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Adaptive backstepping control for maximizing marine current power generation based on uncertainty and disturbance estimation
Electrical Power and Energy Systems 117 (2020) 105329
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Adaptive backstepping control for maximizing marine current power generation based on uncertainty and disturbance estimation
T
Xiuxing Yin The School of Engineering, University of Warwick, Coventry CV4 7AL, UK
A R T I C LE I N FO
A B S T R A C T
Keywords: Marine energy Marine current turbine Maximum power generation Adaptive backstepping control Uncertainty and disturbance estimation
The renewable energy from marine currents is highly promising as a clean, reliable and predictable energy source for the next-generation electricity generation. Therefore, the paper focuses on the design and implementation of the high-efficiency control for maximizing the marine current power generation and hence an adaptive backstepping controller with uncertainty and disturbance estimation is proposed for a generic horizontal marine turbine. The turbine design principle and dynamics modelling are presented and then the control problem is formulated. Consequently, the controller is designed to be composed of a marine turbine speed control loop and a q-axis current control loop while the uncertainty and disturbance is estimated and compensated. A swell filter is also incorporated into the control loop to smooth generator power fluctuations. The stability of the proposed control is verified via the Lyapunov synthesis and all the tracking errors are guaranteed to converge to zero asymptotically. The proposed control is verified in an experimental test bench and the test results indicate that the generator produces obviously more power (up to 30% more power) when using the proposed control in comparison with a conventional backstepping control.
1. Introduction As a vast and important energy source, the marine renewable energy is playing a pivotal role in addressing the issues of rising costs and growing security concerns of the worldwide energy consumption [1]. The marine energy can also provide a predictable, stable and reliable solution in meeting the challenging targets of a massive energy increase from renewables in comparison with wind and wave energies. At present, the marine energy technologies are advancing fast, but most of them are still in the demonstration phase, or are just reaching the commercial viability [2]. Until recently, the most common marine energy technologies involve erecting tidal barrages and installing underwater marine turbines. Due to the use of physical barriers for marine energy extraction, the tidal barrages suffer from the disadvantages of environmental populations and adverse effects on marine ecosystems. Unlike tidal barrages, the marine turbines are designed without using tidal barrages and can be deployed in estuaries or offshore such that they can be rotated by strong marine currents to generate electricity. Even though there exist various configurations of marine turbines, the most promising and common configuration is a horizontal axis marine turbine that resembles the commonly deployed wind turbines for capturing hydrokinetic energy. There are several commercial-sized marine turbines in operation, which mainly include the Verdant Power turbine
[3], the open center turbine Open Hydro [4], the marine current turbines Seaflow and SeaGen [5]. Although marine turbines share some similarities with wind turbines, fundamental differences still exist between them. The marine turbines are generally operated underwater in harsh marine environment that poses unique technical challenges including the potential for cavitation, the free surface effects, the different stall characteristics, and the different range of Reynolds numbers [2]. When considering these fundamental differences, it is necessary to design and implement specific and robust control approaches to maintain the stable operation and high-quality power production of marine turbines under different ocean conditions. One of particular interest is the maximum marine power extraction control that can be used to generate the optimal electrical power from marine currents. By considering the above fundamental differences and the fact that the seawater density is about 832 times higher than air, the maximum marine power extraction control needs to be particularly designed by compensating for relatively large marine loads, unexpected uncertainty and disturbance due to these differences from wind turbines. The design of various controllers for extracting the maximum marine power has recently been a hot topic and some control designs have been reported in the literature. The high-order sliding mode control was proposed in [6] to deal with the maximum marine energy generation. The control is highly dependent on the marine
E-mail address: lixingfi[email protected]. https://doi.org/10.1016/j.ijepes.2019.05.066 Received 8 September 2018; Received in revised form 12 April 2019; Accepted 26 May 2019 0142-0615/ © 2019 Published by Elsevier Ltd.
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support structures on the seabed. The turbine consists of a rotor, a geared transmission, and an electrical generator and is placed in the path of incoming marine currents to extract the kinetic energy from the moving seawater by turning the rotor blades. The extracted mechanical energy is then transmitted to the generator through the geared transmission for electricity production. The generated marine electricity is then transmitted onshore for grid integration via submarine cables. The generator is electrically controlled by power converters in order to achieve the maximum marine power extraction or for the grid connection purpose at different current speed. The floating marine current turbine has prominent advantages of easy installation, easy accessibility, low environmental impact and cost effective maintenance. The marine current turbine has the self-orientation ability, particularly useful in reversing marine currents and can be easily lifted out of the water for maintenance without costly steel supports or alterations to the ocean floor. The turbine can also be installable almost everywhere in sites with 0.5 m/s to 5 m/s current speed with reduced cost compared to fixed structures, enabling the use of gravity anchors. In order to further enhance the marine energy conversion efficiency, a diffuser can be placed around the turbine to increase the mass flowing though the turbine swept area, thereby increasing the marine energy capture [11].
resource and the turbine models. The control was then analyzed regarding the swell effects and validated based on the marine current data from the Raz de Sein (Brittany, France). The high-order sliding mode control was proposed in [7,8] to implement the speed control of a variable speed doubly-fed induction generator (DFIG)-based marine current turbine (MCT). The sensitivity of the control design was analyzed regarding the swell effects. Simulation results were also presented and fully analyzed. The high-order sliding mode control was proposed in [9] for the marine energy maximization of the permanent magnet synchronous generator (PMSG)-based marine current turbine. The controller was incorporated into a global simulation tool with marine resource and turbine models. The control validation was conducted based on the tidal data from the Raz de Sein (Brittany, France) and experiments on a 7.5-kW real-time simulator. The second-order sliding mode control was proposed in [10] to increase the generated power and the turbine efficiency by adequately handling the marine resource characteristics including swell effects and turbulence. The control has also taken the PMSG parameter variations into account. The controller was validated by using marine current data from the Raz de Sein (Brittany, France) and experiments on a 7.5-kW real-time simulator. However, the above-mentioned sliding mode control methods suffer from the inherent chattering problems and switching oscillations that will result in frequent marine power and torque fluctuations. This paper presents a new adaptive backstepping controller for a generic horizontal marine turbine to explore the maximum marine power generation control. The design principle and dynamics modelling of the horizontal axis marine turbine are presented. The control problem is then formulated and the controller is proposed to be composed of a marine current speed tracking loop and a q-axis current tracking loop. In each control loop, the uncertainty and disturbance term is estimated, and the bounds of the estimation errors are estimated and updated online. A swell filter is also incorporated in the control loops to attenuate swell effects. The stability of the overall control system is verified via the Lyapunov theory and all the tracking errors can be guaranteed to converge to zero. The parameter tuning procedure is also detailed to choose the appropriate control parameters for the controller and parameter update law. An experimental test bench is used to test the proposed control in the efficiency for maximizing the marine power generation and a conventional backstepping controller is designed for comparison. The remainder of the paper is organized as follows: in Section 2, the design principle and dynamics modelling of a generic horizontal marine turbine are presented, and the control problem is formulated. In Section 3, the controller design and the stability analysis are detailed. In Section 4, the validation and discussions of the proposed control in comparison with a conventional controller are presented. The Section 5 concludes the paper.
2.2. The dynamics of the marine turbine The marine current turbine can only convert a fraction of the kinetic energy of the incoming marine currents into electrical energy, and the captured kinetic energy is directly proportional to the marine fluid density and the cube of the marine current speed. Providing that the marine current velocity is uniform across the cross-sectional area at any instant, the amount of the available kinetic energy can be expressed by
Pr =
πρR2 3 v (t ) Cp (λ, β ) 2
(1)
The turbine torque acting on the rotor can be calculated accordingly as
Tr =
πρR3v 2 (t ) Pr = Cp (λ, β ) 2λ ωr
(2)
where Pr and Tr are respectively the kinetic marine power and the turbine torque, v(t) is the incoming marine current speed, ωr is the turbine rotation speed, R is the turbine radius, ρ is the sea water density (1025 kg/m3), β is the turbine pitch angle, λ is the tip speed ratio, Cp denotes the power coefficient which indicates the power extraction performance of the turbine. As shown in Eq. (1), the mechanical power generated by the marine current turbine is dependent on the cube of the marine velocity v(t), the swept cross-sectional area πR2 perpendicular to the marine flow direction, and the density of the seawater density ρ. Therefore, even small changes in the marine velocity can cause substantial changes in the amount of available kinetic power and hence, smaller and faster rotating marine turbines can be used in comparison with larger wind turbines for producing the same amount of electricity. The power coefficient is a nonlinear function of the tip speed ratio and the blade pitch angle and can be defined as [12]
2. The marine current turbine and dynamics modelling In this section, a generic horizontal marine turbine is chosen as the control object for investigation and control implementation. The design principle and dynamics modelling of the horizontal axis marine turbine are presented. The control problem is then formulated for facilitating the controller design accordingly. 2.1. The design principle of the marine current turbine
(
)
( )
c c ⎧ c1 α2 − c3 β − c4 β c5 − c6 ⎤ exp − α7 ⎪Cp = ⎡ ⎣ ⎦ ⎨ 1 = 1 − c9 ⎪α λ + c8 β β3 + 1 ⎩
The marine turbine is designed as a floating type horizontal axis marine current turbine whose axis of rotation is horizontal to the incident marine current direction. The three bladed turbine is designed to reach a high efficiency within a relatively large operation conditions range by using a properly designed airfoil section shape. As shown in Fig. 1, the marine current turbine is connected to the buoyant barge that is tethered to the seabed with a mooring rope, and operates essentially as an upside down horizontal turbine without requiring extra
(3)
where c1 ∼ c9 are constant coefficients, c1 = 0.18, c2 = 85, c3 = 0.38, c4 = 0.25, c5 = 0.5, c6 = 10.2, c7 = 6.2, c8 = 0.025, c9 = −0.043. The tip speed ratio (TSR) can be expressed as
λ= 2
λ opt v (t ) ωropt R ωr R ⇒ λ opt = ⇒ ωropt = R v (t ) v (t )
(4)
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Fig. 1. The floating marine current turbine.
where ωropt and λopt are respectively the optimal rotation speed and the optimal tip speed ratio under which the marine current turbine can extract the maximum available power, or the power coefficient can attain the maximum value Cpmax. In the maximum marine power extraction mode, the maximum kinetic marine power can be rearranged based on Eq. (1) as follows
Prmax =
Cpmax 3 1 πρR5· 3 ·ωropt 2 λopt
swell period. The geared transmission is essentially a speed increasing mechanism and the gear ratio can be defined as
ng =
(5)
(Jr + Jg ng2 ) ω̇ r = Tr − br ·ωr − ng Te + δ
(7)
(8)
where Jr denotes the rotor inertia, Jg denotes the high-speed side inertia, br denotes the torsion damping coefficient, Te denotes the reaction electromagnetic torque of the generator, the term δ denotes the lumped uncertainty and disturbance. In practice, the lumped uncertainty and disturbance term is used to model un-modelled dynamics and other uncertainty that are caused by gear meshing, gearbox backlash, the cavitation or scour, the biofouling and obstructions due to marine growth build-up. All the effects included in the uncertain term can be readily estimated and compensated by the proposed controller design. The employed electrical generator is a non-salient pole PMSG whose dynamics can be electrically described in the d/q axis rotation reference frame [13]. Since the maximum power extraction control in the paper is only directly related to the q-axis current dynamics, the focus is on the q axis current dynamics which can be described as
ωropt τf s + 1
ωr
where ωg is the generator rotation speed and ng is the gear ratio. By representing the geared transmission by using a two lumped mass model, it is routine to derive that
where Prmax denotes the maximum extractable mechanical power. As indicated in Eq. (5) the maximum available kinetic marine current power is only directly related with the optimal turbine rotation speed ωropt since the optimal tip speed ratio λopt and the value Cpmax are all constants. Therefore, the turbine rotation speed can be readily controlled to track the optimal speed value ωropt for the maximum marine power extraction while the optimal value ωropt can be used as the reference input for the controller. However, the optimal value ωropt may be highly polluted by the swell effects in the marine currents and will cause unexpected fluctuations and inaccuracy if it is directly used in the control design. Therefore, a swell filter is involved to generate the smooth values of the optimal rotation speed for control reference input in the case of swell effects. Therefore,
ωroptf =
ωg
(6)
where τf denotes the time constant of the swell filter, s denotes the Laplace operator, ωroptf is the filtered smooth turbine rotation speed for control reference input. The time constant τ in Eq. (6) can be empirically determined as a trade-off between the control response speed and the power generation quality, and the practical value can be selected as half of the typical
iq̇ = −
(Ls id + φf ) uq Rs iq + np ωg − Ls Ls Ls
(9)
where φf denotes the permanent magnet flux linkage, Rs and Ls denote respectively the resistance and inductance of the stator winding, uq, iq 3
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z q (t ) = iqopt − iq
are respectively the q axis voltage and current, np is the number of pole pairs, id denotes the d axis current. The generator electromagnetic torque Te can be described as follows
Te =
3 np φf iq 2
⇒ z ṙ (t ) = ω̇ ropt − ω̇ r = ω̇ ropt − fr + gr z q (t ) − gr iqopt − dr
Based on the Eq. (16), the control input or the optimal q-axis current iqopt is defined as
(10)
iqopt = iqopt1 + iqopt 2 + iqopt 3
By combining Eqs. (8) and (10), one obtains
⎧iqopt1 =
3 = Tr − br ·ωr − 2 np ng φf iq + δ (Jr + 3np ng φf Tr br δ ω̇ r = − iq + ·ωr − Jr + Jg ng2 Jr + Jg ng2 2(Jr + Jg ng2 ) Jr + Jg ng2
Jg ng2 ) ω̇ r
⇒ ω̇ r = fr + gr iq + dr
T
b
(Ls id + φf ) uq Rs iq + np ωg − ⇒ iq̇ = fq + gq uq + dq Ls Ls Ls
φf np ωg Ls
(13)
1 1 + τs
(20)
where the parameter τ is the time constant that can be chosen as a small enough positive number such that the spectrum of dr can be covered by the bandwidth of Gf (s). Based on the Eq. (20), it is natural to derive that
;
(14)
1
1 ; τs
⎧ 1 − Gf (s) = 1 + ⎪ ⎪ sGf (s) 1 = τ; ⎨ 1 − Gf (s) ⎪ Gf (s) 1 ⎪ 1 − Gf (s) = τs ; ⎩
By observing Eqs. (11) and (13), it is obvious that the marine current turbine model can be formulated as two strict-feedback affine dynamic equations and the controller can be readily designed based on the Eqs. (11) and (13) accordingly. Therefore, the main control objectives are to make the marine current extract the maximum kinetic power by tracking the smooth turbine rotation speed ωroptf defined in Eq. (6), and concurrently compensate for the uncertainty and disturbances represented by dr and dq.
(21)
By defining
⎧ dr̃ = dr̂ − dr ; ⎨∥dr̃∥ ⩽ Dr . ⎩
3. The controller design
(22)
where dr̃ denotes the estimation error of dr, and Dr denotes the unknown positive upper bound of dr̃ .
Based on the Eqs. (11) and (13), the adaptive back-stepping controller for the maximum marine power extraction can be readily designed while the uncertainty and disturbance can be well estimated and compensated by using an online update law. The controller mainly consists of a marine turbine speed control loop and a q-axis current control loop. The design principle of the controller is detailed below.
Therefore, by estimating the constant bound Dr, the q-axis current control component iqopt3 can be designed as
iqopt 3 =
Dr̂ sign[z r (t )] gr
(23)
where Dr̂ denotes the estimate of Dr. By combining Eqs. (17), (18), (21) and (23), one obtains the optimal q-axis current control input as
3.1. The marine turbine speed control loop The marine turbine speed control loop aims to track the filtered optimal reference speed ωroptf in Eq. (6) such that the maximum marine power generation can be maintained. Therefore, by defining the tracking error zr(t) and considering Eq. (11), it is routine to derive that
iqopt =
1 z (t ) D̂ (ω̇ ropt + kr z r (t ) − fr ) + r + r sign gr τgr gr t
∫ (kr z r (t ) + Dr̂ sign[z r (t )])dt
z r (t ) = ωroptf − ωr ⇒ z ṙ (t ) = ω̇ roptf − ω̇ r = ω̇ roptf − fr − gr iq − dr
(19)
where ℓ−1 (Gf (s )) is the inverse Laplace transformation of the filter Gf(s), “*” is the convolution operator. The filter Gf(s) can be represented as
Gf (s ) = R
(18)
dr̂ = (ω̇ r − fr − gr iq ) ∗ ℓ−1 (Gf (s ))
(12)
where
⎧ fq = − Ls iq + s ⎪ 1 g = − ; Ls ⎨ q ⎪d = i n ω . d p g ⎩ q
+ kr z r (t ) − fr )
where kr is control gain in the q-axis current control loop, dr̂ is the estimate of dr. The control input in Eq. (18) can be readily achievable since the incoming marine flow speed v(t) is highly predictable and measurable by using a Doppler Current Meter for calculating the term fr in Eq. (18). The known dynamic term dr is estimated to be dr̂ in Eq. (18) by using low pass filter Gf (s) [14]. Therefore,
(11)
By arranging Eq. (9), one obtains
iq̇ = −
1 (ω̇ roptf gr
dr̂ ⎨i qopt 2 = − g r ⎩
where
⎧ fr = J + Jr n 2 − J + Jr n 2 ·ωr ; r g g r g g ⎪ 3np ng φf ⎪ ; gr = − 2(Jr + Jg ng2 ) ⎨ ⎪ δ ⎪ dr = Jr + Jg ng2 . ⎩
(17)
where iqopt1, iqopt 2, iqopt 3 denote the q-axis current control components of which iqopt1, iqopt 2 are
2.3. The control problem formulation
⇒
(16)
[z r (t )] +
(15)
By defining the optimal q-axis current iqopt that corresponds to the optimal reference speed ωroptf and the tracking error zq(t), the Eq. (15) can be reformulated as
0
τgr
(24)
Then, by substituting Eq. (24) into (16), one obtains
z ṙ (t ) = −kr z r (t ) + dr̃ − Dr̂ sign[z r (t )] + gr z q (t ) 4
(25)
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Fig. 2. The implementation of the proposed control.
Fig. 3. The marine current speed.
3.2. The q-axis current control loop
represented as
The q-axis current control loop can be designed by using the derived current control signal iqopt as the reference input by following the backstepping approach. By using Eq. (13), the q-axis current tracking error dynamics can be defined as follows
̇ + kq z q (t ) − fq ); ⎧uq1 = g (iqopt q ⎪ ⎨u = − dq̂ . ⎪ q2 gq ⎩
̇ − iq̇ = iqopt ̇ − fq − gq uq − dq z q̇ (t ) = iqopt
1
where kq is positive constant control gain, dq̂ is the estimate of the term d q. By using the Eq. (13), the estimate dq̂ can be represented as
(26)
Based on the Eq. (26), the voltage control input for the q-axis current control loop can be defined as
uq = uq1 + uq2 + uq3
(28)
dq̂ = (iq̇ − fq − gq uq ) ∗ ℓ−1 (Gf (s ))
(27)
Then, by defining
where uq1, uq2, uq3 are the voltage control components which can be 5
(29)
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⎧ dq̃ = dq̂ − dq; ⎨∥dq̃ ∥ ⩽ Dq . ⎩
⎧ Dṙ ̂ = γr [|z r (t )| − ηr (Dr̂ − Dr 0 )]; ⎨ D ̇ ̂ = γ [|z (t )| − η (D ̂ − D )]. q q q0 q q ⎩ q
(30)
It is natural to derive the voltage control component uq3 as follows
uq3 =
Dq̂ gq
sign[z q (t )]
(38)
where Dr 0, Dq 0 are constant initial values for estimating the parameters Dr̂ and Dq̂ , respectively, γr , ηr , γq , and ηq are constant coefficients. According to the Young’s inequality [15], there exist
(31) 2 ηr Dr̃ (Dr̂ − Dr 0) = ηr Dr̃ (Dr − Dr̃ − Dr 0) ⩽ −ηr Dr̃ +
where dq̃ denotes the estimation error, Dq denotes the constant upper bound of the estimation error, and Dq̂ denotes the estimate of Dq. Then, by combining Eqs. (28) and (31) and using Eq. (21), the q-axis voltage control input can be designed as
2
= −
ηr Dr̃ 2
+
2 ηr Dr̃ 2
+
ηr (Dr − Dr 0 )2 2
ηr (Dr − Dr 0)2 2
(39) In the similar way, the following inequality can be derived
uq =
2
z q (t ) Dq̂ 1 ̇ (iqopt + kq z q (t ) − fq ) + sign + τgq gq gq
ηq Dq̃ (Dq̂ − Dq0) ⩽ −
2
ηq (Dq − Dq0 )2
+
(40)
2
By substituting Eqs. (38)–(40) into Eq. (37), it is natural to derive that
t
∫ (kq z q (t ) + Dq̂ sign[z q (t )])dt [z q (t )] +
ηq Dq̃
0
τgq
(
V̇ (t ) ⩽ − kr −
(32)
2
By substituting Eq. (32) into Eq. (26), the q-axis current tracking error dynamics can be represented as
z q̇ (t ) = −kq z q (t ) + dq̃ − Dq̂ sign[z q (t )]
−
ηq Dq̃ 2
|gr | 2
2
−
ηr Dr̃ 2
+
)z
2 r (t )
(
− kq −
ηq (Dq − Dq0 )2 2
+
|gr | 2
)z
2 q (t )
ηr (Dr − Dr 0 )2 2
(41)
By defining two positive constants α and β as
(33)
⎧ α = min{(2kr − |gr |), (2kq − |gq|), γr ηr , γq ηq}; ⎨β = ⎩
3.3. The stability analysis
(34)
V̇ (t ) = z r (t ) z ṙ (t ) + z q (t ) z q̇ (t ) −
(36)
By considering Eqs. (25) and (33), it is routine to derive that
V̇ (t ) = −kr z r2 (t ) − kq z q2 (t ) + gr z r (t ) z q (t ) + dr̃ z r (t ) − Dr̂ |z r (t )| + dq̃ z q (t ) − Dq̂ |z q (t )| 1 − γ Dr̃ Dṙ ̂ − r
⩽ − kr z r2 (t ) − kq z q2 (t ) +
|gr | 2
z r2 (t ) +
1 D ̃ Ḋ ̂ γq q q |gr | 2
z q2 (t ) + Dr |z r (t )| − Dr̂ |z r (t )|
+ Dq |z q (t )| − Dq̂ |z q (t )| 1 − γ Dr̃ Dṙ ̂ − r
|gr | 2
)z
2 r (t )
(
− kq −
|gr | 2
1 − γ Dr̃ Dṙ ̂ − r
1 D ̃ Ḋ ̂ γq q q
)z
2 q (t )
(43)
β [1 − exp(−αt )] α
(44)
By observing Eqs. (35) and (44), it is clear that all signals in the closed-loop system are uniformly ultimate bounded and can converge to a small region when time t→∞. The tracking errors zr(t) and zq (t) and the parameter estimation errors Dr̃ (∞) , Dq̃ (∞) are all bounded. When time t→∞, the error signals are bounded as |z r (∞)| < 2β / α , |z q (∞)| < 2β / α , |Dr̃ (∞)| < 2β / α and |Dq̃ (∞)| < 2β / α . The boundedness of all the signals can be readily adjusted within an arbitrarily small compact set by properly changing the control parameters of kr, kq, τ, γr, ηr, γq, and ηq. By properly increasing the parameters kr and kq, the error bounds for |z r (∞)| and |z q (∞)| can be decreased to arbitrarily small. Therefore, the parameters kr and kq determine the tracking error convergence speed and can be set small at the beginning and then increased online based on a trial-and-error method to stabilize the tracking error dynamics system. The larger values of kr and kq will lead to faster error convergence, but result in aggressive control actions. Therefore, selection of the parameters kr and kq also needs to consider the upper bounds of the control actions such that the q-axis current and voltage do not exceed the maximum available limits. In addition, the parameter estimation errors |Dr̃ (∞)| and |Dq̃ (∞)| can be determined by the parameters γr, ηr, γq, and ηq, which can be set initially large and then reduced in the subsequent tuning. The filter Gf(s) can also be appropriately designed to reduce the estimation and tracking errors by using small time constant τ. Then, the proper choice of these two parameters needs to make a tradeoff between the reduced tracking/estimation errors and the fast dynamic response of the actual control system.
(35)
1 ̃ ̇̂ 1 Dr Dr − Dq̃ Dq̇ ̂ γr γq
(42)
By solving Eq. (43), one obtains
The time derivative of Eq. (35) can be obtained as follows. Therefore,
(
ηr (Dr − Dr 0 )2 2
0 ⩽ V (t ) ⩽ V (0) exp( −αt ) +
1 2 1 1 2̃ 1 2̃ z r (t ) + z q2 (t ) + Dr + Dq 2 2 2γr 2γq
= − kr −
+
V̇ (t ) ⩽ −αV (t ) + β
where Dr̃ , Dq̃ are estimation errors for the parameters Dr and Dq, respectively. Consider the following quadratic Lyapunov function candidate V(t)
V (t ) =
2
The Eq. (41) can be expressed as
The stability and control performance of the overall closed-loop system can be readily proved based on the Eqs. (25) and (33) while the parameter update law for Dr̂ and Dq̂ can be derived accordingly. Define the estimation errors of the two parameters Dr̂ and Dq̂ as
⎧ Dr̃ = Dr − Dr̂ ; ⎨ Dq̃ = Dq − Dq̂ . ⎩
ηq (Dq − Dq0 )2
+ Dr̃ |z r (t )| + Dq̃ |z q (t )|
1 D ̃ Ḋ ̂ γq q q
4. Validations and discussions
(37) Based on Eq. (37), the adaptive update law for the parameters Dr̂ and Dq̂ can be derived as follows
In order to verify the effectiveness of the proposed adaptive backstepping control for maximizing marine current power generation, an 6
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Fig. 4. The turbine rotation speed.
Fig. 5. The tracking errors of the turbine rotation speed zr(t).
traditional proportional integral controller has been implemented to achieve the d-axis current control loop (not shown in Fig. 2). The generated control voltages from the control loops are then transformed into the αβ-axis voltages through the dq/αβ transformation block which are consequently employed to control the power converter and the PMSG through pulse width modulation and by using necessary feedbacks from the generator. As shown in Fig. 2, the PMSG has been embedded in the marine current turbine which has the maximum mechanical output power of 60 kW at the marine current speed of 2.5 m/s. The turbine blade radius is designed as 2.25 m, the turbine rotation speed is rated at 12 rad/s, and the speed increasing gear ratio is designed as 13. Therefore, the generator speed is rated at 1500 rpm which is common for practical electrical generator applications. The resistance and inductance of the generator are set as 0.27 O and 35 mH, respectively. The active gen3 erator power Pg is calculated as Pg = 2 (uq iq + ud id ) , which can be obtained by using the measurements of the d and q axis currents and
experimental test bench has been set up and the designed controller has been implemented. As shown in Fig. 2, the test bench was designed for the development and test of various control strategies and for emulating marine energy patterns injected into a DC bus or grid. All power interfaces, measurements together with low-level control and data logging software are provided in this test set up, which enables researchers to prototype algorithms for PMSG motor drive control. The proposed backstepping control has been implemented in an outer and inner control loops as shown in Fig. 2. The outer control loop has been implemented by using the marine turbine speed controller designed in Section 3.1 while the inner control loop has been designed based on the q-axis current controller in Section 3.2. The additional disturbance estimations for dr̂ and dq̂ , and the parameter update law in Eq. (38) have also been implemented accordingly within the two control loops. Since the focus is on the maximum marine current power generation that is directly related with the q-axis current control, the desired d-axis current has been set as zero for minimizing the generator power loss and a 7
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Fig. 6. The estimation of the term dr.
Fig. 7. The updates of Dr and Dq.
been designed by following the standard backstepping design procedure [17] and hence only the terms iqopt1 and uq1 have been used in the conventional backstepping controller without uncertainty/disturbance estimation and parameter updates. As illustrated in Fig. 3, the marine current speed varies between 0 m/s and 2.5 m/s with clear turbulence. The marine current speed data has been obtained from the real world applications of the marine current turbine in the East China sea. The real world data of the marine current speed represents the below rated operational condition of the marine current turbine. The filtered data of the marine current speed by using the swell filter in Eq. (6) exhibit clear attenuation of the turbulence, which indicates the effectiveness of the designed swell filter in eliminating the swell effects. Therefore, the filtered marine current speed is smooth enough and can be readily used to evaluate the control performance of the proposed control for the maximum marine energy generation. As shown in Figs. 4 and 5, by using the proposed control, the
voltages [16]. All the controller parameters have been tuned through a heuristic tuning approach. The controller gains kr and kq, and initial values Dr0, Dq0 are adjusted from small values to large values until obvious oscillations and overshoot can be observed. The values of ηr and ηq can be set relatively large such that the control loops are in sensitive to noise and high-frequency oscillations in the control system. All the control parameters need to be set such that a tradeoff between the steady-state/ stability performance and transient response including overshoot, tracking error and settling time is accomplished in the implementation. The control parameters are tuned as kr = 0.25, kq = 1.8, Dr0 = 50, Dq0 = 12, γr = 2, ηr = 100, γq = 2 × 10−6, and ηq = 1 × 105, and the time constants τ and τf are all set as 0.02 in the filter implementations. In order to further evaluate the higher control effectiveness of the proposed control in marine energy generation, the proposed control has been compared with a conventional backstepping controller for the sake of fair comparison. The conventional backstepping controller has
8
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Fig. 8. The q-axis current.
Fig. 9. The tracking errors of the q axis current zq(t).
verify the effectiveness of the update law in Eq. (38). The fast transient of the update process provides timely and accurate control signals for implementing the two control loops. As illustrated in Figs. 8 and 9, the q axis current is effectively maintained around the optimal value by using the proposed control whereas the q axis current fluctuates dramatically around the optimal values when the conventional control is used. The q axis current tracking error zq(t) is also regulated to be around zero by using the proposed control while the tracking error varies significantly when the conventional control is applied. Therefore, the proposed control can be employed to achieve accurate current regulation for achieving the maximum marine energy extraction than the conventional control. Fig. 10 shows the estimation results of the uncertainty and disturbance term dq in the q axis current control loop by using the estimator in Eq. (29). As shown in this figure, the estimates closely track the real values of the term dq and the residual tracking error can be kept within a relatively low value regardless of the obvious oscillations in the term dq. Therefore, the uncertainty and disturbance term dq can be
tracking error of the turbine rotation speed zr(t) is significantly reduced to remain around 0 whereas the tracking error varies significantly between [−3.8, 2] rad/s when the conventional controller is applied. It is also clear that the turbine rotation speed tracks the optimal speed more accurately by using the proposed control in comparison with the conventional control. Therefore, the results reveal that the proposed control is more effective in tracking the optimal turbine rotation speed for the maximum marine energy extraction than the conventional control. Fig. 6 shows the estimation performance of the term dr by using the low pass filter based estimator in Eq. (19). It can be found that fairly smooth estimation is achieved. In particular, with the help of the designed estimator in Eq. (19), the lumped uncertainty and disturbance term dr can be readily estimated and then be compensated in the controller design, which exhibits high potential in real world applications. As shown in Fig. 7, the parameters Dr and Dq can update quickly to the designed real values during the control transient. There also exist no overshoot or oscillations in the updates of the two parameters, which 9
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Fig. 10. The estimation of the term dq.
Fig. 11. The generator power.
5. Conclusion
precisely estimated by using the proposed estimator in Eq. (29) which contributes to attenuating the unknown disturbances and uncertainties when the estimator is incorporated in the control loop. As shown in Fig. 11, the generator generates obviously more power (up to 30% more power) when using the proposed control in comparison with the conventional control. The power generation is relatively stable around 20 kW, which indicates that the swell effects in the marine current speed are filtered and attenuated significantly. The results demonstrate that improvements in generating marine current power by using the proposed control. Therefore, the proposed control is readily better in tracking the optimal marine power points and hence in improving the power generation efficiency under the same operation condition than the conventional control.
The adaptive backstepping controller with uncertainty and disturbance estimation has been proposed and designed in the paper for maximizing the marine current power generation of a generic horizontal marine turbine. The generic horizontal marine current turbine has been designed and modelled for the control implementation. The control problem has been formulated and the controller has been designed accordingly to be composed of the marine turbine speed control loop and the q-axis current control loop while the uncertainty and disturbance are estimated and compensated in the control. The swell filter has also been involved in the proposed controller to attenuate any swell effects on the power generation. The proposed control has been implemented and evaluated in an experimental test bench for maximizing marine current power generation and the control performance has been compared with that of a conventional backstepping controller.
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The test results indicate the proposed control is readily better in tracking the optimal marine power points and hence in improving the power generation efficiency under the same operation condition than the conventional control. By using the proposed control, the generator generates obviously more power (up to 30% more power) than that with the conventional control. The lumped uncertainty and disturbance can also be readily estimated and then be compensated in the proposed controller design, which exhibits high potential in real world applications.
[8] Elghali SEB, Benbouzid MEH, Charpentier JF. Modelling and control of a marine current turbine-driven doubly fed induction generator. IET Renew Power Gener 2010;4(1):1–11. [9] Elghali SEB, Benbouzid MEH, Charpentier JF, et al. High-order sliding mode control of a marine current turbine driven permanent magnet synchronous generator. 2009 IEEE international electric machines and drives conference. IEEE; 2009. p. 1541–6. [10] Benelghali S, Benbouzid MEH, Charpentier JF, et al. Experimental validation of a marine current turbine simulator: Application to a permanent magnet synchronous generator-based system second-order sliding mode control. IEEE Trans Ind Electron 2011;58(1):118–26. [11] Jafari SAH, Kosasih B. Flow analysis of shrouded small wind turbine with a simple frustum diffuser with computational fluid dynamics simulations. J Wind Eng Ind Aerodyn 2014;125:102–10. [12] Wang L, Chen SS, Lee WJ, et al. Dynamic stability enhancement and power flow control of a hybrid wind and marine-current farm using SMES. IEEE Trans Energy Convers 2009;24(3):626–39. [13] Fantino R, Solsona J, Busada C. Nonlinear observer-based control for PMSG wind turbine. Energy 2016;113:248–57. [14] Zhang Q, Liu HHT. UDE-based robust command filtered backstepping control for close formation flight. IEEE Trans Ind Electron 2018;65(11):8818–27. [15] Dragomir SS. A note on Young’s inequality. Revista de la Real Academia de Ciencias Exactas. Físicas y Naturales. Serie A. Matemáticas 2017;111(2):349–54. [16] Pican E, Omerdic E, Toal D, et al. Direct interconnection of offshore electricity generators. Energy 2011;36(3):1543–53. [17] Freeman RA, Kokotović PV. Backstepping design of robust controllers for a class of nonlinear systems. Nonlinear Control Systems Design 1992. Pergamon 1993:431–6.
References [1] Borthwick AGL. Marine renewable energy seascape. Engineering 2016;2(1):69–78. [2] Hussain A, Arif SM, Aslam M. Emerging renewable and sustainable energy technologies: State of the art. Renew Sustain Energy Rev 2017;71:12–28. [3] https://www.verdantpower.com/. [4] http://www.emec.org.uk/about-us/our-tidal-clients/open-hydro/. [5] http://www.tidalenergy.eu/marinecurrenturbines_seaflow.html. [6] Elghali SEB, Benbouzid MEH, Ahmed-Ali T, et al. High-order sliding mode control of DFIG-based marine current turbine. IEEE 2008;2008:1228–33. [7] Elghali SEB, Benbouzid MEH, Ahmed-Ali T, et al. High-order sliding mode control of a marine current turbine driven doubly-fed induction generator. IEEE J Oceanic Eng 2010;35(2):402–11.
11
Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Adaptive backstepping control for maximizing marine current power generation based on uncertainty and disturbance estimation
T
Xiuxing Yin The School of Engineering, University of Warwick, Coventry CV4 7AL, UK
A R T I C LE I N FO
A B S T R A C T
Keywords: Marine energy Marine current turbine Maximum power generation Adaptive backstepping control Uncertainty and disturbance estimation
The renewable energy from marine currents is highly promising as a clean, reliable and predictable energy source for the next-generation electricity generation. Therefore, the paper focuses on the design and implementation of the high-efficiency control for maximizing the marine current power generation and hence an adaptive backstepping controller with uncertainty and disturbance estimation is proposed for a generic horizontal marine turbine. The turbine design principle and dynamics modelling are presented and then the control problem is formulated. Consequently, the controller is designed to be composed of a marine turbine speed control loop and a q-axis current control loop while the uncertainty and disturbance is estimated and compensated. A swell filter is also incorporated into the control loop to smooth generator power fluctuations. The stability of the proposed control is verified via the Lyapunov synthesis and all the tracking errors are guaranteed to converge to zero asymptotically. The proposed control is verified in an experimental test bench and the test results indicate that the generator produces obviously more power (up to 30% more power) when using the proposed control in comparison with a conventional backstepping control.
1. Introduction As a vast and important energy source, the marine renewable energy is playing a pivotal role in addressing the issues of rising costs and growing security concerns of the worldwide energy consumption [1]. The marine energy can also provide a predictable, stable and reliable solution in meeting the challenging targets of a massive energy increase from renewables in comparison with wind and wave energies. At present, the marine energy technologies are advancing fast, but most of them are still in the demonstration phase, or are just reaching the commercial viability [2]. Until recently, the most common marine energy technologies involve erecting tidal barrages and installing underwater marine turbines. Due to the use of physical barriers for marine energy extraction, the tidal barrages suffer from the disadvantages of environmental populations and adverse effects on marine ecosystems. Unlike tidal barrages, the marine turbines are designed without using tidal barrages and can be deployed in estuaries or offshore such that they can be rotated by strong marine currents to generate electricity. Even though there exist various configurations of marine turbines, the most promising and common configuration is a horizontal axis marine turbine that resembles the commonly deployed wind turbines for capturing hydrokinetic energy. There are several commercial-sized marine turbines in operation, which mainly include the Verdant Power turbine
[3], the open center turbine Open Hydro [4], the marine current turbines Seaflow and SeaGen [5]. Although marine turbines share some similarities with wind turbines, fundamental differences still exist between them. The marine turbines are generally operated underwater in harsh marine environment that poses unique technical challenges including the potential for cavitation, the free surface effects, the different stall characteristics, and the different range of Reynolds numbers [2]. When considering these fundamental differences, it is necessary to design and implement specific and robust control approaches to maintain the stable operation and high-quality power production of marine turbines under different ocean conditions. One of particular interest is the maximum marine power extraction control that can be used to generate the optimal electrical power from marine currents. By considering the above fundamental differences and the fact that the seawater density is about 832 times higher than air, the maximum marine power extraction control needs to be particularly designed by compensating for relatively large marine loads, unexpected uncertainty and disturbance due to these differences from wind turbines. The design of various controllers for extracting the maximum marine power has recently been a hot topic and some control designs have been reported in the literature. The high-order sliding mode control was proposed in [6] to deal with the maximum marine energy generation. The control is highly dependent on the marine
E-mail address: lixingfi[email protected]. https://doi.org/10.1016/j.ijepes.2019.05.066 Received 8 September 2018; Received in revised form 12 April 2019; Accepted 26 May 2019 0142-0615/ © 2019 Published by Elsevier Ltd.
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support structures on the seabed. The turbine consists of a rotor, a geared transmission, and an electrical generator and is placed in the path of incoming marine currents to extract the kinetic energy from the moving seawater by turning the rotor blades. The extracted mechanical energy is then transmitted to the generator through the geared transmission for electricity production. The generated marine electricity is then transmitted onshore for grid integration via submarine cables. The generator is electrically controlled by power converters in order to achieve the maximum marine power extraction or for the grid connection purpose at different current speed. The floating marine current turbine has prominent advantages of easy installation, easy accessibility, low environmental impact and cost effective maintenance. The marine current turbine has the self-orientation ability, particularly useful in reversing marine currents and can be easily lifted out of the water for maintenance without costly steel supports or alterations to the ocean floor. The turbine can also be installable almost everywhere in sites with 0.5 m/s to 5 m/s current speed with reduced cost compared to fixed structures, enabling the use of gravity anchors. In order to further enhance the marine energy conversion efficiency, a diffuser can be placed around the turbine to increase the mass flowing though the turbine swept area, thereby increasing the marine energy capture [11].
resource and the turbine models. The control was then analyzed regarding the swell effects and validated based on the marine current data from the Raz de Sein (Brittany, France). The high-order sliding mode control was proposed in [7,8] to implement the speed control of a variable speed doubly-fed induction generator (DFIG)-based marine current turbine (MCT). The sensitivity of the control design was analyzed regarding the swell effects. Simulation results were also presented and fully analyzed. The high-order sliding mode control was proposed in [9] for the marine energy maximization of the permanent magnet synchronous generator (PMSG)-based marine current turbine. The controller was incorporated into a global simulation tool with marine resource and turbine models. The control validation was conducted based on the tidal data from the Raz de Sein (Brittany, France) and experiments on a 7.5-kW real-time simulator. The second-order sliding mode control was proposed in [10] to increase the generated power and the turbine efficiency by adequately handling the marine resource characteristics including swell effects and turbulence. The control has also taken the PMSG parameter variations into account. The controller was validated by using marine current data from the Raz de Sein (Brittany, France) and experiments on a 7.5-kW real-time simulator. However, the above-mentioned sliding mode control methods suffer from the inherent chattering problems and switching oscillations that will result in frequent marine power and torque fluctuations. This paper presents a new adaptive backstepping controller for a generic horizontal marine turbine to explore the maximum marine power generation control. The design principle and dynamics modelling of the horizontal axis marine turbine are presented. The control problem is then formulated and the controller is proposed to be composed of a marine current speed tracking loop and a q-axis current tracking loop. In each control loop, the uncertainty and disturbance term is estimated, and the bounds of the estimation errors are estimated and updated online. A swell filter is also incorporated in the control loops to attenuate swell effects. The stability of the overall control system is verified via the Lyapunov theory and all the tracking errors can be guaranteed to converge to zero. The parameter tuning procedure is also detailed to choose the appropriate control parameters for the controller and parameter update law. An experimental test bench is used to test the proposed control in the efficiency for maximizing the marine power generation and a conventional backstepping controller is designed for comparison. The remainder of the paper is organized as follows: in Section 2, the design principle and dynamics modelling of a generic horizontal marine turbine are presented, and the control problem is formulated. In Section 3, the controller design and the stability analysis are detailed. In Section 4, the validation and discussions of the proposed control in comparison with a conventional controller are presented. The Section 5 concludes the paper.
2.2. The dynamics of the marine turbine The marine current turbine can only convert a fraction of the kinetic energy of the incoming marine currents into electrical energy, and the captured kinetic energy is directly proportional to the marine fluid density and the cube of the marine current speed. Providing that the marine current velocity is uniform across the cross-sectional area at any instant, the amount of the available kinetic energy can be expressed by
Pr =
πρR2 3 v (t ) Cp (λ, β ) 2
(1)
The turbine torque acting on the rotor can be calculated accordingly as
Tr =
πρR3v 2 (t ) Pr = Cp (λ, β ) 2λ ωr
(2)
where Pr and Tr are respectively the kinetic marine power and the turbine torque, v(t) is the incoming marine current speed, ωr is the turbine rotation speed, R is the turbine radius, ρ is the sea water density (1025 kg/m3), β is the turbine pitch angle, λ is the tip speed ratio, Cp denotes the power coefficient which indicates the power extraction performance of the turbine. As shown in Eq. (1), the mechanical power generated by the marine current turbine is dependent on the cube of the marine velocity v(t), the swept cross-sectional area πR2 perpendicular to the marine flow direction, and the density of the seawater density ρ. Therefore, even small changes in the marine velocity can cause substantial changes in the amount of available kinetic power and hence, smaller and faster rotating marine turbines can be used in comparison with larger wind turbines for producing the same amount of electricity. The power coefficient is a nonlinear function of the tip speed ratio and the blade pitch angle and can be defined as [12]
2. The marine current turbine and dynamics modelling In this section, a generic horizontal marine turbine is chosen as the control object for investigation and control implementation. The design principle and dynamics modelling of the horizontal axis marine turbine are presented. The control problem is then formulated for facilitating the controller design accordingly. 2.1. The design principle of the marine current turbine
(
)
( )
c c ⎧ c1 α2 − c3 β − c4 β c5 − c6 ⎤ exp − α7 ⎪Cp = ⎡ ⎣ ⎦ ⎨ 1 = 1 − c9 ⎪α λ + c8 β β3 + 1 ⎩
The marine turbine is designed as a floating type horizontal axis marine current turbine whose axis of rotation is horizontal to the incident marine current direction. The three bladed turbine is designed to reach a high efficiency within a relatively large operation conditions range by using a properly designed airfoil section shape. As shown in Fig. 1, the marine current turbine is connected to the buoyant barge that is tethered to the seabed with a mooring rope, and operates essentially as an upside down horizontal turbine without requiring extra
(3)
where c1 ∼ c9 are constant coefficients, c1 = 0.18, c2 = 85, c3 = 0.38, c4 = 0.25, c5 = 0.5, c6 = 10.2, c7 = 6.2, c8 = 0.025, c9 = −0.043. The tip speed ratio (TSR) can be expressed as
λ= 2
λ opt v (t ) ωropt R ωr R ⇒ λ opt = ⇒ ωropt = R v (t ) v (t )
(4)
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Fig. 1. The floating marine current turbine.
where ωropt and λopt are respectively the optimal rotation speed and the optimal tip speed ratio under which the marine current turbine can extract the maximum available power, or the power coefficient can attain the maximum value Cpmax. In the maximum marine power extraction mode, the maximum kinetic marine power can be rearranged based on Eq. (1) as follows
Prmax =
Cpmax 3 1 πρR5· 3 ·ωropt 2 λopt
swell period. The geared transmission is essentially a speed increasing mechanism and the gear ratio can be defined as
ng =
(5)
(Jr + Jg ng2 ) ω̇ r = Tr − br ·ωr − ng Te + δ
(7)
(8)
where Jr denotes the rotor inertia, Jg denotes the high-speed side inertia, br denotes the torsion damping coefficient, Te denotes the reaction electromagnetic torque of the generator, the term δ denotes the lumped uncertainty and disturbance. In practice, the lumped uncertainty and disturbance term is used to model un-modelled dynamics and other uncertainty that are caused by gear meshing, gearbox backlash, the cavitation or scour, the biofouling and obstructions due to marine growth build-up. All the effects included in the uncertain term can be readily estimated and compensated by the proposed controller design. The employed electrical generator is a non-salient pole PMSG whose dynamics can be electrically described in the d/q axis rotation reference frame [13]. Since the maximum power extraction control in the paper is only directly related to the q-axis current dynamics, the focus is on the q axis current dynamics which can be described as
ωropt τf s + 1
ωr
where ωg is the generator rotation speed and ng is the gear ratio. By representing the geared transmission by using a two lumped mass model, it is routine to derive that
where Prmax denotes the maximum extractable mechanical power. As indicated in Eq. (5) the maximum available kinetic marine current power is only directly related with the optimal turbine rotation speed ωropt since the optimal tip speed ratio λopt and the value Cpmax are all constants. Therefore, the turbine rotation speed can be readily controlled to track the optimal speed value ωropt for the maximum marine power extraction while the optimal value ωropt can be used as the reference input for the controller. However, the optimal value ωropt may be highly polluted by the swell effects in the marine currents and will cause unexpected fluctuations and inaccuracy if it is directly used in the control design. Therefore, a swell filter is involved to generate the smooth values of the optimal rotation speed for control reference input in the case of swell effects. Therefore,
ωroptf =
ωg
(6)
where τf denotes the time constant of the swell filter, s denotes the Laplace operator, ωroptf is the filtered smooth turbine rotation speed for control reference input. The time constant τ in Eq. (6) can be empirically determined as a trade-off between the control response speed and the power generation quality, and the practical value can be selected as half of the typical
iq̇ = −
(Ls id + φf ) uq Rs iq + np ωg − Ls Ls Ls
(9)
where φf denotes the permanent magnet flux linkage, Rs and Ls denote respectively the resistance and inductance of the stator winding, uq, iq 3
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z q (t ) = iqopt − iq
are respectively the q axis voltage and current, np is the number of pole pairs, id denotes the d axis current. The generator electromagnetic torque Te can be described as follows
Te =
3 np φf iq 2
⇒ z ṙ (t ) = ω̇ ropt − ω̇ r = ω̇ ropt − fr + gr z q (t ) − gr iqopt − dr
Based on the Eq. (16), the control input or the optimal q-axis current iqopt is defined as
(10)
iqopt = iqopt1 + iqopt 2 + iqopt 3
By combining Eqs. (8) and (10), one obtains
⎧iqopt1 =
3 = Tr − br ·ωr − 2 np ng φf iq + δ (Jr + 3np ng φf Tr br δ ω̇ r = − iq + ·ωr − Jr + Jg ng2 Jr + Jg ng2 2(Jr + Jg ng2 ) Jr + Jg ng2
Jg ng2 ) ω̇ r
⇒ ω̇ r = fr + gr iq + dr
T
b
(Ls id + φf ) uq Rs iq + np ωg − ⇒ iq̇ = fq + gq uq + dq Ls Ls Ls
φf np ωg Ls
(13)
1 1 + τs
(20)
where the parameter τ is the time constant that can be chosen as a small enough positive number such that the spectrum of dr can be covered by the bandwidth of Gf (s). Based on the Eq. (20), it is natural to derive that
;
(14)
1
1 ; τs
⎧ 1 − Gf (s) = 1 + ⎪ ⎪ sGf (s) 1 = τ; ⎨ 1 − Gf (s) ⎪ Gf (s) 1 ⎪ 1 − Gf (s) = τs ; ⎩
By observing Eqs. (11) and (13), it is obvious that the marine current turbine model can be formulated as two strict-feedback affine dynamic equations and the controller can be readily designed based on the Eqs. (11) and (13) accordingly. Therefore, the main control objectives are to make the marine current extract the maximum kinetic power by tracking the smooth turbine rotation speed ωroptf defined in Eq. (6), and concurrently compensate for the uncertainty and disturbances represented by dr and dq.
(21)
By defining
⎧ dr̃ = dr̂ − dr ; ⎨∥dr̃∥ ⩽ Dr . ⎩
3. The controller design
(22)
where dr̃ denotes the estimation error of dr, and Dr denotes the unknown positive upper bound of dr̃ .
Based on the Eqs. (11) and (13), the adaptive back-stepping controller for the maximum marine power extraction can be readily designed while the uncertainty and disturbance can be well estimated and compensated by using an online update law. The controller mainly consists of a marine turbine speed control loop and a q-axis current control loop. The design principle of the controller is detailed below.
Therefore, by estimating the constant bound Dr, the q-axis current control component iqopt3 can be designed as
iqopt 3 =
Dr̂ sign[z r (t )] gr
(23)
where Dr̂ denotes the estimate of Dr. By combining Eqs. (17), (18), (21) and (23), one obtains the optimal q-axis current control input as
3.1. The marine turbine speed control loop The marine turbine speed control loop aims to track the filtered optimal reference speed ωroptf in Eq. (6) such that the maximum marine power generation can be maintained. Therefore, by defining the tracking error zr(t) and considering Eq. (11), it is routine to derive that
iqopt =
1 z (t ) D̂ (ω̇ ropt + kr z r (t ) − fr ) + r + r sign gr τgr gr t
∫ (kr z r (t ) + Dr̂ sign[z r (t )])dt
z r (t ) = ωroptf − ωr ⇒ z ṙ (t ) = ω̇ roptf − ω̇ r = ω̇ roptf − fr − gr iq − dr
(19)
where ℓ−1 (Gf (s )) is the inverse Laplace transformation of the filter Gf(s), “*” is the convolution operator. The filter Gf(s) can be represented as
Gf (s ) = R
(18)
dr̂ = (ω̇ r − fr − gr iq ) ∗ ℓ−1 (Gf (s ))
(12)
where
⎧ fq = − Ls iq + s ⎪ 1 g = − ; Ls ⎨ q ⎪d = i n ω . d p g ⎩ q
+ kr z r (t ) − fr )
where kr is control gain in the q-axis current control loop, dr̂ is the estimate of dr. The control input in Eq. (18) can be readily achievable since the incoming marine flow speed v(t) is highly predictable and measurable by using a Doppler Current Meter for calculating the term fr in Eq. (18). The known dynamic term dr is estimated to be dr̂ in Eq. (18) by using low pass filter Gf (s) [14]. Therefore,
(11)
By arranging Eq. (9), one obtains
iq̇ = −
1 (ω̇ roptf gr
dr̂ ⎨i qopt 2 = − g r ⎩
where
⎧ fr = J + Jr n 2 − J + Jr n 2 ·ωr ; r g g r g g ⎪ 3np ng φf ⎪ ; gr = − 2(Jr + Jg ng2 ) ⎨ ⎪ δ ⎪ dr = Jr + Jg ng2 . ⎩
(17)
where iqopt1, iqopt 2, iqopt 3 denote the q-axis current control components of which iqopt1, iqopt 2 are
2.3. The control problem formulation
⇒
(16)
[z r (t )] +
(15)
By defining the optimal q-axis current iqopt that corresponds to the optimal reference speed ωroptf and the tracking error zq(t), the Eq. (15) can be reformulated as
0
τgr
(24)
Then, by substituting Eq. (24) into (16), one obtains
z ṙ (t ) = −kr z r (t ) + dr̃ − Dr̂ sign[z r (t )] + gr z q (t ) 4
(25)
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Fig. 2. The implementation of the proposed control.
Fig. 3. The marine current speed.
3.2. The q-axis current control loop
represented as
The q-axis current control loop can be designed by using the derived current control signal iqopt as the reference input by following the backstepping approach. By using Eq. (13), the q-axis current tracking error dynamics can be defined as follows
̇ + kq z q (t ) − fq ); ⎧uq1 = g (iqopt q ⎪ ⎨u = − dq̂ . ⎪ q2 gq ⎩
̇ − iq̇ = iqopt ̇ − fq − gq uq − dq z q̇ (t ) = iqopt
1
where kq is positive constant control gain, dq̂ is the estimate of the term d q. By using the Eq. (13), the estimate dq̂ can be represented as
(26)
Based on the Eq. (26), the voltage control input for the q-axis current control loop can be defined as
uq = uq1 + uq2 + uq3
(28)
dq̂ = (iq̇ − fq − gq uq ) ∗ ℓ−1 (Gf (s ))
(27)
Then, by defining
where uq1, uq2, uq3 are the voltage control components which can be 5
(29)
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⎧ dq̃ = dq̂ − dq; ⎨∥dq̃ ∥ ⩽ Dq . ⎩
⎧ Dṙ ̂ = γr [|z r (t )| − ηr (Dr̂ − Dr 0 )]; ⎨ D ̇ ̂ = γ [|z (t )| − η (D ̂ − D )]. q q q0 q q ⎩ q
(30)
It is natural to derive the voltage control component uq3 as follows
uq3 =
Dq̂ gq
sign[z q (t )]
(38)
where Dr 0, Dq 0 are constant initial values for estimating the parameters Dr̂ and Dq̂ , respectively, γr , ηr , γq , and ηq are constant coefficients. According to the Young’s inequality [15], there exist
(31) 2 ηr Dr̃ (Dr̂ − Dr 0) = ηr Dr̃ (Dr − Dr̃ − Dr 0) ⩽ −ηr Dr̃ +
where dq̃ denotes the estimation error, Dq denotes the constant upper bound of the estimation error, and Dq̂ denotes the estimate of Dq. Then, by combining Eqs. (28) and (31) and using Eq. (21), the q-axis voltage control input can be designed as
2
= −
ηr Dr̃ 2
+
2 ηr Dr̃ 2
+
ηr (Dr − Dr 0 )2 2
ηr (Dr − Dr 0)2 2
(39) In the similar way, the following inequality can be derived
uq =
2
z q (t ) Dq̂ 1 ̇ (iqopt + kq z q (t ) − fq ) + sign + τgq gq gq
ηq Dq̃ (Dq̂ − Dq0) ⩽ −
2
ηq (Dq − Dq0 )2
+
(40)
2
By substituting Eqs. (38)–(40) into Eq. (37), it is natural to derive that
t
∫ (kq z q (t ) + Dq̂ sign[z q (t )])dt [z q (t )] +
ηq Dq̃
0
τgq
(
V̇ (t ) ⩽ − kr −
(32)
2
By substituting Eq. (32) into Eq. (26), the q-axis current tracking error dynamics can be represented as
z q̇ (t ) = −kq z q (t ) + dq̃ − Dq̂ sign[z q (t )]
−
ηq Dq̃ 2
|gr | 2
2
−
ηr Dr̃ 2
+
)z
2 r (t )
(
− kq −
ηq (Dq − Dq0 )2 2
+
|gr | 2
)z
2 q (t )
ηr (Dr − Dr 0 )2 2
(41)
By defining two positive constants α and β as
(33)
⎧ α = min{(2kr − |gr |), (2kq − |gq|), γr ηr , γq ηq}; ⎨β = ⎩
3.3. The stability analysis
(34)
V̇ (t ) = z r (t ) z ṙ (t ) + z q (t ) z q̇ (t ) −
(36)
By considering Eqs. (25) and (33), it is routine to derive that
V̇ (t ) = −kr z r2 (t ) − kq z q2 (t ) + gr z r (t ) z q (t ) + dr̃ z r (t ) − Dr̂ |z r (t )| + dq̃ z q (t ) − Dq̂ |z q (t )| 1 − γ Dr̃ Dṙ ̂ − r
⩽ − kr z r2 (t ) − kq z q2 (t ) +
|gr | 2
z r2 (t ) +
1 D ̃ Ḋ ̂ γq q q |gr | 2
z q2 (t ) + Dr |z r (t )| − Dr̂ |z r (t )|
+ Dq |z q (t )| − Dq̂ |z q (t )| 1 − γ Dr̃ Dṙ ̂ − r
|gr | 2
)z
2 r (t )
(
− kq −
|gr | 2
1 − γ Dr̃ Dṙ ̂ − r
1 D ̃ Ḋ ̂ γq q q
)z
2 q (t )
(43)
β [1 − exp(−αt )] α
(44)
By observing Eqs. (35) and (44), it is clear that all signals in the closed-loop system are uniformly ultimate bounded and can converge to a small region when time t→∞. The tracking errors zr(t) and zq (t) and the parameter estimation errors Dr̃ (∞) , Dq̃ (∞) are all bounded. When time t→∞, the error signals are bounded as |z r (∞)| < 2β / α , |z q (∞)| < 2β / α , |Dr̃ (∞)| < 2β / α and |Dq̃ (∞)| < 2β / α . The boundedness of all the signals can be readily adjusted within an arbitrarily small compact set by properly changing the control parameters of kr, kq, τ, γr, ηr, γq, and ηq. By properly increasing the parameters kr and kq, the error bounds for |z r (∞)| and |z q (∞)| can be decreased to arbitrarily small. Therefore, the parameters kr and kq determine the tracking error convergence speed and can be set small at the beginning and then increased online based on a trial-and-error method to stabilize the tracking error dynamics system. The larger values of kr and kq will lead to faster error convergence, but result in aggressive control actions. Therefore, selection of the parameters kr and kq also needs to consider the upper bounds of the control actions such that the q-axis current and voltage do not exceed the maximum available limits. In addition, the parameter estimation errors |Dr̃ (∞)| and |Dq̃ (∞)| can be determined by the parameters γr, ηr, γq, and ηq, which can be set initially large and then reduced in the subsequent tuning. The filter Gf(s) can also be appropriately designed to reduce the estimation and tracking errors by using small time constant τ. Then, the proper choice of these two parameters needs to make a tradeoff between the reduced tracking/estimation errors and the fast dynamic response of the actual control system.
(35)
1 ̃ ̇̂ 1 Dr Dr − Dq̃ Dq̇ ̂ γr γq
(42)
By solving Eq. (43), one obtains
The time derivative of Eq. (35) can be obtained as follows. Therefore,
(
ηr (Dr − Dr 0 )2 2
0 ⩽ V (t ) ⩽ V (0) exp( −αt ) +
1 2 1 1 2̃ 1 2̃ z r (t ) + z q2 (t ) + Dr + Dq 2 2 2γr 2γq
= − kr −
+
V̇ (t ) ⩽ −αV (t ) + β
where Dr̃ , Dq̃ are estimation errors for the parameters Dr and Dq, respectively. Consider the following quadratic Lyapunov function candidate V(t)
V (t ) =
2
The Eq. (41) can be expressed as
The stability and control performance of the overall closed-loop system can be readily proved based on the Eqs. (25) and (33) while the parameter update law for Dr̂ and Dq̂ can be derived accordingly. Define the estimation errors of the two parameters Dr̂ and Dq̂ as
⎧ Dr̃ = Dr − Dr̂ ; ⎨ Dq̃ = Dq − Dq̂ . ⎩
ηq (Dq − Dq0 )2
+ Dr̃ |z r (t )| + Dq̃ |z q (t )|
1 D ̃ Ḋ ̂ γq q q
4. Validations and discussions
(37) Based on Eq. (37), the adaptive update law for the parameters Dr̂ and Dq̂ can be derived as follows
In order to verify the effectiveness of the proposed adaptive backstepping control for maximizing marine current power generation, an 6
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Fig. 4. The turbine rotation speed.
Fig. 5. The tracking errors of the turbine rotation speed zr(t).
traditional proportional integral controller has been implemented to achieve the d-axis current control loop (not shown in Fig. 2). The generated control voltages from the control loops are then transformed into the αβ-axis voltages through the dq/αβ transformation block which are consequently employed to control the power converter and the PMSG through pulse width modulation and by using necessary feedbacks from the generator. As shown in Fig. 2, the PMSG has been embedded in the marine current turbine which has the maximum mechanical output power of 60 kW at the marine current speed of 2.5 m/s. The turbine blade radius is designed as 2.25 m, the turbine rotation speed is rated at 12 rad/s, and the speed increasing gear ratio is designed as 13. Therefore, the generator speed is rated at 1500 rpm which is common for practical electrical generator applications. The resistance and inductance of the generator are set as 0.27 O and 35 mH, respectively. The active gen3 erator power Pg is calculated as Pg = 2 (uq iq + ud id ) , which can be obtained by using the measurements of the d and q axis currents and
experimental test bench has been set up and the designed controller has been implemented. As shown in Fig. 2, the test bench was designed for the development and test of various control strategies and for emulating marine energy patterns injected into a DC bus or grid. All power interfaces, measurements together with low-level control and data logging software are provided in this test set up, which enables researchers to prototype algorithms for PMSG motor drive control. The proposed backstepping control has been implemented in an outer and inner control loops as shown in Fig. 2. The outer control loop has been implemented by using the marine turbine speed controller designed in Section 3.1 while the inner control loop has been designed based on the q-axis current controller in Section 3.2. The additional disturbance estimations for dr̂ and dq̂ , and the parameter update law in Eq. (38) have also been implemented accordingly within the two control loops. Since the focus is on the maximum marine current power generation that is directly related with the q-axis current control, the desired d-axis current has been set as zero for minimizing the generator power loss and a 7
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Fig. 6. The estimation of the term dr.
Fig. 7. The updates of Dr and Dq.
been designed by following the standard backstepping design procedure [17] and hence only the terms iqopt1 and uq1 have been used in the conventional backstepping controller without uncertainty/disturbance estimation and parameter updates. As illustrated in Fig. 3, the marine current speed varies between 0 m/s and 2.5 m/s with clear turbulence. The marine current speed data has been obtained from the real world applications of the marine current turbine in the East China sea. The real world data of the marine current speed represents the below rated operational condition of the marine current turbine. The filtered data of the marine current speed by using the swell filter in Eq. (6) exhibit clear attenuation of the turbulence, which indicates the effectiveness of the designed swell filter in eliminating the swell effects. Therefore, the filtered marine current speed is smooth enough and can be readily used to evaluate the control performance of the proposed control for the maximum marine energy generation. As shown in Figs. 4 and 5, by using the proposed control, the
voltages [16]. All the controller parameters have been tuned through a heuristic tuning approach. The controller gains kr and kq, and initial values Dr0, Dq0 are adjusted from small values to large values until obvious oscillations and overshoot can be observed. The values of ηr and ηq can be set relatively large such that the control loops are in sensitive to noise and high-frequency oscillations in the control system. All the control parameters need to be set such that a tradeoff between the steady-state/ stability performance and transient response including overshoot, tracking error and settling time is accomplished in the implementation. The control parameters are tuned as kr = 0.25, kq = 1.8, Dr0 = 50, Dq0 = 12, γr = 2, ηr = 100, γq = 2 × 10−6, and ηq = 1 × 105, and the time constants τ and τf are all set as 0.02 in the filter implementations. In order to further evaluate the higher control effectiveness of the proposed control in marine energy generation, the proposed control has been compared with a conventional backstepping controller for the sake of fair comparison. The conventional backstepping controller has
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Fig. 8. The q-axis current.
Fig. 9. The tracking errors of the q axis current zq(t).
verify the effectiveness of the update law in Eq. (38). The fast transient of the update process provides timely and accurate control signals for implementing the two control loops. As illustrated in Figs. 8 and 9, the q axis current is effectively maintained around the optimal value by using the proposed control whereas the q axis current fluctuates dramatically around the optimal values when the conventional control is used. The q axis current tracking error zq(t) is also regulated to be around zero by using the proposed control while the tracking error varies significantly when the conventional control is applied. Therefore, the proposed control can be employed to achieve accurate current regulation for achieving the maximum marine energy extraction than the conventional control. Fig. 10 shows the estimation results of the uncertainty and disturbance term dq in the q axis current control loop by using the estimator in Eq. (29). As shown in this figure, the estimates closely track the real values of the term dq and the residual tracking error can be kept within a relatively low value regardless of the obvious oscillations in the term dq. Therefore, the uncertainty and disturbance term dq can be
tracking error of the turbine rotation speed zr(t) is significantly reduced to remain around 0 whereas the tracking error varies significantly between [−3.8, 2] rad/s when the conventional controller is applied. It is also clear that the turbine rotation speed tracks the optimal speed more accurately by using the proposed control in comparison with the conventional control. Therefore, the results reveal that the proposed control is more effective in tracking the optimal turbine rotation speed for the maximum marine energy extraction than the conventional control. Fig. 6 shows the estimation performance of the term dr by using the low pass filter based estimator in Eq. (19). It can be found that fairly smooth estimation is achieved. In particular, with the help of the designed estimator in Eq. (19), the lumped uncertainty and disturbance term dr can be readily estimated and then be compensated in the controller design, which exhibits high potential in real world applications. As shown in Fig. 7, the parameters Dr and Dq can update quickly to the designed real values during the control transient. There also exist no overshoot or oscillations in the updates of the two parameters, which 9
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Fig. 10. The estimation of the term dq.
Fig. 11. The generator power.
5. Conclusion
precisely estimated by using the proposed estimator in Eq. (29) which contributes to attenuating the unknown disturbances and uncertainties when the estimator is incorporated in the control loop. As shown in Fig. 11, the generator generates obviously more power (up to 30% more power) when using the proposed control in comparison with the conventional control. The power generation is relatively stable around 20 kW, which indicates that the swell effects in the marine current speed are filtered and attenuated significantly. The results demonstrate that improvements in generating marine current power by using the proposed control. Therefore, the proposed control is readily better in tracking the optimal marine power points and hence in improving the power generation efficiency under the same operation condition than the conventional control.
The adaptive backstepping controller with uncertainty and disturbance estimation has been proposed and designed in the paper for maximizing the marine current power generation of a generic horizontal marine turbine. The generic horizontal marine current turbine has been designed and modelled for the control implementation. The control problem has been formulated and the controller has been designed accordingly to be composed of the marine turbine speed control loop and the q-axis current control loop while the uncertainty and disturbance are estimated and compensated in the control. The swell filter has also been involved in the proposed controller to attenuate any swell effects on the power generation. The proposed control has been implemented and evaluated in an experimental test bench for maximizing marine current power generation and the control performance has been compared with that of a conventional backstepping controller.
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The test results indicate the proposed control is readily better in tracking the optimal marine power points and hence in improving the power generation efficiency under the same operation condition than the conventional control. By using the proposed control, the generator generates obviously more power (up to 30% more power) than that with the conventional control. The lumped uncertainty and disturbance can also be readily estimated and then be compensated in the proposed controller design, which exhibits high potential in real world applications.
[8] Elghali SEB, Benbouzid MEH, Charpentier JF. Modelling and control of a marine current turbine-driven doubly fed induction generator. IET Renew Power Gener 2010;4(1):1–11. [9] Elghali SEB, Benbouzid MEH, Charpentier JF, et al. High-order sliding mode control of a marine current turbine driven permanent magnet synchronous generator. 2009 IEEE international electric machines and drives conference. IEEE; 2009. p. 1541–6. [10] Benelghali S, Benbouzid MEH, Charpentier JF, et al. Experimental validation of a marine current turbine simulator: Application to a permanent magnet synchronous generator-based system second-order sliding mode control. IEEE Trans Ind Electron 2011;58(1):118–26. [11] Jafari SAH, Kosasih B. Flow analysis of shrouded small wind turbine with a simple frustum diffuser with computational fluid dynamics simulations. J Wind Eng Ind Aerodyn 2014;125:102–10. [12] Wang L, Chen SS, Lee WJ, et al. Dynamic stability enhancement and power flow control of a hybrid wind and marine-current farm using SMES. IEEE Trans Energy Convers 2009;24(3):626–39. [13] Fantino R, Solsona J, Busada C. Nonlinear observer-based control for PMSG wind turbine. Energy 2016;113:248–57. [14] Zhang Q, Liu HHT. UDE-based robust command filtered backstepping control for close formation flight. IEEE Trans Ind Electron 2018;65(11):8818–27. [15] Dragomir SS. A note on Young’s inequality. Revista de la Real Academia de Ciencias Exactas. Físicas y Naturales. Serie A. Matemáticas 2017;111(2):349–54. [16] Pican E, Omerdic E, Toal D, et al. Direct interconnection of offshore electricity generators. Energy 2011;36(3):1543–53. [17] Freeman RA, Kokotović PV. Backstepping design of robust controllers for a class of nonlinear systems. Nonlinear Control Systems Design 1992. Pergamon 1993:431–6.
References [1] Borthwick AGL. Marine renewable energy seascape. Engineering 2016;2(1):69–78. [2] Hussain A, Arif SM, Aslam M. Emerging renewable and sustainable energy technologies: State of the art. Renew Sustain Energy Rev 2017;71:12–28. [3] https://www.verdantpower.com/. [4] http://www.emec.org.uk/about-us/our-tidal-clients/open-hydro/. [5] http://www.tidalenergy.eu/marinecurrenturbines_seaflow.html. [6] Elghali SEB, Benbouzid MEH, Ahmed-Ali T, et al. High-order sliding mode control of DFIG-based marine current turbine. IEEE 2008;2008:1228–33. [7] Elghali SEB, Benbouzid MEH, Ahmed-Ali T, et al. High-order sliding mode control of a marine current turbine driven doubly-fed induction generator. IEEE J Oceanic Eng 2010;35(2):402–11.
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