1. ECE 530 – Analysis Techniques for Large-Scale Electrical Systems
Lecture 24: Numerical Integration;
Power System Equivalents
Cecilia Klauber (Guest Lecturer)
Dept. of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign
12/2/2015

2. Numerical Integration
Power system application: transient stability
Power systems are dynamic state-space systems, represented by DAEs and ODEs
Typically the power flow solution represents an equilibrium point
At some point a contingency occurs, perturbing the state away from the steady-state equilibrium point
Time domain simulation is used to determine whether the system returns to the equilibrium point

3. Methods One-step methods Multistep methods Multi-Rate Methods
Forward Euler’s Method
Runge-Kutta Methods
Multistep methods
Adams-Bashforth Method
Multi-Rate Methods
All these methods have numerical stability concerns depending on the stepsize
Implicit Methods

4. Multistep Methods
Euler's and Runge-Kutta methods are single step approaches, in that they only use information at x(t) to determine its value at the next time step
Multistep methods take advantage of the fact that using we have information about previous time steps x(t-Dt), x(t-2Dt), etc
These methods can be explicit or implicit [dependent on x(t+Dt) values]; we'll just consider the explicit Adams-Bashforth approach

5. Multistep Motivation
In determining x(t+Dt) we could use a Taylor series expansion about x(t)
(note Euler's is just the first two terms on the right-hand side)

6. Adams-Bashforth
What we derived is the second-order Adams-Bashforth approach. Higher-order methods are also possible, by approximating subsequent derivatives. Here we also present the second- and third-order Adams-Bashforth

7. Adams-Bashforth Versus Runge-Kutta
The key Adams-Bashforth advantage is the approach only requires one function evaluation per time step while the RK methods require multiple evaluations
A key disadvantage is when discontinuities are encountered, such as with limit violations;
Another method needs to be used until there are sufficient past solutions
They also have difficulties if variable time steps are used

8. Numerical Instability
All explicit methods can suffer from numerical instability if the time step is not correctly chosen for the problem eigenvalues
Values are scaled by the time step; the shape for RK2 has similar dimensions but is closer to a square. Key point is to make sure the time step is small enough relative to the eigenvalues
Image source:

9. Stiff Differential Equations
Stiff differential equations are ones that exhibit a wide-range of time varying dynamics from “very fast” to “very slow.”
Stiffness is associated with solution efficiency: in order to account for these fast dynamics we need to take quite small time steps

10. Multi-Rate Methods
Multi-rate methods can be used with sets of differential equations in which different parts of the system have different speeds
Use small time steps for the fast parts of the system
Use larger time steps for the slower parts of the system
Subsystems need to be sufficiently decoupled
A good power system reference: J. Chen and M. L. Crow, "A Variable Partitioning Strategy for the Multirate Method in Power Systems," IEEE Trans. Power Systems, vol. 23, pp , 2008

11. Multi-Rate Methods
At each macro step the slow variables are integrated, at each micro step the fast variables are integrated
Macro variables can be interpolated during the micro steps
Source: J. Chen and M. L. Crow, "A Variable Partitioning Strategy for the Multirate Method in Power Systems," IEEE Trans. Power Systems, vol. 23, pp , 2008

12. Multi-Rate Example: Transient Stability
The power system transient stability problem is usually solved with a time step of ¼ or ½ cycle
Some subsystems can have much faster time constants
When starting induction machines can exhibit very fast (relative to the time step) transients
Some types of exciters can have very fast time constants, in which the dynamics only come into play during close by faults

13. Implicit Methods
Implicit solution methods have the advantage of being numerically stable over the entire left half plane
Only methods considered here are the is the Backward Euler and Trapezoidal

14. Implicit Methods
The obvious difficulty associated with these methods is x(t) appears on both sides of the equation
Easiest to show the solution for the linear case:

15. Backward Euler Example
Returning to the cart example

16. Backward Euler Example
Results with Dt = 0.25 and 0.05
time
actual x1(t)
x1(t) with Dt=0.25
x1(t) with Dt=0.05 1
0.25
0.9689
0.9411
0.9629
0.50
0.8776
0.8304
0.8700
0.75
0.7317
0.6774
0.7185
1.00
0.5403
0.4935
0.5277
2.00
-0.416
-0.298
Note: Just because the method is numerically stable doesn't mean it is doesn't have errors! RK2 is more accurate than backward Euler for this case.

17. Trapezoidal Linear Case
For the trapezoidal with a linear system we have

18. Trapezoidal Example
Results with Dt = 0.25, comparing between backward Euler and trapezoidal
time
actual x1(t)
Backward Euler
Trapezoidal 1
0.25
0.9689
0.9411
0.9692
0.50
0.8776
0.8304
0.8788
0.75
0.7317
0.6774
0.7343
1.00
0.5403
0.4935
0.5446
2.00
-0.416
-0.298

19. Overview of Commercial Transient Stability Algorithm
Most commercial packages use an explicit approach, such as a second order Runge-Kutta because of 1) the large number of nonlinear models and 2) the number of limit violations that are encountered
Power flow solution provides the initial starting point
Initial state variables are determined by back solving from the power flow conditions
Example: TGOV1 governor initialization

20. Overview of Commercial Transient Stability Algorithm
For simplicity showing Euler's, which is not used
While (t <= tend) Do Begin
Any events? If so, apply event and solve algebraic equations, g(x,y) = 0
Solve differential equations:
Solve algebraic equations, g(x,y) = 0
Output results
End while
Many complications to consider, including events that occur in the middle of a time step

21. Power System Equivalents
For many power system applications it is not necessary to study the entire interconnected network
Usually we are only concerned with a portion of the network
For real-time operations, real-time information is only available for a portion of the network
System is partitioned into study system, for which a detailed model is desired, and an external system, for which an equivalent model is used
Boundary buses (within the study system) connect the two

22. Power System Equivalents
For decades power system network models have been equivalenced using the approach originally presented by J.B. Ward in 1949 AIEE paper “Equivalent Circuits for Power-Flow Studies”
Paper’s single reference is to 1939 book by Gabriel Kron, so this also known as Kron’s reduction
Additional classical techniques are discussed in S. Deckmann, A. Pizzolante, A. Monticelli, B. Stott, and O. Alsac,“Studies on power system load flow equivalencing,” IEEE Trans. Power App. Syst., vol. PAS-99, no. 6, pp. 2301–2310, Nov./Dec

23. Ward Equivalents (Kron Reduction)
Equivalent is performed by doing a reduction of the bus admittance matrix
Done by doing a partial factorization of the Ybus
Computationally efficient
Yee matrix is never explicitly inverted!
Similar to what is done when fills are added, with new equivalent lines eventually joining the boundary buses

24. Ward Equivalents (Kron Reduction)
Prior to equivalencing constant power injections are converted to equivalent current injections, the system equivalenced, then they are converted back to constant power injections
Tends to place large shunts at the boundary buses
This equivalencing process has no impact on the non-boundary study buses
Various versions of the approach are used, primarily differing on the handling of reactive power injections
The equivalent embeds information about the operating state when the equivalent was created

25. PowerWorld Example
Ward type equivalents can be created in PowerWorld by going into the Edit Mode and selecting Tools, Equivalencing
Use Select the Buses to determine buses in the equivalent
Use Create the Equivalent to actually create the equivalent
When making equivalents for large networks the boundary buses tend to be joined by many high impedance lines; these lines can be eliminated by setting the Max Per Unit for Equivalent Line field to a relatively low value (say 2.0 per unit)
Loads and gens are converted to shunts, equivalenced, then converted back

26. PowerWorld B7Flat_Eqv Example
In this example the B7Flat_Eqv case is reduced, eliminating buses 1, 3 and 4. The study system is then 2, 5, 6, 7, with buses 2 and 5 the boundary buses
For ease of comparison system is modeled unloaded

27. PowerWorld B7Flat_Eqv Example
Original Ybus

28. PowerWorld B7Flat_Eqv Example
Note Yes=Yse' if no phase shifters

29. PowerWorld B7Flat_Eqv Example
Comparing original and equivalent
Only modification was a change in the impedance between buses 2 and 5, modeled by adding an equivalent line

30. Contingency Analysis Application of Equivalencing
One common application of equivalencing is contingency analysis
Most contingencies have a rather limited affect
Much smaller equivalents can be created for each contingent case, giving rapid contingency screening
Contingencies that appear to have violations in contingency screening can be processed by more time consuming but also more accurate methods
W.F. Tinney, J.M. Bright, “Adaptive Reductions for Power System Equivalents,” IEEE. Trans Power Systems, May 1987, pp

31. New Applications in Equivalencing
Models in which the entire extent of the network is retained, but the model size is greatly reduced
Often used for economic studies
Mixed ac/dc solutions, possibly with an equivalent as well
Internal portion is modeled with full ac power flow, more distant parts of the network are retained but modeled with a dc power flow, rest might be equivalenced
Attribute preserving equivalents
Retain characteristics other than just impedances, such as PTDFs; also new research looking at preserving line limits