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352 Power systems electromagnetic transients simulation

RungeKutta (fourth-order):

x(t + t) = x(t) + t

16

k1 +13

k2 +13

k3 +16

k4

(C.4)

k1 = f(x(t),t)

k2 = f

x(t) +

t

2k1,x(t) +

t

2

k3 = f

x(t) +

t

2k2,x(t) +

t

2

k4 = f (x(t) + t k3,x(t) + t)

(C.5)

AdamsBashforth (third-order):

x(t + t) = x(t) + t

2312

f(x(t),t) 1612

f (x(t t),t t)

+ 512

f (x(t 2t),t 2t)

(C.6)

AdamsMoulton (fourth-order):

x(t + t) = x(t) + t

924

f (x(t+ t),t+ t) + 1924

f(x(t),t)

524 f (x(t t),t t) + 124 f (x(t 2t),t 2t)

(C.7)

The method proposed by Gear is based on the equation:

x(t + t ) =

ki=0

i x(t + (1 i)t) (C.8)

This method was modified by Shichman for circuit simulation using a variable time

step. The Gear second-order method is:

x(t + t) =4

3x(t)

1

3x(t t) +

2t

3f (x(t+ t),t+ t) (C.9)

Numerical integration can be considered a sampled approximation of continu-

ous integration, as depicted in Figure C.1. The properties of the sample-and-hold

(reconstruction) determine the characteristics of the numerical integration formula.

Due to the phase and magnitude errors in the process, compensation can be applied

to generate a new integration formula. Consideration of numerical integration from a

sample data viewpoint leads to the following tunable integration formula [4]:

yn+1 = yn + t(fn+1 + (1 )fn) (C.10)

where is the gain parameter, is the phase parameter and yn+1 represents the

y value at t + t.

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Numerical integration 353

s

1

s

Zero-order

hold

1

s

Continuous

Discrete approximation

x (s)

x (s)

x(Z) x(S) x(S) x(Z)

1 est

.

x (s) x (s).

x(t).

x(t+ t) x(t+ t) x(nt + t).

...

x (nt).

Sample Reconstruction Compensation SampleIntegration

e st

Figure C.1 Numerical integration from the sampled data viewpoint

Table C.1 Classical integration formulae as special cases of thetunable integrator

Integration Rule Formula

0 Forward Euler yn+1 = yn + tfn

1

2

Trapezoidal yn+1 = yn +t

2fn+1 + fn

1 Backward Euler yn+1 = yn + tfn+13

2AdamsBashforth 2nd order yn+1 = yn +

t

2

3fn+1 fn

Tunable yn+1 = yn + t

fn+1 + (1 )fn

If = 1 and = (1 + )/2 then the trapezoidal rule with damping is obtained

[5]. The selection of integer multiples of half for the phase parameter produces theclassical integration formulae shown in Table C.1. These formulae are actually the

same integrator, differing only in the amount of phase shift of the integrand.

With respect to the differential equation:

yn+1 = f ( y , t ) (C.11)

Table C.2 shows the various integration rules in the form of an integrator and

differentiator.

Using numerical integration substitution for the differential equations of an induc-

tor and a capacitor gives the Norton equivalent values shown in Tables C.3 and C.4

respectively.

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Numerical integration 355

Table C.4 Linear capacitor

Integration rule Geff IHistory

Trapezoidal2C

t

2C

tvn in

Backward EulerC

t

C

tvn

Gear 2nd order3C

2t

2C

tvn +

C

2tvn1

TunableC

t

(1 y)

yin

C

+ vn

where O(t4) represents fourth and higher order terms. The derivative at n + 1 can

also be expressed as a Taylor series, i.e.

dyn+1

dt=

dyn

dt+ t

d2yn

dt2+

t2

2

d3yn

dt3+ O(t4) (C.13)

If equation C.12 is used in the trapezoidal rule then the trapezoidal estimate is:

yn+1 = yn +t

2

dyn

dt+

dyn+1

dt

= yn +t

2

dyn

dt+

t

2

dyn

dt+ t

d2yn

dt2+

t2

2

d3yn

dt3+ O(t4)

= yn + tdyn

dt+

t2

2

d2yn

dt2+

t3

4

d3yn

dt3+

t

2O(t4) (C.14)

The error caused in going from yn to yn+1 is:

n+1 = yn+1 yn+1 =t3

6

d3yn

dt3

t3

4

d3yn

dt3

t

2O(t4)

=t3

12

d3yn

dt3

t

2O(t4) =

t3

12

d3y

dt3

where tn tn+1.

The resulting error arises because the trapezoidal formula represents a truncation

of an exact Taylor series expansion, hence the term truncation error.

To illustrate this, consider a simple RC circuit where the voltage across the resistoris of interest, i.e.

vR(t) = RCdvC (t)

dt= RC

d(vS(t) RiR(t))

dt(C.15)

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356 Power systems electromagnetic transients simulation

Table C.5 Comparison of numerical integration algorithms(T = /10)

Step Exact F. Euler B. Euler Trapezoidal Gear second-order

1 45.2419 45.0000 45.4545 45.2381

2 40.9365 40.5000 41.3223 40.9297 40.9273

3 37.0409 36.4500 37.5657 37.0316 37.0211

4 33.5160 32.8050 34.1507 33.5048 33.4866

5 30.3265 29.5245 31.0461 30.3139 30.2891

Table C.6 Comparison of numerical integration algorithms (T = )

Step Exact F. Euler B. Euler Trapezoidal Gear second-order

1 18.3940 0.0000 25.0000 16.6667 18.3940

2 6.7668 0.0000 12.5000 5.5556 4.7152

3 2.4894 0.0000 6.2500 1.8519 0.0933

4 0.9158 0.0000 3.1250 0.6173 0.8684

5 0.3369 0.0000 1.5625 0.2058 0.7134

If the applied voltage is a step function at t = 0 then dvS(t)/dt = 0 and equation C.15

becomes:

vR (t) = RCdvR (t)

dt=

dvR (t)

dt(C.16)

The results for step lengths of/10 and (vs = 50 V) are shown in Tables C.5 and C.6

respectively. Gear second-order is a two-step method and hence the value at T is

required as initial condition.

For the trapezoidal rule the ratio (1 t/(2 ))/ (1 + t/(2 )) remains less than

1, so that the solution does tend to zero for any time step size. However for t > 2

it does so in an oscillatory manner and convergence may be very slow.

C.3 Stability of integration methods

Truncation error is a localised property (i.e. local to the present time point and time

step) whereas stability is a global property related to the growth or decay of errors

introduced at each time point and propagated to successive time points. Stability

depends on both the method and the problem.Since general stability analysis is difficult the normal approach is to compare the

stability of different methods for a single test equation, such as:

y = f ( y , t ) = y (C.17)

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Numerical integration 357

Table C.7 Stability region

Integration rule Formula Region of stability

Trapezoidal vR (t + t) =(1 t/(2 ))

(1 + t/(2 ))vR (t ) 0