2. Validated Computing 2002 by Eustaquio A. Martínez 1, Tiaraju Asmuz Diverio 2 & Benjamín Barán 3 1 amartinez@politec.une.edu.py 2 diverio@inf.ufrgs.br 3 bbaran@cnc.una.py Facultad Politécnica - UNE Dpto. de Informática - UFRGS CNC - UNA Paraguay Brazil Paraguay Solving Electrical Power Load Flow Problems using Intervals

3. Validated Computing 2002 Summary Motivation Electrical Power Load Flow Problem Interval approach Solving Sequentially Solving Parallely Experimental Results Conclusions

4. Validated Computing 2002 Motivation Load Flow Problem Unknown: Electrical System    1 2 3 4 5 6

5. Validated Computing 2002 Electrical System Model Proposition Interval Arithmetic All solutions in a domain (operating points)

6. Validated Computing 2002 Electrical Power Load Flow Problem Generally, the problem may be written as: is the group of the bus bars adjacent to and itself. The Electrical Power Load Flow Problem can be formulated as a quasilinear equation system is the admittance matrix (problem’s parameters) and the electric current vector and the unknown n is the problem’s size

7. Validated Computing 2002 Interval Approach and Interval Newton Method where The system can be written as a linear interval system : is the interval vector where the solutions is expected to be found is an inner vector of is the unknown interval vector which is expected to contain the solutions is the interval extension of Jacobian matrix of in.

8. Validated Computing 2002 Computed, the iterative formula of the interval Newton Method for a system with n variables is: If there are not a solutions in The problem’s matrix form is: where  10º  < 1 heuristic for feasible solution Domain for a known solution where

9. Validated Computing 2002 Solving Sequentially Interval Newton/Generalized Bisection Algorithm Interval Newton/Generalized Bisection Algorithm Self Validated Results Low Flow Problem

10. Validated Computing 2002 Interval Newton/Generalized Bisection Algorithm

11. Validated Computing 2002 Solving Parallely Self Validated Results Low Flow Problem

12. Validated Computing 2002 Partition Algorithm Algorithm 1

13. Validated Computing 2002 Paralleling Scheme Master Esclavo 1 Slave 1 Esclavo 2 Slave 2 Esclavo 3 Slave 3 Slave 4

14. Validated Computing 2002 Master’s Process Algorithm

15. Validated Computing 2002 Slaves Process Interval Newton/Generalized Bisection Algorithm - Modified

16. Validated Computing 2002 Computing Environment 10 Mbps local area network; 5 personal computers (Pentium II, 400MHz, 32 MB RAM, Linux SO) ; One acts as the master (NFS, NIS and MPI) ; Four work as slaves.

17. Validated Computing 2002 Sequential - Punctual (N-R) Experimentals Results Sequential - Interval (IN/GB)

18. Validated Computing 2002 Parallel - 3 processors Parallel - 5 processors

19. Validated Computing 2002 Speed - Up

20. Validated Computing 2002 Conclusions Though computationally more expensive, this interval solution of the electrical load flow problem has advantages if compared to traditional methods: It proves the inexistence of solutions (feasible solutions) in a given domain without a solution. If there are several feasibles solutions in a given interval, the method can find all the solutions. It allows to control the precision of each solution, directly on the unknown value, rather than through related variables (such as power mismatch).

21. Validated Computing 2002 ¡¡Thank you Very Much!! Tiaraju Asmuz Diverio diverio@inf.ufrgs.br U.N.E. - Paraguay