ECE 576 – Power System Dynamics and Stability Lecture 8: Synchronous Machine Modeling Prof. Tom Overbye Dept. of Electrical and Computer Engineering University of Illinois at Urbana-Champaign overbye@illinois.edu

Announcements Read Chapter 5 and Appendix A Homework 2 is posted on the web; it is due on Tuesday Feb 16 Midterm exam is moved to March 31 (in class) because of the ECE 431 field trip on March 17 We will have extra classes on Mondays Feb 8, 15 and March 7 from 1:30 to 3pm in 5070 ECEB No class on Tuesday Feb 23, Tuesday March 1 and Thursday March 3

Sinusoidal Steady-State Here we consider the application to balanced, sinusoidal conditions

Transforming to dq0

Simplifying Using d Recall that Hence These algebraic equations can be written as complex equations, The conclusion is if we know d, then we can easily relate the phase to the dq values!

Summary So Far The model as developed so far has been derived using the following assumptions The stator has three coils in a balanced configuration, spaced 120 electrical degrees apart Rotor has four coils in a balanced configuration located 90 electrical degrees apart Relationship between the flux linkages and currents must reflect a conservative coupling field The relationships between the flux linkages and currents must be independent of qshaft when expressed in the dq0 coordinate system

Two Main Types of Synchronous Machines Round Rotor Air-gap is constant, used with higher speed machines Salient Rotor (often called Salient Pole) Air-gap varies circumferentially Used with many pole, slower machines such as hydro Narrowest part of gap in the d-axis and the widest along the q-axis

Assuming a Linear Magnetic Circuit If the flux linkages are assumed to be a linear function of the currents then we can write The rotor self- inductance matrix Lrr is independent of qshaft

Inductive Dependence on Shaft Angle L12 = + maximum L12 = - maximum

Stator Inductances The self inductance for each stator winding has a portion that is due to the leakage flux which does not cross the air gap, Lls The other portion of the self inductance is due to flux crossing the air gap and can be modeled for phase a as Mutual inductance between the stator windings is modeled as The offset angle is either 2p/3 or -2p/3

Conversion to dq0 for Angle Independence

Conversion to dq0 for Angle Independence For a round rotor machine LB is small and hence Lmd is close to Lmq. For a salient pole machine Lmd is substantially larger

Convert to Normalized at f = ws Convert to per unit, and assume frequency of ws Then define new per unit reactance variables

Example Xd/Xq Ratios for a WECC Case

Normalized Equations

Key Simulation Parameters The key parameters that occur in most models can then be defined the following transient values These values will be used in all the synchronous machine models In a salient rotor machine Xmq is small so Xq = X'q; also X1q is small so T'q0 is small

Example X'q/Xq Ratios for a WECC Case About 75% are Clearly Salient Pole Machines!

Key Simulation Parameters And the subtransient parameters These values will be used in the subtransient machine models. It is common to assume X"d = X"q

Internal Variables Define the following variables, which are quite important in subsequent models Hence E'q and E'd are scaled flux linkages and Efd is the scaled field voltage

Dynamic Model Development In developing the dynamic model not all of the currents and fluxes are independent In the book formulation only seven out of fourteen are independent Approach is to eliminate the rotor currents, retaining the terminal currents (Id, Iq, I0) for matching the network boundary conditions

Rotor Currents Use new variables to solve for the rotor currents

Rotor Currents

Final Complete Model

Final Complete Model TFW is the friction and windage component that we'll consider later

Final Complete Model

Single-Machine Steady-State Previous derivation was done assuming a linear magnetic circuit We'll consider the nonlinear magnetic circuit (section 3.5) but will first do the steady-state condition (3.6) In steady-state the speed is constant (equal to ws), d is constant, and all the derivatives are zero Initial values are determined from the terminal conditions: voltage magnitude, voltage angle, real and reactive power injection

Single-Machine Steady-State The key variable we need to determine the initial conditions is actually d, which doesn't appear explicitly in these equations!

Field Current The field current, Ifd, is defined in steady-state as However, what is usually used in transient stability simulations for the field current is So the value of Xmd is not needed

Determining d without Saturation In order to get the initial values for the variables we need to determine d We'll eventually consider two approaches: the simple one when there is no saturation, and then later a general approach for models with saturation To derive the simple approach we have

Determining d without Saturation Recalling the relation between d and the stator values We then combine the equations for Vd and Vq and get

Determining d without Saturation Then in terms of the terminal values

D-q Reference Frame Machine voltage and current are “transformed” into the d-q reference frame using the rotor angle,  Terminal voltage in network (power flow) reference frame are VS = Vt = Vr +jVi

A Steady-State Example Assume a generator is supplying 1.0 pu real power at 0.95 pf lagging into an infinite bus at 1.0 pu voltage through the below network. Generator pu values are Rs=0, Xd=2.1, Xq=2.0, X'd=0.3, X'q=0.5

A Steady-State Example, cont. First determine the current out of the generator from the initial conditions, then the terminal voltage

A Steady-State Example, cont. We can then get the initial angle and initial dq values Or

A Steady-State Example, cont. The initial state variable are determined by solving with the differential equations equal to zero.

PowerWorld Two-Axis Model Numerous models exist for synchronous machines, some of which we'll cover in-depth. The following is a relatively simple model that represents the field winding and one damper winding; it also includes the generator swing eq. For a salient pole machine, with Xq=X'q, then E'd would rapidly decay to zero

PowerWorld Solution of 11.10

Nonlinear Magnetic Circuits Nonlinear magnetic models are needed because magnetic materials tend to saturate; that is, increasingly large amounts of current are needed to increase the flux density Linear

Saturation

Saturation Models Many different models exist to represent saturation There is a tradeoff between accuracy and complexity Book presents the details of fully considering saturation in Section 3.5 One simple approach is to replace With

Saturation Models In steady-state this becomes Hence saturation increases the required Efd to get a desired flux Saturation is usually modeled using a quadratic function, with the value of Se specified at two points (often at 1.0 flux and 1.2 flux) A and B are determined from the two data points

Saturation Example If Se = 0.1 when the flux is 1.0 and 0.5 when the flux is 1.2, what are the values of A and B using the

Implementing Saturation Models When implementing saturation models in code, it is important to recognize that the function is meant to be positive, so negative values are not allowed In large cases one is almost guaranteed to have special cases, sometimes caused by user typos What to do if Se(1.2) < Se(1.0)? What to do if Se(1.0) = 0 and Se(1.2) <> 0 What to do if Se(1.0) = Se(1.2) <> 0 Exponential saturation models have been used We'll cover other common saturation approaches in Chapter 5

Reduced Order Models Before going further, we will consider a formal approach to reduce the model complexity Reduced order models Idea is to approximate the behavior of fast dynamics without having to explicitly solve the differential equations Essentially all models have fast dynamics that not explicitly modeled Goal is a more easily solved model (i.e., a reduced order model) without significant loss in accuracy

Manifolds Hard to precisely define, but "you know one when you see one" Smooth surfaces In one dimensions a manifold is a curve without any kinks or self-intersections (line, circle, parabola, but not a figure 8)

Two-Dimensional Manifolds Images from book and mathworld.wolfram.com

Integral Manifolds Suppose we could find z = h(x) Desire is to express z as an algebraic function of x, eliminating dz/dt Suppose we could find z = h(x)

Integral Manifolds Replace z by h(x) Chain rule of differentiation If the initial conditions satisfy h, so z0 = h(x0) then the reduced equation is exact

Integral Manifold Example Assume two differential equations with z considered "fast" relative to x

Integral Manifold Example For this simple system we can get the exact solution

Solve for Equilibrium (Steady-state ) Values

Solve for Remaining Constants Use the initial conditions and derivatives at t=0 to solve for the remaining constants

Solve for Remaining Constants