Digsilent Powerfactory: Technical Reference Documentation

 

 

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)

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Contents

Contents 1 Genera Generall Des Descri cripti ption on

 

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

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1.1.1 Reactive Reactive Powe Power/V r/Voltag oltage e Control Control  . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Active Active Powe Powerr Control Control and and Balancing Balancing   . . . . . . . . . . . . . . . . . . . . .

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

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

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1.2.1 Complete Complete Short Circuit Circuit Metho Method d . . . . . . . . . . . . . . . . . . . . . . . .

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

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

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

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

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1.3.1 1.3 .1 OPF Con Contro trols ls   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.3.2 Constraints Constraints:: Active / Reactiv Reactive eP Power ower L Limits imits   . . . . . . . . . . . . . . . . .

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1.3.3 1.3 .3 Operat Operating ing Cost Cost   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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1.4.1 1.4 .1 Standa Standard rd Model Model   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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

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

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1.5.2 Equations Equations with stator stator and rotor flux state variabl variables es in stator-s stator-side ide 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 flu flux x variables variables as as used in the the  Pow-  erFactory model   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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1.5.6 1.5 .6 Satura Saturatio tion n  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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1.5.8 1.5 .8 Satura Saturatio tion n  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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

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Contents

2 Input/Output Input/Output Definition Definition of Dynamic Dynamic Models

 

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

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

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3 Refe Refere renc nces es

 

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List of Figures

 

32

List of Tables

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1 Genera Generall Descrip Descriptio tion n

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Gen Genera erall Des Descri cripti ption on

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 ana analysis lysis (section (section 1.1  1.1)) - Short Circui Circuitt Analysis Analysis (section 1.2 (section 1.2)) - Optimal P Power ower Flow (section 1.3 (section 1.3)) - Harmoni Harmonics cs Analysis Analysis (sectio (section n 1.4) - Stability/Electromagnetic T Transients ransients Analysis (section (section 1.5  1.5)) This document describes describes the model equations that are implemented implemented in  PowerFactory . A list of input and output variables can be found in 2.

1.1 Loa Load d Flo Flow w Ana Analy lysis sis

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

1.1.1 Reacti Reactive ve Power/ Power/V Voltage Control Control

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

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• Power Power factor factor control. Large synchronous generators at large power stations typically operate in voltage control mode (“PV” mode). Smaller Small er synchronous synchronous generators, generators, e.g. embedded embedded in distributi distribution on grids typicall typically y keep the power power factor constant (”PQ”-mode). When enabling the Voltage control   option 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  =  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 reacti reactive ve power power limits in the element • Fix reacti reactive ve power power limits in the type • UserUser-defined defined capability capability curve 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   is enabled. If this option of the Load Flow command command is disabled and the specified reactive reactive power power limits are exceeded, exceeded, Power-  Factory  just  just generates a warning message but doesn’t apply any actual limit to the generator’s reactive power output. In th the e ca case se th that at it is diffi difficu cult lt to achi achiev eve e a well well bala balanc nced ed load load flo flow w st stat ate, e, an addi additi tion onal al sc scal alin ing g facto actorr can be applied to the reactive power limits. This scaling factor is more for “debugging reasons” and doesn’t have any physical physical interpretation. interpretation. The reactive reactive scaling scaling factor is only considered 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   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 1.2). ). Us User er-defined capability diagrams are defined using the object  IntQlim , which is stored in the Operational Library. Library.

Synchronous Machine (ElmSym)

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1 Genera Generall Descrip Descriptio tion n

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)  1.3).. If this pointer pointer is assigned assigned,, all othe otherr 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   object (IntQlim ) at nominal voltage.

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Figure 1.3: Load flow page of the synchronous machine object

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

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

Reference Machine The option  Reference Machine  has   has two consequences:

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

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1 Genera Generall Descrip Descriptio tion n

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

Primary Frequency Bias The primary frequency bias is consi considered dered if:

• Paramet Parameter er Kpf (in MW/Hz) MW/Hz) is  > 0 and • Load Fl Flow ow Option Option According to Primary Control  is   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   =  P 0  + K  pf  · ∆F  With: • P: Actua Actuall active active power in MW • P0: Active po power wer setpoint setpoint in MW • Kpf: Primary frequ frequency ency bias bias in MW/Hz • dF: Freq Frequency uency deviati deviation on in Hz The  Primary Controlled Load Flow   represents represents that state of a power power system following following an active 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 flow s of an interc interconn onnect ected ed power power sys system tem,, sec second ondary ary contro controlle lled d power power plants plants take take ov over er the act activ ive e 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 Po Power wer Frequency Controlled Controlled has to be specified and assigned to the machine • The Load F Flow low Option Option According to Secondary Control  has   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)

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1 Genera Generall Descrip Descriptio tion n

• P: Actual act active ive power power of the machine in MW • P0: Active po power wer setpoint setpoint in MW • K: Participation factor (to be sp specified ecified in the power-frequency control object) • dPsco dPsco:: Total active active power power deviation deviation of all units control controlled led by the respective respective power power frequency controller. For more information information related 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, loa d, before before the primary primary contro controll tak takes es over over,, the activ active e pow power er balanc balance e of the system system is establ establish ished ed 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 balanc ed according to the inertia of all generators generators (defined by the Accel Accelerati eration on Time Constant, 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 Operationa Operationall Limits (Min. and Max.). These limits are considered when: • Generator participates in the activ active e power balanci balancing ng • Load Fl Flow ow Option Option Consider Active Power Limits  is   is selected.

1.1.3 Spinni Spinning ng if if circuit circuit breaker is open 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 dynami dynamic c simulation of a synchronisation e event. vent. • Island around this generator shall form a supplied island-grid when disconne disconnected cted 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 respon ding island can continue continue operating operating after having been islanded. Otherwise Otherwise,, the load flow calculation will assume a complete black out in the corresponding island (all loads and generators unsupplied).

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1.2 Short-C Short-Circu ircuit it Anal Analysis ysis 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 induc ed currents in the damper windings, windings, rotor mass and field winding. winding. In the case of a fault fault near to a generator the stator current can increase so that the resulting magnetic field weakens the rotor field considerably considerably.. In steady steady state short circuit circuit analysis, analysis, this field-weaken field-weakening ing effect is represented repres ented by the corresponding corresponding equivalent equivalent source voltage and reactance. reactance. The associated associated positive sequence model of a synchronous machine is shown in Figure  1.4. The  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-phas Singl e-phase e equivalent equiv alent circuit diagram m of a generator generator for short-circuit short-circuit current calculations which include the modelling of thediagra field attenuation

1.2.1 Comple Complete te Short Circuit 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 relevan rele vantt short-circuit short-circuit indices, such as peak short circuit circuit curren current, t, peak-break peak-break current, AC-break AC-break current, equivalent thermal short circuit current by applying the relevant methods according to IEC60909 (see next section).

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1.2.2 Short-Ci Short-Circui rcuitt According According to IEC IEC 60909 60909 or VDE VDE 102/103 102/103

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

Figure 1.5: Short-circuit model for a synchronous machine

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

 

Z S 1  = R  =  R S  + j  +  jd

(3)

Negative sequence system: 

 

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

Synchronous Machine (ElmSym)





 1  · X d   + X q 2









 

(5)

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Zero sequence system: 

 

Z SS  0  =  R S 0  + jX   +  jX S 0

(6)

For the subtransien subtransientt reactance, reactance, the saturated saturated value has to be used leading leading to highest highest possible 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 max  sin n ϕrG U rrG G 1 + χd  si

(7)



with C max max  : Voltage factor, see IEC 60909-0, item 2.3.2, page 41, Table 1.1. Al Alll ot othe herr sh short ort ci circ rcui uitt indi indice ces s are are calc calcul ulat ated ed prec precis isel ely y acco accord rdin ing g to th the e IE IEC6 C609 0909 09 (V (VDE DE 102/ 102/10 103) 3)-standard.

1.2.3 ANSIANSI-C37 C37 Short-Circui Short-Circuitt

Besides IEC60909,   PowerFactory supports short circuit calculation according to ANSI C-37. Similar Simil ar to short circuit circuit calcu calculatio lations ns according according to IEC60909, IEC60909, only subtransie subtransient nt fault fault curren currents ts 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-Circui Short-Circuitt

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:

√ 

 

iK (t) = 2I ac ac (t) + idc (t)

(8)

The a.c. component  I aacc (t)  is calculated with: 



Synchronous Machine (ElmSym)



t/T d

−

I aacc (t) = (I kd − I kd )e



+ (I  (I kd − I kkdd )e



t/T d

−

+ I kkdd

 

(9)

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









 

(10)











 

(11)

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

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

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

E q0  =  U 0 / 3 + I 0 ∗ Z d



√ 



 

(12)



√ 



 

(13)

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

√ 2(I I kd − I 0   sin   φ0)e I dc dc (t) = 2( 

Synchronous Machine (ElmSym)

dc c t/T d

−

(14)

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1.3 Opt Optima imall Pow ower er Flo Flow w The OPF (Optimal Power Flow) function in   PowerFactory allows the user to calculate optimal operational opera tional conditions conditions,, e.g. the minimization minimization of losses or production costs by adjust adjusting ing the active and reactive power dispatch of the generators. To consider insynchronous the OPF calculation following options have to be assigned on the the synchronous “Optimization”machine tab of the machinethe element.

1.3.1 1.3 .1 OPF Contro Controls ls

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.   calculation. On the other hand, the reactive power flag allows the voltage reference of the machine machi ne to be adapted according 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 Constra Constraints: ints: Active Active / Reactive Reactive Power 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 1.3 .3 Ope Operati rating ng Cost 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.

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1.4 Har Harmon monic ic Ana Analys lysis is 1.4.1 1.4 .1 Sta Standa ndard rd Model 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 = 



Ld  + 2

Lq

 

(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  =  k((f ) f ) · r

  with

b

  k(f ) f ) = (1 − a) + a · (f /f nom nom )

(16)

1.4.2 Consid Consider er Transient ransient Parameter Parameters s

When enabling the option  Consider Transient Parameters  the  the harmonic inductance is calculated from xd”, xd’ and xd, as entered for the  RMS-simulation   or  EMT-simulation   functions 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) 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.

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

1.5

Stability/Electroma Stability/Electromagnetic gnetic Transients (RMS- a and nd EM EMT T-Simulation)

Figure 1.8 to Figure 1.8  to Figure 1.10 Figure 1.10 show  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 Mathema Mathematical tical Description Description

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

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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 Equati Equations ons with stator stator and rotor rotor flux state variable variables s in stator-side stator-side p.u. p.u.-syste -system m

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   + nψd uq   =  r s iq  + ωn dt

ud  = r  =  r s id  +

(17)

u0  =  r s i0  +   1 dψ0 ωn dt Rotor voltage equations, d-axis:

  dψe ωn dt   dψD 0 = r D iD  + ωn dt ue  = r  =  r e ie +

(18)

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 Figure  1.8 to  to Figure 1.10 Figure  1.10 showing  showing the actual orientation of currents of the  PowerFactory  model  model

Synchronous Machine (ElmSym)

17

 

1 Genera Generall Descrip Descriptio tion n

  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  =  x md id  + (x ( xmd  + xrl  + xle )ie + (x ( xmd + xrl )iD ψD   =  x md id  + (x ( xmd + xrl )ie + (x ( xmd  + xrl  + xlD )iD

(21)

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

(22)

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

(23)

q-axis, round-rotor:

q-axis, salient rotor:

Electrical torque  t e  in [p.u.]:

te  =  ψ d iq − ψq id

 

(24)

1.5.3 1.5 .3 Mec Mechani hanics cs

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 generato generatorr can then be expressed expressed as:

2 J ttot ot ωn dn

dn

 p2z P r dt   =  T a,tot dt   =  t m  + te dϑ   = ω n n dt Synchronous Machine (ElmSym)

(25)

18

 

1 Genera Generall Descrip Descriptio tion n

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

H ttot  + H m ot  =  H  + me e  =

2   1 J ω02   1 J m me e ω0 2 + · gratio 2  p2z P r 2  pz2 P r

 

(26)

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

T a,tot a,tot  =  T a  + T a,me a,me  = 2 · H tot tot

 

(27)

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

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

For obtaining maximum effectiveness with regard to the numerical accuracy and robustness of th the e mode model, l, th the e mult multip iple le ti time me-s -sca cale le prop propert ertie ies s of th the e equa equati tion on sy syst stem em sh shal alll be used used by parti partion onni ning ng 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  + (x ( xmd + xrl )( )(x xle + xlD ) xq 2  = x  =  x lx xlQ  + (x ( xmq  + xrl )( )(x xlx  + xlQ ) Synchronous Machine (ElmSym)

 

(30)

19

 

1 Genera Generall Descrip Descriptio tion n

The stator flux linkage equations can now be expressed by:









ψd  = x  =  x d id  + ψd ψq   =  x q iq  + ψq

(31)

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

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





(32)







Stator equations with stator currents and subtransient voltages:



  xd did   − nxq iq  + ud ωn dt xq diq uq   = r s iq  + ωn dt   − nxd id  + ud   x0 di0 u0  =  r s i0  + ωn dt

ud  = r  =  r s id  +











(33)

1.5.5 Para Parameter meter Definition Definition

Table 1.1: Set of internal parameters Parameter rs xl xr l xmd xmq xl D xl Q xl e xl x rD rQ re rx

Name in PF rstr xl xr l – – –

–

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 p..u. p.u. p.u. p.u. p.u. p.u. p.u.

The parameters according to Table  Table   1.1 that 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. Table  1.2. For converting the set of input parameters according to Table 1.2 Table  1.2 into  into the set of internal parameters according to Table 1.1 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

 

1 Genera Generall Descrip Descriptio tion n

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

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

Description Stator resistance Stator leakage reactance Rotor leakage reactance synchronous reactance (d- and q-axis) Transient reactance (d- and q-axis) Sub ubtr tra ansie nsient nt reac reacta tanc nce e (d (d-- an and d qq-ax axis is)) Transient time constant (s (sh hort rt-c -ciircuit) Su Subt btra rans nsie ient nt ti time me co cons nsta tant nt (s (sho hortrt-ci circ rcui uit) t)

Unit p.u. p.u. p.u. p.u. p.u. p.u .u.. sec se sec c

PowerFactory  applies a highly accurate parameter conversion method, as described in [1] 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 x xdd 

  

xd  xd   xd + T d  + 1 − T 1  = xd xd xd 











T d



T 2  =  = T   T d  + T d T 3  =

  x2 T 1 − x1 T 2 x1 − x2   x3 T 2 b  = x3 − x2 3

(35)

T d T d 



a  =

T llee

   −  

a2 b 4 a2  − b 4

 − a  +  =

2  − a  + T llD D  = 2

(36)

(37)

Calculation of internal model parameter:

xle  =

  T llee − T llD D T  T    T lD lD x x + x 1−

2

1−

2

3

  T llD D − T lle e xlD  = T  T    T le le x x + x   xle e r  = ωn T llee   xlD rD   = ωn T llD D Synchronous Machine (ElmSym)

1−

2

1−

2

3

(38)

21

 

1 Genera Generall Descrip Descriptio tion n

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:



xlQ  = (xq − xl )( )(x xq  − xl ) xq − xq xq xq − xl  + xlQ r  = 

(39)



Q

xq

ωn T q



1.5.6 1.5 .6 Sat Satura uratio tion n

The model described in the previous section was a purely linear model not considering any saturation effects. Generally,, there exists saturation Generally saturation for all reactances of the synchronous synchronous machine 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  =  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 + (ψ (ψq  + xl iq )2

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

If  ψ m  > Ag :

csat  =

  Bg (ψm − Ag )2 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 Satur ation in d-axis can be measured measured by no-load field tests. Howe However ver,, the saturation saturation of the

  xmq mutual   in the q- axis cannot be measured easily and therefore assumptions have to reactance be taken for q-axis saturation:

Synchronous Machine (ElmSym)

22

 

1 Genera Generall Descrip Descriptio tion n

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

  1 1 + csat   1 ksatq   = mq 1 +   xxmd csat ksatd  =

(44)

0 0

Saturated magnetizing reactances apply to all formulas   (21), (21),   (25), (25), (23 23)) and (28) (28),,   (29), (29),   (30). (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 short-circuits will lead to a saturation effect effect of the leakage reactances. reactances. Because 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 Simpli Simplificatio fication n for RMS-S RMS-Simul imulation ation

For RMS-simulations, stator flux transients are generally not considered. Neglecting stator flux transients in (33) in  (33) leads  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  =

 

Synchronous Machine (ElmSym)

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

(47)

23

 

1 Genera Generall Descrip Descriptio tion n

1.5.8 1.5 .8 Sat Satura uratio tion n

Figure 1.11 shows Figure 1.11  shows the definition of the main flux saturation curve. The linear line represents the air-gap air-ga p line indicating indicating the excitation excitation curren currentt required required ove overcomi rcoming ng 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 respective respectively ly 1.2 p.u. p.u. of the rated generator generator voltage under no-load conditions 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 (1..0 p.u  p.u)) i0   ie (1 (1..2 p.u  p.u)) sg 1.2  = 1.2i0

sg 1.0  =

−1

(48)

−1

For quadratic saturation function

1 .2 Ag   =

1.2

− 1−

  

Bg   =

sg1.2 1 0

sg

.

1.2 ssgg11..02

(49)

  sg1.0 (1 − Ag )2

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

Synchronous Machine (ElmSym)

24

 

1 Genera Generall Descrip Descriptio tion n

1.6 Input Input-, -, Ou Output tput and S Statetate-V Variab ariables les o off 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 following p.u. definitions definitions,, which are also known as “no load p.u.-system”: Rotor currents:

   

ie  = x  =  x md0 ie

iD   =  x md0 iD

(50)

ix  = x  =  x mq0 ix

iQ  = x  =  x mq0 iQ

Rotor-flux:

ψe  =

  xmd0 ψe xe0

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

  

(51)

With

xe0  = x  =  x md0  + xlr  + xle xD0  =  x md0  + xlr  + xlD xx0  = x  =  x mq0  + xlr  + xlx xQ0  = x  =  x mq 0 + xlr  + xlQ

(52)

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 0

0 = ix  + T x dt dψQ 0 = iQ  + T Q0 dt



Synchronous Machine (ElmSym)



(54)

25

 

1 Genera Generall Descrip Descriptio tion n

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 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 Rot Rotor or Ang Angle le Defi Definit nition ion PowerFactory  defines sever several al rotor angles based on different different references. references. The rotor angle is defined as the position of the d-axis. The following variables are available:

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

All rotor angles are shown in Figure 1.12 Figure  1.12..

Synchronous Machine (ElmSym)

26

 

1 Genera Generall Descrip Descriptio tion n

Figure 1.12: Rotor Angle Definition

Synchronous Machine (ElmSym)

27

 

2 Input/Output Input/Output Definition Definition of Dy Dynamic namic Models Models

2

Input Input/Outp /Output ut Defin Definition ition of Dyna Dynamic mic Models Models

2.1 Sta Stabil bility ity Mod Model( el(RMS RMS))

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

Table 2.1: Input Definition of the RMS-Model Par aram amet eter er

V Pet Xmdm

Symbo ymbol/ l/E Equat quatio ion n

Des Desc cri ript ptio ion n

Unit nit

tm/(25), (24)

Euxrcbitin ae tioPnoV T woelrtage Torque Input

p..u u.. p p.u.

Synchronous Machine (ElmSym)

28

 

2 Input/Output Input/Output Definition Definition of Dy Dynamic namic Models Models

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 par t Terminal Voltage, imaginar y par t

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 EMT-Mo -Mode dell

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

Synchronous Machine (ElmSym)

29

 

2 Input/Output Input/Output Definition Definition of Dy Dynamic namic Models Models

Table 2.3: Input Definition of the EMT-Model Par aram amet eter er Ve Pt Xmdm

Symbo ymbol/ l/E Equat quatio ion n

tm/(25), (24)

Des Desc cri ript ptio ion n Excitation Voltage Turbine Power Torque Input

Unit nit 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)

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)

Xme Xmt cur1 cur1r cur1i P1 Q1 Utr Uti

te/(25)

Electrical Torque Mechanical Torque Positive-sequence current Positive-sequence current Positive-sequence current Positive-sequence active power Positive-sequence reactive power Terminal Voltage, real par t Terminal Voltage, imaginar y par t

Synchronous Machine (ElmSym)

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 Refe Refere renc nces es

3

Refe Refere renc nces es

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

Synchronous Machine (ElmSym)

Vorgnge

in 

31

 

List of Figures

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

4

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

6

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

7

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

10

1.5 Short-circuit Short-circuit model for for a synchron synchronous ous ma machine chine   . . . . . . . . . . . . . . . . . . .

11

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

15

1.7 Frequenc Frequency y domain representa representation tion of synchronou synchronous s machine machine (Consider Transient  Parameters )   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

1.8 d-axis d-axis equivalent equivalent circuit circuit for the synchron synchronous ous machi machine ne representatio representation n . . . . . . .

16

1.9 q-axis q-axis equivalent equivalent circuit circuit for the synchronous synchronous machine machine representation representation (round (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/Outpu Input/Outputt Definition Definition of the synchronous synchronous machine model ffor or stabili stability ty analysis analysis (RMS-simulation)   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (RMS-simulation)

28

2.2 Input/Outpu Input/Outputt Definition Definition of the synchronous synchronous machine model ffor or stabili stability ty analysis analysis (EMT-simulation)   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

Synchronous Machine (ElmSym)

32

 

List of Tables

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

20

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

21

2.1 Input Definition Definition of tthe he RMS-Mo RMS-Model del   . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.2 Output Output Definitio Definition n of the RMS-Mod RMS-Model el   . . . . . . . . . . . . . . . . . . . . . . . . .

29

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

30

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

30

Synchronous Machine (ElmSym)

33

 

List of Tables

Synchronous Machine (ElmSym)

34