TechRef_SynchronousMachine

DIgSILENT PowerFactory Technical Reference Documentation

Synchronous Machine ElmSym

DIgSILENT GmbH

Heinrich-Hertz-Str. 9 72810 - Gomaringen Germany T: +49 7072 9168 00 F: +49 7072 9168 88 http://www.digsilent.de

[email protected] r1017

Copyright ©2011, DIgSILENT GmbH. Copyright of this document belongs to DIgSILENT GmbH. No part of this document may be reproduced, copied, or transmitted in any form, by any means electronic or mechanical, without the prior written permission of DIgSILENT GmbH. Synchronous Machine (ElmSym)

1

DIgSILENT GmbH

Heinrich-Hertz-Str. 9 72810 - Gomaringen Germany T: +49 7072 9168 00 F: +49 7072 9168 88 http://www.digsilent.de

[email protected] r1017

Copyright ©2011, DIgSILENT GmbH. Copyright of this document belongs to DIgSILENT GmbH. No part of this document may be reproduced, copied, or transmitted in any form, by any means electronic or mechanical, without the prior written permission of DIgSILENT GmbH. Synchronous Machine (ElmSym)

1

Contents

Contents 1 Genera Generall Des Descri cripti ption on

4

1.1 Load Load Flo Flow w Anal Analysi ysis s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.1.1 Reactive Reactive Powe Power/V r/Voltag oltage e Control Control . . . . . . . . . . . . . . . . . . . . . . . .

4

1.1.2 Active Active Powe Powerr Control Control and and Balancing Balancing . . . . . . . . . . . . . . . . . . . . .

7

1.1.3 Spinning Spinning ifif circuit circuit breaker breaker is open . . . . . . . . . . . . . . . . . . . . . . .

9

1.2 Short-Circuit Short-Circuit Analysis Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1.2.1 Complete Complete Short Circuit Circuit Method Method . . . . . . . . . . . . . . . . . . . . . . . .

10

1.2.2 Short-Circuit Short-Circuit Accordi According ng to IEC IEC 60909 60909 or VDE VDE 102/103 102/103 . . . . . . . . . . .

11

1.2.3 ANSI-C37 ANSI-C37 Short-Circuit Short-Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.2.4 IEC 61363 Short-Circuit Short-Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.3 Optimal Optimal Power Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

1.3.1 1.3.1 OPF Contro Controls ls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

1.3.2 Constraints Constraints:: Active / Reactiv Reactive e Power Power Limits Limits . . . . . . . . . . . . . . . . .

14

1.3.3 1.3.3 Operat Operating ing Cost Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

1.4 Harmoni Harmonic c Analys Analysis is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

1.4.1 1.4.1 Standa Standard rd Model Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

1.4.2 Consider Consider Transi Transient ent Paramet Parameters ers . . . . . . . . . . . . . . . . . . . . . . . .

15

1.5 Stability/Electromagnetic Transients (RMS- and EMT-Simulation) EMT-Simulation) . . . . . . . . .

16

1.5.1 Mathematica Mathematicall Description Description . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

1.5.2 Equations Equations with stator stator and rotor flux state variabl variables es in stator-side stator-side p.u.-sys p.u.-system tem 17 1.5.3 1.5.3 Mechan Mechanics ics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

1.5.4 Equations Equations with stator stator currents currents and and rotor flux flux variables variables as as used in the the Pow- erFactory model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

1.5.5 Parameter Parameter Definition Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

1.5.6 1.5.6 Satura Saturatio tion n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

1.5.7 Simplificat Simplification ion for RMS-Simula RMS-Simulation tion

. . . . . . . . . . . . . . . . . . . . . .

23

1.5.8 1.5.8 Satura Saturatio tion n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

1.6 Input-, Input-, Output Output and State-V State-Variab ariables les of the PowerFactory Model . . . . . . . . . . .

25

1.7 Rotor Angle Definition Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

Synchronous Machine (ElmSym)

2

Contents

2 Input/Output Definition of Dynamic Models

28

2.1 Stability Model(RMS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.2 EMT-Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

3 References

31

List of Figures

32

List of Tables

33


3

1 General Description

1

General Description

This document describes the PowerFactory synchronous machine models, as used for the various steady state and dynamic power system analysis functions supported by PowerFactory , as there are: - Load flow analysis (section 1.1) - Short Circuit Analysis (section 1.2) - Optimal Power Flow (section 1.3) - Harmonics Analysis (section 1.4) - Stability/Electromagnetic Transients Analysis (section 1.5) This document describes the model equations that are implemented in PowerFactory . A list of input and output variables can be found in 2.

1.1 Load Flow Analysis

Figure 1.1: Load flow model of the synchronous machine For steady state load flow calculations a synchronous machine can be modelled by an equivalent voltage source behind the synchronous reactance. However, in actual load flow calculations, the controlled operation of a synchronous generator is typically modelled. Figure 1.1 shows the basic concept of a controlled synchronous machine modelled for load flow analysis

1.1.1 Reactive Power/Voltage Control

The PowerFactory model allows selecting between: • Voltage control Synchronous Machine (ElmSym)

4


• Power factor control. Large synchronous generators at large power stations typically operate in voltage control mode (“PV” mode). Smaller synchronous generators, e.g. embedded in distribution grids typically keep the power factor constant (”PQ”-mode). When enabling the Voltage control option of the generator element, the generator will control the voltage directly at its terminals. For more complex control schemes, e.g. controlling the voltage at a remote bus bar or controlling the voltage at one bus bar using more than one generator, a Station Controller model needs to be defined. In this case, the Station Controller adds an offset to the reactive power operating point specified in the synchronous generator element:

Q = Q 0 + K · ∆QSC O

(1)

For more details, please refer to the Technical Reference Document of the Station Controller.

Reactive Power Limits There are various ways for specifying reactive power limits:

• Fix reactive power limits in the element • Fix reactive power limits in the type • User-defined capability curve Generally, reactive power limits are only considered, if the synchronous machine is in voltage control and the load flow option Consider reactive power limits is enabled. If this option of the Load Flow command is disabled and the specified reactive power limits are exceeded, Power- Factory just generates a warning message but doesn’t apply any actual limit to the generator’s reactive power output. In the case that it is difficult to achieve a well balanced load flow state, an additional scaling factor can be applied to the reactive power limits. This scaling factor is more for “debugging reasons” and doesn’t have any physical interpretation. The reactive scaling factor is only considered if the load flow option Consider Reactive Power Limits Scaling Factor is enabled. Fix reactive power limits. Fix reactive power limits can either be specified at Element or Type level of the synchronous machine. Type limits are used when the option Use Limits Spec- ified in Type is enabled, otherwise, the model takes the Element limits that can either be defined on a p.u.-basis or using actual units (Mvar). User-defined Capability Curve. The user-defined capability curve allows specifying a complete, active power and voltage dependent capability diagram (see Figure 1.2). Userdefined capability diagrams are defined using the object IntQlim , which is stored in the Operational Library.


5


Figure 1.2: Capability curve object For more information, please refer to the Technical Reference of the Capability Diagram For assigning a capability diagram to a Synchronous Machine Element, the corresponding reference (pQlim) must be set (see Figure 1.3). If this pointer is assigned, all other attributes relating to reactive power limits are hidden and the local capability diagram of the Synchronous Machine Element displays the reactive power limits defined by the Ca- pability Curve object (IntQlim ) at nominal voltage.


6


Figure 1.3: Load flow page of the synchronous machine object

1.1.2 Active Power Control and Balancing Fix Active Power Active power will be set to a fix value if

• Option Reference Machine is disabled • No Secondary Controller object is selected • No Primary Frequency bias is defined Besides these local settings, the corresponding options on the “Active Power Control”-tab of the Load Flow Command either activate or deactivate the influence of Secondary Controller or Primary Frequency Bias.

Reference Machine The option Reference Machine has two consequences:

• Voltage angle at the machine’s terminal is fixed. Synchronous Machine (ElmSym)

7


• Machine balances active power if the Load Flow option As Dispatched is selected and the Balancing -Option by Reference Machine is enabled, or the Load Flow Option Secondary Control is selected and no Secondary Controller is specified in the network.

Primary Frequency Bias The primary frequency bias is considered if:

• Parameter Kpf (in MW/Hz) is > 0 and • Load Flow Option According to Primary Control is selected In this case, PowerFactory considers in all isolated grids a common frequency deviation df and establishes an active power balance through this variable and the primary frequency bias of the individual generators:

P = P 0 + K pf · ∆F With: • P: Actual active power in MW • P0: Active power setpoint in MW • Kpf: Primary frequency bias in MW/Hz • dF: Frequency deviation in Hz The Primary Controlled Load Flow represents that state of a power system following an active power disturbance, in which the primary governors have settled and the system finds a “quasi steady-state” before the secondary controlled power plants take over the active power balancing task. During the “primary frequency controlled” state, there is a deviation from nominal frequency.

External Secondary Controller For bringing frequency back to nominal frequency and/or for re-establishing area exchange flows of an interconnected power system, secondary controlled power plants take over the active power balancing task from the primary control after a few minutes (typically five minutes).

For simulating the “Secondary Controlled” state, which is an (artificial) steady state following the settling of the secondary control system: • A Power Frequency Controlled has to be specified and assigned to the machine • The Load Flow Option According to Secondary Control has to be activated. The actual active power of each generator is the defined by:

P = P 0 + K · ∆P SC O

(2)

with: Synchronous Machine (ElmSym)

8


• P: Actual active power of the machine in MW • P0: Active power setpoint in MW • K: Participation factor (to be specified in the power-frequency control object) • dPsco: Total active power deviation of all units controlled by the respective power frequency controller. For more information related to the Power Frequency Controller object, please refer to the corresponding Technical Reference Manual.

Inertial Power Flow During the first seconds following an active power disturbance such as a loss of generation or load, before the primary control takes over, the active power balance of the system is established by releasing energy from the rotating masses of all electrical machines.

This situation can be modelled by enabling the Load Flow Option According to Inertias . In this case, the variable dF represents an equivalent frequency rate of change and active power will be balanced according to the inertia of all generators (defined by the Acceleration Time Constant, to be found on the RMS-/EMT-Simulation page).

Active Power: Operational Limits, Ratings The active power rating can be entered as a Rating Factor on basis of the Nominal Active Power, which is calculated by the rated Apparent Power times the Rated Power Factor (typelevel). De-rating of generators can be considered by entering a rating factor < 1.

When considering active power limits in a load flow calculation, PowerFactory makes reference to the Operational Limits (Min. and Max.). These limits are considered when: • Generator participates in the active power balancing • Load Flow Option Consider Active Power Limits is selected.

1.1.3 Spinning if circuit breaker is open

This option decides whether a synchronous machine can be used for driving an island-network. Typical applications are: • Load flow set-up for a dynamic simulation of a synchronisation event. • Island around this generator shall form a supplied island-grid when disconnected from the main-grid (e.g. during Contingency Analysis). In case of contingencies that split the system two of more isolated areas, PowerFactory requires at least one synchronous generator with this option being enabled for assuming that the corresponding island can continue operating after having been islanded. Otherwise, the load flow calculation will assume a complete black out in the corresponding island (all loads and generators unsupplied).


9


1.2 Short-Circuit Analysis For short-circuit analysis, synchronous machines are represented by their • Subtransient equivalent • Transient equivalent • Synchronous equivalent depending on the considered time phase following grid fault. The distinction of the time dependence is due to the effect of increased stator currents on the induced currents in the damper windings, rotor mass and field winding. In the case of a fault near to a generator the stator current can increase so that the resulting magnetic field weakens the rotor field considerably. In steady state short circuit analysis, this field-weakening effect is represented by the corresponding equivalent source voltage and reactance. The associated positive sequence model of a synchronous machine is shown in Figure 1.4. The delayed effect of the stator field on the excitation and damping field is modelled by switching between the source voltage E”, E’ and E depending on the time frame of the calculation.

Figure 1.4: Single-phase equivalent circuit diagram of a generator for short-circuit current calculations which include the modelling of the field attenuation

1.2.1 Complete Short Circuit Method

In the “complete short circuit method”, the internal voltage source is initialized by a preceding load flow calculation. The “complete short circuit method” calculates subtransient and transient fault currents using subtransient and transient voltage sources and impedances. Based on the calculated subtransient and transient (AC-) currents, PowerFactory derives other relevant short-circuit indices, such as peak short circuit current, peak-break current, AC-break current, equivalent thermal short circuit current by applying the relevant methods according to IEC60909 (see next section).


10


1.2.2 Short-Circuit According to IEC 60909 or VDE 102/103

The IEC 60909 (equivalent to VDE 102/103) series of standards only calculates the subtransient time phase. Short circuit currents of longer time phases are assessed based on empirical methods by multiplying the subtransient fault current with corresponding factors. Figure 1.5 shows the basic IEC 60909 short circuit model of a synchronous machine.

Figure 1.5: Short-circuit model for a synchronous machine

When calculating initial symmetrical short-circuit currents in systems fed directly from generators without unit transformers, for example in industrial networks or in low-voltage networks, the following impedances have to be used Positive sequence system: 

(3)

Z S 1 = R S + jd Negative sequence system: 

Z S 2 = R S + jX 2 = RS + jX 2

(4)

Normally it is assumed that X 2 = X d . If X d and X q differ significantly the following can be used: 



X 2 = X 2 =








1 · X d + X q 2 





(5)

11


Zero sequence system: 

(6)

Z S 0 = RS 0 + jX S 0

For the subtransient reactance, the saturated value has to be used leading to highest possible fault currents. IEC 60909 makes no provision of the pre-fault state. It always considers a voltage factor cmax of 1,1 (or 1,05 in LV-networks). Because this approach would lead to overestimated fault currents, the impedance is corrected by a correction factor K G :

K G =

U n C max U rG 1 + χd sin ϕrG

(7)



with C max : Voltage factor, see IEC 60909-0, item 2.3.2, page 41, Table 1.1. All other short circuit indices are calculated precisely according to the IEC60909 (VDE 102/103)standard.

1.2.3 ANSI-C37 Short-Circuit

Besides IEC60909, PowerFactory supports short circuit calculation according to ANSI C-37. Similar to short circuit calculations according to IEC60909, only subtransient fault currents are actually calculated. For further details related to ANSI C-37, please refer to the original ANSI C-37 standard and corresponding literature.

1.2.4 IEC 61363 Short-Circuit

The IEC 61363 standard describes procedures for calculating short-circuits currents in threephase ac radial electrical installations on ships and on mobile and fixed offshore units. The calculation of the short-circuit current for a synchronous machine is based on evaluating the envelope of the maximum values of the machine’s actual time-dependent short-circuit current. The resulting envelope is a function of the basic machine parameters (power, impedance, etc.) and the active voltages (E”, E’, E) behind the machine’s subtransient, transient and steady-state impedance. The impedance are dependent upon the machine operating conditions immediately prior to the occurrence of the short-circuit condition. When calculating the short-circuit current, only the highest values of the current are considered. The highest values vary as a function of time along the top envelope of the complex timedependent function. The current defined by this top envelope is calculated from the equation:

√

(8)

iK (t) = 2I ac (t) + idc (t) The a.c. component I ac (t) is calculated with: 






t/T d

−

I ac (t) = (I kd − I kd )e



+ (I kd − I kd )e



t/T d

−

+ I kd

(9)

12


The subtransient, transient and steady-state currents are evaluated using equations: 









(10)











(11)

I kd = E q0 /Z d with Z d = (Ra + jX d ) and

I kd = E q 0 /Z d with Z d = (Ra + jX d )

Internal voltages considering terminal voltage and pre-load conditions are calculated using equations:

E q0 = U 0 / 3 + I 0 ∗ Z d



√



(12)



√



(13)

E q 0 = U 0 / 3 + I 0 ∗ Z d The d.c. component can be evaluated from equation:

√



I dc (t) = 2(I kd − I 0


sin

t/T dc

−

φ0 )e

(14)

13


1.3 Optimal Power Flow The OPF (Optimal Power Flow) function in PowerFactory allows the user to calculate optimal operational conditions, e.g. the minimization of losses or production costs by adjusting the active and reactive power dispatch of the generators. To consider the synchronous machine in the OPF calculation the following options have to be assigned on the “Optimization” tab of the synchronous machine element.

1.3.1 OPF Controls

It is possible to enable and disable the active and reactive power optimization of the machine. The active power flag allows the active power dispatch of the machine to be optimized in the OPF calculation. On the other hand, the reactive power flag allows the voltage reference of the machine to be adapted according to the OPF optimization function. When these options are disabled, the synchronous machine is treated as in a conventional load flow calculation during the execution of the OPF .

1.3.2 Constraints: Active / Reactive Power Limits

For every machine a minimum and maximum active and reactive power limit can be defined. For the reactive power limits there is also the possibility to use the limits which are specified in the synchronous machine type (enable the flag Use limits specified in type ). The active and reactive power limits will be considered in the OPF if and only if the individual constraint flag is checked in the synchronous machine element and the corresponding global flag is enabled in the OPF dialogue.

1.3.3 Operating Cost

The table Operating Costs specifies the costs ($/h) for the produced active power (MW) of the units. The representation of the data is shown automatically on the diagram below the table for checking purposes. The cost curve of a synchronous machine is calculated as the interpolation of the predefined cost points.


14


1.4 Harmonic Analysis 1.4.1 Standard Model

The equivalent circuits of the synchronous machine model for harmonics are shown in Figure 1.6.

Figure 1.6: Synchronous machine models for positive, negative and zero sequences The average inductance experienced by harmonic currents, which involve both the direct axis and quadrature axis reactances, is approximated by

L + Lq L = d 2 





(15)

At harmonic frequencies the fundamental frequency reactance can be directly proportioned. The influence of the skin effect on the resistance can be defined in a Frequency Polynomial Characteristic (ChaPol).

R = k(f ) · r

with

k(f ) = (1 − a) + a · (f /f nom )b

(16)

1.4.2 Consider Transient Parameters

When enabling the option Consider Transient Parameters the harmonic inductance is calculated from xd”, xd’ and xd, as entered for the RMS-simulation or EMT-simulation functions. Only in a very narrow band around nominal frequency, the effect of the transient and synchronous reactance is visible (see also Figure 1.7). Because of the highly accurate representation around nominal frequency this model can increase the accuracy of subsynchronous resonance studies based on frequency domain analysis.


15


Figure 1.7: Frequency domain representation of synchronous machine (Consider Transient Pa- rameters )

1.5

Stability/Electromagnetic Transients (RMS- and EMT-Simulation)

Figure 1.8 to Figure 1.10 show the equivalent circuit diagrams of the PowerFactory synchronous machine models, which are represented in a rotor-oriented reference system (Park coordinates, dq-reference frame). The d-axis is always modelled by 2 rotor loops representing the damping and the excitation winding. For the q-axis, PowerFactory supports two models, a model with one rotor loop (for Salient Pole machines) and a model with two rotor loops (for Round Rotor Machines).

Figure 1.8: d-axis equivalent circuit for the synchronous machine representation

1.5.1 Mathematical Description

Based on the equivalent circuit diagrams according to Figure 1.8 to Figure 1.10, the following differential equations can be derived describing the PowerFactory synchronous machine model for time domain simulations.


16


Figure 1.9: q-axis equivalent circuit for the synchronous machine representation (round rotor)

Figure 1.10: q-axis equivalent circuit for the synchronous machine representation (salient rotor) 1.5.2 Equations with stator and rotor flux state variables in stator-side p.u.-system

Using stator and rotor flux as state variables for the description of the synchronous machine model the following set of equations is resulting 1 . The stator voltage equations can be described as follows:

1 dψd − nψq ωn dt 1 dψq uq = r s iq + + nψd ωn dt 1 dψ0 u0 = r s i0 + ωn dt

(17)

dψe ωn dt dψD 0 = r D iD + ωn dt

(18)

ud = r s id +

Rotor voltage equations, d-axis:

ue = r e ie +

Rotor voltage equations, q-axis, round rotor: 1 The

equations of this section are expressed in load orientation for all currents, which is in contrast to the orientation of currents in Figure 1.8 to Figure 1.10 showing the actual orientation of currents of the PowerFactory model


17


dψk CO n dt dψQ 0 = r Q iQ + CO n dt 0 = r k ik +

(19)

Rotor voltage equations, q-axis, salient pole:

0 = r Q iQ +

dψQ ωn dt

(20)

For completing the model, the flux linkage equations are required: d-axis:

ψd = (xl + xmd )id + xmd ie + xmd iD ψe = x md id + (xmd + xrl + xle )ie + (xmd + xrl )iD

(21)

ψD = x md id + (xmd + xrl )ie + (xmd + xrl + xlD )iD q-axis, round-rotor:

ψq = (xl + xmq )iq + xmq ix + xmq iQ ψx = x mq iq + (xmq + xrl + xlx )ix + (xmq + xrl )iQ ψQ = x mq iq + (xmq + xrl )ix + (xmq + xrl + xlQ )iQ

(22)

ψq = (xl + xmq )iq + xmq iQ ψQ = x mq iq + (xmq + xrl + xlQ )iQ

(23)

q-axis, salient rotor:

Electrical torque t e in [p.u.]:

te = ψ d iq − ψq id

(24)

1.5.3 Mechanics

The accelerating torque is the difference between the input torque (mechanical torque) t m and the output torque (electromechanic torque) t e of the generator. The equations of motion of the generator can then be expressed as:

J tot ωn2 dn dn = T a,tot = t m + te 2 pz P r dt dt dϑ = ω n n dt Synchronous Machine (ElmSym)

(25)

18


The inertia of the generator and the turbine, plus the inertia of the mechanical load, can then be expressed in a normalized per unit form as the inertia time constant H tot in [s], with

H tot = H + H me =

1 J ω02 1 J me ω02 2 + · gratio 2 p2z P r 2 p2z P r

(26)

where p z is the number of pole pairs of the machine. The inertia time constant H can be given based on the rated apparent generator power, as shown in the equation above, or based on the rated active generator power. The mechanical starting time or acceleration time constant T a,tot in [s] is then

T a,tot = T a + T a,me = 2 · H tot

(27)

Both H and T a can be entered in PowerFactory based on S r or P r .

1.5.4 Equations with stator currents and rotor flux variables as used in the PowerFac- tory model

For obtaining maximum effectiveness with regard to the numerical accuracy and robustness of the model, the multiple time-scale properties of the equation system shall be used by partionning the equations into “fast” and “slow” equations. PowerFactory uses rotor flux and stator currents as state variables because this choice of state variables leads to the best possible multiple time-scale separation and hence to the best numerical properties.

Introducing the subtransient Flux:



ψd = k e ψe + kD ψD ψq = k x ψx + kQ ψQ

(28)



with the following definition of the factors k:

xmd xlD xd2 xmd xle kD = xd2 xmq xlQ kk = xq 2 xmq xlx kQ = xq 2 ke =

(29)

and:

xd2 = x le xlD + (xmd + xrl )(xle + xlD ) xq 2 = x lx xlQ + (xmq + xrl )(xlx + xlQ ) Synchronous Machine (ElmSym)

(30)

19


The stator flux linkage equations can now be expressed by:









ψd = x d id + ψd ψq = x q iq + ψq

(31)

With these definitions, the subtransient voltage can be introduced as follows:

1 dψd − nψq ωn dt 1 dψq + nψd uq = ωn dt 





ud =

(32)







Stator equations with stator currents and subtransient voltages:

xd did ωn dt xq diq uq = r s iq + ωn dt 

ud = r s id +









− nxq iq + ud



(33)

− nxd id + ud

u0 = r s i0 +

x0 di0 ωn dt

1.5.5 Parameter Definition

Table 1.1: Set of internal parameters Parameter rs xl xrl xmd xmq xlD xlQ xle xlx rD rQ re rx

Name in PF rstr xl xrl – – –

–

Description Stator resistance Stator leakage reactance Rotor leakage reactance d-axis magnetizing reactiance q-axis magnetizing reactance Leakage reactance of d-axis damper winding Leakage reactance of q-axis damper winding Leakage reactance of excitation winding Leakage reactance of x-winding Resistance of d-axis damper winding Resistance of q-axis damper winding Resistance of excitation winding Resistance of x-winding

Unit p.u. p.u. p.u. p.u. p.u. p.u. p.u. p.u. p.u. p.u. p.u. p.u. p.u.

The parameters according to Table 1.1 that have been used in the equation systems and the equivalent circuit diagrams are typically not available for synchronous machines. The classical input parameters of a synchronous machine, as they can be entered directly into the PowerFactory synchronous machine model are depicted in Table 1.2. For converting the set of input parameters according to Table 1.2 into the set of internal parameters according to Table 1.1, there are several methods described in the literature. Some of them are more accurate, some of them are highly simplified but easier to realize. Synchronous Machine (ElmSym)

20


Table 1.2: Standard input parameters of the synchronous machine Parameter rs xl xrl xd, xq xd’, xq’ xd”, xq” Td’, Tq’ Td”, Tq”

Name in PF rstr xl xrl xd, xq xds, xqs xdss, xqss Tds, Tqs Tdss, Tqss

Description Stator resistance Stator leakage reactance Rotor leakage reactance synchronous reactance (d- and q-axis) Transient reactance (d- and q-axis) Subtransient reactance (d- and q-axis) Transient time constant (short-circuit) Subtransient time constant (short-circuit)

Unit p.u. p.u. p.u. p.u. p.u. p.u. sec sec

PowerFactory applies a highly accurate parameter conversion method, as described in [1]. This method consists of th following formulas for the d-axis: Auxiliary variables:

x1 = x d − x1 + xrl x2 = x 1 − x3 =

(xd − xl )2 xd x2 − 1−



(34)



x1 xd xd xd xd 

 

xd xd xd T 1 = T d + 1 − + xd xd xd 











T d



T 2 = T d + T d T 3 =

(35)

T d T d 

x2 T 1 − x1 T 2 x1 − x2 x3 b = T 32 x3 − x2



a =

T le

 −  a2 4 a2 4

− a + =

2 − a + T lD = 2

(36)

b (37)

−b

Calculation of internal model parameter:


xle =

T le T 1 −T 2 x1 −x2

− T lD

xlD =

T lD T 1 −T 2 x1 −x2

+

T lD x3

− T le

+ T xle xle re = ωn T le xlD rD = ωn T lD 3

(38)

21


The q-axis parameters can be calculated analogously to the d-axsis parameters in case of a round rotor machine (2 rotor-loops). For a salient pole machine (1 rotor loop), the internal parameters can be calculated as follows:

(xq − xl )(xq − xl ) xq − xq 

xlQ =



(39)

xq xq − xl + xlQ rQ = xq ωn T q 



1.5.6 Saturation

The model described in the previous section was a purely linear model not considering any saturation effects. Generally, there exists saturation for all reactances of the synchronous machine model. However, for the purpose of system analysis, only main flux saturation has to be considered in the model by considering saturation of the magnetizing reactiances x m d and x m q . In the PowerFactory model saturation is considered in d- and q-axis:

xmd = k satd xmd0 xmq = k satq xmq0

(40)

whereas the level of saturation depends on the magnitude of the magnetizing flux:

ψm =



(41)

(ψd + xl id )2 + (ψq + xl iq )2

PowerFactory supports different approximations for saturation. In case of the saturation model 1, based on the two parameters SG10/SG12, a quadratic approximation is applied:

If ψ m > Ag :

Bg (ψm − Ag )2 csat = ψm

(42)

else:

csat = 0

(43)

In case of a tabular input, csat is calculated based on a spline approximation of the sampled values. Saturation in d-axis can be measured by no-load field tests. However, the saturation of the mutual reactance xmq in the q- axis cannot be measured easily and therefore assumptions have to be taken for q-axis saturation:


22


• In case of a round rotor machine, it is assumed that saturation in q-axis is equal to d-axis saturation. • In case of a salient rotor machine the saturation characteristic in q-axis is weighted by the ratio x q /xd .

1 1 + csat 1 = xmq 1 + xmd csat

ksatd = ksatq

(44)

0 0

Saturated magnetizing reactances apply to all formulas (21), (25), (23) and (28), (29), (30). Saturation in subtransient reactances is not considered, which represents a valid approximation because the subtransient reactance of a generator is only very weakly influenced by main flux saturation. The saturation of stator leakage reactances is a current-dependent saturation, i.e. high currents after short-circuits will lead to a saturation effect of the leakage reactances. Because the use of unsaturated subtransient reactances would therefore lead to underestimated maximum short circuit currents, it is recommended to use saturated values for xd” and xq” (“saturated” refers here to current saturation). For all other parameters (transient and synchronous reactance), unsaturated values shall be entered. The influence of main flux saturation is considered by the model as described above.

1.5.7 Simplification for RMS-Simulation

For RMS-simulations, stator flux transients are generally not considered. Neglecting stator flux transients in (33) leads to the following simplified stator voltage equations for RMS-simulations:









uq = r s id − xq iq + ud

ud = r s iq − xd id + uq

(45)

with the subtransiert voltages:





ud = −nψq 



uq = nψ d

(46)

Assumption that magnetizing voltage is approx. equal to magnetizing flux (for saturation) leads to the following approximation:

ψm ≈ um =




(ud + rs id − xl iq )2 + (uq + rs iq + xl id )2

(47)

23


1.5.8 Saturation

Figure 1.11 shows the definition of the main flux saturation curve. The linear line represents the air-gap line indicating the excitation current required overcoming the reluctance of the air-gap. The degree of saturation is the deviation of the open loop characteristic from the air-gap line.

Figure 1.11: Open loop saturation The characteristic is given by specifying the excitation current I 1.0 pu and I 1.2 pu needed to obtain 1 p.u respectively 1.2 p.u. of the rated generator voltage under no-load conditions. With these values the parameters s g 1.0 (= csat (1.0pu) ) and s g1.2 (= csat (1.2pu) ) can be calculated. Calculation of internal coefficients based on

ie (1.0 p.u) i0 ie (1.2 p.u) sg 1.2 = 1.2i0 sg 1.0 =

−1

(48)

−1

For quadratic saturation function

Ag =

  −

1.2 − 1.2 ssgg 1

Bg =

1.2 1.0

1.2 ssgg11..20

(49)

sg1.0 (1 − Ag )2

Alternatively, a sampled excitation current vs. voltage curve can be entered into the PowerFac- tory model.


24


1.6 Input-, Output and State-Variables of the PowerFactory Model Rotor current and rotor flux of the PowerFactory model is not expressed in a stator per-unit system as it has been used in section 1.5. PowerFactory uses the following p.u. definitions, which are also known as “no load p.u.-system”: Rotor currents:

   

ie = x md0 ie

iD = x md0 iD

(50)

ix = x mq0 ix

iQ = x mq0 iQ

Rotor-flux:

xmd0 ψe xe0 xmd0 ψD = ψD xD0 xmq0 ψx = ψx xx0 xmq0 ψQ = ψQ xQ0



ψe =

  

(51)

With

xe0 = x md0 + xlr + xle xD0 = x md0 + xlr + xlD xx0 = x mq0 + xlr + xlx

(52)

xQ0 = x mq 0 + xlr + xlQ Rotor voltage equations, d-axis:

 

dψe dt dψD 0 = iD + T D0 dt

  

ue = ie + T e0

(53)

Rotor voltage equations, q-axis, round rotor:



dψx dt dψQ 0 = iQ + T Q0 dt

 

0 = ix + T x0


(54)

25


Rotor voltage equations, q-axis, salient pole:



0 = iQ + T Q0

dpsiQ dt

(55)

with

xe0 re ωn xD0 T D0 = rD ωn xx0 T x0 = rx ωn xQ0 T Q0 = rQ ωn T e0 =

(56)

In the p.u.-system used for rotor variables of the PowerFactory model, there will be 1 p.u. stator voltage in case of no load conditions and 1 p.u. excitation voltage (and no saturation).

1.7 Rotor Angle Definition PowerFactory defines several rotor angles based on different references. The rotor angle is defined as the position of the d-axis. The following variables are available:

• fipol / [deg]: Rotor angle with reference to the local bus voltage of the generator (terminal voltage) • firot / [deg]: Rotor angle with reference to the reference voltage of the network (slack bus voltage) • firel / [deg]: Rotor angle with reference to the reference machine rotor angle (slack generator) • dfrot / [deg]: identical to firel • phi / [rad]: Rotor angle of the q-axis with reference to the reference voltage of the network (=firot-90 ) ◦

All rotor angles are shown in Figure 1.12.


26


Figure 1.12: Rotor Angle Definition


27


2

Input/Output Definition of Dynamic Models

2.1 Stability Model(RMS)

Figure 2.1: Input/Output Definition of the synchronous machine model for stability analysis (RMS-simulation)

Table 2.1: Input Definition of the RMS-Model Parameter Ve Pt Xmdm

Symbol/Equation

tm/(25), (24)


Description Excitation Voltage Turbine Power Torque Input

Unit p.u. p.u. p.u.

28


Table 2.2: Output Definition of the RMS-Model Parameter Psie psiD Psix psieQ Xspeed Phi Fref Ut Pgt Outofstep

Symbol/Equation ψe /(21), (51) ψD /(21), (51) ψx /(21), (51) ψQ /(21), (51) n/(25)

Xme Xmt cur1 cur1r cur1i P1 Q1 Utr Uti

te/(25)

Description Excitation Flux Flux in Damper Winding, d-axis Flux in x-Winding Flux in Damper Winding, q-axis Speed Rotor Angle Reference Frequency Terminal Voltage Electrical Power Out of step signal (=1 if generator is out of step, =0 otherwise) Electrical Torque Mechanical Torque Positive-sequence current Positive-sequence current Positive-sequence current Positive-sequence active power Positive-sequence reactive power Terminal Voltage, real part Terminal Voltage, imaginary part

Unit p.u.

p.u. p.u. p.u. rad p.u. p.u. p.u.

p.u. p.u. p.u. p.u. p.u. MW Mvar p.u p.u.

2.2 EMT-Model

Figure 2.2: Input/Output Definition of the synchronous machine model for stability analysis (EMT-simulation)


29


Table 2.3: Input Definition of the EMT-Model Parameter Ve Pt Xmdm

Symbol/Equation

tm/(25), (24)

Description Excitation Voltage Turbine Power Torque Input

Unit p.u. p.u. p.u.

Table 2.4: Output Definition of the EMT-Model Parameter Psie psiD Psix psieQ Xspeed Phi Fref Ut Pgt Outofstep

Symbol/Equation ψe /(21), (51) ψD /(21), (51) ψx /(21), (51) ψQ /(21), (51) n/(25)

Xme Xmt cur1 cur1r cur1i P1 Q1 Utr Uti

te/(25)


Description Excitation Flux Flux in Damper Winding, d-axis Flux in x-Winding Flux in Damper Winding, q-axis Speed Rotor Angle Reference Frequency Terminal Voltage Electrical Power Out of step signal (=1 if generator is out of step, =0 otherwise) Electrical Torque Mechanical Torque Positive-sequence current Positive-sequence current Positive-sequence current Positive-sequence active power Positive-sequence reactive power Terminal Voltage, real part Terminal Voltage, imaginary part

Unit p.u.

p.u. p.u. p.u. rad p.u. p.u. p.u.

p.u. p.u. p.u. p.u. p.u. MW Mvar p.u p.u.

30

3 References

3

References

[1] B. Oswald. Netzberechnung 2: Berechnung transienter Elektroenergieversorgungs-netzen . VDE-Verlag, 1 edition, 1996.


Vorgnge

in

31

List of Figures

List of Figures 1.1 Load flow model of the synchronous machine . . . . . . . . . . . . . . . . . . . .

4

1.2 Capability curve object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3 Load flow page of the synchronous machine object . . . . . . . . . . . . . . . . .

7

1.4 Single-phase equivalent circuit diagram of a generator for short-circuit current calculations which include the modelling of the field attenuation . . . . . . . . . .

10

1.5 Short-circuit model for a synchronous machine . . . . . . . . . . . . . . . . . . .

11

1.6 Synchronous machine models for positive, negative and zero sequences . . . .

15

1.7 Frequency domain representation of synchronous machine (Consider Transient Parameters ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

1.8 d-axis equivalent circuit for the synchronous machine representation . . . . . . .

16

1.9 q-axis equivalent circuit for the synchronous machine representation (round rotor) 17 1.10 q-axis equivalent circuit for the synchronous machine representation (salient rotor) 17 1.11 Open loop saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

1.12 Rotor Angle Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.1 Input/Output Definition of the synchronous machine model for stability analysis (RMS-simulation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.2 Input/Output Definition of the synchronous machine model for stability analysis (EMT-simulation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29


32

List of Tables

List of Tables 1.1 Set of internal parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

1.2 Standard input parameters of the synchronous machine . . . . . . . . . . . . . .

21

2.1 Input Definition of the RMS-Model . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.2 Output Definition of the RMS-Model . . . . . . . . . . . . . . . . . . . . . . . . .

29

2.3 Input Definition of the EMT-Model . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

2.4 Output Definition of the EMT-Model . . . . . . . . . . . . . . . . . . . . . . . . . .

30


33

TechRef_SynchronousMachine

Recommend Documents