by Eustaquio A. Martínez 1 , Tiaraju Asmuz Diverio 2 & Benjamín Barán 3

1. Solving Electrical Power Load Flow Problems using Intervals • by • Eustaquio A. Martínez1, • Tiaraju Asmuz Diverio2 • & • Benjamín Barán3 1amartinez@politec.une.edu.py 2diverio@inf.ufrgs.br 3bbaran@cnc.una.py Facultad Politécnica - UNE Dpto. de Informática - UFRGS CNC - UNA Paraguay Brazil Paraguay Validated Computing 2002

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

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

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

5. Electrical Power Load Flow Problem 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 Generally, the problem may be written as: n is the problem’s size is the group of the bus bars adjacent to and itself. Validated Computing 2002

6. Interval Approach Interval Newton Method where and 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 . Validated Computing 2002

7. 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 Domain for a known solution  10º  < 1 where heuristic for feasible solution Validated Computing 2002

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

9. Interval Newton/Generalized Bisection Algorithm Validated Computing 2002

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

11. Partition Algorithm Algorithm 1 Validated Computing 2002

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

13. Master’s Process Algorithm Validated Computing 2002

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

15. 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. Validated Computing 2002

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

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

18. Speed - Up Validated Computing 2002

19. 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). Validated Computing 2002

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