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190 8 Time Domain Analysis

which lead to Δvh(i) = 0. In other words, if vh(i) = 0 at the iteration i, it

will remain zero for all the following iterations. A very small value of vh(i) would also show a similar numerical issue. Furthermore, v = 0 is a solution of algebraic equations in (8.20). Thus, one has to carefully avoid that voltages become zero (or very small values) at any iteration, otherwise, the Newton’s method is not able to recover voltage magnitudes.

In (8.20), the only algebraic variables are bus voltage magnitudes and phase angles. A more general and flexible model includes additional algebraic variables yˆ, algebraic equations gˆ and controllable parameters η:

The following conclusive remarks are relevant:

1.Both (8.13) and (8.24) are nonlinear. In fact, (8.13) is nonlinear at least in the di erential equations of synchronous machines. Thus, if an implicit solution method is used for the numerical integration, both (8.13) and (8.24) requires a Newton’s method for solving a set of nonlinear equations at each integration step.

2.Equations (8.24) have the advantage of requiring the same model for power flow, continuation power flow and time domain analyses. In other words, (8.24) allow using an unique structure for all devices (there is no di erence between power flow and time domain analysis models of the same device) and writing more compact code (the same algebraic equations are used for both static and dynamic analysis).

Example 8.1 OMIB Di erential Algebraic Equations

This example provides the power-injection as well as the current-injection model (8.24) and (8.13), respectively, for the 2-bus system depicted in Figure 8.6. Bus 0 is an infinite bus where the voltage v¯0 = v0 θ0 is constant, while the machine is a two-order machine model. Assuming that the voltage at the infinite bus is a parameter, the system power-injection model is:

j=0

Both x(t + Δt) and xˆ(t + Δt) approximate the exact solution and xˆ(t + Δt) is an approximation of higher order than x(t + Δt). The di erence x(t + Δt) − xˆ(t + Δt) allows estimating the error with respect to the exact solution and adjusting the step length Δt.

The m-stage formula (8.31) evaluates the new value of the state variables x(t + Δt) using a weighted sum of m values of x˙ at suitable points between t and t + Δt.

A convenient way of visualizing the formulæ (8.31) for a given method is through the Butcher’s tableau [34], as follows:

Using (8.31), a huge variety of explicit methods can be defined, including the large family of Runge-Kutta’s formulæ.

Multi-step or predictor-corrector methods are another class of explicit methods. The general formula of a multi-step method is:

k=0

A drawback of multi-step methods is that they are not self-starting, since the first m steps has to be known to compute the generic step (8.34) for t+Δt. The well-known Adams-Bashforth’s method, Milne-Simpson’s method and Hamming’s method belong to the family of multi-step methods [34]. However, these methods have proved to be less accurate and e cient then Runge-Kutta’s formulæ, at least for power system applications [163].

Example 8.2 Runge-Kutta’s Formulæ

Example 4.6 presented the classical 4th order Runge-Kutta’s formula:

f {1} = f (x(t) + 0.5Δtf {0}) f {2} = f (x(t) + 0.5Δtf {1}) f {3} = f (x(t) + Δtf {2})

x(t + Δt) = x(t) + Δt(f {0} + 2f {1} + 2f {2} + f {3})/6

The RK4 is represented by the following Butcher’s tableau:

0

11

22

Several more sophisticated schemes have been proposed [117]. For example, the Runge-Kutta-Fehlberg’s formula has the following Butcher’s tableau:

Example 8.3 Modified Euler’s Method

The modified Euler’s method is the simplest multi-step method and is the only that has been widely used in power system analysis [163, 291]. It is composed of a predictor and a corrector step, as follows:

x(t + Δt) = x(t) + 12 Δt(f (x(t), t) + f (x˜(t + Δt), t))

The accuracy of this method can be improved by assigning x(t) ← x(t + Δt) and repeating the two-step formula (8.38).

8.3.2 Implicit Methods

In transient stability analysis, one of the possible issues is that time constant can span various time scales. For example if both transient stability and long term dynamics are considered together, time constants vary between 10−2 and 103 s (see Figure 1.6 of Chapter 1). Similarly, if one considers both sub-synchronous resonance phenomena and transient stability, the time scale range is between 10−4 and 101 s. If the time scale range of a ODE problem is “big”, the ODE problem is said to be sti .

The behavior of numerical methods on sti ODE problems can be analyzed

The solution of (8.39) is x(t) = ekt that approaches zero as t → ∞ when{k} < 0, i.e., the left-half of the complex plane is the stability region of the test equation (8.39). If the numerical method also exhibits this behavior, then the method is said to be absolute stable or A-stable according to the Dahlquist’s definition [69]. For example, applying the Runge-Kutta’s formulæ to the test equation (8.39), one has:

where n is the number of steps and x(t0) is the initial value. The function χ(kΔt) is called stability function and must be |χ(kΔt)| < 1 to satisfy x(t) → 0 for n → ∞.

An interesting result of Dahlquist’s theorems is that an explicit multistep method cannot be A-stable [69, 165, 316]. Thus, explicit methods are expected to provide poor behavior for sti ODE problems. On the other hand, implicit methods can be A-stable. For this reason, and because implicit methods allow a simultaneous solution of both state and algebraic variables in DAE problems, implicit methods are of particular relevance for power system analysis.

When using implicit methods, each step of the numerical integration is obtained as the solution of a set of nonlinear equations. Thus, implicit methods are particularly suited for nonlinear DAE systems, since the algebraic equations can be included in the nonlinear system to be solved at each iteration.

For a generic time t, and assumed a step length Δt, one has to solve the following problem [30]:

where f and g are the di erential and algebraic equations and qˆ is a function that depends on the implicit numerical method. Equations (8.41) are nonlinear and their solution is obtained by means of a Newton’s method, which in turn consists of computing iteratively the increments x(i) and y(i) of the state and algebraic variables and updating the actual variables:

where A(ci) is a matrix depending on the algebraic and state Jacobian matrices of the system. Some examples of implicit method formulæ are given in the following examples. Most techniques described in Chapter 4 can be applied to (8.42). For example, the dishonest or very dishonest Newton’s methods can be useful to reduce the number of factorizations of A(ci) and speed up the simulation.

Example 8.4 Backward Euler’s Method

The backward Euler’s method is a first order implicit method. It is generally faster but less accurate than the trapezoidal method that is discussed in the next example. At a generic time step t + Δt and a generic iteration i, A(ci)

ˆ(i)

and q are as follows:

where Inx is the identity matrix of the same dimension of the dynamic order

of the DAE system and all Jacobian matrices and f (i), are computed at the current point (x(i)(t + Δt), y(i)(t + Δt), t + Δt).

Example 8.5 Trapezoidal Method

The Crank-Nicolson’s or trapezoidal method is the workhorse solver for electro-mechanical DAE, and is widely used, in a variety of flavors, in most commercial and non-commercial power system software packages. The implicit version of the trapezoidal method has proved to be very robust and

reliable for a variety of sti ODE and DAE systems. At a generic iteration i,

where the notation is the same as in (8.43).

Example 8.6 Rosenbrock’s Semi-Implicit Method

Dahlquist’s theorems discourage from setting up implicit methods of order greater than 2 [165]. However, an interesting method for improving the accuracy of an implicit method is proposed in [316]. This reference proposes semi-implicit methods for solving ODE systems which are A-stable and avoid the need of iterating. However, semi-implicit methods applied to nonlinear DAE systems still require iterating and factorizing the matrix A(ci) at each iteration.

A well-known semi-implicit method is the one based on Rosenbrock’s formulæ. For example, a 4th order Rosenbrock’s formula is as follows. At a generic time t and iteration i, the matrix A(ci) is the same as in (8.44). The variables x(i) and y(i) are determined by means of a 4th order approximation:

z(i) = z(t) + b1qˆ(1i) + b2q(2i) + b3qˆ(3i) + b4qˆ(4i)

3 Butcher’s tableaux can be defined also for implicit methods. The only di erence with explicit methods is that the coe cient matrix bkj is not necessarily lower triangular. For example, the Butcher’s tableau of the trapezoidal method is:

1 1 1

2 2

0 0 0

1 1

2 2

However, for implicit methods, the determination of each f {k} is generally involved and, hence, the Butcher’s tableau representation is not practical.