1. ECE 530 – Analysis Techniques for Large-Scale Electrical Systems
Lecture 22: SVD; Numeric Solution
of Differential Equations
Prof. Hao Zhu
Dept. of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign
11/13/2014

2. Announcements HW 7 is due Thursday, November 20
Final exam on Monday Dec 15 from 1:30 to 4:30pm in this room (ECEB-4026)
Closed book, closed notes; you can bring in two note sheets (one new note sheet and exam 1 note sheet), along with simple calculators

3. Singular Value Decomposition
An extremely useful matrix analysis technique is the singular value decomposition (SVD), which takes an m by n real matrix A and represents it as where U is an m by n orthogonal matrix (UTU = I), S is an n by n diagonal matrix whose elements are the non-negative singular values of A, and V is an n by n orthogonal matrix
Note, there is an other formulation with U as m by m, and S as m by n
Computational order is O(n2m); ok if n is small

4. Matrix Singular Values
The singular values of a matrix A are the square roots of the eignenvalues of ATA
The singular values are real, nonnegative numbers, usually listed in decreasing order
Each singular value s satisfies where u (dimension m) and v (dimension n) are both unit length and called respectively the left-singular and right-singular vectors for singular value s

5. SVD Applications
SVD applications come from the property that A can be written as where each one of the matrices is known as a mode
More of the essence of the matrix is contained in the modes associated with the larger singular values
An immediate application is data compression in which A represents an image; often a quite good representation of the image is available from just a small percentage of the modes

6. SVD Image Compression Example
Image source

7. SVD Applications
Another application is removing noise. If the columns of A are signals, since noise often affects more the smaller singular values, then noise can be removed by taking the SVD, and reconstructing A without the small singular-value modes
Noise strength is uniform across all modes
Another application is principal component analysis (PCA) in which the idea is to take a data set with a number of variables, and reduce the data and determine the data associations
The principal components correspond to the largest singular values when data is appropriately normalized

8. Pseudo-inverse of a Matrix
The pseudo-inverse of a matrix generalizes concept of a matrix inverse to an m by n matrix, in which m >= n
Specifically talking about a Moore-Penrose Matrix Inverse
Notation for the pseudo-inverse of A is A+
Satisfies AA+A = A
If A is a square matrix, then A+ = A-1
Quite useful for solving the least squares problem since the least squares solution of Ax = b is x = A+ b

9. Pseudo-inverse and SVD
pseudo-inverse can be directly determined from the SVD in which S+ is formed by replacing the non-zero diagonal elements by its inverse, and leaving the zeros
Numerically small values in S are assumed zero
V is n by n
S+ is n by n
UT is n by m
A+ is therefore n by m
Computationally doing the SVD dominates

10. Simple Least Squares Example
Assume we which to fix a line (mx + b = y) to three data points: (1,1), (2,4), (6,4)
Two unknowns, m and b; hence x = [m b]T
Setup in form of Ax = b

11. Simple Least Squares Example
Doing the SVD
Computing the pseudo-inverse

12. Simple Least Squares Example
Computing x = [m b]T gives
With the pseudo-inverse approach we immediately see the sensitivity of the elements of x to the elements of b

13. Switching to Dynamic Systems
The analysis we've done so far in the class has been associated with power system static analysis
Determining characteristics of the power system quasi-steady state equilibrium
Now we're going to do a brief coverage of techniques for analysis of power system dynamics, with fuller coverage detailed in ECE 576
Appropriate models depend on time period of interest
Faster dynamics can be represented as algebraic constraints
Slower dynamics can be represented as constants

14. Power System Time Frames
Lightning Propagation
Switching Surges
Stator Transients and
Subsynchronous Resonance
Transient Stability
Governor and Load Frequency Control
Boiler/Long-Term Dynamics
10-7
10-5
10-3
0.1 10
103
105
Time (Seconds)
Voltage Stability
Power Flow
Image source: P.W. Sauer, M.A. Pai, Power System Dynamics and Stability, 1997, Fig 1.2, modified

15. Power Grid Disturbance Example
Figures show the frequency change as a result of the sudden loss of a large amount of generation in the Southern WECC
Time in Seconds
Frequency Contour

16. Frequency Response for Gen. Loss
In response to rapid loss of generation, in the initial seconds the system frequency will decrease as energy stored in the rotating masses is transformed into electric energy
Solar PV has no inertia, and for most new wind turbines the inertia is not seen by the system
Within seconds governors respond, increasing power output of controllable generation
Solar PV and wind are usually operated at maximum power so they have no reserves to contribute

17. Solution Considerations
In ECE 530 we introduce several solution methods that are more fully considered in ECE 576
A wide variety of different solution methods are possible, with different classes of problems (such as power system transient stability) having customized solutions
There is a balance between the problem to be solved and the solution method
Can we bound the dynamics considered, with fast dynamics represented as algebraic constraints, and slow as constants

18. Differential Algebraic Equations
Many problems, including many in the power area, can be formulated as a set of differential algebraic equations (DAE) of the form
A power example is transient stability, in which f represents (primarily) the generator dynamics, and g (primarily) the bus power balance equations
We'll initially consider the simpler problem of just

19. Ordinary Differential Equations (ODEs)
Assume we have a problem of the form
This is known as an initial value problem, since the initial value of x is given at some time t0
We need to determine x(t) for future time
Initial value, x0, must be either be given or determined by solving for an equilibrium point, f(x) = 0
Higher-order systems can be put into this first order form
Except for special cases, such as linear systems, an analytic solution is usually not possible – numerical methods must be used

20. Equilibrium Points An equilibrium point x* satisfies
An equilibrium point is stable if the response to a small disturbance remains small
This is known as Lyapunov stability
Formally, if for every e > 0, there exists a d = d(e) > 0 such that if ||x(0) – x*|| < d, then ||x(t) – x*|| < e for t  0
An equilibrium point has asymptotic stability if there exists a d > 0 such that if ||x(0) – x*|| < d, then

21. Power System Application
A typical power system application is to assume the power flow solution represents an equilibrium point
Back solve to determine the initial state variables, x(0)
At some point a contingency occurs, perturbing the state away from the equilibrium point
Time domain simulation is used to determine whether the system returns to the equilibrium point

22. Initial value Problem Examples

23. Numerical Solution Methods
Numerical solution methods do not generate exact solutions; they practically always introduce some error
Methods assume time advances in discrete increments, called a stepsize (or time step), Dt
Speed accuracy tradeoff: a smaller Dt usually gives a better solution, but it takes longer to compute
Numeric roundoff error due to finite computer word size
Key issue is the derivative of x, f(x) depends on x, the value we are trying to determine
A solution exists as long as f(x) is continuously differentiable

24. Numerical Solution Methods
There are a wide variety of different solution approaches, we will only touch on several
One-step methods: require information about solution just at one point, x(t)
Forward Euler
Runge-Kutta
Multi-step methods: make use of information at more than one point, x(t), x(t-Dt), x(t-D2t)…
Adams-Bashforth
Predictor-Corrector Methods: implicit
Backward Euler

25. Error Propagation
At each time step the total round-off error is the sum of the local round-off at time and the propagated error from steps 1, 2 , … , k − 1
An algorithm with the desirable property that local round-off error decays with increasing number of steps is said to be numerically stable
Otherwise, the algorithm is numerically unstable
Numerically unstable algorithms can nevertheless give quite good performance if appropriate time steps are used
This is particularly true when coupled with algebraic equations

26. Forward Euler’s Method
The simplest technique for numerically integrating such equations is known as the Euler's Method (sometimes the Forward Euler's Method)
Key idea is to approximate
In general, the smaller the Dt, the more accurate the solution, but it also takes more time steps

27. Euler’s Method Algorithm

28. Euler’s Method Example 1

29. Euler’s Method Example 1, cont’d
xactual(t)
x(t) Dt=0.1
x(t) Dt=0.05 10
0.1
9.048 9
9.02
0.2
8.187
8.10
8.15
0.3
7.408
7.29
7.35 …
1.0
3.678
3.49
3.58
2.0
1.353
1.22
1.29

30. Euler’s Method Example 2

31. Euler's Method Example 2, cont'd

32. Euler's Method Example 2, cont'd
x1actual(t)
x1(t) Dt=0.25 1
0.25
0.9689
0.50
0.8776
0.9375
0.75
0.7317
0.8125
1.00
0.5403
0.6289 …
10.0
-3.129
100.0
0.8623
-151,983
Since we know from the exact solution that x1 is bounded between -1 and 1, clearly the method is numerically unstable

33. Euler's Method Example 2, cont'd
Below is a comparison of the solution values for x1(t)
at time t = 10 seconds Dt
x1(10)
actual
0.25
-3.129
0.10
0.01
0.001