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86 Power systems electromagnetic transients simulation

0 0.5 1 1.5 2 2.5 3 3.5 4

103

103

Time (ms)

Actual step

Simulated step

i simulated

i exact

0

10

20

30

40

50

60

70

80

90

100

0 0.5 1 1.5 2 2.5 3 3.5 4

Time (ms)

Actual step

Simulated step

i simulated

i exact

0

10

20

30

40

50

60

70

80

90

100

110

Current(amps)

Current(amps)

(a)

(b)

Figure 4.18 Step response of an RL branch for step lengths of: (a) t= /10 and(b) t=

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Numerical integrator substitution 87

103

103

Current(amps)

Time (ms)

Actual step

Simulated step

i simulated

i exact

0

20

40

60

80

100

120

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.5 1 1.5 2 2.5 3 3.5 4

Current(amps)

Time (ms)

Actual step

Simulated step i simulated

i exact

0

20

40

60

80

100

120

(a)

(b)

Figure 4.19 Step response of an RL branch for step lengths of: (a) t = 5 and(b) t = 10

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The following data is used for this test system: t = 50s, R = 1.0 ,

L = 0.05 mH and RSwitch = 1010 (OFF) 1010 (ON) and V1 = 100 V.

Initially IHistory = 0

1.000000000 1.000000000 .1000000000E09

1.000000000 1.500000000 0.000000000

v2v3v1

=

0.000000000

0.000000000

The multiplier is0.999999999900000. After forward reduction using this multiplierthe G matrix becomes:

1.000000000 1.000000000 .1000000000E09

0.000000000 0.5000000001 .9999999999E10

v2v3

v1

=

0.000000000

0.000000000

Moving the known voltage v1 to the right-hand side gives1.000000000 1.000000000

0.000000000 0.5000000001

v2v3

=

0.000000000

0.000000000

.1000000000E09

.9999999999E10

v1

Back substitution gives: i = 9.9999999970E009 or essentially zero in the offstate. When the switch is closed the G matrix is updated and the equation becomes:

0.1000000000E+11 1.000000000 .1000000000E+11

1.000000000 1.500000000 0.000000000

v2v3v1

=

0.000000000

0.000000000

After forward reduction:

0.1000000000E+11 1.000000000 .1000000000E+11

1.000000000 1.500000000 .9999999999

v2v3

v1

=

0.000000000

0.000000000

Moving the known voltage v1 to the right-hand side gives1.000000000 1.000000000

0.000000000 1.500000000

v2v3

=

0.000000000

0.000000000

.1000000000E+11

.9999999999

v1

=

0.1000000000E+13

99.99999999

Hence back-substitution gives:

iL = 33.333 A

v2 = 66.667 V

v3 = 33.333 V

4.5 Non-linear or time varying parameters

The most common types of non-linear elements that need representing are induc-

tances under magnetic saturation for transformers and reactors and resistances of

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surge arresters. Non-linear effects in synchronous machines are handled directly in

the machine equations. As usually there are only a few non-linear elements, modifi-

cation of the linear solution method is adopted rather than performing a less efficient

non-linear solution method for the entire network. In the past, three approaches have

been used, i.e.

current source representation (with one time step delay)

compensation methods

piecewise linear (switch representation).

4.5.1 Current source representation

A current source can be used to model the total current drawn by a non-linear com-

ponent, however by necessity this current has to be calculated from information at

previous time steps. Therefore it does not have an instantaneous term and appearsas an open circuit to voltages at the present time step. This approach can result in

instabilities and therefore is not recommended. To remove the instability a large fic-

titious Norton resistance would be needed, as well as the use of a correction source.

Moreover there is a one time step delay in the correction source. Another option

is to split the non-linear component into a linear component and non-linear source.

For example a non-linear inductor is modelled as a linear inductor in parallel with a

current source representing the saturation effect, as shown in Figure 4.20.

4.5.2 Compensation method

The compensation method can be applied provided there is only one non-linear ele-

ment (it is, in general, an iterative procedure if more than one non-linear element is

ikm

iCompensation

ikm (t)

iCompensation

(vk(t)vm (t))

m

k

0

Linear inductor

3

2

Figure 4.20 Piecewise linear inductor represented by current source

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present). The compensation theorem states that a non-linear branch can be excluded

from the network and be represented as a current source instead. Invoking the super-

position theorem, the total network solution is equal to the value v0(t) found with the

non-linear branch omitted, plus the contribution produced by the non-linear branch.

v(t) = v0(t) RTheveninikm(t) (4.51)

where

RThevenin is the Thevenin resistance of the network without a non-linear branch

connected between nodes k and m.

v0(t) is the open circuit voltage of the network, i.e. the voltage between nodes

k and m without a non-linear branch connected.

The Thevenin resistance, RThevenin, is a property of the linear network, and iscalculated by taking the difference between the mth and kth columns of [GU U]

1.

This is achieved by solving [GU U]v(t ) = IU with I

U set to zero except 1.0 in the

mth and 1.0 in the kth components. This can be interpreted as finding the terminal

voltage when connecting a current source (of magnitude 1) between nodes k and m.

The Thevenin resistance is pre-computed once, before entering the time step loop

and only needs recomputing whenever switches open or close. Once the Thevenin

resistance has been determined the procedure at each time step is thus:

(i) Compute the node voltages v0(t) with the non-linear branch omitted. From this

information extract the open circuit voltage between nodes k and m.(ii) Solve the following two scalar equations simultaneously for ikm :

vkm (t) = vkm0(t) RTheveninikm (4.52)

vkm (t) = f (ikm, dikm / d t , t , . . . ) (4.53)

This is depicted pictorially in Figure 4.21. If equation 4.53 is given as an analytic

expression then a NewtonRaphson solution is used. When equation 4.53 is

defined point-by-point as a piecewise linear curve then a search procedure is

used to find the intersection of the two curves.

(iii) The final solution is obtained by superimposing the response to the current source

ikm using equation 4.51. Superposition is permissible provided the rest of the

network is linear.

The subsystem concept permits processing more than one non-linear branch,

provided there is only one non-linear branch per subsystem.

If the non-linear branch is defined by vkm = f (ikm) or vkm = R(t) ikm the

solution is straightforward.

In the case of a non-linear inductor: = f (ikm), where the flux is the integralof the voltage with time, i.e.

(t) = (t t) +

ttt

v(u) du (4.54)

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vkm (t) =f(t,ikm,dikm /dt,)

vkm (t) = vkm0 (t) RThevenin ikm

vkm

ikm

Figure 4.21 Pictorial view of simultaneous solution of two equations

The use of the trapezoidal rule gives:

(t) =t

2v(t) + History(t t) (4.55)

where

History = (t t ) +t

2v(t )

Numerical problems can occur with non-linear elements if t is too large. The

non-linear characteristics are effectively sampled and the characteristics between the

sampled points do not enter the solution. This can result in artificial negative damping

or hysteresis as depicted in Figure 4.22.

4.5.3 Piecewise linear method

The piecewise linear inductor characteristic, depicted in Figure 4.23, can be repre-

sented as a linear inductor in series with a voltage source. The inductance is changed

(switched) when moving from one segment of the characteristic to the next. Although

this model is easily implemented, numerical problems can occur as the need to change

to the next segment is only recognised after the point exceeds the current segment

(unless interpolation is used for this type of discontinuity). This is a switched model

in that when the segment changes the branch conductance changes, hence the system

conductance matrix must be modified.

A non-linear function can be modelled using a combination of piecewise lin-

ear representation and current source. The piecewise linear characteristics can be

modelled with switched representation, and a current source used to correct for the

difference between the piecewise linear characteristic and the actual.

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vkm

ikm

1

2

3

Figure 4.22 Artificial negative damping

km

ikm

k m

Figure 4.23 Piecewise linear inductor

4.6 Subsystems

Transmission lines and cables in the system being simulated introduce decoupling into

the conductance matrix. This is because the transmission line model injects current at

one terminal as a function of the voltage at the other at previous time steps. There is

no instantaneous term (represented by a conductance in the equivalent models) that

links one terminal to the other. Hence in the present time step, there is no dependency

on the electrical conditions at the distant terminals of the line. This results in a block

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diagonal structure of the systems conductance matrix, i.e.

Y =

[Y1] 0 0

0 [Y2] 0

0 0 [Y3]

Each decoupled block in this matrix is a subsystem, and can be solved at each time

step independently of all other subsystems. The same effect can be approximated by

introducing an interface into a coupled network. Care must be taken in choosing the

interface point(s) to ensure that the interface variables must be sufficiently stable from

one time point to the next, as one time step old values are fed across the interface.

Capacitor voltages and inductor currents are the ideal choice for this purpose as neither

can change instantaneously. Figure 4.24(a) illustrates coupled systems that are to be

separated into subsystems. Each subsystem in Figure 4.24(b) is represented in the

other by a linear equivalent. The Norton equivalent is constructed using informationfrom the previous time step, looking into subsystem (2) from bus (A). The shunt

connected at (A) is considered to be part of (1). The Norton admittance is:

YN = YA +(YB + Y2)

Z (1/Z + YB + Y2)(4.56)

the Norton current:

IN = IA(t t) + VA(t t)YA (4.57)

IA

IBA

YA YB

Y2Y1

1

subsystem

2

subsystem

1

subsystem

2

subsystemZTh

IN

YN

VTh

A B

A B

Z

(a)

(b)

Figure 4.24 Separation of two coupled subsystems by means of linearised equivalentsources

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the Thevenin impedance:

ZTh =1

YB

Z + 1/(Y1 + YA)

Z + 1/(Y1 + YA) + 1/YB

(4.58)

and the voltage source:

VTh = VB (t t) + ZT hIBA (t t) (4.59)

The shunts (YN and ZT h) represent the instantaneous (or impulse) response of each

subsystem as seen from the interface busbar. If YA is a capacitor bank, Z is a series

inductor, and YB is small, then

YN YA and YN = IBA (t t) (the inductor current)

ZTh Z and VTh = VA(t t) (the capacitor voltage)

When simulating HVDC systems, it can frequently be arranged that the subsys-

tems containing each end of the link are small, so that only a small conductance

matrix need be re-factored after every switching. Even if the link is not terminated at

transmission lines or cables, a subsystem boundary can still be created by introducing

a one time-step delay at the commutating bus. This technique was used in the EMTDC

V2 B6P110 converter model, but not in version 3 because it can result in instabilities.

A d.c. link subdivided into subsystems is illustrated in Figure 4.25.

Controlled sources can be used to interface subsystems with component models

solved by another algorithm, e.g. components using numerical integration substitutionon a state variable formulation. Synchronous machine and early non-switch-based

Subsystem 1 Subsystem 2 Subsystem 3 Subsystem 4

AC

system 2

AC

system 2

AC

system 1

AC

system 1

(a)

(b)

Figure 4.25 Interfacing for HVDC link

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SVC models use a state variable formulation in PSCAD/EMTDC and appear to their

parent subsystems as controlled sources. When interfacing subsystems, best results

are obtained if the voltage and current at the point of connection are stabilised, and if

each component/model is represented in the other as a linearised equivalent around

the solution at the previous time step. In the case of synchronous machines, a suitable

linearising equivalent is the subtransient reactance, which should be connected in

shunt with the machine current injection. An RC circuit is applied to the machine

interface as this adds damping to the high frequencies, which normally cause model

instabilities, without affecting the low frequency characteristics and losses.

4.7 Sparsity and optimal ordering

The connectivity of power systems produces a conductance matrix [G] which is largeand sparse. By exploiting the sparsity, memory storage is reduced and significant solu-

tion speed improvement results. Storing only the non-zero elements reduces memory

requirements and multiplying only by non-zero elements increases speed. It takes a

computer just as long to multiply a number by zero as by any other number. Finding

the solution of a system of simultaneous linear equations ([G]V = I) using the inverse

is very inefficient as, although the conductance matrix is sparse, the inverse is full.

A better approach is the triangular decomposition of a matrix, which allows repeated

direct solutions without repeating the triangulation (provided the [G] matrix does not

change). The amount of fill-in that occurs during the triangulation is a function of thenode ordering and can be minimised using optimal ordering [7].

To illustrate the effect of node ordering consider the simple circuit shown in

Figure 4.26. Without optimal ordering the [G] matrix has the structure:

X X X X X

X X 0 0 0

X 0 X 0 0

X 0 0 X 0

X 0 0 0 X

After processing the first row the structure is:

1 X X X X

0 X X X X

0 X X X X

0 X X X X

0 X X X X

When completely triangular the upper triangular structure is full

1 X X X X0 1 X X X

0 0 1 X X

0 0 0 1 X

0 0 0 0 1

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Z2 Z3

Z4

Z1

Z5

2

5 4

3

1

0

Figure 4.26 Example of sparse network

If instead node 1 is ordered last then the [G] matrix has the structure:

X 0 0 0 X

0 X 0 0 X

0 0 X 0 X

0 0 0 X X

X X X X X

After processing the first row the structure is:

1 0 0 0 X

0 X 0 0 X

0 0 X 0 X

0 0 0 X X

0 X X X X

When triangulation is complete the upper triangular matrix now has less fill-in.

1 0 0 0 X

0 1 0 0 X

0 0 1 0 X

0 0 0 1 X

0 0 0 0 1

This illustration uses the standard textbook approach of eliminating elements below

the diagonal on a column basis; instead, a mathematically equivalent row-by-row

elimination is normally performed that has programming advantages [5]. Moreover

symmetry in the [G]matrix allows only half of it to be stored. Three ordering schemes

have been published [8] and are now commonly used in transient programs. There

is a tradeoff between the programming complexity, computation effort and level of

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optimality achieved by these methods, and the best scheme depends on the network

topology, size and number of direct solutions required.

4.8 Numerical errors and instabilities

The trapezoidal rule contains a truncation error which normally manifests itself as

chatter or simply as an error in the waveforms when the time step is large. This is

particularly true if cutsets of inductors and current sources, or loops of capacitors

and voltage sources exist. Whenever discontinuities occur (switching of devices, or

modification of non-linear component parameters, ) care is needed as these can

initiate chatter problems or instabilities. Two separate problems are associated with

discontinuities. The first is the error in making changes at the next time point after the

discontinuity, for example current chopping in inductive circuits due to turning OFFa device at the next time point after the current has gone to zero, or proceeding on a

segment of a piecewise linear characteristic one step beyond the knee point. Even if

the discontinuity is not stepped over, chatter can occur due to error in the trapezoidal

rule. These issues, as they apply to power electronic circuits, are dealt with further in

Chapter 9.

Other instabilities can occur because of time step delays inherent in the model.

For example this could be due to an interface between a synchronous machine model

and the main algorithm, or from feedback paths in control systems (Chapter 8).

Instabilities can also occur in modelling non-linear devices due to the sampled nature

of the simulation as outlined in section 4.5. Finally bangbang instability can occur

due to the interaction of power electronic device non-linearity and non-linear devices

such as surge arresters. In this case the state of one influences the other and finding

the appropriate state can be difficult.

4.9 Summary

The main features making numerical integration substitution a popular method for

the solution of electromagnetic transients are: simplicity, general applicability andcomputing efficiency.

Its simplicity derives from the conversion of the individual power system ele-

ments (i.e. resistance, inductance and capacitance) and the transmission lines into

Norton equivalents easily solvable by nodal analysis. The Norton current source rep-

resents the component past History terms and the Norton impedance consists of a

pure conductance dependent on the step length.

By selecting the appropriate integration step, numerical integration substitution

is applicable to all transient phenomena and to systems of any size. In some cases,

however, the inherent truncation error of the trapezoidal method may lead to oscilla-

tions; improved numerical techniques to overcome this problem will be discussed in

Chapters 5 and 9.

Efficient solutions are possible by the use of a constant integration step length

throughout the study, which permits performing a single conductance matrix

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triangular factorisation before entering the time step loop. Further efficiency is

achieved by exploiting the large sparsity of the conductance matrix.

An important concept is the use of subsystems, each of which, at a given time

step, can be solved independently of the others. The main advantage of subsystems

is the performance improvement when multiple time-steps/interpolation algorithms

are used. Interpolating back to discontinuities is performed only on one subsystem.

Subsystems also allow parallel processing hence real-time applications as well as

interfacing different solution algorithms. If sparsity techniques are not used (early

EMTDC versions) then subsystems also greatly improve the performance.

4.10 References

1 DOMMEL, H. W.: Digital computer solution of electromagnetic transients insingle- and multi-phase networks, IEEE Transactions on Power Apparatus and

Systems, 1969, 88 (2), pp. 73441

2 DOMMEL, H. W.: Nonlinear and time-varying elements in digital simulation of

electromagnetic transients, IEEE Transactions on Power Apparatus and Systems,

1971, 90 (6), pp. 25617

3 DOMMEL, H. W.: Techniques for analyzing electromagnetic transients, IEEE

Computer Applications in Power, 1997, 10 (3), pp. 1821

4 BRANIN, F. H.: Computer methods of network analysis, Proceedings of IEEE,

1967, 55, pp. 178718015 DOMMEL, H. W.: Electromagnetic transients program reference manual: EMTP

theory book (Bonneville Power Administration, Portland, OR, August 1986).

6 DOMMEL, H. W.: A method for solving transient phenomena in multiphase

systems, Proc. 2nd Power System Computation Conference, Stockholm, 1966,

Rept. 5.8

7 SATO, N. and TINNEY, W. F.: Techniques for exploiting the sparsity of the net-

work admittance matrix, Transactions on Power Apparatus and Systems, 1963,

82, pp. 94450

8 TINNEY, W. F. and WALKER, J. W.: Direct solutions of sparse network equationsby optimally ordered triangular factorization, Proceedings of IEEE, 1967, 55,

pp. 18019

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Chapter 5

The root-matching method

5.1 Introduction

The integration methods based on a truncated Taylors series are prone to numerical

oscillations when simulating step responses.

An interesting alternative to numerical integration substitution that has already

proved its effectiveness in the control area, is the exponential form of the differ-

ence equation. The implementation of this method requires the use of root-matching

techniques and is better known by that name.The purpose of the root-matching method is to transfer correctly the poles and

zeros from the s-plane to the z-plane, an important requirement for reliable digital

simulation, to ensure that the difference equation is suitable to simulate the continuous

process correctly.

This chapter describes the use of root-matching techniques in electromagnetic

transient simulation and compares its performance with that of the conventional

numerical integrator substitution method described in Chapter 4.

5.2 Exponential form of the difference equation

The application of the numerical integrator substitution method, and the trapezoidal

rule, to a series RL branch produces the following difference equation for the branch:

ik =(1 tR/(2L))

(1 + tR/(2L))ik1 +

t/(2L)

(1 + tR/(2L))(vk + vk1) (5.1)

Careful inspection of equation 5.1 shows that the first term is a first order approx-

imation ofex , where x = tR/L and the second term is a first order approximation

of (1 ex )/2 [1]. This suggests that the use of the exponential expressions in the

difference equation should eliminate the truncation error and thus provide accurate

and stable solutions regardless of the time step.

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