Round Rotor Synchronous Generator

Introduction

The Round Rotor Synchronous Generator model in the DS Engine has been developed with the help of Mr. F. Paul deMello, a noted machines and dynamic stability expert. Material written on the subject of classic sub-transient machine models can be found in many textbooks and papers.

For EasyPower, this model was developed using classic materials written by Mr. deMello, as well as detailed development and critical review time with him. Suffice it to say, the models developed here fall in line with methods that are not new, and which have been in use for decades to simulate the detailed action of a synchronous machine.

The Circuit Model

The classic block diagram of the round rotor generator is shown in Figure 147. From Figure 147, we see that the model includes transient and sub-transient effects, d and q axis modeling, saturation on both axes, and excitation field effects as an input from an external excitation system. The model produces as its output air-gap flux, which then can be used to produce a Thevenin driving voltage behind an internal machine impedance for connection with the network. The EasyPower model also produces (as will be seen later) machine speed from a separate inertia model. In Figure 147, we also see that the model has an input from the terminal current of the machine (Id and Iq). These values, of course, must be converted from terminal form to d and q-axis form, for input into the model.

In Table 5, all of the parameters necessary for the model are tabulated, and given brief descriptions. These correspond to parameters seen in the Figure 147 blocks, as well as the machine rating (for proper scaling in the DS Engine), and the inertia component. In “ALL” cases in the DS Engine (generators and motors), inertia is a combined value of inertia. For generators, it represents the total inertia of the generator rotor and prime mover (turbine or engine). For a motor, it represents the total inertia of the motor rotor and load (pump, fan, etc.).

Another important note about Figure 147, is that the circuit shown is, in reality, only for the rotor of the generator. We can thus conclude that the majority of the time constant effects, etc.

take place in the rotor. Now, there are effects that take place in the stator, but given that DS simulations are typically not used to simulate any transient effects in the stator (DC offset, transient Ldi/dt effects, etc.), and the network is solved as a simple set of network equations (i.e.

not including transient Ldi/dt and C(dv/dt) effects), the rotor is where the predominant effects are located. Thus, the generator model presented here does not include any transient stator effects.

Why a D-Q Axis Model

As can be seen in Figure 147, all modeling of the rotor is performed using a form broken into a d-axis section and q-axis section. The d and q-axis formulation is simply a mathematical technique used to simplify modeling so that equations can readily be formed on the rotor independent of rotor position (angle), and so that parameters can be created that are easy to

determine. A quick look at the simple 2-pole rotor shown in Figure 147 shows why a d and q-axis formulation is appropriate and why it is an excellent method to simplify modeling.

 

Figure 147. Round Rotor Generator model block diagram.

In Figure 148, we see that for definition purposes, the d-axis (direct axis) is defined along the center of the rotor, and the q-axis (quadrature axis) is defined 90 degrees away in an orthogonal relationship. Thus, effects on the d-axis do not affect the q-axis, and vice versa, lending to a

mathematical simplification. Each axis can be analyzed independently. Now, in reality, any relationships between the rotor and the stator are linked across the air-gap. Thus, if we were to write coupling equations, we would need to write them in a distributed manner all along the circumference of the air-gap. This would necessitate a very complex and detailed set of equations, based on construction and the physical relationships between the rotor and stator

Center Line of

Figure 148. Synchronous generator rotor positional definitions.

To simplify this work, engineers (notably R. H. Park in 1929) determined that the reaction of the rotor could be broken into two orthogonal terms, denoted as the d and q-axis. The construction of the rotor worked well with this simplification, where the physical differences in the rotor from top to bottom are accumulated into two quadrature effects. Notice in Figure 148 how, due to the need for adding windings on the rotor, the rotor mechanically breaks somewhat into two pieces.

The air-gap on the d-axis is somewhat consistent all along the top of the pole piece, and the q-axis air-gap has a deeper air-gap (due to windings). The quadrature d and q-q-axis formulation method is the method used in the EasyPower round rotor generator model. That model uses a separate d and q-axis set of equations, and relates them to the stator through the angle of the rotor relative to the centerline of Phase A, which we denote as top dead center of the machine. Thus, as the rotor rotates relative to the stator, the flux linkages across the air-gap are constantly changing based on the angle of the rotor. In essence, we have a transformer with an air-gap in its core, where the one winding is constantly rotating relative to the other. As the winding rotates, the air-gap changes due to the physical construction of the rotor.

It is unfortunate that we only have time and space in this manual for a brief overview of the reasons and methods involved in creating formulations for the synchronous machine model.

There are many texts that can help you in this area, if you choose to do further study. There is also much technical content and history that one would be wise to at least perform a casual read to gain insight into the behavior of the synchronous machine. As an overall method, the self and mutual inductances of the rotor and stator are built into a set of equations. Those equations are then formulated into a block diagram as shown in Figure 147, for simulation in a time step oriented integration method. Machine reactances represent the mutual and self-inductances of the machine, and the time constants include the effect of resistance.

Table 6. Round Rotor Generator Model Parameters.

Parameter Units Description

Rated MVA MVA

Rated Efficiency Percent Rated Speed RPM Rated Voltage Volts LL Rated Current Amps Rated PF

Ra pu Stator winding resistance (armature resistance)

Xl pu Stator winding leakage reactance

Xd pu d-axis unsaturated synchronous reactance

Xq pu q-axis unsaturated synchronous reactance

X’d pu d-axis unsaturated transient reactance

X’q pu q-axis unsaturated transient reactance

X’’d = X’’q pu d & q-axis unsaturated sub-transient reactance T’do Seconds d-axis transient OC time constant

T’qo Seconds q-axis transient OC time constant T’’do Seconds d-axis sub-transient OC time constant T’’qo Seconds q-axis sub-transient OC time constant

E1 pu First voltage to define saturation

E2 pu Second voltage to define saturation

S( E1 ) pu Saturation at E1

S( E2 ) pu Saturation at E2

H kW-Sec / kVA Combined machine and prime mover inertia

D pu Machine damping, normally = 0

Windage pu Machine friction and windage

Note: OC means Open Circuit

Finally (and to illustrate the importance of further study into synchronous machine modeling), we supply some discussions of interest in regards to the round rotor generator and machines in general.

Transient. The transient terms on the d-axis correspond to the field winding.

Transient is used, since the field winding supplies a changing reaction as the field voltage is changed. In fact, the transient open circuit time constant T’do, can be measured by switching a DC source on the field of the generator. The current in the field will gradually grow according to an exponential time constant, over 2 to 10 seconds. The plot in Figure 149 shows an actual test, where 12 V DC was applied to the field of a generator. The time constant is the time it takes the current to grow to 63% of its final value.

Sub-Transient. Note that the sub-transient nature of the model on the d-axis is due solely to amortisseur effects in the rotor. In the round rotor generator, these are due to the solid nature of the rotor. Currents flow in a distributed fashion in the rotor iron, when there is slip between the stator rotating mmf (created by the stator currents) and the rotor’s rotating mmf (the field). Under a dynamic response, currents can be generated in the rotor iron (a reluctance effect), and thus can impact the response of the generator. In a salient pole machine, amortisseur windings or bars are added in and through the pole face of the rotor (parallel to the shaft), to supply needed damping (induction motor reluctance effect). Thus, the sub-transient nature of the model has been created to specifically model physical amortisseur components of the generator.

The name sub-transient is also a bit misleading, as we have come to re-write its definition in terms of short circuit current magnitude and the “smallest machine impedance” in mind. In reality, the sub-transient term on the d-axis is the amortisseur term of the machine, and due to construction simply has the shortest time constant and lowest machine reactance. As noted in Concordia’s text, “The name sub-transient is used in order to distinguish these reactances from the transient reactances, which are defined in the same way except that the presence of the amortisseur windings is ignored. Historically, the machine without amortisseur was analyzed first and the name transient appropriated for that case.”

Q-Axis. The q-axis circuit in the round rotor generator looks like a symmetrical version of the d-axis. This is due to actually simulating two equivalent amortisseur windings for the q-axis. Since the q-axis does not involve the field, there is actually no transient effect on the q-axis. However, due to solid rotor effects of the round rotor generator, an equivalent transient effect still occurs, but is due solely to amortisseur effects. As is noted for the salient pole generator in the next section, that machine does not have any transient q-axis effect, as there are no solid rotor effects, and no field linkage on the q-axis.

X’’d = X’’q. Refer to “Synchronous Motor Modeling” for more detail on synchronous machine data, and for specific reasons why X’’d and X’’q are

specified equal. This is a requirement for the all generator models used in the EasyPower.

(Applied DC Voltage VR and Measured Current Ife)

Figure 149. Test measurement of field time constant.

Some Linking Relationships

The model displayed in Figure 147 shows field voltage Efd, Id and Iq as inputs, and air-gap flux (ψ’’d and ψ’’q) as outputs. These seem all well and good, but a few defining relationships will help in understanding the interconnected nature of the model.

Field voltage, Efd, is an input from an excitation system model, and thus is generated externally.

This represents an actual separate piece of equipment that is producing a field voltage to supply current to the windings of the field. Without the input field voltage, the generator cannot be excited, and thus it will not produce any voltage. The EasyPower round rotor model assumes that if Efd is zero, that no terminal voltage will be created in an open circuit condition. This is the theoretical response. In reality, some residual magnetism will exist in the field, which will then create a small terminal voltage on the machine when rotating with no field applied.

In the next section we will discuss the unit’s inertia model, which necessitates an input of electrical gap torque from the generator model. The gap torque is calculated using the air-gap flux and current in the stator windings of the generator. This calculation can be performed as a d and q-axis calculation on rotor convention, or after conversion of the air-gap flux onto the stator reference. Due to the fact that we already have stator currents on a d and q-axis reference (needed as input into the model), we will calculate air-gap torque using d and q-axis components.

This calculation is:

" "

Airgap q d d q

T I I

This equation may not seem logical, as there is an intermixing of d and q-axis components;

however, it is correct when the detail of the formulations are included. This equation is supplied in per unit (all EasyPower machine modeling is internally performed in per unit).

Finally, as noted in the first sections in this manual, the Thevenin equivalent voltage is needed for the network model. As noted earlier, the network includes a Thevenin equivalent voltage source, where the source impedance is defned as RA + jX’’d. This source then needs a voltage to drive the network. This voltage is the airgap flux times the speed of the machine:

 

For these equations, the per unit speed (1+ p) corresponds to 1.0 at rated speed of the generator (see definition of p below). Again, the mixture of d and q-axis components is seen, which is the correct implementation. To complete the calculation, a conversion from the d and q-axis domain back to the stator refererence is needed as well, which as seen from Figure 148, simply involves knowing the position of the rotor, relative to the stator. As with most DS machine models that are simulating dynamics without transient effects (network and stator Ldi/dt etc. effects), rotor position at rated speed (which is the defined condition at initialization of the model) is assumed to be stationary and equal to the power angle, determined by machine loading. This is due to the fact that the network and rotor speed are the same, and makes modeling easier to formulate.

Internally, all EasyPower machine models assume that initialization occurs at rated speed (and thus frequency), and that the system is operating at rated frequency. This simplification generates a fixed angle (i.e. power angle is not changing) assuming there are no imbalances in delivered and generated power.

Inertia Model

The inertia modeling used in the round rotor generator is formulated according to classical inertia modeling in machines, and is shown in Figure 150. In Figure 150, we see the speed of the machine (p) being controlled by torque difference across the machine’s shaft, and being integrated (1/s) via the machine inertia (H). All values in Figure 150 are in per unit.

We also see that we have elected to include friction and windage losses, modeled as a constant torque. Without modeling friction and windage, if the unit is tripped, it will not be able to have a speed recovery. Simply put, when the valve on the governor system is fully shut (PMech = 0.0), the only way to slow the unit down is via friction and windage. This is typically compensated for in other stability packages by allowing the governor to output a slight negative power. This is the case if you ever notice a governor having a small negative limit specified instead of zero power.

Figure 151 is supplied to illustrate this. That figure shows generator frequency for an actual unit trip (Meas) overlaid with a simulated response (Sim). This test is typically called a partial load rejection, as the unit was loaded to a small fraction of its rated output before tripping the unit.

After the steam valve goes fully closed, the frequency of the unit turns around (starts to drop).

The flat line slope during the speed reduction is the slowing down of the unit from friction and windage losses, where the slope of the response can be used to calculate the percent friction and windage losses of the machine. The simulated response in Figure 151 was created using the Steam Turbine Governor model in EasyPower, with a matching generator with properly set inertia (H), and friction and windage torque.

Finally, load damping (D) is included, if desired, as a feedback from speed if there is no damping from system load. In most simulations, this value is left equal to zero, the default value in

Figure 150. Inertia vs. Speed Integral Model.

By definition, p is the per unit difference in speed from rated synchronous speed. For example, if the rated speed of the machine were 3600 RPM, and the actual speed were 3600 RPM, then p

would be:

3600 1.0 0.0 p 3600  If the actual speed were 3590 RPM, then p would be:

3590 1.0 0.00277 p 3600  

If the actual speed were 3630 RPM, then p would be:

3630 1.0 0.00833 p 3600 

You can also see that PMech from the governor system (prime mover and governor combination) is divided by speed to convert its output to torque.

0.995

Figure 151. Partial load rejection of actual machine measurement with simulated response.

Inertia Constant

The inertia constant H, is defined in kW-sec/kVA. With this per unit form of definition, we can directly simulate speed change according to the inertia equations just discussed. This value, however, is rarely supplied; instead, other forms (units) of inertia are supplied. The most typical English units value is WK² or WR², both in lb-ft². The constant H can easily be calculated from lb-ft² using the following equation:

 

^{2 2}



⁶ generator rotor, rotor (or engine) of the prime mover, and all connecting shafts and equipment.

Though machine speed is calculated within the generator model, it represents the speed of the combined generator and prime mover.

For those accustomed to having inertia specified in metric units, the following equation can be used:

Calculation of Inertia Constant from Test

To calculate the inertia constant from test, one must monitor machine speed or output frequency while subjecting the unit to a partial load rejection. Between 10 and 20% of rated load is typical.

If the test shown above in Figure 151 is looked at much closer from 2 to 5 seconds, one will discern a clear initial slope as shown in Figure 152. This initial slope is the units change in frequency vs. time with no governor operation, or with the input shaft power remaining constant.

Using this slope (df/dt), the Rated MVA of machine, the initial power Po before the rejection test and an equation that relates these, we can determine the machine inertia constant.

For this example test, the real and reactive power before tripping was 10.681 MW and –5.53 MVar. The frequency was 60 Hz before the trip and the final frequency oscillated between 60.18 and 60.23 Hz after the trip. From the figures, we derive the machine inertia from the initial slope of the speed response to be:

pu

Figure 152. Zoom in of partial load rejection to discern slope.

Initialization

Initialization for generator models should only be performed in an online condition. No provision is made in the EasyPower DS Engine to start a generator from an offline condition. The reason for this is simple; to actually start up a generator, there are a host of automatic controls and operator actions that spin the unit up and synchronize it with the grid. These are not being simulated. The process is not like a simple, “close the breaker and go” motor start simulation. If the generator is offline it will initialize with all states, Efd, and PMech set to zero. Closing the generator breaker will start the unit like an induction motor; however, no field will ever be applied, and thus doing such a simulation is basically unusable. We suggest only closing into a generator that is already online to simulate closing actions.

When the round rotor generator model is initialized, steps are taken to set all of the generator’s internal States (see Dynamics 101 tutorial for a clear definition of a State), so that the generated Thevenin voltage (discussed above) combined with the terminal voltage conditions and the source impedance of the generator (RA + jX’’d) will create the same power conditions from the

When the round rotor generator model is initialized, steps are taken to set all of the generator’s internal States (see Dynamics 101 tutorial for a clear definition of a State), so that the generated Thevenin voltage (discussed above) combined with the terminal voltage conditions and the source impedance of the generator (RA + jX’’d) will create the same power conditions from the

In document Reference Manual of Dynamic Voltage Stability (Page 154-167)