm what stated above, the $G^{j}$ coefficients have been determined by means of a Fuzzy Logic mapping (Dambrosio, 2017; Mendel, 1995; Zadeh et al., 2011) where the estimation error and its first time derivative have been used as inputs. The reason why the fuzzy mapping is employed relies its easy-to-design character that allows one to readily determine the $G^{j}$ coefficients on the basis of the estimation error and its first time derivative by using a thumb-rule approach. Thus, the consequent fuzzy logic base of rules results to be not optimized, assuming a quite simple form. Tables 1 and 2 show fuzzy logic base of rules employed for the test case presented in the Results section; in particular, Table 1 contains the $G^{j}$ recursive updates according to the estimation error ( $e_{st}$ ) and its first time derivative ( ${\overset{\cdot}{e}}_{st}$ ) values, whereas Table 2 refers to the $G^{j}$ thresholds. One can observe that fuzzy logic bases of rules specified in Tables 1 and 2 are not j-dependent which, in other words, means that both the controlled variables (output voltage and frequency of the electric generator) share the same bases of rules, confirming the unoptimized nature of the adopted fuzzy bases of rules. Such an issue must not be considered as a weakness, but, on the contrary, it demonstrates the robustness of the proposed parameter estimator method.

Table 1.

$G^{j}$ recursive updates.

$e_{st}$ → ${\overset{\cdot}{e}}_{st}$	$- 4 \times 10^{- 4}$	$- 2.5 \times 10^{- 4}$	$- 1 \times 10^{- 4}$	0	$1 \times 10^{- 4}$	$2.5 \times 10^{- 4}$	$4 \times 10^{- 4}$
$2 \times 10^{- 4}$	$- 2 \times 10^{- 3}$	$- 1.5 \times 10^{- 3}$	$- 1 \times 10^{- 3}$	0	$- 1 \times 10^{- 3}$	$- 1.5 \times 10^{- 3}$	$- 2 \times 10^{- 3}$
$2.1 \times 10^{- 4}$	$1 \times 10^{- 9}$	$1 \times 10^{- 9}$	$1 \times 10^{- 9}$	$1 \times 10^{- 9}$	$1 \times 10^{- 9}$	$1 \times 10^{- 9}$	$1 \times 10^{- 9}$

Table 2.

$G^{j}$ thresholds.

$e_{st}$ → ${\overset{\cdot}{e}}_{st} ↓$	$- 4 \cdot 10^{- 4}$	$- 2.5 \times 10^{- 4}$	$- 1 \times 10^{- 4}$	0	$1 \times 10^{- 4}$	$2.5 \times 10^{- 4}$	$4 \times 10^{- 4}$
$2 \times 10^{- 4}$	$1.5 \times 10^{- 1}$	$8 \times 10^{- 2}$	0	0	0	$8 \times 10^{- 2}$	$1.5 \times 10^{- 1}$
$2.1 \times 10^{- 4}$	1	1	1	1	1	1	1

Moreover, the OSA controller, featured by the ordinary LSA (equation (14)), provides the control law $U_{i}$ , equation (18), that could assume large values leading to undesired oscillations. This aspect is due to the poor parameters estimation accuracy, especially with regard to the $B_{i - 1}^{0}$ parameter that may take small values, resulting in an excessive control effort. To solve this issue it is possible to set a minimum threshold the $B_{i - 1}^{0}$ term. Such a solution causes a non-zero residual control tracking error which, in turns, requires an integral correction for the control law determined in (18) (Dambrosio et al., 2003). This approach generates two more variables to be tuned, $B^{0}$ minimum threshold and the integral correction gain, which are quite problem-dependent, reducing the algorithm generality. On the other hand, by using the OSA controller, with the proposed LSA (equation (20)), since the $B_{i - 1}^{0}$ estimation is more accurate, the $B^{0}$ minimum threshold is very small resulting in no appreciable residual control tracking error.

Finally, since the proposed LSA parameter estimator does not consider any resetting strategy, the constant $k$ in equation (19) is no longer a further variable to be tuned, increasing the algorithm generality.

Results

The wind power plant configuration considered in the present paper is composed of a three bladed horizontal axis wind turbine with variable blade pitch connected to a three-phase synchronous generator by means of a gear box, as shown in Figure 1. Such a wind power plant supplies an isolate electrical load disconnected from the power supply grid. For this reason, the output voltage and frequency (controlled variables) must track some prescribed values; this is achieved by acting on the manipulating variables which are represented by the wind turbine blade pitch and voltage of synchronous generator rotor winding. The wind system starts from rest and, after a starting time interval, it reaches a steady state condition. During the steady state time interval, an appropriate disturbance pattern has been hypothesized, which perturbs the system status. Such disturbances are represented by the instantaneous changes of the wind speed ( $V_{0} = 10 m / s \to 11 m / s$ ) at $t = 200 s$ and the resistance load ( $R_{L} = 10 Ω \to 17 Ω$ ) at $t = 300 s$ , as shown in Figure 5. During the starting time interval, the control system must guarantee a prescribed speed up phase of the wind system (linaer frequency increase from the initial 0 to $50 Hz$ at $100 s$ ), whereas, during the steady state time interval, it must counteract the disturbance effects to preserve the desired values of both voltage and frequency of the electric generator. The control variables are represented by the blade pitch, $β$ , and voltage of rotor winding, $v_{f}$ . Two tests using the same disturbances and the same set point time variation will be presented: the first one will consider the OSA controller featured by the proposed LSA parameter estimator, whereas, at the end of the present section, some results from Dambrosio (2018) will be shown for comparison.

Figure 5.

Disturbances time variation.

Figure 6 shows the time history of the output frequency, $f$ , of the synchronous generator, while Figure 7 presents the corresponding percentage control error $Δ f %$ . In Figure 6 dashed and solid lines represent the target and the actual output frequency, respectively. From a quick glance at these figures, it is possible to state that the OSA controller equipped with the proposed LSA parameter estimator is able to maintain small percentage control errors, except for the very beginning of the starting period, where, not only the parameter estimation is quite poor but small absolute errors lead to large percentage errors. Moreover, the only evidence of the disturbance actions is represented by the negligible step-like variations in Figure 7 ( $Δ f % \approx 0.3 %$ ) at the leading edge of the step variation of the disturbance variables. These step-like variations are due to the lack of any integral correction in the control law and their tiny amplitude confirms high accuracy of the parameter estimation.

Figure 6.

Output frequency (controlled variable).

Figure 7.

Percentage error of the output frequency.

Figure 8 presents the output voltage, $V$ , while Figure 9 shows the corresponding percentage error, $Δ V %$ . Again, the actual and the reference output voltages are practically coincident for most of the considered time interval, except for the starting period, where the electric generator presents a too low winding excitation. Moreover, Figure 9 shows neither significant spikes nor residual control tracking errors during the steady state time interval; this aspect indicates no practical effect of the instantaneous changes of wind speed and the resistance load. Finally, Figures 10 and 11 show the time variations of the control variables, i.e. the blades variable pitch, $β$ , and the voltage of the rotor windings, $v_{f}$ . Such control variables vary in order to counterbalance the disturbance effects.

Figure 8.

Output voltage (controlled variable).

Figure 9.

Percentage error of the output voltage.

Figure 10.

Blades pitch (control variable).

Figure 11.

Voltage of the rotor windings (control variable).

Finally Figure 12 shows the time variations of the gains which enable the $P^{j}$ update modulation. In particular dashed line refers to the gain variation for the frequency regulation, while the solid line indicates the gain variation for the voltage regulation. As earlier stated, even if the gains computations do not come from an optimization process, their time histories allow an accurate parameter estimation for both the controlled variables. In particular, the frequency control problem gain variation (dashed line) presents, during the steady state period two spikes that occur at the disturbances step variation leading edge: the reason for such variations lies in the fact that, though LSA presents a good estimation accuracy, due to the disturbances step variation, it needs a higher model estimation convergence rate.

Figure 12.

Gains for the $P^{j}$ update modulation.

At the end, some results coming from Dambrosio (2018) where the OSA algorithm is applied by using a smart resetting procedure of the LSA covariance matrix will be presented for comparison. For the sake of conciseness, only the results regarding the output frequency and voltage percentage errors of the synchronous generator will be shown (Figures 13 and 14), respectively. From a quick glance, comparing these figures with the corresponding ones obtained by using the proposed LSA algorithm (Figures 7 and 9), one can note that they provide similar performance. Nevertheless, during the staring period, by using OSA controller featured by the proposed LSA, both the controlled variables reach the neighborhood of the zero-control tracking error condition earlier with respect to those obtained in Dambrosio (2018). On the other hand, during the steady state condition, the frequency percentage error from Dambrosio (2018) reports two evident spikes due the disturbances step variations; on the contrary, no significant impact seems to affect the frequency percentage error presented in this paper. Such aspect is important, as it is the evidence of the robustness and accuracy of the OSA controller equipped with the proposed LSA parameter estimator.

Figure 13.

Percentage error of the output frequency from Dambrosio (2018).

Figure 14.

Percentage error of the output voltage from Dambrosio (2018).

Conclusions

In the present paper, an enhanced version of the One Step Ahead control algorithm has been proposed to regulate a wind system designed as an isolated electrical power generation station. The wind system is composed of a horizontal-axis three bladed wind turbine connected to an electrical synchronous generator through a gear box. The wind system has been numerically simulated and the mathematical models of its components have been presented. The proposed enhancement of the One Step Ahead controller considers an appropriate modulation of the LSA covariance matrix, instead of its resetting to a starting value. Such change permits, on one hand, to improve the estimation accuracy and the model convergence rate only when genuinely necessary; on the other hand, it reduces drastically the number of problem-dependent parameters necessary to define the One Step Ahead controller. Finally, achieving a good estimation accuracy, no integral correction to the control law is needed.

The results of the application of the enhanced controller algorithm to regulate a wind system designed as an isolated electrical power generation station have been presented. Comparing such results with those obtained by means of the same control scheme but featured by a LSA with covariance matrix resetting procedure, it appears quite evident that the proposed parameter estimator enhancement achieves a faster and more accurate estimation which, in turns, translates into smaller control tracking errors and disturbances rejections. Moreover, such One Step Ahead scheme effectively eliminates all the local instabilities due to the former resetting procedures. All these considerations have proven the immediate application of the proposed procedure and its robustness.

Footnotes

Declaration of conflicting interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Lorenzo Dambrosio

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