Resilient and Robust Control of Time-Delay Wind

Xin Wang1 Assistant Professor, Department of Electrical and Computer Engineering, Southern Illinois University, Edwardsville, IL 62026 e-mail: [email protected]

Mohammed Jamal Alden Department of Electrical and Computer Engineering, Southern Illinois University, Edwardsville, IL 62026 e-mail: [email protected]

Resilient and Robust Control of Time-Delay Wind Energy Conversion Systems Wind energy is the fastest growing and the most promising renewable energy resource. High efficiency and reliability are required for wind energy conversion systems (WECSs) to be competitive within the energy market. Difficulties in achieving the maximum level of efficiency in power extraction from the available wind energy resources warrant the collective attention of control and power system engineers. A strong movement toward sustainable energy resources and advances in control system methodologies make previously unattainable levels of efficiency possible. In this paper, we design a general resilient and robust control framework for a time-delay variable speed permanent magnet synchronous generator (PMSG)-based WECS. A linear matrix inequality-based control approach is developed to accommodate the unstructured model uncertainties, L2 type of external disturbances, and time delays in input and state feedback variables. Computer simulation results have shown the efficacy of the proposed approach of achieving asymptotic stability and H∞ performance objectives. [DOI: 10.1115/1.4034661]

Introduction Motivated by the growing environmental concerns, the depletion of fossil fuels, and the world’s expanding population, WECSs have drawn increasing attention from both academia and industry over the last decade. Wind energy is considered as the most promising and fastest growing renewable energy resource [1–3]. It is expected that wind energy will contribute around 12% of the world’s total energy production by 2020 [4]. Difficulties in designing a resilient and robust WECS in achieving the maximum level of efficiency in power extraction from available wind resources warrant the collective attention of modern control and power system engineers. Driven by a stochastic wind input, a nonlinear WECS is controlled not only to maximize output power production, but also to meet power and voltage quality requirements for electrical power grid connection, while minimizing acoustic noise emissions, and being robust and resilient against modeling uncertainties, external disturbances, and feedback delays. A strong movement toward sustainable energy resources and advances in control system methodologies make previously unattainable levels of efficiency possible. Depending on the utilized control schemes, various types of WECSs can be found in Refs. [2–20]. The variable speed control scheme is the most popular WECS type, which provides significant controllability as the generator is decoupled from the utility power grid. Therefore, optimal energy extraction can be used to achieve maximum power point tracking [2,11]. Moreover, a growing trend away from the use of asynchronous machines such as doubly fed induction generators (DFIGs), brought on by the implementation of PMSGs further compounds the need for new control strategies. The PMSG-based variable speed WECSs show higher efficiency, greater reliability, larger power factor (close to one), lower copper losses, and larger power to weight ratio [1–4,6,7], compared with other types of generators. PMSGs eliminate the necessities of a gearbox, which further reduces costs associated with maintenance by allowing for direct coupling of generator shaft and wind turbine. Meanwhile, time-delay is considered as the major source of instability and performance degradation of electrical energy systems. Time delays exist in WECSs due to the time lag in feedback 1 Corresponding author. Manuscript received February 25, 2016; final manuscript received September 6, 2016; published online November 21, 2016. Assoc. Editor: Konstantin Zuev.

measurements, which drastically affects not only the output power and voltage quality, but also may lead to instability or even “black-out” of the entire power system [14]. This work aims to expand existing research into linear matrix inequality architecture, in an attempt to design a more robust and resilient control framework for time-delay WECSs, with delays in state variables and input signals. WECSs inherently exhibit nonlinear dynamics, motivating the use of advanced control techniques such as gain-scheduled control to continuously adapt the model dynamics. To address the effect of time-delay, the introduction of a linear matrix inequality-based control approach is considered as a suitable way of improving reliability of wind turbines and lowering costs of repairs. Finally, the lack of accurate models must be countered by robust control strategies capable of securing stability and satisfactory performance despite model uncertainties. This paper proposes a novel linear matrix inequality-based resilient and robust control framework with H ∞ performance objective for WECSs with fixed state and input delays. The unstructured bounded uncertainties are assumed to exist in the system model. And the system is exposed to external L2 type of disturbances and noises. Additionally, controller gain perturbations are assumed to be of the additive type. By formulating the control problem into convex optimization, the linear matrix inequalities can provide resilient and robust solutions. The structure of this paper is organized as follows: First, the dynamic model of a PMSG-based WECS is developed. Second, the design of a resilient and robust control framework is proposed for time-delay WECSs, which guarantees asymptotic stability in the sense of Lyapunov, along with the H∞ performance objective. Computer simulation studies have shown the efficacy of the proposed approach. Last, the final section concludes this work with comments on the effectiveness of the proposed control approach.

Wind Turbine and Generator Modeling The architecture considered is of variable speed, fixed pitch, rigid drive train, and WECS employing a surface-mounted PMSG. Ideal Actuator Disk Model. The aerodynamic behavior of an ideal wind turbine is modeled through an actuator disk used to extract mechanical power from the stochastic wind power input.

ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems Part B: Mechanical Engineering

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Fig. 2 Simplified diagram of a PMSG-WECS

Fig. 1 Model of actuator disk interaction with wind

Figure 1 shows the actuator disk model, where the variables with subscript u indicate conditions (velocity and pressure) in front of the disk, subscript 0 indicates conditions at the disk, and subscript w indicates conditions behind the disk. Wind of air mass m, pressure p, density ρ, and velocity v transfers momentum H ¼ mðvu − vw Þ to a disk of cross-sectional area A. The resulting force is F¼

ΔH Δmðvu − vw Þ ¼ ¼ ρAv0 ðvu − vw Þ Δt Δt

Pwt ¼ 4að1 − aÞ2 Pwind

Cp ¼

From Eq. (10), Cp achieves its maximum value Cp;max ¼ 0.59, when a ¼ 1=3, which is known as the Betz limit and represents the maximum power extraction of a WECS in the most optimal case. Performance of Non-Ideal Wind Turbine. For most WECS, as shown in Fig. 2, the maximum achievable power extraction is approximately 70–80% of the Betz limit (41.5–47.4% in total efficiency from power extraction from wind), therefore

ð1Þ

Cp ¼

Equivalently, the force on the actuator disk can be written as − F ¼ Aðpþ 0 − p0 Þ

ð2Þ

Pwt < 0.5 Pwind

−

p− 0Þ

1 ¼ ρðv2u − v2w Þ 2

1 F ¼ ρAðv2u − v2w Þ 2

ð4Þ

From Eqs. (1)–(4), the wind velocity at the actuator disk can be found in terms of input and output wind speed in the form of 1 v0 ¼ ðvu þ vw Þ 2

λ¼

ð3Þ

which means the wind force can be expressed as

ð5Þ


ð6Þ

The kinetic energy of the air mass traveling at speed v is E ¼ 12 mv2 . As the air mass that passes the actuator disk in 1 sec. of time can be expressed as m ¼ ρAv0 , where A is the crosssectional area, the input wind power is given in the form 1 Pwind ¼ ρAv30 2

ð7Þ

The power extracted by the actuator disk, i.e., the power supplied to the wind turbine, can be obtained as Pwt

1 1 ¼ ρAv0 ðv2u − v2w Þ ¼ ρAv30 4að1 − aÞ2 ; 2 2

ð8Þ

where a¼1−

v0 vu

ð9Þ

Denote the power coefficient Cp as the power ratio of power extracted by the actuator disk to the power input from wind. Cp can be obtained as 011005-2 / Vol. 3, MARCH 2017

ωr Rt v

ð12Þ

where Rt is the blade length of the turbine and ωr is the rotor mechanical speed of rotation. The Cp power coefficient describes the power extraction efficiency of a wind turbine. A commonly used Cp power coefficient is calculated as a function of the tip speed ratio λ and the blade pitch angle β, and it is given by the following mathematical approximation:
 −c 5 c ð13Þ Cp ðλ; βÞ ¼ c1 2 − c3 β − c4 e λi λi

Equivalently, we have vu − vw ¼ 2ðvu − v0 Þ

ð11Þ

Denote tip speed ratio, λ as the ratio between the peripheral blade speed and the corresponding wind speed v (Note: v ¼ v0 ) as

Based on Bernoulli’s equation, the pressure difference is ðpþ 0

ð10Þ

λi ¼

1 c7 − λ þ c6 β β 3 þ 1

−1

ð14Þ

where c1 through c7 are wind turbine constants. Considering β ¼ 0o , c1 ¼ 0.39, c2 ¼ 116, c3 ¼ 0.4, c4 ¼ 5, c5 ¼ 16.5, c6 ¼ 0.089, and c7 ¼ 0.035, we have Cp;max ¼ 0.4953 and the optimal tip speed ratio λo ¼ 7.2, which falls within the range of realistic expectations for a wind turbine. The power coefficient Cp curve is shown in Fig. 3. The aerodynamic torque applied to the wind turbine is given as: τm ¼

Pwt 1 ρCp πR2t v3 ¼ 2 ωr ωr

ð15Þ

where Rt is the blade length of the turbine and v ¼ v0 is the wind speed. Since the extracted mechanical power from a wind turbine can be characterized as follows: 1 Pwt ¼ ρπR2t v3 Cp ðλ; βÞ; 2

ð16Þ

substituting Eq. (16) in Eq. (15) yields Transactions of the ASME

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uq ¼ −Rs iq − Lq

ð21Þ

For a surface-mounted PMSG, Ld ¼ Lq , which we hereby denote as L for both quantities. Rearrangement of Eq. (20) and Eq. (21) yields the following system model dynamics:

0.8 0.6

Cp

diq þ ð−Ld id þ Ψm Þωe dt

0.4

did R 1 ¼ − s id þ ωe iq − ud L dt L

ð22Þ

diq R 1 1 ¼ − s iq − ωe id − ud þ Ψm ωe L L dt L

ð23Þ

0.2 0 0

15 10

beta

10 20

5 30

The third state variable is introduced based on the high-speed shaft rotational speed equation as:

lambda

0

C p power coefficient versus tip speed versus pitch angle

dωr τ m τ e Bωr ¼ − − dt J J J

Fig. 3

τm ¼

ρπR3t Cp ðλ; βÞv2 1 ρAv3 Cp ðλ; βÞ ¼ 2ωr 2λ

ð17Þ

For torque assessment and control purpose, the torque coefficient Cq , which characterizes the rotor output torque, is derived from the power coefficient simply by dividing it by the tip speed ratio as Cq ¼

Cp λ

ð18Þ

The resultant Cq curve is shown in Fig. 4. Hence, the extracted mechanical torque from the wind can also be written as: τm ¼

ρπR3 Cq ðλ; βÞv2 2

ð19Þ

ð24Þ

where Bωr denotes the damping friction torque. If we neglect the damping friction torque, we may set B ¼ 0. The electrical speed of rotation ωe ¼ P2 ωr , where P is the number of stator poles. In order to maximize the electrical power extraction, the field-oriented control approach is applied by independently controlling the electromechanical torque and rotor flux. The electrical torque τ e in Eq. (24) has the following form τe ¼

3P Ψ i ¼ kt iq 22 m q

ð25Þ

where the torque coefficient kt ¼ 34 PΨm . Thus, the overall PMSGbased WECS model can be summarized as follows Ref. [21]: 2 3 −Rs P 2 3 −1 id þ ωr iq 3 6 2 7 ˙id ðtÞ 0 2 L 6 7 6 L 7  7 7 6 −R 7 ud 6 Ψm P2 ωr 7 6 P s 6 6 ˙iq ðtÞ 7 ¼ 6 −1 7 7þ 5 6 7 u 4 0 6 L − 2 ωr id þ L 7 6 5 q 6 7 4 L ω˙ r ðtÞ 4 5 τ m 3P 0 0 Ψ i − 4J m q J ð26Þ

PMSG Model. Park’s transform is used to transfer the abc coordinate frame PMSG model to the dq coordinate frame model. This yields the following equations for the direct and quadrature axis voltages: ud ¼ −Rs id − Ld

did þ Lq iq ωe dt

Linearized WECS Model. Based on Eq. (12), the optimal mechanical speed of rotation is given in the form

ð20Þ

ωr ¼

λo v Rt

ð27Þ

where optimal tip speed ratio λo ¼ 7.2. Substituting the optimal rotor shaft speed and optimal tip speed ratio in Eq. (17) yields the maximum mechanical power input τ m;max ¼

ρπR5t ðωr Þ2 Cp;max 2λ3o

ð28Þ

ρπR5t Cp;max 2λ3o

ð29Þ

Denote k ¼

then the maximum mechanical power input can be written in the form τ m ¼ k ðωr Þ2 Fig. 4 C q torque coefficient versus tip speed versus pitch

angle

ð30Þ

Neglecting losses, the input mechanical power equals the developed electrical power output, i.e., τ e ¼ τ m . Therefore, the optimal value for quadrature axis current iq is given as

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iq ¼

k  2 ðω Þ kt r

ð31Þ

This section presents the main result of this paper. The general time-delay system model is given as

As mentioned earlier, the torque coefficient kt is given by kt ¼

3Ψm P 4

ð32Þ

By utilizing a field-oriented control approach, the optimal direct axis current id is set to zero, since it is desirable to have the stator current space vector to be aligned with the quadrature axis (q-axis) in order to maximize the developed electromechanical torque τ e , as shown in Eq. (25) [5,22,23]. Therefore, we have id ¼ 0

Design of Resilient and Robust Controller with Linear Matrix Inequalities

ð33Þ

Based on the optimal operating condition in Eqs. (27), (31), and (33), the system Eq. (26) is linearized by taking the partial derivatives at this desired equilibrium point. The linearized model for the PMSG-based WECS is obtained in the form of 2

3 3 2 Piq −Rs Pωr −1 2 3 2 3 6 7 0 7" # ˙id 2 2 6 L 6 L 7 id 7 ud   76 7 6 6 7 6 Pω −R 4P Pi 6 6 7 s r d 6 ˙iq 7 ¼ 6 −1 7 þ6 iq 7 7 76 − 4 5 4 5 6 2 0 7 uq 6 2L L 2 7 5 6 4 7 ω L ω˙ r 4 5 r −PΨm 0 0 0 0 4J

x˙ ðtÞ ¼ ðA0 þ ΔA0 ÞxðtÞ þ ðA1 þ ΔA1 Þxðt − dÞ þ ðB0 þ ΔB0 ÞuðtÞ þ ðB1 þ ΔB1 Þuðt − hÞ þ DwðtÞ

ð40Þ

zðtÞ ¼ ExðtÞ

ð41Þ

where xðtÞ is the state variable and zðtÞ is the controlled performance output. ΔA0 , ΔA1 , ΔB0 , ΔB1 are the modeling parameter variation matrices. d, h are the time delays in state and input variables, respectively. wðtÞ is the L2 type of external disturbance and noise. It is required to design a state feedback controller of the form uðtÞ ¼ ðK þ ΔKÞxðtÞ

where ΔK is the uncertainty in control feedback gain. Therefore, the closed-loop control system dynamics is given as x˙ ðtÞ ¼ ðAc þ ΔAc ÞxðtÞ þ ðA1 þ ΔA1 Þxðt − dÞ þ ½ðB1 þ ΔB1 ÞðK þ ΔKÞxðt − hÞ þ DwðtÞ

2

˙id ðtÞ

3

2

g1 7 6 6 6 ˙iq ðtÞ 7 ¼ 6 −g2 5 4 4 0 ω˙ r ðtÞ 2 −1 6 L 6 6 þ6 6 0 4 0

g2 0

32

3

2

id ðtÞ 0 0 76 7 6 6 7 6 7 g1 0 54 iq ðtÞ 5 þ 6 0 0 4 g3 0 ωr ðtÞ 0 0 3 2 3 0 7" # 1 7 ud 6 7 7 7 −1 7 þ6 4 1 5w 7 u L 5 q 1 0

3 Piq 2

Ac ¼ A0 þ B0 K

ð44Þ

ΔAc ¼ ΔA0 þ B0 ΔK þ ΔB0 K þ ΔB0 ΔK

ð45Þ

Before proceeding to state the main theorem, the following lemma and assumption are stated [13]. LEMMA 1. ABt þ BAt ≤ αAAt þ α−1 BBt

3

i ðt − τ Þ 76 q 7 76 74 iq ðt − τ Þ 7 5 5 ωr ðt − τ Þ 0

2 g4

ð35Þ

ð43Þ

where

ð34Þ Typically, the actual mechanical speed of rotation ωr is measured by a tachometer in real-time; and the data are fed-back to the control system through communication channels. Considering feedback delay in ωr , model uncertainties and external disturbances, the linearized dynamical model for a PMSG-based WECS is derived as follows:

ð42Þ

ð46Þ

To prove this inequality, we can consider the following equivalent inequality that always holds, given arbitrary α > 0: ðα1=2 A − α−1=2 BÞðα1=2 A − α−1=2 BÞt ≥ 0 ð47Þ h ti h i a 0 Furthermore, if A and B are chosen to be and , bt 0 respectively, we get 

0

at b

bt a

0



 ≤

ζat a

0

0

ζ −1 bt b

 ð48Þ

ASSUMPTION 1.

where g1 ¼

−Rs þ ΔRs L þ ΔL

ð36Þ

Pωr 2

ð37Þ

g2 ¼

−3PΨm g3 ¼ 4 g4 ¼

4P pi − d 2L þ ΔL 2

ð38Þ

ð39Þ

where ΔRs and ΔL are the resistance and inductance variations, and w is the external disturbance term. 011005-4 / Vol. 3, MARCH 2017

ΔAt0 ΔA0 ≤ γ A0 I

ð49Þ

ΔAt1 ΔA1 ≤ γ A1 I

ð50Þ

ΔBt1 ΔB1 ≤ γ B1 I

ð51Þ

ΔBt0 ΔB0 ≤ γ B0 I

ð52Þ

ΔK t ΔK ≤ γ k I

ð53Þ

THEOREM 1. Under the feedback control law Eq. (42), the system defined by Eq. (43) is asymptotically stable for all delays satisfying d, h ≥ 0. And the H ∞ performance objective kT zw k∞ ≤ γ 2 can be satisfied, if there exist symmetric positive-definite matrices X, Y, Qt , Qs satisfies the following linear matrix inequality Transactions of the ASME

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2

Φ1

6 6 X 6 6 6 Y 6 6 6 XAt1 6 6 t t 6 Y B1 4 D

t

X

Yt

A1 X

B1 Y

D

Φ2

0

0

0

0

0

−½α2 þ α6 I

0

0

0

0

0

α−1 5 γ A1 I − Q t

0

0

0

0

0

Φ3

0

0

0

0

2

0

3

Substituting Eq. (43) into Eq. (62) yields

7 7 7 7 7 7 7 0 and V˙ < 0. In order to satisfy the H∞ performance objective, the following condition needs to be satisfied Z

∞ 0

ðzt z − γ 2 wt wÞdt < 0:

Z

∞ 0

˙ ðzt z − γ 2 wt w þ VÞdt