2) PI-controller (Kp=2.5, TI=5.0 ) and PID-controller (Kp=2.5, TI=5.0, K D =1,T D =0.01 )

1. PSERC PSERC PSERC PSERC PSERC PSERC PSERC PSERC PSERC PSERC Load Exciter PSERC ProjectContribution Allocation for Voltage Stability Assessment In Deregulated Power Systems----A new application of Bifurcation Analysis Dr. Huang (PI), Kun. Men, N.Nair (Students)Texas A&M University PSerc Review Meeting Albany, NY, May 28, 2002 Our research focus: Topic 2-continued: 3) b) The impacts of limits of the Generator 1.How to allocate the contribution/responsibility by bifurcation analysis. 2. How the capacity limits and parameters of the system components influence the bifurcation points? • Considered capacity limits and parameters 1) The parameters of the generator, the controller 2) The parameters of the transmission line 3) The impact of the reserve limits on the voltage stability a. The impact of the size of the exciter b. The impact of other limits of the system • Three typical bifurcation points are found and considered: 1) Hopf bifurcation point, we denote it as A 2) Saddle node, we denote it as B 3) Singularity induced bifurcation point, we denote is as C · Section E–F–G of the curve shows limit due to stator (armature) current. This section is a portion (arc) of a circle that has its center in the origin of – (MW-Mvar) coordinates of the generator. · Section D–E of the curve shows limit due to field current limitations. This is a portion (arc) of a circle that has its center on the Y axis (Mvar) and shifted from the origin by a value proportional with the machine short-circuit ratio (SCR). · Section H–G of the curve shows limit due to over-heating of the stator core end when the generator is under-excited in conditions of leading PF, when the generator is absorbing Mvars A simple one generator and one load bus system is used to demonstrate our analysis. (It include three basic part: Exciter & Regulator, Generator and Transmission part.) In the above system, we assume that the power factor of the load is constant as the load changes. We also assume that the voltage dynamic is decoupled from the angle dynamic, which is well behaved, and the angle dynamic can be approximated away. Topic 1: Three widely used controllers: Topic 2-continued: We can observe three basic patterns by the ordering of A, B, C : 3) The impacts of the capacity limits 1) P-controller ( , Efd0 is rescheduled as load change to keep Eg as Er) Three Basic Patterns characterized by points A B C inside or outside of DEFGH: a) The impact of the size of the exciter When KP=2.5, 5, 10, the below figures shows that how the eigenvalue vary with the change of KP When Efd hit the limit Efd max, the input Efd of the exciter will keep as Efd max, there are two basic patterns by upper or lower intersections: 1)A<B<C. When P∈(A,B), both EigC and EigT are positive; when P∈(B,C), only the EigT is positive. From the parameter analysis, we can conclude that the voltage collapse is due to both controller and transmission when P∈(A,B). The voltage collapse is only caused by transmission part when P∈(B,C). In this case, [A, C] is the unstable area, and A defines the dynamic stability margin boundary—the valid stability margin of the system. 2) A<C<B. When P∈(A,C), both EigC and EigT are positive; when P∈(C,B), only the EigC is positive. From the parameter analysis we can conclude that the voltage collapse is due to both controller and transmission when P∈(A,C). The voltage collapse is caused by controller when P∈(C,B). In this case, [A, B] is the unstable area, and A defines the dynamic stability margin boundary —the valid stability margin of the system. 3) A disappears and B<C, only the EigT is positive when P∈(B,C). Thus, the voltage collapse is only due to transmission when P∈(B,C). In this case, [B, C] is the unstable area, and B is the dynamic stability margin boundary —the valid stability margin of the system. (Note:To our experience, pattern 1 usually appears with big ratio of xd/x, and pattern2 usually appears with small ratio of xd/x. Also note that when a regulator is not functioning, C1 is always at the lower curve and there is no A1) Lower intersection pattern 1: . Right figure show that how the voltage changes with P when there is a limit Efd max=2.0. In this basic pattern, the valid stability margin depends on the ordering. i) Three bifurcation points A, B and C are all located outside the boundary D-E-F-G-H, so F is the stability margin (Here we neglect the limit of the prime motor—Pm, it is clear that if Pm is less then F, then Pm will be stability margin). ii) A or B appear within the boundary D-E-F-G-H, C is out of this boundary, so F replace C to be the steady-state stability margin, the point A or B (when A disappear) will be the dynamic stability margin—the valid stability margin. iii) A,B and C are all within the boundary D-E-F-G-H, the point A or B (when A disappear) will be the dynamic stability margin, and bifurcation point C or B(When C is in the lower part of the PV curve) may be the steady-state stability margin.  In these figures, note that Kp has little influence on one of the eigenvalues (denoted by EigT), while Kp has a substantial impact on the other eigenvalue (denoted by EigC). We should also note that B will vary with the change of KP; B Pmax with KP ; and B>C when KP>5.25; B A when KP=1.895; when KP<1.895, A will disappear; and B 0.735 with KP 0 . Basic pattern i) usually appears with three commonly used regulator when they are well tuned and the x and xd are not very big. When the regulator is not well tuned, the size of exciter is too small or the transmission line is too long, basic pattern ii) and iii) may appear).Basic pattern iii) usually appear with P- regulator (Efd0 is constant), even the transmission line is not very long and the exciter size is not too small. A<D, then the dynamic stability margin is A  A>D and C1<D, then the dynamic stability margin is D  A>D and C1>=D, then the dynamic stability margin is C1. For the steady stability margin, there are also several possibilities: C<D, then the steady-state margin is C C>D and C1<D, then the steady-state margin is D C>D and C1>=D, then the steady-state margin is C1. 2) PI-controller (Kp=2.5, TI=5.0 ) and PID-controller (Kp=2.5, TI=5.0, KD=1,TD=0.01 ) Conclusion: Upper intersection pattern 2: Topic 2: Influence Of Capacity Limits and Other System Parameters On Stability: PSERC From topic 1, we know that A or B(When A disappears) will be the valid stability margin when the system dose not hit the limits, and dynamic analysis must be applied. From topic 2, we know that D, B1 or F may be the valid stability margin instead of A or B when the system hits the limits; in this case, steady state analysis is good enough. 1) The parameter of the generator (xd) (KP=2.5, TI=5.0, KD=1.0, TD=0.01) In this basic pattern, there are also several possibilities for the dynamic stability margin:  A<D, then the valid (dynamic) stability margin is A  A>D ,then the valid stability margin is B1 For the steady stability margin: C<D, then the steady-state margin is C C>D, then the steady-state margin is B1 Note that the stability margin of P-regulator and PID-regulator are nearly the same, both bigger than the margin of the PI-controller. But for the dynamic response, PID and PI have no steady-state error after a small disturbance; the response speed of PI and PID are faster than P-regulator; and PID have less oscillation than PI-regulator, so PID is the best regulator. a) P-controller: When all of below three conditions are satisfied, the steady-state analysis is good enough. Otherwise the dynamic analysis should be applied. b) PI-controller: 1) The value of x+xd is not very big . 2) The regulator is not P-regulator with a constant Efd0. 3) The regulator is well tuned. The detailed algorithm is under development. c) PID-controller: When A determines the valid stability margin, the regulator controller is responsible for the voltage collapse; Otherwise the size of the exciter determines the valid stability margin and the exciter (or a device such as SVC size) is responsible. When the valid stability margin is C, the regulator controller determines the steady-state stability margin, and the regulator is responsible; Otherwise (B1 determines the margin) the the size of the exciter determines the stability margin, and the exciter is responsible. It is clear that D and B1 will increase with bigger Efd max, thus, bigger size of exciter contributes to both the increased dynamic and steady-state stability margins.To keep the same stability margin, bigger xd need bigger exciter size. 2) The parameter of the transmission (x) When P∈(A,C), both the eigenvalue EigT and EigC are positive; when P∈(C,B), only the eigenvalue EigC is positive. We can see that the shorter transmission line will increase the stability margin for all regulators. We can see that the bifurcation locations of PI and PID controller are very similar to the P controller as Kp→∞ , which is included in the basic pattern 2 we mentioned later. P-regulator(Efd0 is rescheduled)