Scheduling and control co-design for Networked Wind Energy Conversion Systems

Volume 2 Number 4 August 2019 (328-335) DOI: 10.1016/j.gloei.2019.11.005

Global Energy Interconnection i

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Scheduling and control co-design for Networked Wind Energy Conversion Systems Zhihong Huo1, Zhixue Zhang2 1. College of Energy and Electrical Engineering, Hohai University, Nanjing 210098, Jiangsu Province, P.R.China 2. Research & Development Center, Nari-Relays Electric Co., LTD, Nanjing 211106, Jiangsu Province,

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P.R.China Abstract: Fieldbus, industrial Ethernet that is simple, reliable, economical, and practical, is widely used in Wind Energy Conversion Systems (WECSs). These techniques belong to the field of networked control systems. Network embedding to Wind Energy Conversion Systems brings many new challenges. Implementing a control system over a communication network causes inevitable time delays that may degrade performance and can even cause instability. This work addresses challenges related to the reliable control of wind energy conversion systems, based on the theoretical framework of networked control systems. A type of WECS with network-induced delay and packet dropout is modeled and adjustable deadbands are explored as a solution to reduce network traffic in WECSs. A method to study the reliable control of WECSs is presented, which takes into account system response as well as the network environment. After detailed theoretical analysis, simulation results are provided, which further demonstrate the feasibility of the proposed scheme. Keywords: Wind Energy Conversion System (WECS), Networked Control System (NCS), Time-delay, Deadband, Package-dropout.

1 Introduction The steadily rising global energy demand and the environmental problems related to fossil energy utilization necessitate the discovery and development of new energy sources to replace traditional fossil fuels. Renewable energy technology is one such solution; it produces energy by transforming the energy from natural phenomena (or natural resources) into useful forms of energy. During the last twenty years, the Wind Energy Conversion System (WECS) Received: 12 March 2019/ Accepted: 15 March 2019/ Published: 25 August 2019 Zhihong Huo [email protected]

Zhixue Zhang [email protected]

has attracted increasing interest. Wind energy has been proven to be a mature electricity production technology, constituting not only an economically attractive option for the constantly increasing global energy demand, but also a sustainable energy solution for global development with very limited environmental impact [1-2]. The dynamical behavior of a WECS is nonlinear, highly dependent on wind speed, and may change over time due to alterations in the blade surfaces [3]. Because of the stochastic, time-variant uncertainty in wind energy conversion systems, it is necessary to develop a design controller that uses a simple algorithm and is robust under system uncertainty parameters. Additional research is required on WECSs, especially with regard to the realization of the control system.

2096-5117/© 2019 Global Energy Interconnection Development and Cooperation Organization. Production and hosting by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/ ).

328

Zhihong Huo et al. Scheduling and control co-design for Networked Wind Energy Conversion Systems

In terms of control, the wind turbine operates in two distinct regions. Below a certain wind speed, in the partial load region, the turbine is controlled to generate as much power as possible. In the full load region, the wind turbine is controlled to produce a rated power output. Fieldbus, industrial Ethernet that is simple, reliable, economical, and practical, is widely used in wind power generation system. These techniques belong to the category of networked control systems (NCSs). Network embedding to wind power systems brings many new challenges. Implementing a control system over a communication network induces inevitable time delays that may degrade performance and can even cause instability. One of the most effective ways to reduce the negative effect of delays on the performance of an NCS is to reduce network traffic [4-5]. Time-delay exists in various chemical, mechanical, and biological systems, either in the state, the control input, or the measurements. As the existence of time-delays usually degrades the performance of closed-loop systems and may be a source of instability, the control problem for systems with time-delays has received increasing attention. WECS has great inertia, and the time delay the existence of this process, previous studies has not on the in-depth study, in fact, the influence of time-delay on wind turbine operation is very important. Furthermore, control systems typically operate under persistent external disturbances. The exogenous disturbances to the system can be primarily classified as deterministic disturbances and uncertain disturbances [6-9]. Deterministic disturbances, which possess known dynamic characteristics, are common forms of disturbance in practical systems, such as WECSs. In this work, a three-bladed horizontal-axis variablespeed variable-pitch wind turbine is considered; adjustable deadbands are explored as a solution to reduce network traffic in the WECS. We present a new modeling method for the WECS, based on the network environment. The robustness and stability of the WECS is analyzed based on networked control theory. A new concept related to the robust and reliable control of a WECS is put forward. This concept links the quality of control performance (QoP) and quality of network service (QoS) by synthesizing both control and scheduling schemes. This paper is organized as follows. Section 2 describes the principle of Networked Control Systems. Section 3 describes the model of the wind turbine under the constraints of the network environment. Section 4 presents a robust and reliable control method for controller design that takes timedelay and parametric uncertainties into account. Section 5 provides the simulation results of networked wind power generation systems under the influence of disturbance.

2  Networked Control Systems Networked control system is a type of distributed control system. Serial communication network is used to exchange system information and control signals between various physical components of the systems that may be physically distributed [11]. Fig. 1. illustrates a typical setup of an NCS. Compared with conventional point-to-point interconnected control systems, the primary advantages of an NCS include modular and flexible system design (e.g., distributed processing and interoperability), simple and fast implementation (e.g., reduced system wiring and powerful configuration tools), ease of system maintenance, and increased system agility [8]. However, implementing a control system over a communication network induces inevitable time delays that degrade system performance and may even lead to system instability. Time delays are a function of device processing times and communication medium shared. Thus, system performance is dependent not only on the performance on its components, but also on their network interactions. Consequently, theoretical interests in the field of networked control systems are continually increasing.

Physical Plant

Actuator 1

Actuator 2

Sensor 1

Control Network

Other Processes

Sensor 2

Other Processes

Controller

Fig. 1  Typical NCS setup and information flows

Networked-induced delay is inevitable during information transmission and receiving that is attributable to limited communication bandwidth and a high number of information sources. It is well known that in control systems, time delays can degrade a system’s performance and even cause system instability [12-13]. The time delays characteristic of an NCS could be constant, bounded, or even random, depending on the network protocols adopted and the chosen hardware. Essentially, there are two kinds of delays in NCS: communication delay between the sensor and the controller τsc, and communication delay between the controller and the actuator τca. Firstly, the following assumptions regarding the networked control system are made: 329

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Assumption 1: S ensor works in a time-driven mode. However, controller and actuator are event-driven. Assumption 2: There are no data dropouts during transmission on the communication network. Assumption 3: The lengths of the past time delays are known to the controller and actuator via timestamp technology. Assumption 4: The communication delays τsc and τca vary randomly and are independent of each other. Fig. 2 illustrates the sequence of networked-induced delay. The total time delay τ = τsc + τca is no larger than (m-1)h, where m, lk, and hk are certain integers, (m≥2), and lk＜hk＜m. h is sampling period of the sensor, and τ is a random variable. k−m−1 Sensor

k−m

k−hk

k−lk

k −1

k

k+1

Controller

Actuator

Fig. 2  Sequence of networked-induced delays

3 Wind turbine model under network environment Modeling and control of wind turbine systems is one of the most challenging aspects in the field of wind energy conservation systems. The better the representation of the dynamic behavior of the system by the model, the better will be the wind turbine control system synthesis and analysis. We consider the following models for wind energy conservation systems.

3.1 Aerodynamic model The rotor of the wind turbine converts energy from the wind to the rotor shaft, rotating at the speed ωr(t). The power from the wind depends on the wind speed, air density ρ, and swept area, A. From the available power in the swept area, the power on the rotor is given based on the power coefficient, Cp (λ(t), β(t)). The power coefficient depends on the pitch angle of the blades, β(t), and the ratio between the speed of the blade tip and the wind speed; the ratio is denoted as tip-speed ratio, λ(t). The aerodynamic torque applied to the rotor by the effective wind speed passing through the rotor, ν(t), is given as:  1 ρAvr3 (t )C p (λ (t ), β (t ) ) Ta (t ) = (1) 2ω r (t ) 330

3.2 Drive train model The schematic representation of a transmission subsystem is given in Fig. 3. Noise w(k) Angular speed ωr Wind speed v Wind Rotor Transimission Torque Tr

Torque Te Angula speed ωg

Generator

P

Pitch angle β

Fig. 3  Block diagram of the wind turbine model

The drive train model consists of a low-speed shaft and a high-speed shaft having inertias Jr and Jg, respectively. The shafts are interconnected by a gear ratio, Ng, combined with torsion stiffness, Kdt, and torsion damping, Bdt, which result in a torsion angle θ(t). The drive train efficiency ηdt drives the loading torque, Tg(t), from the generator at a speed ωg(t). The linear model is given as: J ω (t ) = T (t ) − K θ (t ) − B θ(t ) (2) r

r

a

dt

dt

 J ω (t ) = η dt K dt θ (t ) − η dt Bdt θ(t ) − T (t ) g g g Ng Ng 1 θ(t ) = ω r (t ) − ω g (t ) Ng

(3) (4)

3.3 Pitch system model The pitch system tracks a reference, βref (t-τ) and is modeled as a first order system. Its time constant is τ, and also includes a communication delay, τ.

1  β (t ) = β (t ) + β ref (t − τ )

τ

(5)

Besides the linear dynamics described by (5), the model also includes constraints on the slew rate and operational range.

3.4 Generator and converter models Electric power is generated by the generator, while a power converter interfaces the wind turbine generator output with the utility grid, and controls the currents in the generator. The generator torque in (6) is adjusted by the reference Tg,ref (t). The dynamics of the converter is approximated by a first order system with time constant τg and communication delay tg,d. Similar to the model of the pitch system, the slew rate and operational range of the converter are limited. 1 1   Tg (t ) = − Tg (t ) + Tg , ref (t − t g , d ) (6)

τg

τg

The power produced by the generator can be approximated from the mechanical power calculated below, where ηg denotes the efficiency of the generator, which is assumed to be constant.

Zhihong Huo et al. Scheduling and control co-design for Networked Wind Energy Conversion Systems

 Pg(t) = ηgωg(t)Tg(t) Consider the following WECS system: ·x (t) = Ax(t) + Bu(t)

y(t) = Cx(t) and a discrete controller u(k) = Kx(k), k = 0,1,2,… p m where x ∈ R n , u ∈ R , y ∈ R , and A, B, C, K are of compatible dimensions, h is sampling period. Aˆ = Φ Ah, k

ti −1 k th Bˆ i ,k = ∫ k e A(h− s ) Bds, t i denote the i data package arriving ti

time. Assume network-induced delay τ is bounded, and the packet is transmitted in sequence. x(t) = [ωr(t) ωg(t) θ(t) β(t) Tg(t)]T u(k) = [βref (k-d) Tg,ref (k-dg)]T The network time-varying delays can be obtained online. The closed-loop networked control system with timevarying delays can be described as n  x(k + 1) = Aˆ + Bˆ K x(k ) + Bˆ Kx(k − i ) (8) ∑ i ,k 0 ,k  i =1 The implementation of deadbands results in network traffic reduction while maintaining acceptable system performance [14]. A node with a deadband compares the value prepared to send to the network, x, to either the last value sent, xsent , or a constant threshold, δt. If the absolute value of the difference between x and xsent is within the deadband, δxsent or δt, then no update is sent to the network. IF |x-xsent|≥δxsent (or δt) THEN Broadcast x xsent = x ELSE No message broadcast For the case when the term δxsent is used, the deadband changes as a function of the node state and is viewed as a relative deadband. When δt is used, the deadband remains constant independent of the state. As the size of the deadband coefficient δ or δt, increases, the node broadcasts fewer messages. Implementing deadbands to reduce network traffic produces uncertainty in the state of the system. Since the controller relies on the broadcast state xb, to compute the control signal, it is important to determine whether this uncertainty could drive the system to instability. At any given time, the true state of the system x is: x = xb±δxb In case of the controller node, the following control signal it sent to the plant:

(

)

u = Kxb

(7)

The discrete-time version of the control signal is given by: u (k ) = K (I + δ ) x(k )



(9)

−1

Substituting (9) into (8), the WECS with communication constraint under networked environment can be described as:

(

)

n

(

)

−1 −1 x(k + 1) = Aˆ + Bˆ 0,k K (I + δ ) x(k ) + ∑ Bˆ i ,k K (I + δ ) x(k − i ) i =1



(10)

4  Networked control for WECS Wind power has great inertia. However, the time delay that exists in the power generation process has not been studied in detail in previous studies, although it is a very important factor influencing wind turbine operation. Timedelay degrades turbine performance and can even cause instability [15-16]. For convenience, we introduce the following useful lemmas. Lemma 1 [17] For any compatible dimension vector y, z, then z T y + y T z ĸy T y + z T z . Lemma 2 [18] For any ε > 0 , and εET E≤I, FT F≤I, then  . Based on these, the closed-loop networked control system can be described as

(

)

n

(

)

−1 −1 x(k + 1) = Aˆ + Bˆ 0,k K (I + δ ) x(k ) + ∑ Bˆ i ,k K (I + δ ) x(k − i ) i =1



(11)

For convenience, let K = K (I + δ )−1, then u (k ) = K x(k ) .

(

)

n

(

)

x(k + 1) = Aˆ + Bˆ 0,k K x(k ) + ∑ Bˆ i ,k K x(k − i ) Theorem 1 Let Bˆ k

2

i =1

(

= max Bˆ i ,k 0ĸiĸn i , k ∈N

2

), if there exists a

positive define matrix P = PT, such that

λmin (S ) > 0

where

M ∈Ω

S = P − 2 Aˆ T PAˆ − 2 K T Bˆ 0T PAˆ − 2 Aˆ T PBˆ 0 K n

(

− 2 K T Bˆ 0T PBˆ 0 K − (n + 1)∑ K T Bˆ kT PBˆ k K i =1

)

then WECS (11) is stable. Here λm(·) denotes the minimum eigenvalue of matrix S. Proof. Consider Lyapunov function n

V (k ) = x T (k )Px(k ) + (n + 1)∑

∑ [x (k )K k −1

i =1 j = k − i

T

T

]

Bˆ Tj , k PBˆ j , k K x(k )

331

Vol. 2 No. 4 Aug. 2019

Global Energy Interconnection

∆V = V (k + 1) − V (k )

= x T (k + 1)Px(k + 1) − x T (k )Px(k )

[

n

]

+ (n + 1)∑ x T (k )K T Bˆ iT, k PBˆ i , k K x(k ) − x T (k − i )K T Bˆ iT, k PBˆ i , k K x(k − i ) i =1

(

 = x T (k ) Aˆ + Bˆ 0 K  n

) P(Aˆ + Bˆ K )− P + (n + 1)∑ (K n

T

0

[

] (

[

]

T

i =1

)

(

)

 Bˆ iT, k PBˆ i , k K  x(k ) 

) P∑ [Bˆ

+ ∑ x T (k − i )K T Bˆ iT, k P Aˆ + Bˆ 0 K x(k ) + x T (k ) Aˆ + Bˆ 0 K i =1

n

T

i =1

i,k

]

K x(k − i )

[

]

n n n   + ∑ x T (k − i )K T Bˆ iT, k P ∑  Bˆ i , k K x(k − i ) − (n + 1)∑ x T (k − i )K T Bˆ iT, k PBˆ i , k K x(k − i )  i =1 i =1  i =1 

By Lemma 1, we have

∑[ n

i =1

and

](

∑ [x (k − i )K n

)

x (k − i )K Bˆ iT, k P Aˆ + Bˆ 0 K x(k ) T

T

i =1

n

i =1

T

n

T

T

n

0

T

i =1 j =1

i,k

i =1

T i,k

n

)

[

]

[

]

thus,

(

) P(Aˆ + Bˆ K ) − P + (n + 1)∑ (K

(

) P(Aˆ + Bˆ K )x(k ) + ∑ [x

+ x T (k ) Aˆ + Bˆ 0 K n

i =1

i,k

 ∆V ĸ x T (k ) Aˆ + Bˆ 0 K 

](

Bˆ iT, k P Aˆ + Bˆ 0 K x(k )

ĸn∑ x T (k − i )K T Bˆ iT, k PBˆ i , k K x(k − i )

0

T

n

T

= ∑ ∑ x T (k − j )K T Bˆ Tj , k PBˆ i , k K x(k − i )

( ) P∑ [Bˆ Kx(k − i )] ĸx (k )(Aˆ + Bˆ K ) P (Aˆ + Bˆ K )x(k ) + ∑ [x (k − i )K Bˆ PBˆ K x(k − i )] + x (k ) Aˆ + Bˆ 0 K T

T

n

T

0

T

i =1

n

T

0

[

]

i =1

T

)

 Bˆ iT,k PBˆ i ,k K  x(k ) 

(k − i )K T Bˆ iT,k PBˆ i ,k K x(k − i )] n

[

]

+ n∑ x T (k − i )K T Bˆ iT,k PBˆ i ,k K x(k − i ) + (n + 1)∑ x T (k − i )K T Bˆ iT,k PBˆ i ,k K x(k − i ) i =1

(

) (

i =1

)

(

n T  = x T (k )2 Aˆ + Bˆ 0 K P Aˆ + Bˆ 0 K − P + (n + 1)∑ K T Bˆ iT,k PBˆ i ,k K i =1  T T T ˆT T ˆ ˆ ˆ ˆ ˆ ĸ− x (k ) P − 2 A PA − 2 K B PA − 2 A PB K − 2 K T Bˆ T PBˆ K

[

n

[

]

0

0

0

0

) x(k ) 

]

− (n + 1)∑ K T Bˆ kT PBˆ k K x(k ) x(k ) i =1

= − x T (k )Sx(k )

ĸ−λ min (S )x T (k )x(k )

If λmin (S ) > 0 , then ∆V < 0 . From Lyapunov stability M ∈Ω

theory, WECS (11) is asymptotically stable. It is difficult to describe a real system with accurate mathematical model. Thus, ΔA, and ΔB are employed to describe the uncertainty with an appropriate dimension, indicating parameter uncertainties of the system. Consider the uncertainty for an NCS model

x (t ) = ( A + ∆A)x(t ) + (B + ∆B )u (t )

332

Considering the state feedback control law u (k ) = Kx(k ), the close-loop control system can be written as

(

)

q

(

)

x(k + 1) = Aˆ + ∆Aˆ x(k ) + ∑ Bˆ i ,k + ∆Bˆ i ,k K x(k − i ) i =0

(

)

where, ∆Aˆ = e ( A+ ∆A )h − e Ah = e Ah e ∆A⋅h − I , t ik−1

[

]

∆Bˆ i , k = ∫ k e ( A+ ∆A )(h − s ) (B + ∆B ) − e A(h − s ) B ds ti

t ik−1

[

((

)

)]

= ∫ k e A(h − s ) e ∆A⋅(h − s ) − I B + e ∆A(h − s ) ∆B ds ti

Zhihong Huo et al. Scheduling and control co-design for Networked Wind Energy Conversion Systems

where ∆Aˆ , ∆Bˆ i ,k are admissible parameter uncertainties of the system that can be described as ∆Aˆ = DF (k )E , ∆Bˆ i ,k = Di Fi (k )Ei (i = 0 ,1,… ,n ) , w h e r e i n D i , E i a r e known constant matrices. F i (k) are time-varying T u n c e r t a i n t i e s m a t r i c e s s a t i s f y i n g F (k ) F (k )ĸI , T Fi (k ) Fi (k )ĸI (i = 0 ,1,… ,n ). The close-loop networked control system model with sensor failures or actuator failures can be written as −1 x(k + 1) = Aˆ + ∆Aˆ + Bˆ + ∆Bˆ K (I + δ ) x(k )

(



n

(

((

0, k

0, k

)

)

)

)

(12)

−1 + ∑ Bˆ i , k + ∆Bˆ i , k K (I + δ ) x(k − i ) i =1

Theorem 2 Let Bˆ k

2

), ε,

(

= max Bˆ i ,k 0ĸiĸn i , k ∈N

)

(

)

T −1

(

Gi = K T Bˆ kT P −1 − ε i Di Di

Bˆ 0 K

)

Bˆ k K + ε i−1 K T EiT Ei K

k −1

T

Let A = Aˆ + ∆Aˆ + Bˆ 0,k K + ∆Bˆ 0,k K , Bi ,k = Bˆ i ,k + ∆Bˆ i ,k . From Theorem 1, we have

∆V = V (k + 1) − V (k )

[

]

[

]

− (n + 1)∑ x T (k − i )Gi x(k − i )

By Lemma 2 and P −1 − ε i Di Di T > 0, we have

[Bˆ

i,k

ĸBˆ

] P[Bˆ + D F (k )E ] (P − ε D D ) Bˆ + ε E E

+ Di Fi (k )E i

T i,k

−1

i

i

T

i,k

i

T −1

T

0, k

(

0, k

ĸ2 Aˆ T P −1 − εD D

i

i,k

)

T −1

(

0, k

Aˆ + 2ε −1 E T E

)

−1

Bˆ 0 K

Thus, n





∆V ĸx T (k )2 A T PA − P + (n + 1)∑ G i  x(k )  ˆT 4 A 

(

(P

−1

− εDD

T

)

i =1

−1

Aˆ + 4ε −1 E T E

)

−1

Bˆ 0 K i =1





If λmin (T ) > 0, then ∆V < 0 . From Lyapunov stability M ∈Ω

i

−1 i

i

T i

Numerical simulation was performed to evaluate the performance of the controller introduced in Theorem 2. Consider the following WECS

]

T

u (t − d ) = β r (t − d ), w(t ) = ω (t ) ,

+ (n + 1)∑ x T (k − i )K T BiT, k PBi , k K x(k − i )

i =1

)

0, k

T

[

  = x T (k )2 A T PA − P + (n + 1)∑ Gi  x(k ) i =1  

n

0, k

x(t ) = ω r (t ) ω g (t ) β θ ,

n

i =1

K

0, k

T

5 Simulation results

j

i =1 j = k −i

n

0, k

theory, system (12) is robust and asymptotically stable.

∑ [x (k )G x(k )].

n

0,k

)

ĸ−λ min (T )x T (k )x(k )

then system (12) is stable. Proof. Consider Lyapunov function

V (k ) = x T (k )Px(k ) + (n + 1)∑

)

= − x T (k )Px(k )

i =1

T −1

K

K + ∆Bˆ 0, k K

T

n

n

T 0

0, k

0, k

+ 4ε 0−1 K T E 0T E 0 K − P + (n + 1)∑ G i  x(k )

− 4ε K E E 0 K − (n + 1)∑ Gi T

0, k

T

T + 4 K T Bˆ 0T P −1 − ε 0 D0 D0

Aˆ − 4ε −1 E T E

− 4 K T Bˆ 0T P −1 − ε 0 D0 D0 −1 0

( K + ∆Bˆ + (Aˆ + ∆Aˆ ) P (Bˆ K + ∆Bˆ K ) + (Bˆ K + ∆Bˆ K ) P (Aˆ + ∆Aˆ ) ĸ2(Aˆ + ∆Aˆ ) P (Aˆ + ∆Aˆ ) + 2(Bˆ K + ∆Bˆ K ) P (Bˆ K + ∆Bˆ + Bˆ 0, k K + ∆Bˆ 0, k

T

M ∈Ω

(

) K ) P (Bˆ

T

= Aˆ + ∆Aˆ P Aˆ + ∆Aˆ

ĸx (k )

λmin (T ) > 0 T = P − 4 Aˆ T P −1 − εDD

) (

) P(Aˆ + ∆Aˆ + Bˆ

+ 2ε 0−1 K T E 0T E 0 K

2

If there exists a positively defined matrix P=PT, such that

T −1

(

T

T + 2 K T Bˆ 0T P −1 − ε 0 D0 D0

ε i (i = 0,1,…, n ) > 0 , P −1 − εD TD > 0 , and P −1 − ε i Di Di T > 0 .

where

(

A T PA = Aˆ + ∆Aˆ + Bˆ 0, k K + ∆Bˆ 0, k K

i

then K T Bˆ iT,k PBi , K K ĸGi.

From the above derivations, and Lemma 1 and Lemma 2, when P −1 − εDD T > 0, P −1 − ε 0 D0 D0 T > 0, we have

 − 137.43 22.3214 − 70.0714 − 1000    0 223.2143   22.3214 − 4.97 A=  , −5 0 0 0     − 0.2232 1 0 0  

0   0 B=  , 5   0  

 214.115     0  D= 0 ,    0   

0 0 0   0.4323   0.63 0 0   0 D= 0 0 0.3276 0 ,    0 0 0 1.6987   333

Vol. 2 No. 4 Aug. 2019

Global Energy Interconnection

0 0.1275 − 3.5782   0.5843   − 0.2598 0 0  1.4963  EA =  , 0 0 0.07843 1.2562     0 0.03876 0 0.0469    2 sin(t )  0.1953    2  0   0 ( ) F EB =  , t =  0 0.0126      0     0

2 2

0 cos(t )

0 0 2 2

0 0

0 –20

2 2

0

   0   sin(t )

The power coefficient function C p (λ (t ), β (t ) ) is a highly non-linear function and therefore, we have the following expression:

 151  C P = 0.73 − 0.58β − 0.002 β 2.14 − 13.2  e  λ1  where

λi =

Power cofficient Cp

60

80

100

120

140

160

180

200

160

180

200

160

180

200

(a) Wind speed (m/s) and direction 100 50 0 –50 –100

0

20

40

60

80

100

120

140

10 0

0.2 0 -0.2 80 60

15 40

10 5

0

20

40

60

80

100

120

140

(c) Rotor speed (rpm)

Fig. 5  The block diagram of the MATLAB simulation

as the network environment. The simulation results further demonstrate the proposed scheme. In this study, adjustable deadbands were explored as a solution to reducing network traffic in WECS. A new modeling method for WECS based on the network environment was explored. The robustness and stability of the WECS was analyzed based on the network control theory. The new model functions by linking the quality of control performance (QoP) and quality of network service (QoS) and synthesizing both control and scheduling schemes.

Tip-speed ratio lamda

Fig. 4  Power coefficient function Cp versus tip-speed ratio and blade pitch angle for the simulated wind turbine

In accordance with Theorem 2, the controller is designed using the robust toolbox MATLAB. The block diagram of this simulation is presented in Fig. 5. The desired and actual pitch angles are β d and β respectively, in Fig. 5. It is clear that β successfully tracks βd. From these graphs, we can see that the controller has good performance and robustness. The torque of the generator controlled by the controller can easily follow the change in the aerodynamic torque. The generator torque is also not affected by time delay and uncertainty. A method to study the reliable control of WECSs is presented in this paper, which takes into account system response as well 334

40

20

0.4

0

20

(b) Pitch command and angle (deg)

0.6

0

0

λi

The coefficient Cp describes the aerodynamic efficiency of the rotor by nonlinear mapping as illustrated in Fig. 4.

Pitch angle beta

–5

−18.4

1 1 0.03 − 3 λ − 0.02 β β + 1

20

Direction

5

0 0

cos(t ) 0

Speed

20

Acknowledgements This work was supported by National Natural Science Foundation of China Research on the Formation Mechanism and Coupled Evolution of Complex Terrain and Wind Turbine Eddy Current, No. U1865101.

References [1] Bonqers PMM (1994) Experimental robust control of a flexible wind turbine system. In: Proceeding of American Control conference. Vol. 6, pp: 3124-3218 [2] Njiri JG, Söffker D (2016) State-of-the-art in wind turbine control: Trends and challenges. Renewable and Sustainable Energy Reviews. Vol. 60, pp: 377-393 [3] Kiviluoma J, Holttinen H, Weir D (2016) Variability in large-

Zhihong Huo et al. Scheduling and control co-design for Networked Wind Energy Conversion Systems

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[15] Rocha R, Martins-Filho LS, Bortolus MV (2005) Optimal Multivariable Control for Wind Energy Conversion System - A comparison between H2 and H∞ controllers. In: Proceedings of the 44th IEEE Conference on Decision & Control, pp:7906-7911 [16] Mei S, Shen T, Liu K (2003) Modern Robust Control Theory and Application. Tsinghua University Press [17] Khargonekar PP, Petersen IR, Zhou K (1990) Robust stabilization of uncertain linear systems: quadratic stabilizability and H/sup infinity /control theory. IEEE Transactions on Automatic Control, 35(3): 356-361 [18] Li X, De Souza CE (1996) Criteria for Robust Stabilization of Uncertain Linear Systems with Time-Varying State Delays. In: Proceedings of IFAC 13th Triennial World Congress, pp:137-142

Biographies Zhihong Huo received her Ph.D. degree from Huazhong University of Science and Technology (HUST) in 2006. Now she is in College of Energy and Electrical Engineering, Hohai University. Her current interests are wind turbine control, fault diagnosis and faulttolerant control of networked control system. Zhixue Zhang received his Ph.D. degree from Huazhong University of Science and Technology (HUST) in 2004. Now he is senior engineer in Research & Development Center Nari-Relays Electric Co., LTD. His current interests are power system monitoring, networked control system, fault diagnosis and fault-tolerant control. (Editor  Chenyang Liu)

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