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Optimal Control with Stochastic Principles for Voltage Decoupling Control of a Wind Turbine

N. Schinas

Department of Electrical Engineering, Technological Educational Institute of Western Greece, Patras, Greece

Abstract: This paper presents new control strategies for a variable speed wind turbine (WT) which consists of a doubly fed asynchronous generator. In this kind of generators, the rotor windings are supplied with an appropriate voltage vr through a dc link. There are 2 converters: the rotor-side and the grid-side converter in the dc link. The right pulsation of the rotor-side converter defines the rotor voltage vr in such a way that both real power maximization from the wind speed and nominal voltage are secured at the same time. With all the flux dynamics inside the machine having been included in the generator model, there is a 2-input-2-output system. Based on the dq electric machines analysis theory, the proposed control scheme manages to create 2 single-input-single-output systems with the rotor voltage component in each axis responsible only for real power or voltage nominal value respectively. This is achieved through optimal control theory with the right definition of the cost function. A new control scheme that contains stochastic control principles is proposed for the grid-side controller in which the decoupling of the loops is achieved by means of the grid frequency being regarded as a ‘random’ variable to the voltage control loop and so it has no impact on it. The nonlinear system has been tested in the SIMULINK software under 3 tests: In the first test the wind speed changes and brings about an analogous alteration of the WT output. In the second test there is a short circuit in the grid. According to the new rules, the WT is not disconnected from the grid during the short circuit and continues to produce energy with a very smooth behavior. In the third test, there is a large load disturbance in the grid. Again the WT manages to supply the grid with electrical energy in a nice way. Under all tests the controllers seem to cooperate very well and the variations of the voltage and the frequency are within acceptable limits.

Keywords: wind turbine, asynchronous motor, doubly fed induction generator, optimal control

Citation: N. Schinas, Optimal Control with Stochastic Principles for Voltage Decoupling Control of a Wind Turbine. American Journal of Electrical and Electronic Engineering, 5 (2). 40-47. January 2017. doi:10.12691/ajeee-5-2-2

1. Introduction

There have been many proposals about the control schemes for both the rotor side and the grid side converter of a doubly fed wind turbine using induction generator (DFIG) but most of them intend to make the WT work satisfactorily in steady state conditions capturing maximum power from the wind. The increased need for wind energy development makes the installation of WTs in ‘weak’ ac grids necessary. Besides, a number of voltage instability incidents have been experienced around the world and so, voltage stability has become a major research area in the field of power systems [1,2]. In order voltage stability to be secured, the grid codes in some countries demand that the WT remains connected to the grid for voltage levels as low as 15% of the nominal voltage for 625 ms [3,4,10].

The control design techniques presented in this paper are based on the theory of the dq analysis of the asynchronous generator and the Direct Current (DC) link in the rotor and intend to make the WT have a smooth performance under steady and transient state when it is connected to a very weak system. Besides, the performance of the designed controllers ensure stability of the system under large voltage sags in order the above grid demands to be met.

The dq model demands two inputs: the voltage vq in the q–axis and the voltage vd in the d-axis respectively. There are actually two outputs (y) of the model: the mechanical speed of the rotor of the generator which determines the real power output of the machine and the stator voltage which is relevant to the reactive power of the WT. If we do not neglect the flux dynamics in each axis, the dq theory based on all the machine equations results in equations of the form in the s-plane for every output. We actually have a 2-input-2-output system and this is depicted in Figure 1. As we can see, there is no real mathematical decoupling between the two main loops in the machine with all fluxes transient states being taken into consideration.

In this paper, a completely new way of handling the inputs vq and vd is proposed. In the proposed control methods, the input rotor voltage vqr regulates only the stator voltage vs and so determines only the reactive power of the system. In the same way, the input rotor voltage vdr is responsible for only the real power of the system. Although these refer to the rotor side converter control, a similar method has been developed for the grid side converter control also.

Figure 1. The induction machine as a system with 2 inputs and 2 outputs

The control methods have been designed based on the linearized models of the systems but the controllers have been tested in the nonlinear system of the WT. The system under study is shown in Figure 2. A wind park (WP) that consists of 3 DFIGs with rated power of 2 MW each, is connected to a weak medium voltage ac grid with short-circuit capacity equal to 150 MVA via a medium voltage line of 20 km. There also is a large asynchronous motor in the grid representing the various inductive loads. We test the system under wind speed changes, short-circuit conditions in the grid and load variations. The recent rules in some grid companies demand that the machine is not disconnected during specific short-circuit conditions but this means that the machine must have a smooth response in this case. The proposed controllers give extremely good performance of the nonlinear system in all transient conditions.

Figure 2. System under study

2. Description and Modeling of the System

2.1. Mathematical Model of the Asynchronous Generator

In Figure 3 a general configuration of the WT using an asynchronous generator is depicted. The afore-mentioned vdr, vqr voltages are the voltages that must be applied to the rotor windings of the generator and this is the scope of the rotor-side converter control scheme.

Figure 3. General configuration of the DFIG

According to [5,11], the nonlinear reduced-order model of the induction generator based on the theory of the d and q axis, is given by:

where: is the generator angular speed, is the synchronous speed, is the rotor resistance, , are the d and q components of the rotor magnetic flux respectively, , are the and components of the rotor voltage as imposed on it from the rotor side converter.

By substitution of the values given at the Table A1 of the Appendix, we get:

[picture] | (1)

The linearization of the above system around an operation point ( λqr0, λdr0, iqr0, idr0, ωslip0, vdr0,vqr0) is:

[picture] | (2)

[picture] | (3)

From the definition of ωslip, it is: . By substitution of the parameters values given at the Appendix in equations (2)-(3) we get:

[picture] | (4)

[picture] | (5)

From the general theory of the machine we also have [5]:

from which we get:

[picture] | (6)

We take into consideration that

[picture] | (7)

From (6), (7) and substitution we have:

[picture] | (8)

We get the linearized equation for the stator voltage vs as:

[picture] | (9)

From (8), (9) it is:

By substitution of the values from the Appendix we have:

[picture] | (10)

2.2. Mathematical Model of the Wind Turbine

The two-mass drive train model for the WT system is used in this paper as the dynamic characteristics of the WT can be reproduced satisfactorily because the low speed axis is relatively soft. The gear box and the high speed axis are assumed to be infinitely stiff. The angular velocity of the rotor is given by [6]:

where: is the low speed inertia constant, is the aerodynamic torque from the wind speed, N is the gear ratio, is the torsion stiffness, is the torsion angle. The torsion damping constants have been neglected. By substitution from the parameters given in the Appendix, we have:

[picture] | (11)

For MPPT operation of the WT, it is [7]: . By substitution:

, so (11) becomes:

[picture] | (12)

By linearization around the operation point ωt0 we get:

[picture] | (13)

and by substitution of the value :

[picture] | (14)

The torsion angle is given by [6]: where is the rotor angular speed of the induction generator. By substitution:

[picture] | (15)

The slip frequency s of the generator is given by [6]: where is the electromechanical torque of the generator. By substitution we have:

[picture] | (16)

The linearized forms of the equations (15)-(16) are:

[picture] | (17)

The electromechanical torque is equal to [5]: By linearization we get:

[picture] | (18)

By substitution we have:

Also, So, the last equation of (17) becomes:

[picture] | (19)

2.3. State Space Form

By setting the above equations (4),(5),(14),(17),(19) are written in state space form where:

[picture] | (20)

The output y of the system is firstly considered to be the stator voltage concerning the reactive power loop. From (10) it is in state space form: , where:

[picture] | (21)

From (20),(21) the output stator voltage in transfer function form is given by:

[picture] | (22)

where:

The output y of the system is now considered to be the WT rotor speed concerning the real power loop. From (14) it is in state space form: , with:

[picture] | (23)

From (20), (23) the output WT rotor speed in transfer function form is equal to:

[picture] | (24)

where:

2.4. Grid - Side Converter Modelling

Figure 5 shows a general schematic configuration of the grid side converter, where ea, eb, ec are the grid phase voltages and va, vb, vc are the converter phase voltages. According to the dq analysis of the quantities, we have the following equations for the d,q components (id, iq) of the line currents ia, ib,ic [6]:

Figure 4. Grid-side configuration of the rotor converter

[picture] | (25)

The first control loop is the reactive power output regulation through the current . We have assumed for convenience that the grid voltage e is located on the d-axis. The huge alterations of the WT real power output especially when it works in a weak system may result in alterations of the grid frequency . For this reason, the parameter is considered to be a disturbance for the reactive power adjustment loop. The reference is achieved through the loop given in the Figure 5a and so the grid voltage e is controlled. The second loop is the real power regulation through the current . The reference id,ref is achieved through the loop given in the Figure 5b and so the right value of is preserved.

The linearization of (25) around an operation point (iq0, id0, vq0, vd0) results in:

[picture] | (26)

The grid voltage e is constant, so . By substitution of the values from the Appendix into (26) we get:

[picture] | (27)

In state space form with, . From (27) the matrices are:

[picture] | (28)

Figure 5a. Id,ref calculation for real power system

Figure 5b. Iq,ref calculation for reactive power system

The reactive power control loop has been designed in this paper. For output , we have:

[picture] | (29)

From the state space model given by (28), (29) we can have the output in transfer function form given by:

[picture] | (30)

where:

3. Control Systems Design

3.1. Rotor-side Converter Control

As it has already been developed, the output ωt (for maximization of real power) and the (for constant stator voltage) depend on the components of the rotor voltage on the dq axis: and which are regarded as inputs to the system. The objective of the control system in rotor side converter is the construction of two stable decoupled control loops with the voltage component to control only and the voltage component to control only the output .

Figure 6. Objective of the control system design

We firstly work on the first loop regarding the reactive power control. We want the output to be controlled only by . In order this to be achieved we work by means of the optimal control theory.

We set the following quadratic cost function [8,9]:

Since we want the output to be controlled only by , we must minimize the effect of u₂ onto the output. We set the matrices , M, R as following:

The value of a is arbitrary, but it must be high and positive. We set . The optimal feedback control law is of the form: . The value of C was found by means of the MATLAB software and is equal to:

The bode diagram for the output from each input is shown in the Figure 7. We can see very clearly that the output actually depends only on the input u₁. So we ignore the input u₂ and we reach the closed-loop system of Figure 6a by finding the equivalent series controller which is:

If we follow the same procedure for the second loop, we shall come to a similar result regarding the control of only from .

Figure 7. Bode diagrams of the output Δvs from the 2 inputs

3.2. Grid-side Converter Control

As it seems from (30), the reactive power loop with as output y, includes the parameter , which actually depends on the change of the active power. The aim of the grid-side converter control is to distinguish the two loops by eliminating the impact of onto . In order this to be achieved, we work with the theory of discrete systems. The nonlinear system has been simulated in SIMULINK software with time step size of 200μsec. The transfer functions of (30) are converted to z-plane with sec and we get:

From (30), if we set , and it is:

[picture] | (31)

where d = deg A’-deg B₁ (or B₂) and general any function X* is the reciprocal of X, that is X*(z⁻¹) = z-nX(z) and z⁻¹ is the backward-shift operator in the z plane. It is assumed that X is of order n.

Supposing that C* and A’* are of the same order, we can write:

then after d situations the output y will be:

From (31) it is:

By substitution to the previous we have:

[picture] | (32)

Now, we must recall that and where is the rated value of phase voltage and is the angle between the voltages and , as these voltages are depicted in Figure 4. So the inputs in the linearized model are:

By substitution to the previous we have:

The output must converge and this is best accomplished when

By setting we have:

The first and the third term are equal to zero because the signal e(k) has zero mean value as it is considered to be an independent random variable and we want it to be irrelevant to the voltage – control loop. So from the previous equation:

The optimal and simpler solution to this demand is:

This is the control law for input u₁. From the previous it is: This is the control law for input u₂.

4. Nonlinear System Simulation Results

The system under study as has been depicted in Figure 1 has been simulated in SIMULINK software. The rated values for the machines are listed in the Appendix. The wind speed has been simulated as a series of random numbers around a mean value. In the first case we consider the wind speed to be altered at t = 9 sec (Figure 8) from an average value of 7 m/s to an average value of 12 m/s. The response of the system is shown in Figure 9. The change of the power from the WT follows the change of the wind speed (Figure 9a), the grid voltage is constant (Figure 9b) and the motor speed (which depends on the grid frequency) is also constant (Figure 9c).

Figure 8. Wind speed for the first test of the system

Figure 9a. WT real power output

Figure 9b. Grid voltage

Figure 9c. Grid frequency

In the second case, (Figure 10), at t = 36 sec there is a 3-phase short-circuit in the grid for a time duration of 400msec. According to new grid rules, the WT is not disconnected. The WP manages to deliver a small amount of real power into the grid during the short circuit. When the fault is cleared, the WP starts to produce power again with no serious transient problems and a very smooth behavior. The response of the system is depicted in Figure 10. The grid voltage is in Figure 10a, the WP output is shown in Figure 10b and the motor speed in Figure 10c.

Figure 10a. Grid voltage with the short-circuit

Figure 10b. WP real power during and after a short-circuit

Figure 10c. Motor speed during and after short-circuit

In the third test, there is a severe load disturbance near the WP. The motor is disconnected at t = 36 sec and reconnected again at t = 37 sec. This situation could change the grid voltage significantly as the rated power of the motor is large. The WT behavior could be affected because the asynchronous machine is very sensitive to the grid voltage. Again, the proposed controllers manage to make the WT behave in a very smooth way despite the large disturbance. In Figure 11a the motor speed is shown and the grid voltage is shown in Figure 11b. The output of the WP is shown in Figure 11c.

Figure 11a. Motor speed for motor disturbance

Figure 11b. Grid voltage under load disturbance

Figure 11c. WP real power for load disturbance

5. Conclusions

A control system that takes into account all the flux dynamics has been proposed for application to a wind turbine using an induction generator. The real decoupling between the real and the reactive power loops inside the machine have as a result an improved performance of the generator especially in transient state. The control systems were designed through optimal control theory with stochastic principles. The results of the nonlinear system are very good under vast changes of the wind speed, short-circuit state and load disturbances. The cooperation of the controllers ensure that the voltage and the frequency are within acceptable limits.

References

Appendix

Table A1. WT PARAMETERS [5,6]

Table A2. MEDIUM VOLTAGE LINE

Table A3. ASYNCHRONOUS MOTOR PARAMETERS

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[4]   | Interconnection for Wind Energy, Tech. rep., Order No. 661, 70 Fed. Reg. 34993 (FERC Stats. & Regs. 31, 186 (2005).
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