ECE 530 – Analysis Techniques for Large-Scale Electrical Systems

1. ECE 530 – Analysis Techniques for Large-Scale Electrical Systems Lecture 15: DC Power Flow Model; Sensitivity Analysis Prof. Hao Zhu Dept. of Electrical and Computer Engineering University of Illinois at Urbana-Champaign haozhu@illinois.edu 10/14/2014

2. Announcements • 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 (before sensitivity)

3. Fast Decoupled Power Flow • Fast decoupled power flow (FDPF) further approximates Jacobian to be independent of the voltage magnitudes/angles • This means the Jacobian need only be built/inverted once per power flow solution • FDPF uses the same mismatch equations as standard power flow (just scaled) so it should have same solution, and so it DPF • The FDPF is widely used, though usually when we only need an approximate solution

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

5. 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

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

7. 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”

8. 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 (D. Jajicic and A. Bose, “A Modification to the Fast Decoupled Power Flow for Networks with High R/X Ratios,” IEEETransactionsonPowerSys., May 1988) • Hence we have

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

10. FDPF Three Bus Example, cont’d

11. FDPF Three Bus Example, cont’d

12. FDPF Region of Convergence

13. 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

14. “DC” Power Flow • The “DC” power flow makes the most severe approximations: • completely ignore reactive power • This makes the power flow a linear set of equations, which can be solved directly (no convergence issue!) • The term dc power flow actually dates from the time of the old network analyzers (going back into the 1930’s) • Not to be confused with the inclusion of HVDC lines in the standard NPF P sign convention is generation is positive

15. DC Power Flow References • A nice formulation is given in the Power Generation and Control book by Wood and Wollenberg (with the 3rd edition out) • The August 2009 paper in IEEE Transactions on Power Systems, “DC Power Flow Revisited” (by Stott, Jardim and Alsac) provides good coverage • T. J. Overbye, X. Cheng, and Y. Sun, “A comparison of the AC and DC power flow models for LMP Calculations,” in Proc. 37th Hawaii Int. Conf. System Sciences, 2004, compares the accuracy of the approach

16. DC Power Flow Example Example from Power System Analysis and Design, by Glover, Sarma, Overbye, fifth edition

17. DC Power Flow in PowerWorld • PowerWorld allows for easy switching between the dc and ac power flows To use the dc approachin PowerWorldselect Tools,Solve, DCPower Flow Notice thereare no losses Notice with the dc power flow all of the voltage magnitudes are 1.0 per unit. PowerWorld case is Bus5_GSO solved with the dc power flow

18. Sensitivity Analysis • System description and notations • Motivation for the sensitivity analysis • From sensitivity to distribution factors • Definitions of the various distribution factors • Analysis of the distribution factors • Distribution factor applications

19. Notation • We consider a system with N buses and L lines given by the set given by the set • Some authors designate the slack as bus zero; an alternative approach, that is easier to implement in cases with multiple islands and hence slacks, is to allow any bus to be the slack, and just set its associated equations to trivial equations just stating that the slack bus voltage is constant • We may denote the kth transmission line or transformer in the system, , as ,), from node to node

20. Notation, cont. • We’ll denote the real power flowing on line from bus to bus as • The vector of active power flows on the L lines is: • The bus real and reactive power injection vectors are(note we use lower-case p/q injection in later analysis) p

21. Notation, cont. • The series admittance of line is , and we define the matrix • We define the LN incidence matrix: where the element j of vector ai is nonzero , whenever line is coincident with bus j. Hence matrix A is quite sparse, with two nonzeros per row

22. Analysis Example: Available Transfer Capability • The power system available transfer capability (ATC) is defined as the maximum additional MW that can be transferred between two specific areas, while meeting all the specified pre- and post-contingency system conditions • ATC impacts measurably the market outcomes and system reliability and, therefore, the ATC values impact the system and market behavior • A useful reference on ATC is Available Transfer Capability Definitions and Determination from NERC, June 1996 (available freely online)

23. ATC and Its Key Components • Total transfer capability (TTC ) • Amount of real power that can be transmitted across an interconnected transmission network in a reliable manner, including considering contingencies • Transmission reliability margin (TRM) • Amount of TTC needed to deal with uncertainties in system conditions; typically expressed as a percent of TTC • Capacity benefit margin (CBM) • Amount of TTC needed by load serving entities to ensure access to generation; typically expressed as a percent of TTC

24. ATC and Its Key Components • Uncommitted transfer capability (UTC) UTC =TTC – existing transmission commitments • Formal definition of ATC is ATC =UTC – CBM – TRM • We focus on determining Um,n, the UTC from node mto node n • Um,n is defined as the maximum additional MW that can be transferred from node m to node nwithout violating any limit in either the base case or in any post-contingency conditions

25. UTC Evaluation no limit violation for the base casej = 0 and each contingency case j = 1,2 … , J

26. Conceptual Solution Algorithm • Solve the initial power flow, corresponding to the initial system dispatch (i.e., existing commitments); Initialize the change in transfer  t(0) = 0, k =0; set step size d ; j is used to indicate either the base case (j=0) or a contingency, j = 1,2,3…J • Compute  t(k+1)=  t(k) + d • Solve the power flow for the new  t(k+1) • Check for limit violations: if violation is found set U(j)m,n =  t(k) and stop; else set k = k+1, and goto 2

27. Conceptual Solution Algorithm, cont. • This algorithm is applied for the base case (j=0) and each specified contingency case, j=1,2,..J • The final UTC, Um,n is then determined by • This algorithm can be easily performed on parallel processors since each contingency evaluation is independent of the other

28. Five Bus Example: Base Case PowerWorld Case: B5_DistFact

29. Five Bus Network Data

30. Five Bus UTC • We evaluate U2,3 using the previous procedure • Gradually increase generation at Bus 2 and load at Bus 3 • We consider the base case and a single contingency with line 2 outaged (between 1 and 3): J=1 • Simulation results show for the base case that • And for the contingency that • Hence

31. UTC: Base Case

32. UTC: Contingency Case

33. Computational Considerations • Obviously such a brute force approach can run into computational issues with large systems • Consider the following situation: • 10 iterations for each case • 6,000 contingencies • 2 seconds to solve each power flow • It will take over 33 hours to compute a single UTC for the specified transfer direction from mto n. • Consequently, there is an urgent need to develop fast tools that can provide satisfactory approximations