1. ECE 530 – Analysis Techniques for Large-Scale Electrical Systems Prof. Hao Zhu Dept. of Electrical and Computer Engineering University of Illinois at Urbana-Champaign haozhu@illinois.edu 10/9/2014 1 Lecture 14: Quasi-Newton Power Flow Methods

2. Announcements HW 4 is due October 10 HW 5 is due October 21 Midterm exam is October 23 in class; closed book and notes, but one 8.5 by 11 inch note sheet and simple calculators allowed – Test covers up to dc power flow (today’s notes) 2

3. Ordering for Shorter Paths The paper 1990 IEEE Transactions on Power Systems paper “Partitioned Sparse A -1 Methods” (by Alvarado, Yu and Betancourt) they introduce ordering methods for decreasing the length of the factorization paths Factorization paths also indicate the degree to which parallel processing could be used in solving Ax = b by LU factorization – Operations in the various paths could be performed in parallel 3 Image from Alvarado 1990 paper

4. Computation with Complex and Blocked Matrices In the previous analysis we have implicitly assumed that the values involved were real numbers (stored as singles or doubles in memory) Nothing in the previous analysis prevents using other data structures for analyzing – Complex numbers would be needed if factoring the bus admittance matrix (Y bus ); this is directly supported in some programming languages and can be easily added to others; all values are complex numbers – Two by two block matrices are common for power flow Jacobian factorization; for this we use 2 by 2 blocks in the matrices and 2 by 1 blocks in the vectors 4

5. 2 by 2 Block Matrix Computation By treating our data structures as two by two blocks, we reduce the computation required to add fills substantially – Half the number of rows, and four times fewer elements Overall computation is reduced somewhat since we have four times fewer elements, but we do have more computation per element 5

6. 2 by 2 Block Matrix Example In the backward substitution we had 6 For i := n downto 1 Do Begin k = rowPerm[i]; p1 := rowDiag[k].next; While p1 <> nil Do Begin bvector[k] = bvector[k] – p1.value*bvector[p1.col]; p1 := p1.next; End; bvector[k] := bvector[k]/rowDiag[k].value; End;

7. 2 by 2 Block Matrix Example We replace the scalar bvector entries by objects with fields.r and.i (for the real and imaginary parts) and we replace the p1.value field with four fields.ul,.ur,.ll and.lr corresponding to the upper left, upper right, lower left and lower right values. The first multiply goes from bvector[k] = bvector[k] – p1.value*bvector[p1.col] to 7

8. 2 by 2 Block Matrix Example The second numeric calculation changes from bvector[k] := bvector[k]/rowDiag[k].value To Which can be coded by directly doing the inverse as 8

9. Sparse Matrix and Vector Methods Previous slides have presented sparse matrix and sparse vector methods commonly used in power system and some circuit analysis applications These methods are widely used, and have the ability to substantially speed up power system computations They will be applied as necessary throughout the remainder of the course 9

10. Quasi-Newton Power Flow Methods In this section we treat some modified versions of the Newton power flow (NPF) Since most of the computation in the NPF is associated with building and factoring the Jacobian matrix, J, the focus is on trying to reduce this computation In a pure NPF J is built and factorized each iteration Over the years pretty much every variation of the NPF has been tried; here we just touch on the most common Whether a method is effective can be application dependent – For example, in contingency analysis we are usually just resolving a solved case with a often small perturbation 10

11. Quasi-Newton Power Flow Methods The simplest modification of the NPF results when J is kept constant for a number of iterations, say k iterations – Sometimes known as the Dishonest Newton The approach balances increased speed per iteration, with potentially more iterations to perform There is also an increased possibility for divergence Since the mismatch equations are not modified, if it converges it should converge to the same solution as the NPF These methods are not commonly used, though some benefit can be achieved, such as not rebuilding/factoring J when the mismatch is low 11

12. Dishonest N-R Example, cont’d We pay a price in increased iterations, but with decreased computation per iteration 12

13. NPF (Honest) Region of for Two Bus Example Convergence Maximum of 15 iterations Red region converges to the high voltage solution, while the yellow region converges to the low voltage solution 13

14. Two Bus Dishonest ROC In this case being honest pays! At least with respect to the region of convergence (ROC) Maximum of 15 iterations 14

15. Quasi-Newton Power Flow Methods A second modification is to modify the step size in the direction given by the NPF – This is one we’ve already considered with the optimal multiplier approach The generalized approach is to solve what is known as the line search (i.e., a one-dimensional optimization) to determine  15

16. THE SINGLE DIMENSIONAL 0 16

17. Line Search We need a cost function, which is usually the Euclidean norm of the mismatch vector The line search is a general optimization problem for which there are many potential solution approaches – Determines a local optimum within some search boundaries – Approaches depend on whether there is gradient information available Aside from the optimal multiplier approach, which can be quite helpful with little additional computation, the convergence gain from determining the “optimal”  is usually more than offset by the line search computation 17

18. Decoupled Power Flow Rather than not updating the Jacobian, the decoupled power flow takes advantage of characteristics of the power grid in order to decouple the real and reactive power balance equations – There is a strong coupling between real power and voltage angle, and reactive power and voltage magnitude – There is a much weaker coupling between real power and voltage magnitude, and reactive power and voltage angle Key reference is B. Stott, “Decoupled Newton Load Flow,” IEEE Trans. Power. App and Syst., Sept/Oct. 1972, pp. 1955-1959 18

19. Decoupled Power Flow Formulation 19

20. Decoupling Approximation 20

21. Off-diagonal Jacobian Terms By assuming ½ the elements are zero, we only have to do ½ the computations (easiest to see with the block approach) 21 <<

22. Decoupled N-R Region of Convergence The high V solution ROC is actually larger than the standard NPF. Obviously this is not a good way to get the low V solution 22

23. Fast Decoupled Power Flow By continuing with our Jacobian approximations we can actually obtain a reasonable approximation that is independent of the voltage magnitudes/angles. This means the Jacobian need only be built/factorized once per power flow solution This approach is known as the fast decoupled power flow (FDPF) 23

24. Fast Decoupled Power Flow, cont. FDPF uses the same mismatch equations as standard power flow (just scaled) so it should have same solution The FDPF is widely used, though usually when we only need an approximate solution Key fast decoupled power flow reference is B. Stott, O. Alsac, “Fast Decoupled Load Flow,” IEEE Trans. Power App. and Syst., May 1974, pp. 859-869 Modified versions also exist, such as D. Jajicic and A. Bose, “A Modification to the Fast Decoupled Power Flow for Networks with High R/X Ratios, “IEEE Transactions on Power Sys., May 1988, pp. 743-746 24

25. FDPF Approximations 25 To see the impact on the real power equations recall

26. FDPF Approximations With the approximations for the diagonal term we get for the off-diagonal terms (k≠i ) Hence the Jacobian for the scaled real equations can be approximated as –B 26

27. FPDF Approximations For the reactive power equations we also scale by V i Similarly, the Jacobian off-diagonals 27 http://nptel.ac.in/courses/108107028/module2/lecture9/lecture9.pdf

28. FDPF Approximations And for the reactive power Jacobian diagonal As derived the real and reactive equations have a constant Jacobian equal to –B – Usually modifications are made to omit from the real power matrix elements that affect reactive flow (like shunts) and from the reactive power matrix elements that affect real power flow, like phase shifters – We’ll call the real power matrix B’ and the reactive B” 28

29. FDPF Approximations It is also common to flip the sign on the mismatch equation, by changing it from (summation – injection) to (injection – summation) – Other modifications on the B matrix have been presented in the literature (such as in the Bose paper) Hence we have 29

30. FDPF Three Bus Example Use the FDPF to solve the following three bus system 30

31. FDPF Three Bus Example, cont’d 31

32. FDPF Three Bus Example, cont’d 32

33. FDPF Region of Convergence 33

34. FDPF Cautions The FDPF works well as long as the previous approximations hold for the entire system With the movement towards modeling larger systems, with more of the lower voltage portions of the system represented (for which r/x ratios are higher) it is quite common for the FDPF to get stuck because small portions of the system are ill-behaved The FDPF is commonly used to provide an initial guess of the solution or for contingency analysis 34