Phase-Shifting TransformerModeling in PSSE by Carlos Grande-

Power Technology Issue 111 Siemens Energy, Inc. Phase-shifting Transformer Modeling in PSS®E Carlos Grande-Moran, Ph.D. Principal Consultant carlos.grande@siemens.com Phase-shifting transformers are often used in power systems to control the active power flow (MW) in branches in meshed networks or to control the active power flow at the interface between two large and stiff independent grids. The control of MW flow is achieved by adjusting the phase angle of the voltages at the phase-shifting transformer terminals. Phase-shifting transformers are also known as phase angle regulating (PAR) transformers. Phase-shifting transformers built for transmission grids are generally a three-phase, two-terminal pair design. The terminal where power is injected into the transformer unit is called the “source terminal” and the power where load is exiting the transformer unit is called the “load terminal.” The change in phase angle between the terminal voltages of the transformer unit is carried out by adding a regulated voltage to the phase-to-neutral voltage at the source terminal. A winding in series with a network branch is used to insert the regulated voltage that, when added with the appropriate phase to the source terminal phase-to-neutral voltage, sets up the desired direction of the active power flow between the transformer terminals. The convention used by manufacturers of these devices regarding an advanced or retarded phase angle is that an advanced phase angle means that the phase-to-neutral voltage at the load terminal leads the phase-to-neutral voltage at the source terminal, and a retarded phase angle means that the phase-to-neutral voltage at the load terminal lags the phase-to-neutral voltage at the source terminal. Two phase-shifting transformer designs are the most prevalent in power systems applications: symmetric phase-shifting transformers and asymmetric phase-shifting transformers. Symmetric phase-shifting transformers are designed such that the amplitudes of the no-load winding voltages do not change during the phase shifting operation. The complex transformer voltage ratio for this type of transformer is then j1.0 e  . The IEEE model for phase-shifting transformers is based on the symmetric phase-shifting transformer where the no-load phase angle Φ is the angle by which the winding 1 voltage (source side) leads the winding 2 voltage (load side). Figure 1 below shows a schematic diagram of a symmetric phase-shifting transformer. VΔ 2V  1VSource 1I Series Winding Exciting Winding Regulating Transformer Regulating Winding Series Transformer Excited Winding 2V 2I Load mVm Δ 1V m mV 2 1 V V φj 21 e0.1VV  and Figure 1 - Symmetric Phase-shifting Transformer Power Technology March 2012 Page 2 The asymmetric phase-shifting transformer can add an in-phase and quadrature regulating voltage with a winding connection angle α to the phase-to-neutral voltage at the source terminal. When the winding connection angle α is 0º or  180º, the quadrature regulating voltage is zero and the phase-shifting transformer operates as a conventional voltage or reactive power control transformer. For any winding connection angle α where the quadrature regulating voltage is not zero, the phase-shifting transformer will control both active and reactive power flows. A particular case where the winding connection angle α is  90º is used widely in power systems and is known as a “quadrature booster” transformer. This transformer type controls mostly active power flow but, because of its asymmetry, it also exerts a small control action on reactive power flow. Asymmetric phase-shifting transformers change not only the phase angle between the winding 1 and winding 2 voltages, but also their magnitudes. Thus, the complex transformer voltage ratio for this type of phase-shifting transformer is jt e  , where t is the transformer turns ratio, 1 2 t t , and t1 and t2 are the winding 1 and winding 2 turns ratios, respectively. Figure 2 below shows a schematic diagram of an asymmetric phase-shifting transformer operating as a quadrature booster phase-shifting transformer. qVΔ 2V  1VSource 1I Series Winding Exciting Winding Regulating Transformer Regulating Winding Series Transformer Excited Winding 2V 2I Load qVΔdVΔ Δ 1V d q Quadrature In-phase φj 2 1 21 etV VVV  and Figure 2 - Asymmetric Phase-shifting Transformer The no-load complex voltage ratio between winding 1 and winding 2 phase-to-neutral terminal voltages in phasor notation is given below: 1 2 V V j j1 2 t e t e t           (per unit) Neglecting winding and core losses and using the principle of conservation of energy, the power flowing into a phase-shifting transformer is equal to the power flowing out of the transformer. Thus, in phasor notation: * 22 * 11 IVIV  (per unit) and jt e   * 1 2 * 2 1 V I= V I (per unit) The asterisk in the equation above represents the complex conjugate operation. Power Technology March 2012 Page 3 The no-load phasor diagram for an asymmetric phase-shifting transformer is shown in Figure 3 below. The following parameters are used in the modeling of asymmetric phase-shifting transformers:  Phase shift angle range [Φmax,Φmin] in degrees  Number of tap positions  Nominal tap position: j01.0 e  (per unit on bus voltage base)  Transformer leakage impedance at nominal tap position in per unit on windings 1 and 2 voltage base and on either system apparent power base or winding 1-2 apparent power base  Winding connection angle α in degrees Applying the Law of Sines to the triangle in the phasor diagram in Figure 3, it is possible to obtain the magnitude of the regulating voltage ΔV in the series winding for a phase shift angle Φ and winding connection angle α. The expression for the magnitude of the regulating voltage is then:   sinV sin      ; Φmin ≤ Φ ≤ Φmax and α≠ Φ (per unit) and the off-nominal transformer turns ratio, t, for a winding angle α≠0º is given by:   sint sin     ; Φmin ≤ Φ ≤ Φmax and α≠ Φ (per unit) The equations above become simpler when applied to the quadrature booster phase-shifting transformer (α=90º). The new expressions for ΔV and t are then: V tan   ; Φmin ≤ Φ ≤ Φmax (per unit) and 1t cos   ; Φmin ≤ Φ ≤ Φmax (per unit) Power Technology March 2012 Page 4 minmax mxV mxS mnS mnV neutralS  je  jet  90 90 0 0 0 0 mnmx  180 qmxVΔ dmxVΔ Figure 3 - Asymmetric Phase-shifting Transformer General Model Phase-shifting Transformer Representation in PSS®E Phase-shifting transformers are modeled in PSS®E as two-winding transformers. Users have multiple choices to enter the model data in PSS®E. The choices of data codes for entering each winding’s leakage impedance and magnetization branch are the same as those for conventional voltage control transformers. These parameters can be entered in per unit or can be from data collected in the short circuit and no-load tests. Only winding 1 is allowed to have an under-load tap changer; winding 2 has an off-load tap changer. Tap position and maximum and minimum phase-shift angles are specified in electrical degrees. The transformer control band is specified in MW. Power Technology March 2012 Page 5 The circuit model used in PSS®E for phase-shifting transformers is shown in Figure 4 below. iie  Source Side jje  Load Side no 2 2Zt nonono jXRZ   j 2 1 e t tt 2 j 1 t:et  je  Figure 4 - Standard PSS®E Phase-shifting Transformer Circuit Model The model is made up of an ideal two-winding transformer with a complex transformer ratio, : jt e 1  , in series with the positive sequence phase-shifting transformer’s leakage impedance. Winding 1 is associated with the source side (“from” bus) of the phase-shifting transformer and winding 2 with the terminal of the ideal two-winding transformer on the load side (“to” bus) of the phase-shifting transformer. The positive sequence magnetizing admittance of the phase-shifting transformer is connected as a shunt to the winding 1 terminal. Voltage at the winding 2 terminal of the two-winding ideal transformer is not available to the user. Only during no-load conditions is the winding 2 terminal voltage ( j'e ) of the ideal transformer equal to the load terminal voltage of the phase-shifting transformer. And so, the phase shift angle range used in the model is the no-load phase shift angle range which relates the source terminal voltage ( ie ) to the winding 2 terminal voltage ( j'e ) of the ideal transformer. For power flow calculations the user must set: (1) the control mode in the transformer data record to either MW control (COD1=3) or asymmetric MW control (COD1=5), (2) the transformer upper RMA1 and lower RMI1 tap limits to the phase-shifting transformer angle limits in electrical degrees, (3) the transformer loading limits in MVA, (4) the winding connection angle α in electrical degrees, and (5) the upper VMA1 and lower VMI1 control band limits to be enforced by the phase-shifting transformer in MW. The MW control mode (COD1=3) is used for symmetric phase-shifting transformers and the asymmetric MW control mode (COD1=5) is used for asymmetric phase-shifting transformers. A negative control mode (COD1= -3 or -5) indicates that the phase shift angle is held fixed in the load flow solution. The winding connection angle default value is 0º. The use of this default value in the load flow computations with asymmetric control mode (COD1=5) results in PSS®E handling this phase-shifting transformer as a symmetric phase shifting transformer. Quadrature booster transformers are specified by selecting the asymmetric control mode and a winding connection angle of ± 90º. Selection of winding connection angles in the range 0º < α < 360º, and an asymmetric control mode, are used to specify asymmetric phase-shifting transformers where both in-phase and quadrature regulating voltage components are considered. The phase-shift angle adjustment is continuous, and if the regulated active power flow of at least one of the phase-shifting transformers falls outside its scheduled MW control band, all phase-shifting transformers and bus voltage phase angles are adjusted simultaneously. An excessively narrow control band may cause non-convergence of the power flow solution. Thus, a reasonable control band must be chosen in proportion to the active power target flow. For example, if the MW target flow is about 500 MW, a control band of  5 MW is a reasonable choice. The upper limit of the control band will be 505 MW and the lower limit will be 495 MW. Power Technology March 2012 Page 6 For short circuit calculations the phase-shifting transformer model for the negative sequence network will employ the same leakage impedance used in the load flow calculations, but the sign of the phase shift angle will be reversed. Hence, if the phase shift angle is +Φ in the positive sequence network, then this angle will be -Φ in the negative sequence network. This change in sign causes the shunt and series components of the negative sequence pi-equivalent network model to be different from those used in the positive sequence network. However, if the classical short circuit solution is selected in the PSS®E short circuit module, both the positive and negative sequence representations of a phase-shifting transformer will be identical, because all phase-shift angles of transformer units are set to 0º. The zero sequence network representation for phase-shifting transformers is dependent on the winding connections of the series and regulating (shunt) transformers, their core design, and whether the regulating transformer has any delta-connected tertiary winding. The connection code CC=9 can be used to represent phase-shifting transformers in the zero sequence network. Winding 1 is the regulating transformer and winding 2 is the series transformer. For the T-equivalent network used for CC=9, the impedance connected to the winding 1 terminals represents the zero sequence impedance of the regulating transformer as seen from its input terminals; the impedance connected to winding 2 terminals represents the zero sequence impedance of the series transformer as seen from its output terminals; and the shunt branch represents the impedance of a tertiary winding or the stray air path of the zero sequence flux through the tank walls. As an example of data preparation for a symmetrical phase-shifting transformer, consider a three phase, two-winding transformer, 230/230 kV, 700 MVA OA rating, and nominal winding phase-to-phase voltage of 230 kV. Its positive sequence leakage impedance at nominal tap is 0.0 +j0.1257 per unit on 700 MVA and 230/230 kV base. The total number of taps is 49 and the no-load phase shift angle range is  32º. The minimum, nominal and maximum tap positions are 1, 25 and 49, respectively. The PSS®E data record for this transformer unit is shown in Figure 5 below. Note that the winding connection angle is set to 0º and the control mode is set to COD1=3. Figure 5 - PSS®E Data Record for the Symmetrical Phase-shifting Transformer If the phase-shifting transformer data record shown in Figure 5 above were to belong to an asymmetric quadrature booster phase-shifting transformer, the only change required to be implemented would be to change the winding connection angle to 90º. Figure 6 below shows the data record for the quadrature booster phase-shifting transformer. Note that for a phase-shift angle Φ of 15º, the required value for winding 1 turns ratio, t1, is 1.0353 per unit on a 230 kV bus voltage base. Power Technology March 2012 Page 7 Figure 6 - PSS®E Data Record for the Asymmetrical Phase-shifting Transformer