A Nonlinear Variable Speed Tracking Control for Wind Turbine Systems In Celebration of the Life, Mathematics and Memories of Chris Byrnes Wei Lin Dept. of Electrical Engineering and Computer Science Case Western Reserve University, Cleveland, Ohio Joint work with Dr. Z. Lu at Emerson Network Power

Introduction  

Components of a typical wind turbine The wind encounters the rotor on the horizontal-axis turbine, causing it to spin. The low-speed shaft transfers energy to the gear box, which steps up in speed and spins the high speed shaft. The high speed shaft causes the generator to spin, hence generating electricity. The yaw system is used to turn the nacelle so that the rotor faces into the wind.

(Figure courtesy of the U.S. Department of Energy)

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Outline            

Introduction Wind Speed Estimation Wind Turbine Modeling Nonlinear Control Design Simulation Study Summary

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Introduction  

Output power of a typical wind turbine operating in different wind speed regions, denoted by 1, 2, 3.

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Turbine Control  

Supervisory Control: Start and shut-down the turbine

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Yaw Control: Turn the turbine to face the wind

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Pitch and Generator Control: Capture the maximum power without exceeding safe rotor speed and stress and convert the wind power into electricity.

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Grid-side Inverter Control: Synchronize with and send the energy to the power grid. 5

Wind Turbine Control System

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Turbine Control in Region 2 Control objective: To control generator torque (and hence, the rotor speed) to achieve the maximum Cp at all wind speed.  

From the graph, it is easy to see that we can keep the pitch angle fixed (at about -1 degree) and control the tip speed ratio (λ) to about 8.5.

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Turbine Control in Region 2  

Wind power that can be captured by a practical wind turbine

   

Cp curve has a unique maximum at With fixed pitch angle, we want to keep the tip speed ratio at its optimal point

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The rotor speed must track the reference signal

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Wind Speed Estimation  

Stochastic approach

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Rotor effect wind speed estimation

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Wind Speed Estimation  

Step 1: Estimation of Wind Turbine Mechanical Power

Wind power (Pm) can be calculated from the power extracted to generator (Pe) and the mechanical power loss (Ploss)

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Wind Speed Estimation

It can be approximated by the discrete time format

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Wind Speed Estimation  

Step 2: Wind Speed Calculation

Newton-Raphson iterative method

The wind speed estimation Rotor speed reference signal 12

Wind Turbine Modeling  

If the rotor speed can track the reference signal, then the tip speed ratio can be maintained at the optimal point. As a result, we have the following relationship

where

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Wind Turbine Modeling  

Two Mass Drive Train

Let

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Wind Turbine Modeling  

If a perfectly rigid low speed shaft is assumed, then the two-mass drive-train model reduces to a one-mass drive-train model.

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Nonlinear Control Design  

Nonlinear Tracking Control Formulation Deign the control input Te to forceω to track the reference signalω*.

 

Challenge: Neither in normal form nor in lower triangular form 16

Nonlinear Control Design  

Introduce a change of coordinates

where

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Nonlinear Control Design Challenges still exists:   Still neither in normal form nor in lower triangular form   Although it is nonminimum-phase, more than one state variables are involved in the zero dynamics.  

 

Solution: Partial Feedback Design Instead of achieving asymptotic tracking for all states, onlyξ1 (→ω*) achieve asymptotic tracking and keep other states bounded.

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Nonlinear Control Design  

Theorem 1 (State Feedback) Consider an equivalent wind turbine system (17)-(20), for a given twice differentiable reference speed signalω*, there exists a state feedback controller such that the rotor speed ξ1 can globally track the desired speed ω* asymptotically.

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Nonlinear Control Design  

Proof of Theorem 1

Step1: Change of coordinates

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Nonlinear Control Design  

Proof of Theorem 1 (cont’d)

Step2: Lyapunov design for e1 and e2

 

z is guaranteed to be bounded since (e1, e2)→0 andδ(·) is bounded.

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Nonlinear Control Design  

Output feedback design motivation: Torsional angleθk (z ) is difficult to measure.

 

Theorem 2 (Output Feedback) Consider an equivalent wind turbine system (17)-(20), for a given twice differentiable reference speed signalω*, there exists a output feedback controller

such that the rotor speed ξ1 can globally track the desired speed ω* asymptotically. 22

Nonlinear Control Design  

Proof of Theorem 2 Step1: Reduced order observer design

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Nonlinear Control Design  

Proof of Theorem 2 (cont’d) Step2: Feedback controller design

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Nonlinear Control Design   Proof

of Theorem 2 (cont’d)

Step3: Closed-loop stability analysis

z is guaranteed to be bounded because of (e1, e2)→0, asymptotic stability of the observer and boundedδ(·).

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Simulation Study  

Reference speed signal

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Simulation Study  

Simulation Results

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Nonlinear Control Design  

Theorem 3 (One-Mass Drive-Train Model) Consider a reduced wind turbine system model for a given twice differentiable reference speed signalω*, there exists a feedback controller such that the rotor speedω can asymptotically track the desired speedω*.

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Nonlinear Control Design  

Proof of Theorem 3

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Summary        

Wind turbine control problem in Region 2 Rotor effect wind speed estimation Two-mass drive-train modeling Nonlinear tracking control for maximum power capture based on two-mass drive-train model

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Acknowledgement  

This work was supported in part by the Robert Herbold Faculty Fellow Award and the funding from Alstom Power Inc.

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