ECE 576 POWER SYSTEM DYNAMICS AND STABILITY Lecture 38 BCU Method Professor M.A. Pai Department of Electrical and Computer Engineering © 2000 University of Illinois Board of Trustees, All Rights Reserved

PEBS and BCU Method Example PEBS For 3-machine, 9-bus system using classical model compute for a fault at Bus 7 followed by clearing of line 7-5. Use . criterion for PEBS crossing. with fault at Bus 7 (faulted system) Construct the matrix and then set For this we delete the row and column corresponding to bus 7. Eliminate all buses except the internal nodes 10, 11 and 12. The result is 12 10 11 Note that since bus 7 is grounded, there is no connection between 10 and 12 to 11. Hence, the zero entries.

PEBS and BCU Method (contd) with lines 7-5 cleared (post-fault system) is first computed with lines 7-5 removed, and the rest of the steps are as before. Buses 1-9 are eliminated, resulting in This is different from the pre-fault 10 11 12

PEBS and BCU Method (contd) The initial rotor angles are The COA is calculated as where Hence, The post-fault s.e.p is calculated as

Computing From the entries in for faulted and post-fault systems, the appropriate and are calculated to put the equations in the form of

Solving for Since can be expressed in terms of the other and substituted in (1) which is then equivalent to Example Substitute (6) in (3) and (4) to get: (7) and (8) are solved for using Newton’s method with initial guess Then get from (6). (Pre-fault state)

Steps to compute Step 1 From entries in for faulted and post-fault system, compute the appropriate and terms to use in (1) and (2) (slide 4). Step 2 is given by . The path dependent integral term is evaluated with either approximately or using trapezoidal rule. Step 3 The faulted system (1) integrated and at each time step as well as are computed. Also the dot product is monitored. The plots of and and are shown. is reached at approximately 0.36 sec. Note that the zero crossing of occurs at approximately the same time.

PEBS crossing The monitoring of the PEBS crossing by PEBS Function zero crossing Time (sec) The monitoring of the PEBS crossing by

Plot of Energy Function Time (sec)

The Boundary Controlling u.e.p (BCU) Method This method provided another breakthrough in applying energy function methods to stability analysis after the work of Athay et al, in 1979, which originally proposed the controlling u.e.p method. The equations of the post-fault system can be put in the sate-space form as Faulted equations are unchanged.

The Boundary Controlling u.e.p (BCU) Method Hence, , is a valid energy function.

The Boundary Controlling u.e.p (BCU) Method The equilibrium points of the post-fault system lie in the subspace such that We have qualitatively characterized the PEBS as the hypersurfaces connecting the u.e.p’s. Consider the gradient system The gradient system has dimension m, which is half the order of the original system. It has been shown by Chiang et al that the region of attraction of the system is the union of the stable manifolds of u.e.p.’s lying on the stability boundary. If this region of attraction is projected onto the angle space , it can be characterized by where is an u.e.p on the stability boundary in the angle space.

The Boundary Controlling u.e.p (BCU) Method The stable manifold is defined as the set of trajectories that converge to Since the gradient is a vector orthogonal to the level surfaces constant in the direction of increased values of , the PEBS in the direction of decreasing values of can be described by the differential equations Hence, when the fault-on trajectory reaches the PEBS at corresponding to , we can integrate the set of equations for as where pertains to the post-fault system.

The Boundary Controlling u.e.p (BCU) Method This will take along the PEBS to the saddle points (u.e.p’s in figure depending on ). The integration of the gradient system requires very small time steps since it is “stiff”. Hence, we stop the integration until is minimum. At this point let If we need the exact , we can solve for using . as an initial guess. The BCU method is next explained in an algorithmic manner.

Algorithm for BCU 1. For a given contingency that involves either line switching or load/generation change, the postdisturbance s.e.p. is calculated as follows. The s.e.p. and u.e.p’s are solutions of the real power equations. Since it is sufficient to solve for with being substituted in (1) in terms of . Generally, the s.e.p. is close to the pre-fault e.p. Hence, using as the starting point (1) can be solved using the Newton-Raphson method.

Algorithm for BCU (contd) 2. Next compute the controlling u.e.p. as follows Integrate the faulted system and compute . at each time step. Determine when the PEBS is crossed at corresponding to using the criterion

Algorithm for BCU (contd) After the PEBS is crossed, the faulted swing equations are no longer integrated. Instead, the gradient system of the post-fault system is used. This is a reduced-order system in that only the dynamics are considered as explained earlier, i.e., for (1) is the Gradient system

Algorithm for BCU (contd) The gradient system is integrated while looking for a minimum of At the first minimum of the norm given by (1), and is a good approximation to the critical energy of the system. The value of is almost the relevant or the controlling u.e.p.

Critical Energy The exact u.e.p. can be obtained by solving and using as a starting point to arrive at Note that since is nonlinear, some type of minimization routine must be used to arrive at Generally, is so close to that it makes very little difference in the value of whether or is used.

Critical Energy (contd) 3. is approximated as Because of the path-dependent integral term in , this computation also involves approximation. Unlike computing from the faulted trajectory where was known, here we do not know the trajectory from the full system. Hence, an approximation has to be used. The most convenient one is the straight-line path of integration. is evaluated as

Critical Energy (contd) To compute we go back to the already-computed value of from the fault-on trajectory in the case of a fault, or the postdisturbance trajectory in the case of load/generation change, and find the time when In the case of a fault, this time instant gives and the system is stable if the fault is set to clear at a time In the case of load/generation change, the system is stable if for all t.

Example Compute using the BCU method for the 3-machine system. Use instead of the exact controlling u.e.p to compute The PEBS crossing is computed as in the earlier example. is obtained as 0.3445 sec and The gradient system for is given by where is given by with the parameters pertaining to the post-fault system. The equations are now integrated until is minimum. The minimum obtained is

Example (contd) At this point, With this, is calculated as 1.0815. Hence, from the fault-on trajectory in figure (slide 23), . is between 0.180 and 0.184 seconds. In this case, by both PEBS and BCU method is about the same.

Plot as a function of time in the gradient system Example (contd) Norm of f Time (sec) Plot as a function of time in the gradient system

Example (contd) Energy Time (sec)