Trajectory Approximations for Direct Energy. Methods that Use Sustained Faults with Detailed. Power System Models. P. W. Sauer, Senior Member, A. K. Behera ...
A recent paper has suggested a method of correcting the first problem, which occurs during the disconnect period. The -approach is to adjust the speed of the revolving reference frame in the motor equations to the prevailing frequency of the isolated bus to which the motor group is connected. This causes terminal voltage to remain fairly constant relative to the reference frame so that the reduced order equations yield accurate results. However, this approach requires solution of a nonexplicit algebraic equation at each integration step to find the prevailing bus frequency. In addition, it is necessary to keep track of the varying reference frame position so that flux linkages can be re-initialized when the motor is reconnected to the synchronous bus. This appears to require a step by step solution similar to an additional integration. In this paper, the standard synchronous reference frame reduced model is retained at all times. Motor group terminal frequency variation during disconnect is corrected by a simple correction term in the stator equations based on the quasi steady state response of the motor. An approximate correction for the torque associated with the stator transient at reconnect is introduced by a change of variable and an additional term in the torque equation. Results The figure shows the slip, current, and voltage response of a group of three induction machines and a resistive load that are transferred between two buses while under load, with a .1 second (6 cycle) transfer time. Two of the machines are identical. The transfer is simulated using three different motor models. The response M1 corresponds to a simulation using fifth, or full order models of the motors. M2 corresponds to the standard reduced order model. M3 is the model presented in the paper, using correction factors for the disconnect period and the reconnect period. Note that the voltage in the M2 simulation is too high during the disconnect period for reasons discussed above. However, a correction factor enables M3 to replicate the full order model, Ml, almost exactly during the disconnect period. The corrected model is also closer to reproducing the full order reponse than the standard model for the speed recovery period after reconnect because of a correction factor added to the torque equation.
88 SM 688-4 May 1989
Trajectory Approximations for Direct Energy Methods that Use Sustained Faults with Detailed Power System Models P. W. Sauer, Senior Member, A. K. Behera, and M. A. Pai, Fellow Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Urbana, Illinois J. R. Winkelman, Member Scientific Research Lab. Ford Motor Co. Dearborn, Ml J. H. Chow, Member Dept. of Electrical Engineering Rensselaer Polytechnic Inst. Schenectady, NY
Abstract-This paper addresses the effects of fast dynamics on direct energy methods that use a sustained fault trajectory to approximate critically cleared trajectories and the corresponding critical energies. Integral manifold concepts are used to explain why fast dynamics can destroy such an approximation. Methods to remedy this problem are given and illustrated with a 10 machine example. 1. Introduction Several major breakthroughs in direct methods were reported by Athay, et al. in [11. In that work they proposed an energy function in integral form which could in principle accommodate virtually any model detail. Their case studies however indicated that the sustained fault Potential Energy Boundary Surface (PEBS) method could not accurately predict critical clearing times for detailed models. This paper proposes essentially the same generalized energy function in differential form, and presents a modification in the sustained fault PEBS method needed to accommodate detailed models. 2. The Power System Model The paper presents an energy method which is applicable to virtually any power sytstem dynamic model. The following general model of an n-machine system is used:
Si=
-
MA )= fi(6,
w,
xj=gi(6,
w,
0 =hi (6,cCo, all for i
0.00
0.80
1.60
2.40
3.20
TIME SEC
xlO'
4.00
4.80
Fig. 1. Voltage, current and slip for the motor group during 6 cycle bus transfer.
=
(2)
x, y)
x, y)
(3)
y)
(4)
x,
1, *,n.
3. The Generalized Potential Energy While "potential" energy has a special meaning for conservative systems, we will work with only two types of energy: Transient Kinetic, and Generalized Potential. This analysis is concerned only with the phenomenon of synchronous stability, so the energies associated with the total system will be discounted by introducing the center of inertia (COI) variables: 1
48
(1)
co
CCI
MT
where MT is the
sum
n
n
1 T
=
M1
of all inertia Mi's.
IEEE Power
Engineering Review, May
1989
isly In this paper, a system will be considered synchronouisly stable if each machine speed tends to the center of ine!rtia speed as times goes to infinity i.e. Wir-+ col as t--oo Consistent with this definition, the transient kinetic energ y is defined as, 1
VTKE=2
n
2= 1
M;@-wCO')
(5)
A Generalized Potential Energy VGPE is now defined such that when it is added to VTKE and evaluated along a cleared (final configuration) system trajectory, the total is constant with respect to time. This simply requires, d VGpE d VTKE
dt +_ dt=O
for all t > t,
or from Equations (2) and (5),
dVGpE dt dt
n
_
i=1
i (w-owco)fi(G, co, x, y)
(6)
where the f; are from the cleared (final configuration system equations. Hence by definition, for all time after clearing (final configuration),
VTKE + VGPE = E (a constant) 4. Summary of Results For a system to remain in synchronism after a fault has been cleared, all of the kinetic energy obtained by the system during the fault must be converted to some other form after the fault is cleared. VGPE as defined above is the sum of all other energies. The challenge of Direct Methods is to find the maximum amount of energy that can be absorbed by the system. Reference [11 proposed essentially the same VGPE and tried using sustained fault trajectories to approximate critically cleared trajectories for evaluation of VGPE. This paper proposes that this was due to fast dynamics in the detailed model, and proposes alternative trajectory approximations for evaluating VGPE. The alternatives are illustrated on a 10 machine example system.
References [11 T. Athay, V. R. Sherkat, R. Podmore, S. Virmani, and C. Puech, "Transient Energy Stability Analysis," Section IV of Systems Engineering for Power: Emergency Operating State Control, U.S. Dept. of Energy Publication CONF790904-P1, Davos, Switzerland, Sept. 30, 1979-Oct. 5, 1979.
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88 SM 689-2
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Algorithms for a Spot Price Responding Residential Load Controller F. C. Schweppe, Fellow, IEEE, B. Daryanian, Student Member, IEEE, and R. D. Tabors, Member, IEEE Massachusetts Institute of Technology Laboratory for Electromagnetic and Electronic Systems
Unbundling of electric service is a major topic of discussion in today's electric utility industry. One approach to unbundling is to charge the customers for electric energy as a function of the marginal cost of providing the energy (with revenue reconciliation to allow for recovery of capital costs). This leads to the establishment of a spot price based energy
marketplace. This paper discusses algorithms for a residential load controller designed to operate in a spot price based energy marketplace. The algorithms are specially designed to be implemented in the EPRI supported hardware/software system called the Load Control Emulator. The Load Control Emulator is a highly versatile, sophisticated tool designed to enable utilities to experiment with a set of demand side management techniques using a single programming system. Use of and response to price based signals is one of the control algorithms available within the emulator. The algorithms presented in this paper are designed around but not limited to the Load Control Emulator. The presentation includes a wide variety of structures at a variety of levels of sophistication. This enables an individual utility to choose the particular algorithms that the utility feels best match its own needs with those of its customers. The explicit subject of the paper is residential energy management. However, many of the concepts and even some specific algorithms presented are equally applicable to the small commercial customer. Important issues related to a price responding demand side managment system which are not discussed in this paper include how to specify the price variations, what class of price variations to offer the various customers, and cost benefit analysis from both the customers' and utility's point of view. The paper assumes one of two types of price signals are being provided o 1 Hour Update: Price changes each hour; specified. 5 minutes before the hour. o 24 Hour Update: Price changes each hour; specified at, say, 3pm of the preceding day for the period 2 am to 2 am starting the next day. The price variation each hour depends on overall utility system conditions related to load, weather, contract constraints, equipment outages (generation through distribution) and fuel cost availability. It is assumed that forecasts of future hourly price changes and weather conditions are also provided. A complete price responding energy management system has the following four main functions: o Tactical Control: Provides the real time control of end use devices o Behavior Modeling: Provides the behavior models that are needed by the Tactical Control o Usage Diagnostics: Provides customers with statistics on how they have been using the end use devices and their associated costs o Strategic Planning: Provides the criteria and limits which are used by Tactical Control Tactical Control, the real time heart of the overall system, is divided into two parts. The State Estimator determines the on-off status of the various devices and measures of the
IEEE Power Engineering Review, May 1989_
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