Extended Linear Factors for Power System Contingency Analysis

1. Extended Linear Factors for Power System Contingency Analysis Henry Louie Power Affiliates Program University of Illinois at Urbana-Champaign May 14, 2004

2. Outline • Purpose • Background (CTDF, PTDF) • Extended Factors • Example • Conclusions and Future Work

3. Purpose • The reclosure of a line after an outage can cause damaging torques to generator shafts. • Current reclosure “rules” for this involve the angle across the open breaker. • Can we develop a rapid way of finding the angle across the open breaker and the torque created by a reclosure?

4. Sequence of Events • A fault occurs on line i-j. • Circuit breakers operate to clear the fault. • Currents instantaneously redistribute as the change in topology occurs. • The generators experience line-outage torque. • Voltages at bus i and bus j are no longer approximately equal. • A standing phase angle (SPA) develops. • The fault clears.

5. Purpose • Is it safe to close the breaker under this SPA level? • Example of acceptable SPA closing levels • 20º for 500 kV • 40º for 230 kV • 60º for 115 kV • There is no universal relationship between SPA and torque.

6. Purpose • Develop a way of rapidly calculating the SPA. • Develop a way of rapidly calculating possible torques based on SPA for a given topology and operating point. • Linear approximations (Distribution Factors) of power systems have been used in contingency analysis for years. • Modify these distribution factors to approximate transient current/SPA. Use the current to calculate torque.

7. What are Distribution Factors? • We are interested in analyzing the impacts of system changes. • The power system is analyzed as constant power loads and sources which are nonlinear in nature. • Nonlinear systems are not easy to analyze. • is linear. • Use an approximate relationship between current and power:

8. Current Transfer Distribution Factors (CTDF) • Calculates how a current injection is distributed throughout the network. • Formulation is exact if all loads and sources are constant-current. • Assume a constant slack bus voltage (V1). • Transmission lines are assumed to be the π equivalent model.

9. CTDF • Consider an n bus system, m generators. • Z is the inverse of the admittance matrix. • Specify the vector I (current injected at each bus). • V1 is constant, so we remove its equation (I1 can no longer be specified).

10. CTDF • The first row and column are eliminated. • Note this Z’ matrix is different from the previous Z matrix.

11. CTDF • Current on line i-j is: • is the primitive line impedance. • Assume an injection at bus k only • is the CTDF • This is the amount of the current injection k on line i-j

12. Line Outage Distribution Factor (LODF) • Approximates the change in line flow due to a line outage. • Find an injection that zeros the current flow in the line to be outaged. • Remove the line while the flow is zero. • Remove the injection from the system without the line.

13. LODF

14. Line Outage Generation Factor (LOGF) • Include shunts. • Add the generator transient reactance to the Y matrix to provide a line for which we want to compute the current change. This expands the original Y matrix to (n+m) x (n+m). • Enforce the constant internal voltage at each generator – this reduces the expanded Y matrix back to n x n as each fixed internal voltage is eliminated. • Compute LODFs using the updated n x n matrix. • Use LODFs to predict the generator currents.

15. LOGF • Three nodes have been added to the system

16. Line Closure Distribution Factor (LCDF) • Approximates the change in line flow due to the addition of a line. • Assume line i-j is already out. • Inject current at bus i to zero the voltage drop across i-j. • Close the line while the voltage drop is zero. • Remove the injection from the system with the line.

17. LCDF • The current injected to zero the voltages on line is • The change in current on line a-b after the line is added and the injection removed is

18. Line Closure Generation Factor (LCGF) • Include shunts. • Add the generator transient reactance to the Y matrix to provide a line for which we want to compute the current change. This expands the original Y matrix to (n+m) x (n+m). • Enforce the constant internal voltage at each generator – this reduces the expanded Y matrix back to n x n as each fixed internal voltage is eliminated. • Compute LCDFs using the updated n x n matrix. • Use LCDFs to predict the generator currents.

19. Line Outage Angle Factor (LOAF) • Approximate the angle difference across an open line • Change in bus voltages due to the line outage • Base case voltage difference plus outage voltage difference

20. LOAF Assumptions No Resistance

21. Example System

22. LOGF Result

23. LOAF Result

24. LCGF Result

25. Conclusions/Future Work • The extension of traditional distribution factors can be applied to generator currents and angles. • LOAF is an accurate approximation. • How does the linearized current compare to the time-domain current? • Correlate torque with the generator current.

26. QUESTIONS