DIgSILENT PowerFactoryTechnical 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
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Synchronous Machine (ElmSym) 1 Contents Contents 1 General Description 4 1.1 Load Flow Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 Reactive Power/Voltage Control . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.2 Active Power Control and Balancing . . . . . . . . . . . . . . . . . . . . . 7 1.1.3 Spinning if circuit breaker is open . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Short-Circuit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.1 Complete Short Circuit Method . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.2 Short-Circuit According to IEC 60909 or VDE 102/103 . . . . . . . . . . . 11 1.2.3 ANSI-C37 Short-Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.4 IEC 61363 Short-Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Optimal Power Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.1 OPF Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.2 Constraints: Active / Reactive Power Limits . . . . . . . . . . . . . . . . . 14 1.3.3 Operating Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Harmonic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4.1 Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4.2 Consider Transient Parameters . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 Stability/Electromagnetic Transients (RMS- and EMT-Simulation) . . . . . . . . . 16 1.5.1 Mathematical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5.2 Equations with stator and rotor flux state variables in stator-side p.u.-system 17 1.5.3 Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.5.4 Equations with stator currents and rotor flux variables as used in the Pow- erFactory model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5.5 Parameter Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.5.6 Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.5.7 Simplification for RMS-Simulation . . . . . . . . . . . . . . . . . . . . . . 23 1.5.8 Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.6 Input-, Output and State-Variables of the PowerFactory Model . . . . . . . . . . . 25 1.7 Rotor Angle 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 Synchronous Machine (ElmSym) 3 1 General Description 1 General Description This document describes the PowerFactory synchronous machine models, as used for the vari- ous 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 equiv- alent voltage source behind the synchronous reactance. However, in actual load flow calcula- tions, 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 1 General Description • 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 = Q0 + K · ∆QSCO (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 com- plete, active power and voltage dependent capability diagram (see Figure 1.2). User- defined capability diagrams are defined using the object IntQlim, which is stored in the Operational Library. Synchronous Machine (ElmSym) 5 1 General Description 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 correspond- ing 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. Synchronous Machine (ElmSym) 6 1 General Description 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 1 General Description • 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 = P0 + Kpf · ∆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 = P0 + K · ∆PSCO (2) with: Synchronous Machine (ElmSym) 8 1 General Description • 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 fre- quency controller. For more information related to the Power Frequency Controller object, please refer to the cor- responding 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 (type- level). 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 cor- responding 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 genera- tors unsupplied). Synchronous Machine (ElmSym) 9 1 General Description 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 cal- culations 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). Synchronous Machine (ElmSym) 10 1 General Description 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 genera- tors without unit transformers, for example in industrial networks or in low-voltage networks, the following impedances have to be used Positive sequence system: Z S1 = RS + jd00 (3) Negative sequence system: Z S2 = RS + jX200 = RS + jX2 (4) Normally it is assumed that X2 = Xd00 . If Xd00 and Xq00 differ significantly the following can be used: 1 00 X200 = X2 = · Xd + Xq00 (5) 2 Synchronous Machine (ElmSym) 11 1 General Description Zero sequence system: 00 ZS0 = RS0 + jXS0 (6) 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 KG : Un Cmax KG = (7) UrG 1 + χ00d sin ϕrG with Cmax : 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 three- phase 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 time- dependent function. The current defined by this top envelope is calculated from the equation: √ iK (t) = 2Iac (t) + idc (t) (8) The a.c. component Iac (t) is calculated with: 00 0 00 0 Iac (t) = (Ikd − Ikd )e−t/Td + (Ikd 0 − Ikd )e−t/Td + Ikd (9) Synchronous Machine (ElmSym) 12 1 General Description The subtransient, transient and steady-state currents are evaluated using equations: 00 Ikd 00 = Eq0 /Zd00 with Z 00d = (Ra + jXd00 ) (10) and 0 Ikd 0 = Eq0 /Zd0 with Z 0d = (Ra + jXd0 ) (11) Internal voltages considering terminal voltage and pre-load conditions are calculated using equations: √ 00 Eq0 = U 0 / 3 + I 0 ∗ Z 00d (12) √ 0 Eq0 = U 0 / 3 + I 0 ∗ Z 0d (13) The d.c. component can be evaluated from equation: √ 00 Idc (t) = 2(Ikd − I0 sin φ0 )e−t/Tdc (14) Synchronous Machine (ElmSym) 13 1 General Description 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. Synchronous Machine (ElmSym) 14 1 General Description 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 L00d + L00q L00 = (15) 2 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). b R = k(f ) · r with k(f ) = (1 − a) + a · (f /fnom ) (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 in- crease the accuracy of subsynchronous resonance studies based on frequency domain analy- sis. Synchronous Machine (ElmSym) 15 1 General Description 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. Synchronous Machine (ElmSym) 16 1 General Description 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 resulting1 . The stator voltage equations can be described as follows: 1 dψd ud = rs id + − nψq ωn dt 1 dψq uq = rs iq + + nψd (17) ωn dt 1 dψ0 u0 = rs i0 + ωn dt Rotor voltage equations, d-axis: dψe ue = re ie + ωn dt (18) dψD 0 = rD iD + ωn dt 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 Synchronous Machine (ElmSym) 17 1 General Description dψk 0 = rk ik + COn dt (19) dψQ 0 = rQ iQ + COn dt Rotor voltage equations, q-axis, salient pole: dψQ 0 = rQ iQ + (20) ωn dt For completing the model, the flux linkage equations are required: d-axis: ψd = (xl + xmd )id + xmd ie + xmd iD ψe = xmd id + (xmd + xrl + xle )ie + (xmd + xrl )iD (21) ψD = xmd id + (xmd + xrl )ie + (xmd + xrl + xlD )iD q-axis, round-rotor: ψq = (xl + xmq )iq + xmq ix + xmq iQ ψx = xmq iq + (xmq + xrl + xlx )ix + (xmq + xrl )iQ (22) ψQ = xmq iq + (xmq + xrl )ix + (xmq + xrl + xlQ )iQ q-axis, salient rotor: ψq = (xl + xmq )iq + xmq iQ (23) ψQ = xmq iq + (xmq + xrl + xlQ )iQ Electrical torque te in [p.u.]: te = ψd iq − ψq id (24) 1.5.3 Mechanics The accelerating torque is the difference between the input torque (mechanical torque) tm and the output torque (electromechanic torque) te of the generator. The equations of motion of the generator can then be expressed as: Jtot ωn2 dn dn = Ta,tot = tm + te p2z Pr dt dt (25) dϑ = ωn n dt Synchronous Machine (ElmSym) 18 1 General Description 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 Htot in [s], with 1 Jω02 1 Jme ω02 2 Htot = H + Hme = + · gratio (26) 2 p2z Pr 2 p2z Pr where pz 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 Ta,tot in [s] is then Ta,tot = Ta + Ta,me = 2 · Htot (27) Both H and Ta can be entered in PowerFactory based on Sr or Pr . 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 nu- merical properties. Introducing the subtransient Flux: ψd00 = ke ψe + kD ψD (28) ψq00 = kx ψx + kQ ψQ with the following definition of the factors k: xmd xlD ke = xd2 xmd xle kD = xd2 xmq xlQ (29) kk = xq2 xmq xlx kQ = xq2 and: xd2 = xle xlD + (xmd + xrl )(xle + xlD ) (30) xq2 = xlx xlQ + (xmq + xrl )(xlx + xlQ ) Synchronous Machine (ElmSym) 19 1 General Description The stator flux linkage equations can now be expressed by: ψd = x00d id + ψd00 (31) ψq = x00q iq + ψq00 With these definitions, the subtransient voltage can be introduced as follows: 1 dψd00 u00d = − nψq00 ωn dt (32) 1 dψq00 u00q = + nψd00 ωn dt Stator equations with stator currents and subtransient voltages: x00d did ud = rs id + − nx00q iq + u00d ωn dt x00q diq uq = rs iq + − nx00d id + u00d (33) ωn dt x0 di0 u0 = rs i0 + ωn dt 1.5.5 Parameter Definition Table 1.1: Set of internal parameters Parameter Name in PF Description Unit rs rstr Stator resistance p.u. xl xl Stator leakage reactance p.u. xrl xrl Rotor leakage reactance p.u. xmd – d-axis magnetizing reactiance p.u. xmq – q-axis magnetizing reactance p.u. xlD – Leakage reactance of d-axis damper winding p.u. xlQ Leakage reactance of q-axis damper winding p.u. xle Leakage reactance of excitation winding p.u. xlx Leakage reactance of x-winding p.u. rD – Resistance of d-axis damper winding p.u. rQ Resistance of q-axis damper winding p.u. re Resistance of excitation winding p.u. rx Resistance of x-winding 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 parame- ters 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 1 General Description Table 1.2: Standard input parameters of the synchronous machine Parameter Name in PF Description Unit rs rstr Stator resistance p.u. xl xl Stator leakage reactance p.u. xrl xrl Rotor leakage reactance p.u. xd, xq xd, xq synchronous reactance (d- and q-axis) p.u. xd’, xq’ xds, xqs Transient reactance (d- and q-axis) p.u. xd”, xq” xdss, xqss Subtransient reactance (d- and q-axis) p.u. Td’, Tq’ Tds, Tqs Transient time constant (short-circuit) sec Td”, Tq” Tdss, Tqss Subtransient time constant (short-circuit) 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 = xd − x1 + xrl (xd − xl )2 x2 = x1 − xd (34) x1 x00 x2 − xd d x3 = x00 1 − xdd xd 0 xd xd T1 = 0 Td + 1 − 0 + 00 Td00 xd xd xd T2 = Td0 + Td00 (35) q T3 = Td0 Td00 x2 T1 − x1 T2 a= x1 − x2 (36) x3 b= T2 x3 − x2 3 r −a a2 Tle = + −b 2 4 r (37) −a a2 TlD = + −b 2 4 Calculation of internal model parameter: Tle − TlD xle = T1 −T2 TlD x1 −x2 + x3 TlD − Tle xlD = T1 −T2 Tle x1 −x2 + x3 (38) xle re = ωn Tle xlD rD = ωn TlD Synchronous Machine (ElmSym) 21 1 General Description 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 )(x00q − xl ) xlQ = xq − x00q (39) x00q xq − xl + xlQ rQ = xq ωn Tq00 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. How- ever, for the purpose of system analysis, only main flux saturation has to be considered in the model by considering saturation of the magnetizing reactiances xm d and xm q. In the PowerFactory model saturation is considered in d- and q-axis: xmd = ksatd xmd0 (40) xmq = ksatq xmq0 whereas the level of saturation depends on the magnitude of the magnetizing flux: q ψm = (ψd + xl id )2 + (ψq + xl iq )2 (41) 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 ψm > Ag : Bg (ψm − Ag )2 csat = (42) ψm 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: Synchronous Machine (ElmSym) 22 1 General Description • 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 xq /xd . 1 ksatd = 1 + csat 1 (44) ksatq = xmq0 1 + xmd0 csat 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 = rs id − x00q iq + u00d (45) ud = rs iq − x00d id + u00q with the subtransiert voltages: u00d = −nψq00 (46) u00q = nψd00 Assumption that magnetizing voltage is approx. equal to magnetizing flux (for saturation) leads to the following approximation: q ψm ≈ um = (ud + rs id − xl iq )2 + (uq + rs iq + xl id )2 (47) Synchronous Machine (ElmSym) 23 1 General Description 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 I1.0pu and I1.2pu 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 sg1.0 (=csat (1.0pu) ) and sg1.2 (=csat (1.2pu) ) can be calculated. Calculation of internal coefficients based on ie (1.0p.u) sg1.0 = −1 i0 (48) ie (1.2p.u) sg1.2 = −1 1.2i0 For quadratic saturation function q s 1.2 − 1.2 sg1.2 g1.0 Ag = q sg1.2 1 − 1.2 sg1.0 (49) sg1.0 Bg = (1 − Ag )2 Alternatively, a sampled excitation current vs. voltage curve can be entered into the PowerFac- tory model. Synchronous Machine (ElmSym) 24 1 General Description 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: eie = xmd0 ie eiD = xmd0 iD (50) eix = xmq0 ix eiQ = xmq0 iQ Rotor-flux: xmd0 ψee = ψe xe0 xmd0 ψeD = ψD xD0 xmq0 (51) ψex = ψx xx0 xmq0 ψeQ = ψQ xQ0 With xe0 = xmd0 + xlr + xle xD0 = xmd0 + xlr + xlD (52) xx0 = xmq0 + xlr + xlx xQ0 = xmq0 + xlr + xlQ Rotor voltage equations, d-axis: dψee u ee = eie + Te0 dt (53) dψeD 0 = eiD + TD0 dt Rotor voltage equations, q-axis, round rotor: dψex 0 = eix + Tx0 dt (54) dψ Q 0 = eiQ + TQ0 dt Synchronous Machine (ElmSym) 25 1 General Description Rotor voltage equations, q-axis, salient pole: dpsiQ 0 = eiQ + TQ0 (55) dt with xe0 Te0 = re ωn xD0 TD0 = rD ωn xx0 (56) Tx0 = rx ωn xQ0 TQ0 = rQ ωn 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 gener- ator) • 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. Synchronous Machine (ElmSym) 26 1 General Description Figure 1.12: Rotor Angle Definition Synchronous Machine (ElmSym) 27 2 Input/Output Definition of Dynamic Models 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 Symbol/Equation Description Unit Ve Excitation Voltage p.u. Pt Turbine Power p.u. Xmdm tm/(25), (24) Torque Input p.u. Synchronous Machine (ElmSym) 28 2 Input/Output Definition of Dynamic Models Table 2.2: Output Definition of the RMS-Model Parameter Symbol/Equation Description Unit Psie ψe /(21), (51) Excitation Flux p.u. psiD ψD /(21), (51) Flux in Damper Winding, d-axis Psix ψx /(21), (51) Flux in x-Winding p.u. psieQ ψQ /(21), (51) Flux in Damper Winding, q-axis p.u. Xspeed n/(25) Speed p.u. Phi Rotor Angle rad Fref Reference Frequency p.u. Ut Terminal Voltage p.u. Pgt Electrical Power p.u. Outofstep Out of step signal (=1 if generator is out of step, =0 otherwise) Xme te/(25) Electrical Torque p.u. Xmt Mechanical Torque p.u. cur1 Positive-sequence current p.u. cur1r Positive-sequence current p.u. cur1i Positive-sequence current p.u. P1 Positive-sequence active power MW Q1 Positive-sequence reactive power Mvar Utr Terminal Voltage, real part p.u Uti Terminal Voltage, imaginary part p.u. 2.2 EMT-Model Figure 2.2: Input/Output Definition of the synchronous machine model for stability analysis (EMT-simulation) Synchronous Machine (ElmSym) 29 2 Input/Output Definition of Dynamic Models Table 2.3: Input Definition of the EMT-Model Parameter Symbol/Equation Description Unit Ve Excitation Voltage p.u. Pt Turbine Power p.u. Xmdm tm/(25), (24) Torque Input p.u. Table 2.4: Output Definition of the EMT-Model Parameter Symbol/Equation Description Unit Psie ψe /(21), (51) Excitation Flux p.u. psiD ψD /(21), (51) Flux in Damper Winding, d-axis Psix ψx /(21), (51) Flux in x-Winding p.u. psieQ ψQ /(21), (51) Flux in Damper Winding, q-axis p.u. Xspeed n/(25) Speed p.u. Phi Rotor Angle rad Fref Reference Frequency p.u. Ut Terminal Voltage p.u. Pgt Electrical Power p.u. Outofstep Out of step signal (=1 if generator is out of step, =0 otherwise) Xme te/(25) Electrical Torque p.u. Xmt Mechanical Torque p.u. cur1 Positive-sequence current p.u. cur1r Positive-sequence current p.u. cur1i Positive-sequence current p.u. P1 Positive-sequence active power MW Q1 Positive-sequence reactive power Mvar Utr Terminal Voltage, real part p.u Uti Terminal Voltage, imaginary part p.u. Synchronous Machine (ElmSym) 30 3 References 3 References [1] B. Oswald. Netzberechnung 2: Berechnung transienter Vorgnge in Elektroenergieversorgungs-netzen. VDE-Verlag, 1 edition, 1996. Synchronous Machine (ElmSym) 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 Synchronous Machine (ElmSym) 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 Synchronous Machine (ElmSym) 33 List of Tables Synchronous Machine (ElmSym) 34