Ronald G. Harley, Senior Member, IEEE, and. Juan C. Balda, Member, IEEE ... Gill G. Richards, Member, IEEE. Department of Electrical and Computer ...
88 SM 676-9 May 1989
Three-Phase Modeling for Transient Stability of Large Scale Unbalanced Distribution Systems
INFINITE I
0I
55
4
T2 6
T1I
Elham B. Makram, Senior Member, IEEE, V. Omar Zambrano, Student Member, IEEE, Ronald G. Harley, Senior Member, IEEE, and Juan C. Balda, Member, IEEE Department of Electrical and Computer Engineering Clemson University Clemson, SC Summary The transient stability of a distribution system containing rotating machines is usuatly evaluated by considering a symmetrical disturbance applied to a balanced network. Hence, most, if not all, available algorithms and stability programs use the bus admittance matrix for the positive sequence only. However, a typical distribution system may contain untransposed feeders, single- and/or three-phase unbalanced static shunt loads, single- and/or three-phase dynamic loads (such as induction motors), co-generators, transformers and capacitor banks. Furthermore, even if the network is balanced, unsymmetrical faults introduce unbal-
3
Ti
v
2
1
3
STATIC LOAD
STATIC LOAD
Fig. 1. Case study of a ten-bus distribution system.
ance.
The effects of unbalances on the transient behavior of a single induction motor in a two-bus system using simultaneous nodal voltage equations has been recently investigated. It was concluded that a significant error may occur by replacing an unbalanced shunt load by an averaged balanced load and by assuming an unbalanced feeder to be an equivalent balanced feeder. This paper extends the previous investigation to large scale networks using a generalized three-phase bus admittance matrix [Ybus] method which takes care of the unbalances referred to above. Developing the three-phase 1 YbusI itself is not new, but the novelty of the proposed method lies in the combination of the three-phase [ Ybus] with the dynamics of rotating machines in transient stability studies of large networks using step-by-step integration of nonlinear differential equations. The method utilizes phase frame representation of network and machine elements. In order to be general, the computer program allows the user to include synchronous generators, induction motors, transformers, feeders and static loads. The paper illustrates how this method can be applied systematically to evaluate the transient response of an unbalanced N-bus network containing synchronous and induction machines and subjected to symmetrical and unsymmetrical shunt faults. The results of several case studies concerning the ten-bus distribution system shown in Figure 1, are presented to illustrate the versatility of the method.
IEEE Power Engineering Review, May 1989
88 SM 685-0 May 1989
Reduced Order Models for an Induction Motor Group During Bus Transfer Gill G. Richards, Member, IEEE Department of Electrical and Computer Engineering Louisiana State University Baton Rouge, LA In power system simulations such as transient stability studies it is desirable to represent induction motor loads with a standard third order reduced model in order to decrease computational effort. The order reduction is achieved by setting the derivatives of stator flux linkages, which appear in the fifth, or full order model, to zero in the stator differential equations, with all equations referred to the synchronously revolving reference frame. This requires two assumptions. First, terminal voltage magnitude cannot continually change at a high rate for any period of time, and it must remain at synchronous frequency. This insures that the stator equation derivatives in the synchronous reference frame remain small. The second assumption is that the oscillating torque component, caused by stator flux linkage transients, can be safely ignored. These assumptions are valid during normal operation. However, the standard reduced model may perform inaccurately during a bus transfer, when groups of induction machines and possibly static loads are isolated from the power source momentarily and then reconnected. Both assumptions for order reduction are violated: the prevailing terminal voltage frequency gradually falls from synchronous frequency as the motor group slows during the disconnect period, causing slip frequency voltage variation on the synchronous reference frame. And when the motor group is reconnected, the natural response of the stator windings causes strong transient torque components which may alter speed recovery. These conditions reduce the accuracy of the standard reduced model, which will tend to show terminal voltages that are too high during disconnect and a speed recovery which ignores the drag effect of transient torque, compared to the full order model. 47
