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 haozhu@illinois.edu 12/2/2015

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

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

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

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)

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

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

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: http://www.staff.science.uu.nl/~frank011/Classes/numwisk/ch10.pdf

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

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. 259-266, 2008

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. 259-266, 2008

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

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

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:

Backward Euler Example Returning to the cart example

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 -0.3944 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.

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

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 -0.4067

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

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

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

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. 1980.

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

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

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

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

PowerWorld B7Flat_Eqv Example Original Ybus

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

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

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. 351-359

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