4. Numerical results

In this section, the performances of the proposed observers are illustrated in simulation. Observers' algorithms have been implemented in MATLAB/SIMULINK software. The doubly-fed induction generator system states which have been used for estimation are expressed into a vector x, this vector includes as parameters to estimate the stator and rotor resistances, as follows:

$$\mathcal{X} = \begin{bmatrix} \Phi\_{ds} & \Phi\_{qs} & \Phi\_{dr} & \Phi\_{qr} & R\_s & R\_r \end{bmatrix}^T \tag{38}$$

Figures 3 and 4 show the generated estimates of the rotor and stator resistances

Initial guess [0; 0:5; 0:5; 1; 0:02; 0:02]

diag([111110–410�4])

diag([11111])

by the HGO, UKF and the MHE in the healthy mode of working of the DFIG. Nevertheless, Figures 5 and 6 show the generated estimates of the rotor and stator resistances by the HGO, UKF and the MHE in the faulty mode of working, let us mention that faulty mode is simply a mode where the DFIG undergoes a fault on its stator and/or rotor resistances during the operation. We just simulated those scenarios to appreciate the estimation performance of different observers in particular the HGO, UKF and MHE for process monitoring or diagnostics purposes. We can observe that the estimates by the MHE converges to the actual parameters in fewer time compared to the HGO and UKF. In Table 3, we notice the total computation time to obtain an estimate for the HGO algorithm is about 1.200 seconds, for the UKF algorithm is also about 1.190 seconds while the MHE algorithm took

Parameters Values Covariance matrix P0 eye(6)

Covariance matrix Q 10�<sup>2</sup>

Covariance matrix R 10�<sup>2</sup>

152.978 seconds to estimate the parameters in the normal mode of working, and in

observers, respectively. We can conclude that when the asynchronous machine has a stator or rotor resistance fault, the estimation time increases. The reason the MHE algorithm takes longer to make an estimate is that in simulation, the optimization of the objective function, through a nonlinear programming algorithm has been performed at each time step, in this case study the nonlinear programming algorithm used is the sequential quadratic programming in the MATLAB in-built func-

<sup>2</sup> such as, n is the size of observation vector, when that number is

θ<sup>i</sup> � θ ^i

^<sup>i</sup> is the estimated value from the filters. Table 8 shows a

<sup>2</sup> (41)

n for the high gain observer [18], for our experiment the value of the gain is θ = 27,

2 L + 1 sigma points and associated weights to represent state of the system. Tables 6 and 7 show the standard deviation and the variance of the estimation error. The comparison of these observers can be made by finding the mean squared

> <sup>N</sup> � <sup>n</sup> <sup>∑</sup> N i¼1

where N is the number of time steps, n is the dimension of state vector, θ<sup>i</sup> is the

comparison of the three observers by finding the mean squared error in the healthy

MSE <sup>¼</sup> <sup>1</sup>

on the other hand, the UKF algorithm has to handle.

error (MSE) value. The MSE can be evaluated as:

the faulty mode, we have about 1.901, 1.666 and 154.234 seconds for those

tion fmincon. For the HGO we can underline this, a big value of θ leads to consolidate the linear part and to guarantee the stability of the nonlinear part through the fact that φ is imposed globally Lipschitz in relation to x [27]. If θ are big enough, the time of convergence decreases, but the observation becomes extremely sensitive to the measurement noises. A small value of θ leads to the reverse effect obviously. In comparison with the extended Kalman filter, this observer contains a lot less of setting variables that facilitates its optimization. Besides the number of equations to solve are a lot weaker and it decreases the time of calculation considerably. To know that the number of differential equations to solve for the Kalman

filter is of <sup>n</sup> <sup>þ</sup> n nð Þ <sup>þ</sup><sup>1</sup>

Table 5. UKF parameters.

Advanced Monitoring of Wind Turbine DOI: http://dx.doi.org/10.5772/intechopen.84840

simulated value and θ

33

The inputs of the system are the rotor angular electrical speed, stator and rotor voltages, as in:

$$\boldsymbol{\mu} = \begin{bmatrix} \boldsymbol{v}\_{ds} & \boldsymbol{v}\_{qs} & \boldsymbol{v}\_{dr} & \boldsymbol{v}\_{qr} & \boldsymbol{o}\_{r} \end{bmatrix}^{T} \tag{39}$$

The d and q axis of stator and rotor currents and the mechanical torque constitute the measurements of the systems,

$$\mathbf{y} = \begin{bmatrix} T\_m & \mathbf{i}\_{ds} & \mathbf{i}\_{qs} & \mathbf{i}\_{dr} & \mathbf{i}\_{qr} \end{bmatrix}^T \tag{40}$$

Table 3 shows a comparison of the running time of high gain observer (HGO), the unscented Kalman filter (UKF), and the moving horizon estimation (MHE) for the DFIG system. The high gain observer being the fastest among the three methods under various modes especially the healthy mode which represents a healthy DFIG and the faulty mode where stator and rotor resistance would have changed value during the operation of the DFIG. Tables 4 and 5 give the parameters of UKF and MHE only. For the UKF, the primary, secondary, and tertiary scaling parameters α, β and κ are chosen as 1, 2, and 0, respectively.


Table 3.

Running time of the three observers for the DFIG (in seconds).


Table 4. MHE parameters. Advanced Monitoring of Wind Turbine DOI: http://dx.doi.org/10.5772/intechopen.84840


#### Table 5.

With Π<sup>0</sup> given. The Moving horizon estimation algorithm is described by the

In this section, the performances of the proposed observers are illustrated in simulation. Observers' algorithms have been implemented in MATLAB/SIMULINK software. The doubly-fed induction generator system states which have been used for estimation are expressed into a vector x, this vector includes as parameters to

x ¼ Φds Φqs Φdr Φqr Rs Rr

u ¼ vds vqs vdr vqr ω<sup>r</sup>

y ¼ Tm ids iqs idr iqr

Healthy mode 1.200 1.190 152.978 Faulty mode 1.901 1.666 154.234

Parameters Values Weight matrix G eye(6) Covariance matrix P0 3eye(6) Covariance matrix Q 0.5eye(6) Covariance matrix R eye(5) Length horizon H 10

Initial guess [0; 0:5; 0:5; 1; 0:02; 0:02]

The inputs of the system are the rotor angular electrical speed, stator and rotor

The d and q axis of stator and rotor currents and the mechanical torque consti-

Table 3 shows a comparison of the running time of high gain observer (HGO), the unscented Kalman filter (UKF), and the moving horizon estimation (MHE) for the DFIG system. The high gain observer being the fastest among the three methods under various modes especially the healthy mode which represents a healthy DFIG and the faulty mode where stator and rotor resistance would have changed value during the operation of the DFIG. Tables 4 and 5 give the parameters of UKF and MHE only. For the UKF, the primary, secondary, and tertiary scaling parameters α,

<sup>T</sup> (38)

<sup>T</sup> (39)

<sup>T</sup> (40)

HGO UKF MHE

diagram in Figure 2.

Wind Solar Hybrid Renewable Energy System

4. Numerical results

voltages, as in:

Table 3.

Table 4. MHE parameters.

32

estimate the stator and rotor resistances, as follows:

tute the measurements of the systems,

β and κ are chosen as 1, 2, and 0, respectively.

Running time of the three observers for the DFIG (in seconds).

UKF parameters.

Figures 3 and 4 show the generated estimates of the rotor and stator resistances by the HGO, UKF and the MHE in the healthy mode of working of the DFIG. Nevertheless, Figures 5 and 6 show the generated estimates of the rotor and stator resistances by the HGO, UKF and the MHE in the faulty mode of working, let us mention that faulty mode is simply a mode where the DFIG undergoes a fault on its stator and/or rotor resistances during the operation. We just simulated those scenarios to appreciate the estimation performance of different observers in particular the HGO, UKF and MHE for process monitoring or diagnostics purposes. We can observe that the estimates by the MHE converges to the actual parameters in fewer time compared to the HGO and UKF. In Table 3, we notice the total computation time to obtain an estimate for the HGO algorithm is about 1.200 seconds, for the UKF algorithm is also about 1.190 seconds while the MHE algorithm took 152.978 seconds to estimate the parameters in the normal mode of working, and in the faulty mode, we have about 1.901, 1.666 and 154.234 seconds for those observers, respectively. We can conclude that when the asynchronous machine has a stator or rotor resistance fault, the estimation time increases. The reason the MHE algorithm takes longer to make an estimate is that in simulation, the optimization of the objective function, through a nonlinear programming algorithm has been performed at each time step, in this case study the nonlinear programming algorithm used is the sequential quadratic programming in the MATLAB in-built function fmincon. For the HGO we can underline this, a big value of θ leads to consolidate the linear part and to guarantee the stability of the nonlinear part through the fact that φ is imposed globally Lipschitz in relation to x [27]. If θ are big enough, the time of convergence decreases, but the observation becomes extremely sensitive to the measurement noises. A small value of θ leads to the reverse effect obviously. In comparison with the extended Kalman filter, this observer contains a lot less of setting variables that facilitates its optimization. Besides the number of equations to solve are a lot weaker and it decreases the time of calculation considerably. To know that the number of differential equations to solve for the Kalman filter is of <sup>n</sup> <sup>þ</sup> n nð Þ <sup>þ</sup><sup>1</sup> <sup>2</sup> such as, n is the size of observation vector, when that number is n for the high gain observer [18], for our experiment the value of the gain is θ = 27, on the other hand, the UKF algorithm has to handle.

2 L + 1 sigma points and associated weights to represent state of the system. Tables 6 and 7 show the standard deviation and the variance of the estimation error. The comparison of these observers can be made by finding the mean squared error (MSE) value. The MSE can be evaluated as:

$$\text{MSE} = \frac{1}{N \times n} \sum\_{i=1}^{N} \left(\theta\_i - \hat{\theta}\_i\right)^2 \tag{41}$$

where N is the number of time steps, n is the dimension of state vector, θ<sup>i</sup> is the simulated value and θ ^<sup>i</sup> is the estimated value from the filters. Table 8 shows a comparison of the three observers by finding the mean squared error in the healthy

Rotor resistance estimation in a healthy mode with HGO, UKF and MHE. (a) Rotor resistance estimation (HGO), (b) Rotor resistance estimation (UKF), (c) Rotor resistance estimation (MHE).

Figure 4.

Advanced Monitoring of Wind Turbine DOI: http://dx.doi.org/10.5772/intechopen.84840

35

Stator resistance estimation in a healthy mode with HGO, UKF and MHE. (a) Stator resistance estimation

(HGO), (b) Stator resistance estimation (UFK), (c) Stator resistance estimation (MHE).

Advanced Monitoring of Wind Turbine DOI: http://dx.doi.org/10.5772/intechopen.84840

Figure 4.

Stator resistance estimation in a healthy mode with HGO, UKF and MHE. (a) Stator resistance estimation (HGO), (b) Stator resistance estimation (UFK), (c) Stator resistance estimation (MHE).

Figure 3.

34

Rotor resistance estimation in a healthy mode with HGO, UKF and MHE. (a) Rotor resistance estimation

(HGO), (b) Rotor resistance estimation (UKF), (c) Rotor resistance estimation (MHE).

Wind Solar Hybrid Renewable Energy System

Figure 5.

Rotor resistance estimation in a faulty mode with HGO, UKF and MHE. (a) Rotor resistance estimation (HGO), (b) Rotor resistance estimation (UKF), (c) Rotor resistance estimation (MHE).

Figure 6.

Advanced Monitoring of Wind Turbine DOI: http://dx.doi.org/10.5772/intechopen.84840

37

Stator resistance estimation in a faulty mode with HGO, UKF and MHE. (a) Stator resistance estimation

(HGO), (b) Rotor resistance estimation (UKF), (c) Rotor resistance estimation (MHE).

Advanced Monitoring of Wind Turbine DOI: http://dx.doi.org/10.5772/intechopen.84840

Stator resistance estimation in a faulty mode with HGO, UKF and MHE. (a) Stator resistance estimation (HGO), (b) Rotor resistance estimation (UKF), (c) Rotor resistance estimation (MHE).

Figure 5.

36

Rotor resistance estimation in a faulty mode with HGO, UKF and MHE. (a) Rotor resistance estimation

(HGO), (b) Rotor resistance estimation (UKF), (c) Rotor resistance estimation (MHE).

Wind Solar Hybrid Renewable Energy System


variations but it is not so, it is just that the statistical difference generated is weak enough compared to others. From these responses, we can conclude that the rotor inductance changes do not affect the performance of the moving horizon estimation

Advanced Monitoring of Wind Turbine DOI: http://dx.doi.org/10.5772/intechopen.84840

Robustness test. Rotor inductance variation (+50%). (a) Rotor resistance estimation, (b) Stator resistance estimation.

Figure 7.

39

#### Table 6.

General statistics of the three observers (healthy mode).


#### Table 7.

General statistics of the three observers (faulty mode).


#### Table 8.

MSE values of nonlinear observers: HGO, UKF and MHE are compared.

and the faulty mode of operation of the DFIG and we can notice that generally, the mean squared error of states and parameters in faulty mode is relatively greater than those in the healthy mode because of the fault occurring suddenly during the operation, but we can always see the high performance of the moving horizon estimation on the others observers.

To verify the robustness, we have performed parametric variation on the observer in relation to the identified values. Figures 7 and 8 show the responses obtained when a rotor inductance variation of +50 and 50% is considered for the observer test. The robustness of the observers'scheme with respect to this parameter changes is clearly shown. In Figures 7 and 8, it is clearly shown that a +50 and 50% rotor inductance variation generates a high statistical difference on rotor and stator resistances for the unscented Kalman filter. For the high gain observer, that variation is much more felt on the rotor resistance on the both figures. Incontestably the moving horizon estimation seems remain insensitive to the parametric

variations but it is not so, it is just that the statistical difference generated is weak enough compared to others. From these responses, we can conclude that the rotor inductance changes do not affect the performance of the moving horizon estimation

Figure 7. Robustness test. Rotor inductance variation (+50%). (a) Rotor resistance estimation, (b) Stator resistance estimation.

and the faulty mode of operation of the DFIG and we can notice that generally, the mean squared error of states and parameters in faulty mode is relatively greater than those in the healthy mode because of the fault occurring suddenly during the operation, but we can always see the high performance of the moving horizon

Std (HGO) <sup>10</sup><sup>5</sup> Std (UKF) <sup>10</sup><sup>5</sup> Std (MHE) <sup>10</sup><sup>5</sup>

Variance (HGO) <sup>10</sup><sup>5</sup> Variance (UKF) <sup>10</sup><sup>5</sup> Variance (MHE) <sup>10</sup><sup>5</sup>

Std (HGO) <sup>10</sup><sup>4</sup> Std (UKF) <sup>10</sup><sup>4</sup> Std (MHE) <sup>10</sup><sup>4</sup>

Variance (HGO) <sup>10</sup><sup>5</sup> Variance (UKF) <sup>10</sup><sup>5</sup> Variance (MHE) <sup>10</sup><sup>5</sup>

HGO UKF MHE Healthy Faulty Healthy Faulty Healthy Faulty

Rs 4.73E06 8.02E06 8.66E04 9.65E04 1.11E07 1.27E07 Rr 1.77E09 1.79E05 9.36E05 1.14E04 5.71E08 7.33E08 Φds 5.70E04 6.31E04 9.02E05 9.23E05 21.0E04 11.0E04 Φqs 1.93E06 2.10E06 1.10E16 1.10E16 39.0E04 27.0E04 Φdr 2.08E08 16.00E04 6.47E06 6.87E06 246E04 210E04 Φqr 1.73E10 4.81E06 1.10E06 1.10E06 67.0E04 70.0E 04

Rs 350 7450 57.75 Rr 7.29 3940 41.38

Rs 1.26 550 0.033 Rr 0.0053 160 0.017

Rs 25 8208 270 Rr 4.86 278 26

Rs 0.62 6737 0.74 Rr 0.024 77.47 0.702

To verify the robustness, we have performed parametric variation on the observer in relation to the identified values. Figures 7 and 8 show the responses obtained when a rotor inductance variation of +50 and 50% is considered for the observer test. The robustness of the observers'scheme with respect to this parameter changes is clearly shown. In Figures 7 and 8, it is clearly shown that a +50 and 50% rotor inductance variation generates a high statistical difference on rotor and stator resistances for the unscented Kalman filter. For the high gain observer, that variation is much more felt on the rotor resistance on the both figures. Incontestably

the moving horizon estimation seems remain insensitive to the parametric

estimation on the others observers.

MSE values of nonlinear observers: HGO, UKF and MHE are compared.

Table 6.

Table 7.

Table 8.

38

General statistics of the three observers (healthy mode).

Wind Solar Hybrid Renewable Energy System

General statistics of the three observers (faulty mode).

5. Conclusion

Advanced Monitoring of Wind Turbine DOI: http://dx.doi.org/10.5772/intechopen.84840

Author details

41

and Adolphe Moukengue Imano

provided the original work is properly cited.

even in the presence of measurement noise.

Steve Alan Talla Ouambo, Alexandre Teplaira Boum\*

\*Address all correspondence to: boumat2002@yahoo.fr

Department of Physics, Faculty of Science, University of Douala, Cameroon

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

In this chapter, a general framework for the doubly fed induction generator has been presented in order to carry out a dynamic estimation of states and parameters of the DFIG. The DFIG parameters are largely influenced by different factors (for instance, temperature, magnetic saturation and eddy current) that is why it is necessary to develop techniques to estimate the changes of parameters. The proposed techniques are performed with high gain observer (HGO), unscented Kalman filter (UKF) and moving horizon estimation algorithms using noisy measurements. A comparison of the three estimation techniques has been made under different aspects notably, computation time and estimation accuracy, in two modes of operation of the DFIG, the healthy mode and the faulty mode. The MHE estimation technique has significantly lower estimation error and converges with fewer samples time than the HGO and the UKF. Whatever the mode of functioning, the simulation results showed that a good standard of performance could be obtained

#### Figure 8.

Robustness test. Rotor inductance variation (50%). (a) Rotor resistance estimation, (b) Stator resistance estimation.

considerably, but in regard to the other observers, the changes disturb their performance a lot as shown in the figures and that the MHE scheme is robust enough under parametric uncertainties.
