Modelling & ctrl of PE converters-01.pptx

Modelling & ctrl of PE converters

1. Session – 09 Classical Averaged Model

2. Objectives At the end of this lecture, student will be able to explain: Basics of averaging methodology and states some theoretical fundamentals. Methodology of obtaining small-signal and large-signal averaged models

3. Contents Analysis of Averaging Errors Exact Sampled-Data Model Exact Sampled-Data Model: Example Relation Between Exact Sampled-Data Model and Exact Averaged Model

4. Analysis of Averaging Errors The error between the average of a given switched model state-space solution and the averaged model state-space solution (on the same time window) is easy to put into light by numerical simulation. The error signal is denoted by ε(t ). An estimation of this kind of error can also be performed analytically by referring to the sampled-data model.

5. Analysis of Averaging Errors Averaged model output vs. average of switched model output

6. Exact Sampled-Data Model Let us consider a linear system that switches between two configurations. α being the duty ratio . It can be described by the following set of differential equations : ---------(4.16) which is valid for each switching period [ kT , (k + 1)T) where k ∈ N . It is proposed that system (4.16 ) be solved by integrating each configuration and postulating the state variables’ continuity.

7. Exact Sampled-Data Model Let us introduce the general notation of the transition matrix φ (t ) = exp (A . t ) The transition matrices associated with the two configurations are: φ 1 (t ) = exp (A 1 . t) and φ 2 (t ) (t ) = exp (A 2 . t ). For the first configuration one can write: Integrating the equation: --(4.17) where I n is the identity matrix of the same dimension as A.

8. Exact Sampled-Data Model For the second configuration the result is given directly (algebra is the same ): --(4.18) By substituting the value of x[(k + α)T] in Eq. (4.17 ) into Eq. (4.18) results in the recurrent equation of the form: Eq. (4.18) and (4.19) are forms of the Exact Sampled-data Model . Eq.(4.19 ) is very complex and difficult to manipulate. In order to avoid cumbersome mathematical developments certain systems having simple representations will be taken into account. This can be obtained from Eq. (4.18) by employing an adequate change of variable.

9. Exact Sampled-Data Model: Example Determine the sampled-data model of the dynamic of system described in the figure:

10. Exact Sampled-Data Model: Example The system of Eq. (4.20) corresponds to the dynamic depicted in Fig . ---Eq.4.20 Integrating the first equation of (4.20), one obtains: ---Eq.4.21 For the second configuration: ---Eq.4.22

11. Exact Sampled-Data Model: Example By substituting Eq. (4.21) into Eq. (4.22), one obtains ---Eq.4.23 (Sampled data topological model) This is the output of the switched model at switching instants in a recurrent form . The computation of matrices φ 1 and φ 2 can be simplified more or less satisfactorily, by their first-order expansions, respectively: ----Eq.4.24 Where I is the identity matrix of the same dimension as A1 and A2.

12. Exact Sampled-Data Model: Example Introducing the simplified expressions (4.24) into Eq. (4.23) yields the first-order approximated sampled-data model . Task

13. Relation Between Exact Sampled-Data Model and Exact Averaged Model The Eq . (4.23), provides the solution of the switched model, contains matrix products . In the general case a product of matrices is not commutative. i.e., the following relation generally holds between the two state-space matrices: A 1 . A 2 ≠ A 2 . A 1

14. Relation Between Exact Sampled-Data Model and Exact Averaged Model Case of commutative matrices: the exceptions where A 1 . A 2 = A 2 . A 1 In such cases the matrix exponentials are also switching, i.e., φ 1 . φ 2 = φ 2 . φ 1 . Therefore, the following relation holds: ---Eq.4.25 if applied to the exact switched model (4.23), gives -----Eq.4.26 where for matrix φ M one obtains successively

15. Relation Between Exact Sampled-Data Model and Exact Averaged Model where for matrix φ M one obtains successively: more synthetically represented as: ---- Eq.4.27 The averaged model of the system Where (the state matrix of the averaged model) .

16. Relation Between Exact Sampled-Data Model and Exact Averaged Model At sampling moments it holds that This is why the model expressed below is called the exact averaged model. -----Eq.4.28 Its dynamic is represented in Fig.  

17. Relation Between Exact Sampled-Data Model and Exact Averaged Model General case: The assumption of matrices being commutative does not hold in the quasi-totality of power converters. In the general case, the matrix product between A 1 and A 2 is not commutative; hence, their exponentials are not commutative : Therefore, relation (4.28) becomes: ---Eq.4.29 This is approximated averaged model

18. Relation Between Exact Sampled-Data Model and Exact Averaged Model Its trajectory is no longer passing through the points of the sampled data model as shown in Fig ., but it will be an averaged trajectory more or less close to the sliding average of the exact trajectory . An issue is to quantify the error introduced by the approximation in this model in relation to the exact sampled-data model. This represents an upper bound of the error between the output of the averaged model and the average of the switched model . Its absolute value is: ---Eq.4.30

19. Relation Between Exact Sampled-Data Model and Exact Averaged Model The complete computation of error expressed by (4.30) is not trivial and can only be done in numerical form. For the sake of simplicity a second-order approximation of the matrix exponential is employed , In order to express the matrix φ m and the product  

20. Relation Between Exact Sampled-Data Model and Exact Averaged Model One obtains successively:

21. Relation Between Exact Sampled-Data Model and Exact Averaged Model One can remark that the matrix error Err expressed by Eq. (4.30) between the second-order developments is reduced to the matrix E as follows : ----Eq.4.31 Equation (4.31) shows that if matrices A1 and A2 are commutative then the sampled-data and averaged models are confused. In the general case, the smaller the norm of the error matrix in relation to the norms of the other state matrices weighted by their associated enabling times within a switching period, αT and (1 - α)T, respectively, the more precise approximate model is obtained.

22. Relation Between Exact Sampled-Data Model and Exact Averaged Model This further requires that time T be small: – The converter to operate at high frequency – The duty ratio α be close to one or zero If these assumptions hold, this leads to smaller ripple of the state variables.

23. Summary The averaged model can be seen as an “ideal”, operating at infinite frequency. The switched model, operating at finite frequency, exhibits, besides the variables’ ripples. An average is different from the averaged model solution.