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VOLTAGE AND VAR OPTIMIZATION
FOR ENERGY CONTROL //
by
JERRY S. HORTON ,.-·,/
Dissertation submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY
Electrical Engineering
APPROVED:
L. L. Griga;y, &airman
l> c 6' '7' t>'e .. "
L. T. Watson A. G. Phadke
/ t111:..~~.__:/~'--==----;/ J. A. Nachlas /
L. C. Frair f
December, ·19s3
Blacksburg, Virginia
VOLTAGE AND VAR OPTIMIZATION
FOR ENERGY CONTROL
by
JERRY S. HORTON
(ABSTRACT)
The topic of voltage optimization has been of recent interest to
many researchers and is being considered by many utilities for
implementation in their Energy Control Centers. Much of the past
research has utilized linear programming incremental models or
strictly gradient techniques. This research combines both linear
programming (LP) and generalized reduced gradient techniques (GRG) for
voltage optimization. The result provides most of the advantages of
both LP and gradient techniques. Further, the research incorporates
important considerations for implementation in an Energy Control
Center.
'.,
ACKNOWLEDGEMENTS
My research extended over a considerable time period and many of
my friends have provided encouragement. The author would first give a
special thanks to his wife and children who have certainly sacrificed
much to allow him to pursue his research studies. The author
certainly could not have achieved his objective without their
understanding and support. I give a special thanks to Dr. L. L.
Grigsby for the encouragement and confidence he extended over the
years I have known him. The author could not wish to have a better
research advisor than he. Another thanks is gratefully extended to
' who, over the years, has been a good friend and has
been extremely helpful with his suggestions. The author also thanks
Energy and Control Consultants and the Energy Research Group at
Virginia Tech, who have supported his research studies, and
of Houston Lighting and Power Company, for his valuable
insights in Energy Control Systems.
Finally, the author is grateful for the dedication and patience of
and in the typing of
the thesis.
iii
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ABSTRACT CHAPTER 1 CHAPTER 2 CHAPTER 3 CHAPTER 4 CHAPTER 5 CHAPTER 6
CHAPTER 7 CHAPTER 8
CHAPTER 9 BIBLIOGRAPHY APPENDIX 1 APPENDIX 2 APPENDIX 3 APPENDIX 4
INTRODUCTION POWER SYSTEM CONTROLS PROBLEM DESCRIPTION • DISCUSSIONS OF PREVIOUS RESEARCH GENERAL SOLUTION METHODOLOGY IMPLEMENTATION OF THE VOLTAGE CONTROL ALGORITHM 6.1 Introduction • • ••• 6.2
6.3 6.4
Feasibility Optimization Step Summary· of the Algorithm •
6.5 Computer Implementation DISCUSSION OF RESULTS • • • • • IMPLEMENTATION CONSIDERATIONS • 8.1 Integration of Voltage Control Into An
Energy Control Center. 8.2 Algorithm Modifications For On-Line
8.3 8.4 8.5 8.6
Operation ••••••• EMS Simulation • • • Simulation Results Dispatcher Man-Machine Interface Extension To Other Controls
CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK. •
. . . . .
iv
1
5 18 26
35 47 47 49 53 63 67 66
91
91
95 97 99
110 122
126 128 135 141
146 151
APPENDIX 5 APPENDIX 6 APPENDIX 7 APPENDIX 8 APPENDIX 9 VITA
TABLE OF CONTENTS (Continued)
v
159 169 179 188
201
223
LIST OF TABLES AND FIGURES
Page
Figure 2-1 Overview of an Energy Control Center . 7 Figure 4-1 GRG Solution Procedure . . . . . . . 31 Table 6-1 Error Analysis - Reactive Sensitivities To Real
Power Losses . . . . . . . . . . . . . . . 57 Table 6-2 Error Analysis - Real Power Sensitivities 58 Figure 6-1 Algorithm Summary . . . . 64 Table 7-1 Feasibility Test Results . . . . . 67 Table 7-2 30 Bus IEEE Modified - Initial Power Flow
Solution . . . . . . . . . . . . 68 Table 7-3 118 Bus IEEE Violation Summary . . . . . . 69 Table 7-4 118 Bus IEEE LP Summary 70 Table 7-5 118 Bus IEEE Initial Power Flow Solution 71 Table 7-6 118 Bus Final Solution . 73 Table 7-7 Solution Summary . . . . . . . . . . . 75 Table 7-8 Convergence Summary . . . . . . . . 79 Table 7-9 IEEE 14 Bus Test Results . . . . . . . . . . 80 Table 7-10 23 Bus Test Results . . . . . 80 Table 7-11 IEE 30 Bus Modified Test Results . 81 Table 7-12 IEEE 30 Bus Test Results . 81 Table 7-13 IEEE 57 Bus Test Results . . . . . 82 Table 7-14 IEEE 18 Bus Test Results . 83 Table 7-15 118 Bus IEEE GRG Summary . . . . . . . . . 84 Table 7-16 14 Bus IEE GRG Summary . . . . . . . . . 85 Table 7-17 23 Bus GRG Summary . . . . . . . 85 Table 7-18 30 Bus IEEE GRG Summary . . . . . . . . 86 Table 7-19 5 Bus Stagg GRG Summary . . . . . . . . . . . . 86 Table 7-20 3 Bus Stagg GRG Summary . . . . . . . . . . . . 87 Table 7-21 30 Bus IEEE Modified GRG Summary 87 Table 7-22 57 Bus IEEE GRG Summary . . . . 88 Table 7-23 Linearization Summary 92
vi
Figure 8-1 Figure 8-2 Table 8-1
Figure 8-3 Table 8-2 Table 8-3 Table 8-4 Table 8-5 Table 8-6 Table 8-7 Table 8-8 Table 8-9 Table 8-10 Table 8-11 Table 8-12 Table 8-13 Table 8-14 Table 8-15 Table 8-16 Table 8-17 Table 8-18 Table 8-19
LIST OF TABLES AND FIGURES (Continued)
Energy Control Center Overview • • • • • Power System - Control Center Simulation • Simulator Test Case - Equal Cost Curves -IEEE 30 Bus Mod. • • • • • • • MW Losses - 30 IEEE Modified • • Production Cost Savings - 30 Bus IEEE 5 Bus Stagg Production Cost Savings($) 14 Bus IEEE - Production Cost Savings($) • 30 Bus IEEE Mod. - Production Cost Savings($) 57 Bus IEEE Mod. - Production Cost Savings($) 30 Bus IEEE Mod. - Production Cost Savings($) Study Voltage Control Voltage Control Display Menu • Voltage Monitor - Control Summary Voltage Monitor - Event Summary Voltage Monitor - Power Plant Summary Voltage Monitor - Power Plant Limits •• Voltage Monitor - Voltage Control Display Voltage Monitor - Transformer Control Display Voltage Monitor - Capacitor Control Display Voltage Monitor - Constraint Summary Voltage Monitor - Violation Summary Voltage Limit Display •••••••
vii
92 98
101 102 104 105 106 107 108 109 111 113 114 114 117 117 119 119 120 120 121 123
CHAPTER 1
INTRODUCTION
This research is concerned with voltage and reactive power
optimization and control for electric power systems. The purpose is
to schedule voltages and reactive generation such that power system
constraints are met and a defined objective is achieved.
The research work required the development of a mathematical
solution to the reactive power and voltage optimization problem and
the study of the engineering design requirements for successful
implementation in an energy control center. Hopefully, the
contribution will aid in the successful application of reactive power
control in today's energy control centers.
This topic is of current interest to many utilities because of
high fuel costs and the difficulty in maintaining power system
security due to improper reactive power coordination. Aldrich,
et.al., [1 J predicted that by properly coordinating reactive
generation and transformer taps, $350,000 per year (1977 fuel costs)
could be saved in fuel costs due to decreased real power transmission
losses. Operational security violations due to poor reactive power
support and voltage are typical in many power systems. The French
network collapse in December, 1978, was basically a voltage-reactive
power phenomena [2]. A total system load of 38 GW was reduced by 75%
for several hours in this disturbance. Moreover, voltage control has
1
2
been identified as one of the most important operational needs [3].
Further, some utilities are implementing voltage control algori thrns
such that reactive sources may be controlled more effectively.
In past developments, the reduced gradient and generalized reduced
gradient ( GRG) techniques [ 4, 5, 6 J were used for power system
optimization problems. These gradient methods have several drawbacks,
including slow convergence, complexity, and difficulty in adapting to
different problems. Others [7,8] have used linear programming (LP)
techniques to solve security dispatch problems in an attempt to
overcome the drawbacks associated with gradient techniques. More
recently, LP techniques [9,10] have been applied to reactive
optimization problems using incremental network models. These
incremental LP techniques have several drawbacks. First, since the
solution is by definition on the boundary of constraints, LP may
result in many controls being scheduled at their limits. For power
system controls, such a control strategy is not desirable.
Oscillatory behavior may also occur if good linearization techniques
are not used. At least one researcher has addressed this second
drawback and has developed a good linearization model [11]. Finally,
and most importantly, LP allows for only linear objective functions.
The research presented here achieved the following objectives:
o development of an LP-based algorithm for scheduling voltages and megavars
o modification of LP algorithm to allow solutions at interior points of the constraints
o use of an extension of LP (i.e. , GRG) for real power loss optimization, thus allowing for nonlinear objective functions in optimization.
3
The algorithm uses reactive power generation and bus voltages as
controls. Tap changing controls and MVA flow constraints are not
included but may be accommodated within the algorithm. The algorithm
developed will:
o minimize real power losses
o schedule selected bus voltages
o schedule reactive power (generators or synchronous condensers)
o include linear constraints on reactive power sources and resources, such as reactive power reserve
o provide fast, reliable means for remedial action and optimization.
Presently, there is a major effort in the electric utility
industry to develop and implement such algorithms on-line. Beyond the
algorithm development there exists practical engineering
considerations that must be considered before the algorithms may be
implemented successfully. These considerations include:
o How is the algorithm to be integrated into an energy control center?
o Under what circumstances should each control be implemented?
o What are the important voltage controls, and how can they be practically applied by power system dispatchers?
o How fast and accurate should the voltage control algorithms be?
Past research has neglected these types of questions. These
questions, however, are some of the first to be asked during
implementation.
4
In the research reported here, it is important to distinguish
between the control of real power controls and reactive power and
voltage controls. The objective here is to study control of reactive
power and voltages. As will be discussed in Chapter 2, control of
real power in energy control centers is independent of reactive power
control, and this research will assume this same independence.
Chapter .2 gives a brief discussion of power system control and
describes how voltage control is done in today's energy control
centers. In Chapters 3 and 4, the optimization problem is defined
mathematically, and the previous research is discussed. Chapters 5
and 6 present the development of the optimization algorithm and
discusses how the algorithm is implemented on a computer. Chapter 7
is a detailed summary of results and a discussion of the performance
of the algorithm. Chapter 8 discusses how the algorithm may be
extended to model additional controls and constraints and how the
algorithm should be implemented into an energy control center.
Finally, in Chapter 9, a summary and conclusions are presented.
CHAPTER 2
POWER SYSTEM CONTROLS
The three major objectives of power control for energy control
centers are the maintenance of power balance, security, and system
economics. Power balance must be maintained such that generation
matches demand plus losses. Since the system demand is constantly
changing, a power system controller must continually monitor the
network and adjust controls to maintain the power balance. The
controls must not result in overloaded equipment and demand must be
served at all times. This is the definition of power system security.
The controller must then be intelligent enough to recognize potential
insecurities and to incorporate security considerations into its
control strategy.
Economics is concerned with the minimization of total fuel costs
and capital costs. Capital costs are not directly considered, but if
equipment loading can be reduced by on-line controls, then, in effect,
capital costs may be reduced. Fuel costs may be reduced in a number
of ways. First, the most economical generator units may be scheduled
first and operated at their most efficient setpoints. Second, the
total amount of generation required to be dispatched may be reduced.
This reduction can be done by either reducing ~ower demand or power
losses. Other means of maintaining system economics include
transacting power interchanges with the neighboring utilities so that
the overall fuel costs are reduced.
5
6
In today's electric utility industry, power controllers have been
implemented in large scale computer control systems. These systems
typically consist of large, real-time computers tied to an extensive
communication system that may be microwave or leased telephone line.
The control computers communicate to controllers located at power
generation plants and to substations located throughout the power
transmission network. Such systems are large scale, typically
controlling thousands of pieces of equipment throughout a large
geographic area on a second by second time frame. Thus, the power
controller, as defined above, must be designed for maximum efficiency
and reliability.
The following pages give a brief summary of how the computer
control system works so that the environment in which voltage control
must be implemented is understood. This research has been done within
the context of the power control described below.
A conceptual overview of a power control system is shown in Figure
2-1. The power control system and the power system are represented by
the large blocks and the communication system is shown between the
blocks. The major subsystems of the control system include Network
Analysis, Generation Scheduler, SCADA, and AGC. The Dispatcher
man-machine interface to the computer is also represented.
Telemetered data sampled from the power system is transmitted to the
power system controller via the communications. After telemetered
data is processed, controls are transmitted back to power system
equipment.
,. -
POWER
SYSTEM
,-
Controls
Telemetered ~
Data I
I
'--
- - - - - - - - - - -- I
~ ""'M"--
SCADA
NETWORK ANALYSIS
AGC
GENERATOR SCHEDULER
~ ~ DISPATCHER
- - - - - - - ....__ - -- - - -- ---- -Energy Control Center
_J
I I I I I
Figure 2-1. - Overview of an Energy Control Center
'-I
8
Telemetered data consists of the following types of data:
o real and reactive power generation for each generator
o real and reactive transmission and transfomer power flows
o real and reactive power demand
o switching device status (breakers and disconnects)
o bus voltages
o transfomer tap positions
o frequency.
The telemetered analog data consisting of power, voltage and
transfomer tap measurements is typically sampled every ten seconds,
and the digital data (switching device status) is sampled every two
seconds. Given this telemetered data, the power control system
transmits power system control signals, including the following:
o open or close a switching device
o raise or lower a transfomer
o raise or lower real power at generating plants
o raise or lower a voltage setpoint or tap position at a transfomer
In today's control systems, no direct controls from the energy
control center are available to generating plants to control reactive
power or voltage.
Within the power control system, the SCADA function processes . telemetered data by converting the data to engineering units, checking
the data for out-of-limit conditions, and fomatting the data into a
9
useful form for additional processing. This processed, telemetered
data is used as a basis for deciding how to control and monitor the
power system.
All power system controls in today's power control systems are
implemented by either the Automatic Generation Control (AGC) or by the
dispatcher via a manual request to the computer system. The AGC
function uses the telemetered data to compute the total real power
generation requirement so that real power balance is maintained. AGC
also computes an economically desired real power setting for each
unit. These desired real power settings are compared with the current
real power setpoints and a series of control signals are transmitted
to the generators to either raise or lower real power generation. The
AGC function is designed to maintain a balance between the sum of real
power demand, real power losses, and desired net real power
interchange with neighboring utilities and at the same time maintain
frequency of the power generation within desired limits. The AGC
function is an automatic closed loop control on the powers system.
That is, the computer sends control signals directly to the generators
as an automatic control system.
The dispatcher typically implements all other power system
controls, and may override the AGC controls. The following types of •
control actions may be implemented by the dispatcher:
o open or close switching device - take equipment in or out of service
10
o raise or lower transformer tap setting or voltage setpoint
o request reactive power generation or voltage setting at a generator via telephone call to the power plant operator
o request real power exchanges with neighboring utilities
o take a generator off line (available or unavailable for service).
The dispatcher interfaces to the computer system via a CRT, as
shown in Figure 2-1. When requesting a control action, he manually
enters the desired control on the CRT and monitors the resulting power
system response. In this design of these man-machine CRT interfaces,
much effort is placed on the human engineering, and, thus, the
controls must be designed such that the dispatcher may easily
interface via man-machine interface to enter controls and monitor the
overall system.
In addition to these on-line control functions, the dispatcher is
performing operational planning, equipment maintenance, evaluation of
future conditions, and monitoring maintenance of system security.
The generation scheduling function is used to plan the commitment
of units on-line or off-line and to forecast system demand. The
outputs of this function are a recommended schedule for unit megawatt
generation plan and the hourly megawatt forecasted demand. The
generation plan and forecasts are performed on a time frame of one to.
seven days and serve as an overall guide for which the minute-by-
minute control must operate within.
11
The network analysis function uses the telemetered data and
forecasted data from the generation scheduling function as inputs and
performs the following analyses:
o detects telemetered data that is erroneous
o computes network total solution from the telemetered data consisting of voltages, phase angles, and power flows
o performs monitoring on estimated quantities
o produces alarms to the dispatcher for out-of-limit quantities
o evaluates system security
o determines remedial action controls to eliminate overloads or even potential overloads
o determines settings for megavar and voltage controls
o evaluates future conditions for equipment maintenance studies
o computes penalty factors for AGC.
The first two functions are critical to the success of the network
analysis function. The total network solution consists of bus voltage
magnitudes and phase angles and all real and reactive equipment power
flows. These estimated quantities are used to perform monitoring and
alarming on out-of-limit data. System security is evaluated by
estimating network solutions for a selected set of probable
contingencies (equipment outages) and determining the severity of the
resulting equipment overloaqs, if any. The dispatcher may review this
security analysis results and assess how insecure the network is and
may, as a result, implement remedial actions.
12
The network analysis function also provides recommended control
actions for relief of a system overload or insecurity. In today's
power control systems, these recommendations are generally in terms of
real power generation to relieve a MVA overload on a transmission line
or transformer. However, reactive power generation and voltage
settings may be recommended to relieve overvol tage or undervol tage
conditions also.
A new function that is being implemented, at this time, is voltage
control. Using this function, the dispatcher is provided with the
recommended voltage and reactive power generation settings to achieve
a predetermined· objective, such as the minimization of real power
losses. The dispatcher would then use these recommendations as a
guide in operating the power system.
A most important function of the network analysis function is the
study of future operating conditions. For example, to plan an
equipment maintenance outage, the dispatcher would simulate the system
conditions and then assess whether the outage should be performed at
the selected time. This function appears to be a major use of the
network analysis function and has a direct impact on operating costs.
Other network analysis studies include "what if" questions as to
taking equipment in or out of service, given the current system
conditions. These studies require the knowledge of current system
conditions and are important in the minute-by-minute control of the
power system.
13
Another function of the network analysis function is to determine
the sensi ti vi ty of real power transmission losses with respect to
real power generation. In order to compute this sensi ti vi ty
accurately, the current system solution, in terms of bus voltage and
bus angles, must be known. These sensitivities are used directly by
the AGC function to compensate the units for losses incurred in the
transmission of power to the load centers. The effect of these
sensitivities are to penalize generators for contribution to the
overall real power transmission loss in transmitting power to the
loads. As a result of this compensation, overall lower production
costs are realized.
The overall sequence of execution of the functions in the power
control computer is an important consideration. The SCADA function
typically executes periodically on a ten-second cycle for scanning and
retrieval of analog data. The AGC function executes on two periods.
On the first period, it executes every two seconds to regulate the
real power generation around a desired real power base point. On a
longer time period, on the order of five minutes, it computes a new
base point for all units. Network analysis functions may execute on a
ten-minute period and, therefore, are on a much longer time period
than other functions. The generation scheduling function is executed
once a day upon dispatcher request and is not executed periodically.
Of all the functions, only AGC and SCADA can be considered real-time
control function and it is only AGC that is an automatic closed loop
control.
14
The above discussion of the execution of these functions has several
important ramifications. Reactive power generation and voltage are
not controlled automatically and megawatt power is controlled
independently of reactive power demand, reactive power flows, or bus
voltages. Reactive power control and scheduling of bus voltages is
done directly by the dispatcher by reviewing the telemetered data or
network analysis results. Reactive power controls, if implemented,
are done on a much slower time frame than real power controls.
In the future, automatic closed loop reactive power and voltage
control may be implemented as done by AGC. However, this is not
likely in the near future • Further, reactive power and voltage
control algorithms have not been available that are acceptable guides
for these controls to be implemented automatically. Because of this,
it will be some time before enough experience has been gained to allow
closed-loop automatic control to be done. The research in this thesis
is then directed toward development of an algorithm that will be an
integral part of the network analysis function and that is acceptable
in such an environment. As a first step, the algorithm need only
produce reliable results and recommended settings for reactive
generation and voltage settings. Important constraints are that
generation unit megavars limits cannot be exceeded and bus voltages
must be maintained within acceptable limits. Further, the algorithm
should be adaptable to a variety of objective functions, as well as
many types of reactive power and voltage controls.
15
In this research, only reactive power generation and controllable
voltages at the generator will be considered. Other controls that
should be considered include transformer positions and setpoints,
phase shifters, and capacitors. Discussions with dispatchers at
several large utilities throughout the United States indicate that of
all these controls, reactive generation and desired bus voltages are
the most practical to implement throughout the power system. The
dispatcher may periodically telephone operators at the power plants
and verbally communicate control of the schedule of reactive power
generation or voltages. Such a procedure is acceptable and is done in
many control centers today. Transformer tap positions or setpoint
controls have to be implemented by the dispatcher entering the desired
settings directly into the computer system. Many transformers exist
within the power system and, thus, the dispatcher would be burdened
with a heavy workload if transformer tap settings have to be adjusted
often. For these reasons, transformer tap control has not been
incorporated into the algorithm developed in this thesis but may be
accommodated if needed, as discussed in Chapter 8. Phase shifters may
be used to reduce real power losses for those utilities who have phase
shifters installed. Again, these controls are not included but, as
discussed in Chapter 8, may be accommodated. Finally, the scheduling
of capacitors, either on or off, is an effective reactive power
control. This scheduling is usually done by operations planning
16
personnel such that a schedule of switching is known before reactive
power generation and voltage setpoints are known. Modeling of
capacitors may be very difficult since they represent discrete
variables (off or on) and thus require integer or mixed integer
mathematical solution techniques for solution.
capacitors are not addressed.
For these reasons,
In summary, this thesis will focus on the network analysis
function of an energy control center. Within this context, an
algorithm is to be developed that results in recommended reactive and
voltage settings that a dispatcher may implement directly. Real power
controls are treated as a separate or decoupled control as currently
done in industry. An important objective function that will be
studied is the minimization of real power transmission losses. This
objective function is the most difficult to model and is of most
interest because of the impact on system production costs. By
reducing real power losses, the real power generation requirement is
reduced and, as a result, lower operating costs are realized.
As a final comment on power system control, it should be
emphasized that the resulting control algorithm for voltage control
must be extremely efficient for on-line computer use. The algorithm
is not to be implemented as a closed loop controller but will have to
be executed periodically in an on-line computer system and must have a
fast response time to be useful to the operator. Moreover, it is
17
expected that the response and reliability is of prime importance with
respect to overall solution accuracy.
CHAPTER 3
PROBLEM DESCRIPTION
In this section the problem to be solved will be developed
mathematically and a general procedure for solution will be outlined.
Conceptually, the problem can be represented as shown in below:
Source (known) demand (unknown)
Two generation power sources exist serving power demand at two
locations via four transmission paths with four network
interconnections (or busses). The power generated and transmitted to
serve the load consists of a real part (real power) and an imaginary
part (reactive power) and is referenced by units of megawatts and
megavars. Throughout this thesis real power will refer to the real
part of the complex power and reactive power will refer to the
imaginary part of complex power. The real and reactive power demand
is known and the objective is to determine the real power, reactive
power, and the voltage magnitude at the power generation sources such
that fuel cost of power generation and associated power losses are
minimum. The constraints that must be considered are:
18
19
o Real and reactive power demand must be served
o The net real and reactive power flows at each junction are zero
o Power transmission paths must not be overloaded
o Real and reactive power generation limits must not be exceeded
o the voltage magnitude at each junction must be within limits
An additional constraint that must be considered in the
implementation is that reactive power generation limits are dependent
on the real power generation as defined by the unit capability curves.
In this research, it is assumed that such limitations are already
incorporated into the reactive power generation limits. -
The mathematical problem to be solved is similar to the
transportation problem in operations research except that the
relationship between voltage magnitude, real power, and reactive power
is nonlinear, as will be shown below.
Define the following electrical variables:
real power injection at bus i (generation minus
demand)
reactive power injection at bus i (generation
minus demand)
v. ,. l.
The voltage magnitude at bus i
Q. ,.
l. phase angle at bus i
P. ,. J.j the real power flowing from bus i to bus j
Qij ,. the reactive power flowing from bus i to bus j
20
gij A transmission line conductance from bus i to bus j
bij A transmission line susceptance from bus i to bus j
s .. A one half .lJ the transmission line charging
susceptance from bus i to bus j
The power injection is defined as the power generation or power
demand at an interconnection (or bus).
The example network can then be re-labeled as:
PG1 and PG2 represent the real power generation at busses 1 and 2,
PD3 and PD4 are the real power demands at busses 3 and 4. The
megawatts transmitted from bus 2 to bus 3 is represented by P23 and
correspondingly the real power flow from bus 3 to bus 2 is P32 .
The corresponding reactive power flows are the same as for the
real power flows shown on the diagram.
It can be shown that the megawatts and megavars transmitted form
bus i to bus j are:
P .. .lJ 2 = giJ.v1. - {g .. cos(Q. - Q.) + b .. sin(Q. - Q.)} v.v. .lJ . .l J .lJ .l J .l J
v.v . .l J
( 3-1 )
(3-2)
21
In these equations the complex branch admittance is assumed to be
gij + jbij"
At each junction of the network the network power flows must be
zero, such that:
PG1 = p12 + p14
PG2 = p21 + p23
-PD3 = p32 + p34
-PD4 = p 41 + p43
A similar set of equations apply to reactive power. In general,
for a N bus network:
N P1. =LP .. 1J j=1
N Q. = L_Q .. 1 . 1 1J J=
(3-3)
i=1, ••• ,N
(3-4)
The equations (3-3) and (3-4) are known as the power flow
equations and represent the conservation of real and reactive power at
each bus.
In the above equations (3-1) through (3-4), the unknowns include
all bus voltage magnitudes (v), all bus phase angles (Q), except for a
reference bus phase angle, real power generation (P1 ,P2 ), and all
reactive power generation (Q1 ,Q2) • The knowns are the bus demands
(P3 ,P4,Q3 ,Q4) and the reference bus phase angle. Nonlinearities occur
in the equation (3~3) and (3-4) due to terms vi2 ' vj 2 ' vivjsin(Qi -
Q.), and v.v.cos(Q. - Q.). J 1 J 1 J
22
The constraints considered in this research include upper and
lower limits on all bus voltages and reactive power generation so that
the other equations of interest include:
vL. ~ v. < vH. i=1 ' ••• '4 1 1 1
QL < Q. < QH k=1,2 ·1 = k k k
The general mathematical programming problem that is of interest
is stated as:
minimize f(,S_,~,Q)
S.T. 4
P1. =L) .. . 1 1J J=
4 Q. =I:Q ..
1 . 1 1J J=
P .. and Q .. are nonlinear functions of Q and v. 1J 1J
i=1, ••• ,4
k=1 ,2
In this research, real power generation (P1 ,P2 ) is assumed
constant and, thus, is not an unknown. Further, this research did not
consider flow limitations on transmission lines. The mathematical
programming problem that is to be solved is then one of minimizing an
objective function subject to the power flow equations with the
reactive generation sources (Q1 ,Q2), bus voltages (v), .
and phase
angles (9) as unknowns. For control purposes, the power system
dispatcher would adjust the voltage (v1 ,v2 ) and the reactive power
23
(Q1 ,Q2). The bus voltages (v3 ,v4), phase angles (9), real power flows
(P .. ), and reactive power flows (Q .. ) would change correspondingly. 1J 1J Several types of objective functions may be defined for this
problem [12]. For reactive power and voltage control, the objective
that is most difficult and the one that is of most interest is the
minimization of real power losses. The objective is defined as:
f = p12 + p21 + p23 + p32
+ p14 + P41 + P34 + P43
The difference in the real flows at each end of the transmission
is the real power loss of the transmission line. Let k represent an
index for each transmission line, then the loss function f is:
~ 2 2 f = L._gk[v. + v. - 2v.v.cos(9. - 9.)] k=1 1 J 1 J 1 J
i and j and the from and to bus numbers at each end of the
transmission line k.
For the example, the optimization problem is stated as follows:
... f ~ [ 2 minimize = 2.___ gk v. k=1 1
S.T. 4 2
P. -~ g .. v. - [g .. cos(9. - IL) +b .. sin(9. - 9.)J v1.vJ. = 0 1 '-:--1 1J 1 1J 1 J 1J 1 J J= #1
N Q1. -c(b ..
. 1 1J J= #1
+ s .. )v. 2 - [g: .sin(9. - 9.) - b .. cos(9. - 9.)]v1.v: = 0 1J 1 1J 1 J 1J 1 J J
vL. < v. < vH. i=1, ••• ,4 1 1 1
QL < Qk < Q~ k=1,2 k
24
As discussed in Chapter 1, other constraints and unkno~ms may be
defined; however, if this problem is solved satisfactorily, extension
of the solution to include additional features should be fairly
straightforward. It should be noted that both the objective function
and the equality constraints are nonlinear. Furthe~, the objective
function is not separable and is not a quadratic function. Only if
the bus phase angles are assumed constant or negligible can the
objective be defined as quadratic.
Define vector x to include voltages, reactive power, and phase
angles. Then the optimization problem may be stated as:
S.T. minimize f(_!)
,!:!(.!,) = 0
l (.!,) ~ .!?. E_ represents the equality and 1 represents inequality constraints.
The Kuhn Tucker necessary conditions for an optimum x* are:
> 0 = -
x*T [Vf(x*) +~*TVl_(x*) +~*TVE(x*)] = 0
,0 * T [l_ ( x*) - .!?_] = 0
/\ *T [h(x*)] - -- = 0
.,Y. * unrestricted
,X > o -= -Sufficient conditions of a global optimum_!* are that if f and h
are convex and 1 is linear then:
Vf(_!*) - ~ *TVh(_!)
~T*E(x*)
t*
= 0
= 0
f 0
25
The constraints 1_(_!) are linear. The objective function is convex
since the Hessian matrix can be written:
where d .. = l.l.
D =
NBR = Z::: 2gkv.v.cos(Q.-Q.).
k=1 l. J l. J Where branch k is
incident to bus i. The Hessian matrix is positive definite since all
leading minors of matrix D are positive and, therefore, the objective
function is convex. The convexity of the equality constraints ~ is
much more difficult to prove. The Hessian of these constraints has
been used successfully in power flow optimization problems [ 13, 14],
but, in general, no conclusions can be made at this time as to
convexity. The expression for the Hessian of these equality
constraints is very complex and would be difficult to evaluate as a
general sense.
As discussed by Luenberger [15], it can be shown that a relative
minimum exists if x* is a regular point satisfying the equations shown
above. A regular point exists if the gradients of the active
constraints form a linearly independent set. As the problem has been
formulated, the constraints will be linearly independent and since the
gradients form a basis for the power flow solution, they also will be
linearly independent. Therefore, a relative minimum exists.
CHAPTER 4
DISCUSSIONS OF PREVIOUS RESEARCH
The general topic of power flow optimization has been researched
extensively in the past and is of great interest to many researchers
today. Many researchers have addressed reactive power optimization
only as part of the solution of the general power flow optimization.
Much of the past research has focused on the mathematical
techniques and little has focused on the engineering problems
associated with implementation. Several research papers have directly
focused on engineering considerations [10,16,17,18].
A review of the mathematical solution techniques is given in this
chapter. General categories of solution techniques are presented and
the most successful of these will be discussed in detail.
The general solution techniques are in the following general
categories:
o Penalty Function Methods
Fletcher Reeves [19] Fiacco, McCormick, Lootsma, Zangwill [20] SUMT - sequential unconstrained minimization Hessian matrix - Newton's method [13,14]
o Approximate Programming
Recursive Quadratic Programming [21,22,23,24,25] Reduced Gradient with Penalty Functions [4,26,27] Generalized Reduced Gradient [5,6] Recursive Linear Programming [7,28,29,30,31,32,33,34,8,35,36,37,38,39,40]
Other algorithms have been tried, but the most popular techniques
are listed above.
26
27
Penalty function methods transform constraints into the objective
function using penalty functions and the solution is obtained using
unconstrained optimization techniques. For optimization problem
defined as: minimize f(.!_)
S.T.
x > 0 - = -
This constrained problem is transferred into an unconstrained
minimization such that:
minimize f(.!_) N 2
+ Lr. g. (x) i=1 1 1 -
This function is minimized and, as a result, deviations from the
feasible region will be penalized. This equation is iteratively
solved and each iteration r and x are moved such that f(.!_) is
minimized and the solution is feasible. r. of the violated 1
constraints are adjusted to each iteration to force the solution to
become feasible.
The penalty function method is very useful when constraints are
highly nonlinear. However, the success of the technique depends on
the appropriate choice of the penalty functions. These functions may
differ between power systems and power system operating conditions.
Poor choices may lead to excessive oscillation of the solution between
the feasible and infeasible regions or to very slow convergence. For ..
electric power systems, a general purpose penalty function is hard to
devise.
28
Approximate programming is an extension of techniques derived from
unconstrained optimization methods. Using this approach, extra steps
in the solution are made to account for constraints. The overall
objective is to find a descent direction that points toward the
constrained minimum and then determine a step size that is restricted
to the feasible set [25].
All the methods under approximate programming solve nonlinear
optimizations by methods resembling the simplex method of linear
programming. The philosophy is to first organize the unknowns into a
dependent and an independent set. The dependent variables are
eliminated and the optimization is done only in terms of independent
variables.
The most widely accepted method for solving the Optimal Power Flow
is the generalized reduced gradient method [4,5]. The Generalized
Reduced Gradient method is an extension of the reduced gradient
techniques and is outlined below. Al though the algorithm has been
developed for some time, it is still the most successful in terms of
actual implementation.
The development follows from that published by Himmelblau [41].
Given the mathematical program:
minimize
S.T.
f(~, ~)
~(~,~) = 0
,!!(~,u) < 0
~L 2 x ~ ~H
~L ~ ~ ~ UH
(4-1)
(4-2)
29
A differential displacement in ~ and ~ is made using the objective
function f(~,~) and equality constraint~(~,~)
df(~,~)
( ) ""I _g(_x ,_u)dx + dg x,u = ~
~~
~~(~,~)du = 0
d~
(4-3)
(4-4)
The objective is to eliminate the dependent variables. Solving
(4-4) for dx will result in:
and substitution of this expression into (4-3) will result in
df(~,~) =
or
df(~,~) = du
One necessary condition for f(~,~) to be minimum is that
df(~,~) = 0 or by analogy to the condition for an unconstrained
minimum, that: df(~,~) = 0
du
This condition may only hold if control variables have not reached
their bounds. If the controls hit a limit, then the control variable
is set at the limit and not allowed to move further.
Equation (4-5) for the reduced gradient is often replaced by the
following expression:
30
where the dual vector is computed from the following equation:
~f --+ ~ .!
The reduced gradient takes account of the fact that a small change
f}2:: creates a small induced changed A_!. The condition that the
reduced gradient is zero is satisfied by selecting points 2::, 2::1 , ••• 2::i•
The inequality constraints are modeled implicitly in the
algorithm. For example, if a state variable .! reaches a limit, the
problem variables are relabeled. The former independent variable x
now becomes an variable u and the former independent variable becomes
a dependent variable. Hence, the GRG operates similar to an LP
algorithm in the way inequality constraints are handled.
The solution procedure is outlined in Figure 4-1. Generally, the
problem is solved in two passes. First, the algorithm shown is solved
to become feasible. The objective function is formulated to drive all
variables into their feasible ranges. Then, a second pass is executed
to optimize once feasibility is attained.
Successful implementation of the GRG algorithm has been reported
several times [5,6,42]. Al though the method has been fairly
successful, the method suffers from several drawbacks. First, the
overall solution time is relatively slow and considerable tuning of
the algorithm is required for different power systems. Further, the
algorithm is not that flexible in that the addition of new types of
constraints and modeling problems require considerable new development
in the algorithm. Overall solution is slow in particular for
obtaining feasibility since the algorithm does not proceed to obtain
convergence in a structured manner as in Linear Programming.
Swap Independent & Dependent Variables.
31
Initialize System Variables x and u
Solve Equality Constraints
Compute Dual Vector
Compute Reduced Gradient
Limit A~ if Limit Exceeded
Compute Induced Changes
Determine Step Size ·s
Update Variables x=x+s*Ax u=u+s*.A u
y
cdf
Solve Power Flow Equations £ (~.~) = ~
of -= +~T [~~] du ~~
Set.6,~ df du
Figure 4-1 - GRG Solution Procedure
32
For the optimization of megawatt generation, Linear Programming
based techniques have been extremely successful. Stott [ 8 J and
Wollenberg [7] have been extremely successful for these problems. LP
techniques improved execution speed, improved convergence, and
resulted in considerably more flexibility. Others [21,22,23,24,39]
have shown that quadratic programming can be adapted to real power
optimization problems successfully.
The LP and QP approaches generally require a three step procedure.
First, the power flow equations are solved to determine the system
operating point. Next, a linearization is performed and finally, an
optimization step is done. The steps will be outlined below.
Step 1. Solve~(_!,~)= 0 using power flow solution techniques and
determine which constraints _!!(~.'~) are violated.
Step 2. Linearize the power flow equality constraints and
inequality constraints that have been violated:
a~(_!,~) A.! + a~(_!,~) A u = o o.! a~
The objective function is approximated as:
for LP + ofA_!+ of f = fo A~ o.! a~
for QP f = f + l::c. 6U. +L_r:_d .. ,AU.AU.
0 • J. J. • . J.J J. J J. J. J
33
Reference [32] shows how the !J.~ terms can be eliminated
in the LP approach and the objective function then can be
expressed as only a function of A ~. Also, reference
[12] shows how the objective function can be approximated
as a quadratic.
Step 3. Solve the resulting equations using LP or QP and update~
by U : U ld +AU -new -o u._
and then iterate back to step 1.
The process is iterated until the solution x of ~(~,~) between
successive steps converges.
Hobson [32] and Mamandar [10] have published results using Linear
Programming for reactive power optimization. Bubenko and Sjelvgren
[38] have reported results using Linear Programming also. These
Linear Programming techniques have several drawbacks. First, since
the nonbasis variables or controls are by definition on the constraint
boundaries, LP will result in all nonbasis variables or controls being
scheduled at their limits. Also, oscillatory behavior may result if
the linear program is solved iteratively with a power flow without a
good linearization of power flow equations. Hobson's approach [32]
did not incorporate power flow equations in the LP and when iterated
with a power flow may result in oscillatory behavior between the LP
and power flow. At least one researcher has addressed this second
drawback and has developed a good linearization model [11]. Finally,
and most important, LP allows for only linear objective functions.
34
The GRG technique does not suffer these drawbacks since the degree
of linearization is not as severe. That is, the gradient and Jacobian
terms of the equality constraints are incorporated directly into the
GRG algorithm and the objective is retained as a nonlinear function.
Quadratic programming overcomes some of the difficulties that LP
has since, at least, a quadratic objective function may be modeled.
CHAPTER 5
GENERAL SOLUTION METHODOLOGY
In the previous section, the development of various optimization
algorithms was reviewed. Most of the past success was in application
of approximate programming techniques. It was noted that strict
linear programming techniques would not be applicable to a nonlinear,
nonseparable objective function of real power losses. Based on this
review of past research, two approaches were thought to have
potential. Quadratic programming was the first and linear
programming, in combination with the reduced gradient, the other. In
this section, both approaches will be discussed and the resulting
algorithm described in detail.
In Chapter 3, the development of the mathematical optimization
problem was presented. These equations are repeated here for
convenience:
minimize f NBR 2 2 (
= L gk [v. +v. -2cos Q.-Q.)v.v.] k=1 1 J 1 J 1 J
S.T. N ~- 2 P. - 1 g .. v.
1 --:--1 1J 1 J= #i
N 2 Q1. - J(b .. +s .. )v. -~ 1J 1J 1
#i vL.
1
QL m
< v. < vH. 1 1
< ~ < QH m
35
(5-1)
= 0 (5-2)
i = 1, ••• , N
m = 1 , ••• , NC
36
NBR represents the number of branches, N the number of busses, and NC
the number of voltage control busses.
In this problem definition it is the dependence of the objective
function on the phase angle that makes the objective nonseparable and
nonquadratic. Thus, in the two methods to be evaluated, if the phase
angle dependence in the objective cannot be approximated, then strict
quadratic programming without some approximation of the objective
would not apply. The resulting algorithm, as indicated in earlier
chapters, should have the following features:
1. Use actual equipment constraints but not necessarily schedule equipment at their limits.
2. Converge under all conditions, especially in emergency system control conditions to relieve overloads.
3. Fast computer execution and use moderate computer memory since the algorithm is to be implemented in an on-line power system control center.
4. Have the capability of modeling other objective functions and constraints in a convenient manner.
5. Provide accurate solutions, but not at the expense of excessive computer resources such that the function would not be useful to the power system dispatcher.
The selection and development of the algorithm will use these
desired features as a basis.
In the general solution of the problem, two control objectives
must be considered. First, the solution process must determine a
feasible solution such that no equipment ~s overloaded and second, the
solution process must optimize the objective such that real power
system losses are minimized. In that regard, the overall solution may
be diagramed as shown below:
Solve Active Constraints
Construct Linearized Model
No
37
Optimize and/or Feasibility
Stop
This is the approximate programming technique diagrammed.
Mamandur [ 10 J used this same solution technique for minimizing real
power losses using Linear Programming. In his development, Linear
Programming was used in the optimization/feasibility step and a
linearized approximation to the real power loss equation was used for
an objective function. The published results indicated oscillatory
behavior occurred. A damping function had to be used to converge and
also most of the variables were driven to their limits. Dayal [24]
used Wolfe's method of quadratic programming for minimization of
production costs using generator megawatts as variables. His research
showed moderate computer memory requirements, however, the number of
iterations for optimization was on the order of fifteen. Biggs and
Laughton [39 J also used Quadratic Programming and their algorithm
required twenty-five to thirty iterations for convergence • . In order to describe the QP and GRG methods more simply, equations
(5-1), (5-2), and (5-3) are written in a more general form as:
38
minimize
S.T.
.!1 ~ .! ~ .!H In both the QP and GRG techniques, the equality constraints .[(_!)
must be solved for a nominal operating point .!o• Then these
constraints may be linearized around the operating point x • The -0
linearization may be done as a truncated Taylor's series:
.[ (_! - x ) = 0 -0 x x = x -0
.!1 ~ .! ~ .!H In the third step, the following optimization problem can be
solved:
S.T. .[ x = .[ .!o -x x - x = x - x = x -0 -o
The linearized equality constraints restrict the movement of x to
be within the neighborhood of x • -0
Given this linearized model, both QP and GRG techniques will be
described below and a recommended approach will be chosen.
In order to use quadratic programming, some approximation to the
objective function must be made. The resulting QP would be as
follows:
39
minimize f(_!) = .£o + cTx + 1/2 ,!TD,!
S.T. A x = b
x > 0
Several approximations to the objective could be made. First, the
cosine terms could be assumed near unity or constant at a nominal
value and, thus, the dependence of real power losses to phase angles
would be disregarded. Second, a Taylor's series expansion of the
objective could be done that would incorporate second order terms.
The first approximation would assume that no cross coupling exists
between bus phase angles and real power losses. A change in voltage
or megavar power will induce changes in bus phase angles according to
the constraint equations (5-2) and (5-3) and in turn will affect real
power losses. Thus, the first approximation is unacceptable. This
second approximation would be most acceptable in terms of accuracy.
The two methods of quadratic programming that have been applied to
power system problems are Beale's method and Wolfe's method. Wolfe's
method has been most popular and successful and for a comparative
analysis to the GRG method it shall be used as an example. Other
methods [25,43] are available for solving QP problems, but for a
comparative analysis, the most successful QP technique that has been
applied to power systems will be used as a basis.
The quadratic programming solution, as developed by Wolfe [44],
requires the following equations (Kuhn-Tucker'conditions) ·to be
solved:
40
Ax = b
D,! - &{+ ATx_ =
xi.~ = o
_!,x_,g f 0
T -c
x_ is a dual vector and .g = ...J::..(_!,x_,g) and x
L is the Langragian: T T L(_!,x._,g) = f(_!) + x_ (Ax-b) - .{! ,!
The conditions (5-4), (5-5), and (5-6) result from the
~L ~L -*= o, -*= M., = 0 CiX. d,!
(5-4)
(5-5)
(5-6)
solutions:
respectively. Equations (5-4) through (5-6) are solved by a
modification of the simplex method of linear programming. The Linear
Programming basis consists of equations (5-4) and (5-5). To insure
condition in (5-6) the simplex method is modified to insure that both
an element l(i and the corresponding xi are nonzero. When the LP is
feasible, then the optimal solution has been found.
This method of solution has the advantage that the conditions for
optimum (Kuhn-Tucker conditions) are solved directly and thus is
expected to yield good results. However, there are several apparent
disadvantages. First, the objective function must be approximated and
cannot be dealt with directly. Secondly, during the solution of the
QP, the coefficients of the simplex tableau D, A, and . c remain
constant throughout the LP solution and are not relinearized each
iteration of the solution. Mamandur [10] has shown that the amount of
41
change in the optimization step should be restricted in order to
maintain feasibility in the case where both feasibility and
optimization is performed without re-linearizing the constraints. The
QP, using Wolfe's method, moves to feasibility and to optimal solution
in one solution, and the coefficients of the tableau remain constant
within the QP. Because of this, the potential exists for oscillatory
behavior such that the amount of change made during each QP would have
to be restricted as in the case of recursive LP. Finally, for Wolfe's
algorithm, the computer memory requirements are relatively high. The
unknowns are x consisting of .2c, !i' and ~; a vector .g_ the size of x
and x._, which is a vector the size of the total number of constraints.
The number of unknowns is 7N+3NC. The number of equations is 5N+2NC.
For a large power system, N may be on the order of 1100 unknowns so
that the number of constraints are greater than 5500.
Beale's method [45,46] of quadratic programming would require less
number of constraints and variables but require cutting plane
techniques to obtain a solution. Further, Beale's method suffers from
many of the disadvantages described above in Wolfe's method.
The GRG method, as explained in Chapter 4, solves the optimization
problem in two passes. First, feasibility solution is found by
iterating between the solution of the equality constraints, the
linearization step, and the GRG step. After a feasible solution is
found, the GRG moves toward the optimum such that feasibility is
retained. The following will describe these GRG solution steps in
more detail such that a good comparison between QP can be made.
42
Using the GRG technique, linear programming may be used to obtain
a feasible solution. In this regard, straight LP is much more
efficient than Wolfe's QP method since only 3N+NC constraints must be
modeled. For 1100 unknowns the number of constraints would be on the
order of 3300. Thus, using the GRG technique, feasibility may be
obtained in an efficient manner iterating between the solution of the
constraint equations, linearization of these constraints, and LP to
become feasible.
Next, the optimization step must be considered. Using GRG, the
objective function does not have to be approximated. Strict linear
programming or quadratic programming does not have this advantage.
The GRG method, however, does have this advantage and, further, is an
extension of linear programming as developed by Philip Wolfe [47]. In
order to understand the method fully, a review of the linear
programming simplex method is described below.
Linear Programming requires that N unknowns be organized into ~B
basic or dependent variables and ~NB nonbasic or independent
variables. The basic variables can then be solved in terms of
nonbasic variables.
x =
The constraint matrix A has been organized into a basic part B and
a nonbasic part N. Then, -1 -1 ~B = B E_ -B N~NB
-1 ' -1 = B b -/-. _B ~jxj J
43
Where a. is a column of N belonging to A, the cost function z can -J
then be written:
where
z = ex
= z 0
-1 zJ. = cBB a. - -J
-1 ) B a.x. + -J J c .x. J J
z. - c. are known as the relative cost factors of the nonbasis J J
variables. The algorithm proceeds by determining which nonbasis
variable to increase in order to decrease the objective function z.
The relative cost factor thus expresses the change in the objective
due to a change in a nonbasis variable. From the expression for the
relative cost factors, it can be seen that xj should be increased when
Zj-C/O•
The simplex method chooses the nonbasic x . J
with the largest
zj-cj that is greater than zero to become basic and increase this xj
as much as possible without violating constraints. In order to
determine how much this nonbasic variable may be increased, the
constraint equations are examined.
Wolfe [44] used these same concepts of the simplex method but
applied to nonlinear problems. Two important changes were made.
Solutions were allowed to exist at points other than constraint
boundaries and all nonbasis variables were allowed to change at the
44
same time. In the simplex method, only one variable exchange occurs
each iteration and the amount of change is computed such that the new
nonbasis variable is at a limit (constraint boundary). The equations
are similar to the simplex method.
-1 -1 .!B = B ..!?_ - B N_!NB
z = f(_!B '.!NB)
The relative cost factor is computed as in the simplex method.
-1 zj - cj = cBB N - cj
= [~!Bl B-1;;j -[~!NBL The relative cost factor describes the objective z changes with
respect to .!NB. Wolfe suggested that all nonbasic variables be
moved at one time such that z is minimized. As part of this process,
the change in basic variables must be considered. The change in basic
variables induced by changes .!NB are: -1
d .!B = B N ~ xNB
with both ~.!B and ll .!NB an optimum step size may be determined such
that z is minimized. During this step size determination, if a basic
variable exceeds a limit, basis exchange is made and the algorithm
proceeds through another simplex iteration. Wolfe designated the
algorithm as an extended simplex method.
It is seen that the GRG optimization is similar to linear
programming in that a simplex method of exchanging basis and nonbasis
variables is possible. The total number of constraints is not more
45
than LP, but an exact representation of the objective function is
possible. Further, after each GRG move, the constraint matrix A may
be updated. The only disadvantage is that a step size evaluation of
the objective must be performed upon each GRG i tera ti on. Another
important consideration is the modeling of other objective functions.
The GRG can model a quadratic objective function, as well as other
objectives that QP cannot model without some approximation. A case in
point may be the requirement to minimize the MVA flow on a particular
transmission line. The objective function is not easily expressed as
a quadratic function. For these reasons and in anticipation that GRG
may be more efficient in computer on-line operation, the GRG in
combination with LP is selected.
As discussed by Gill, et. al. [43], the GRG method has been very
successful in general optimization problems where constraints are
nearly linear. When the starting point is far from the optimal and
constraints are highly non-linear, the GRG algorithm can converge very
showly. This is because, in order to maintain feasibility, only
smal 1 steps may be taken toward the optimum. For power system' s
problem, this is not the case. The starting solution is relatively
near the optimum and the non-linear constraints may be linearized
effectively. For these reasons, it is expected that the algorithm
will perform well. QP techniques may solve the optimization problem
without having to determine a step size per se. That is, both
direction and step are determined at one time such that feasibility is
46
maintained. Thus, QP may offer some advantage over GRG in this
regard. However, the price paid for this advantage is that the
objective function must be approximated and, thus, some accuracy may
be lost.
In summary, linear programming is used directly to obtain a
feasible solution and an extension of linear programming (GRG) may be
used to optimize a nonlinear objective. The number of constraints and
unknowns are small compared to Wolfe's quadratic programming and,
further, the algorithm may not be limited to a quadratic objective.
It is believed that other quadratic programming algorithms [see ref.
43] may be much more efficient than Wolfe's, but given the experience
to-date, as well as other considerations discussed above, it was
decided to use the GRG. The resulting algorithm is a composite
algorithm; one that uses linear programming to become feasible and
uses the reduced gradient (extended linear programming) to optimize
the objective function. The resulting approach, as will be shown in
the following chapters, is efficient and results in convergence
rapidly and reliably.
CHAPTER 6
IMPLEMENTATION OF THE VOLTAGE CONTROL ALGORITHM
6.1 Introduction
In this chapter the voltage control algorithm is discussed in
detail. In the preceding chapter, the justification for the approach
was given and this chapter will re-emphasize the advantages of this
method over others, in addition to providing a detailed development.
The general solution is performed in three steps. First, the
power flow equality constraints are solved to obtain a nominal
solution. This solution is used as a basis for a linearization as a
second step. An optimization step (GRG) or a linear programming step
(LP) is then performed using the linearized equations. The solution
procedure is:
o Solve equations (1) and (2) holding controllable voltages or reactive power generation constant.
o Perform linearization of the power flow equations.
o Use the linearized constraints and perform GRG or LP.
o Return to the first step if objective is still decreasing or if the problem is still infeasible.
The first step may be done using standard solution techniques,
such as Newton method or equivalent. The fast decoupled solution
method [48] was chosen for this first step and was very satisfactory.
The second step is critical. If a good linearization is obtained,
then a feasible solution may be retained during the LP and GRG steps.
47
48
This would in effect allow for greater moves to the optimum without
violating constraints. Several types of linearized models are
possible. Each of these will be discussed in this chapter.
The third step in the solution process is done either by using LP
to gain feasibility or the GRG technique to move in the optimal
direction after feasibility is obtained. As will be discussed later
in this chapter, the LP simplex method was modified to accommodate the
solution. The overall solution is thus:
FDPF LINEARI-ZATION
LP
YES GRG
YES STOP
After each LP or GRG, the power flow equality constraints are
solved by the fast decoupled power flow (FDPF) and relinearized.
Within the LP, iterations occur to gain feasibility without having to
re-linearize the power flow equation. Similarly, in the GRG step,
iterations are performed before returning to the power flow for
re-linearization. It is important to note that a converged solution
that is feasible is obtained before an optimization step is taken.
Once this feasibility is attained, the optimization step is made such
that feasibility is maintained and the objective is minimized.
49
6.2 Feasibility
In the feasibility step, the objective is to eliminate reactive
power generation constraint violations and bus voltage violations by
rescheduling of either reactive generation or bus voltages. A
truncated Taylor series expansion of the equality constraints yields
the following equations:
N Opij bQj + aPij 0 (6-1) r bVj = a Qj a v. j=1 J
i=1 , ••• , N N a Qij /).Q. + a Qij ~QC. 0 (6-2) L AVj + =
a Qj J av. j=1 1
J
If these linearized equations were used, the partial derivations
would be constants in the LP tableau. In these linearized equations
the term a Q .. will ag~J
be small compared to a Q. . and the corresponding --1J av. a p .. will ire much --1J av. tertils involving Q,Q
smaller than o P. . • The~efore, the off-diagonal oG .1J and P, v can beJneglected. These approximations
have been implemented in solution of the power flow equality
constraints and are well accepted throughout the industry [48]. The
resulting equations are:
N opij ~Qj 0 (6-3) = aQ. j=1 J i=1 , ••• , N
N a Qij AQc. 0 (6-4) ~ vj + = j=1 a vj 1
50
Since the partial derivatives will remain constant in the LP and
the objective is to reschedule voltages and reactive power, the
equations are independent for purposes of the Linear Programming step.
Therefore, only the reactive power flow equations (6-4) are needed.
Only one set of linear equations relating voltage, -:!._, to controllable
reactive power, Q , are needed • ...:c
The resulting equality constraint is modelled in a similar manner
as the fast decoupled power flow equations and the result is:
or
c j=1
N
a .. 1J /JV. + J
L:"a .. v. + . 1 1J J J=
b Q = 0 c. 1
i = 1, ••• ,N
N L:"a .. vo.+Qo j=1 1J J Ci
The term aij is determined as follows:
(6-5)
(6-6)
= - [g1.J.sin(Q1. - Q.) - b .. cos(Q. - Q.)]v. i#j J 1J 1 J 1
N = L 2(b.k + s .k)v. i=j
k=1 1 1. 1
These partial derivatives are evaluated at the power flow solution
v = v0 and Q = and remain constant throughout the Linear
Programming step.
The resulting LP requires N equality constraints and N + NC
variables. The equality constraints have been developed in a manner
51
similar to the decoupled power flow equations, except that busses
containing controllable injections are retained. The inequality
constraints that must also be modeled include upper and lower limits
shown below:
g_L~~~g_H
The resulting constraint equations (6-6) are written as follows:
Ax= b
If the variables are separated into basis variables, .!.B' and
nonbasis variables, x , the resulting equations are: NB
where
and v
.!c is the voltage at controllable busses. The voltage vector has been
partitioned into a basis voltage vB and a control voltage v • -c
Generator terminal voltages are taken as controls (nonbasis variables)
if the generator vars are not at a limit. Controllable reactive power
injections and all other bus voltages are defined as independent and,
therefore, are initially in the LP basis. B is the basis matrix and N
is the nonbasis matrix. The equation represents the equality
constraint that must be satisfied throughout the LP step.
52
In the simplex method, the nonbasis variables, xNB' must exist at
either an upper or lower bound. Using the simplex method directly,
they would require setting controllable bus voltages, v , either to an -c
upper bound vH or lower bound v1 • It is not desirable to operate
controllable bus voltages or reactive power generation at their limits
unless it is absolutely necessary. Further, since the optimal
solution as performed in the GRG step may not have voltages scheduled
at their limit, it is not desirable to use the simplex method without
some modification. For these reasons, a modification of the simplex
method to allow nonbasis variables to exist at points other than their
bounds was devised. The solved voltages or reactive power from the
power flow equality constraint solution are used as initial values of
these nonbasis variables. As the simplex method is performed, these
nonbasic variables are exchanged for basis variables to eliminate
constraint violations; thus, the new exchanged non basis basic
variables were scheduled at a limit. By initially scheduling the
nonbasis variables at the solution provided by the solution of the
equality constraints, the minimum change in controls were required to
become feasible. If, on the other hand, nonbasis variables were taken
at a limit, the LP solution may result in a larger unnecessary change
from the solution provided of the equality constraints. This
modification to the simplex method is similar to that proposed by
Wolfe in the extended simplex method. This technique requires two
changes to the simplex methodology.
0
53
Upon initialization of LP set nonbasic controllable voltage used in the FDPF. voltage is violated, set it to a limit.
variables to the If a controllable
o In the exchange process of the simplex method, a small modification must be incorporated to recognize that a nonbasic variable can exist at a value other than an upper or lower bound.
The second bullet means that in the computation of how much a
nonbasic variable is increased or decreased, the algorithm must
consider that the nonbasic variable is not necessarily at a limit.
In summary, it was shown that to obtain a feasible solution only
linearized reactive power equality constraints must be incorporated in
the LP. Also, it was found that to effectively reschedule voltages
and reactive power, a minor modification of the LP simplex method was
needed. The result is that for feasibility, a LP solution is required
with moderate computer storage requirement of N equations and N+NC
unknowns.
6.3 Optimization Step
The objective is to minimize real power losses, PL' as represented
by the following objective function:
NBR =C (6-6)
k=1
gk is the transmission line conductance of branch k. i and j
represent the from and to busses of branch k. The real power losses
are expressed as a nonlinear function of bus voltage and bus pha:3e
angle. This expression for real power losses is the most simplistic
to model mathematically. The objective function is not, however,
54
expressed directly as a function of controllable reactive power, Q • -""C
As will be shown in this section, the sensitivity of PL with respect
to Q is required in the GRG method will cause some difficulty in ~
computation.
Real power losses may, also, be expressed in other ways
mathematically. One method is to express the losses in terms of the
Zbus matrix, as developed by Neuenswander [49]. This second method
results in an expression of losses in terms of bus voltage, phase
angle, real power injection, and reactive power injections and can be
used for loss minimization. This second method was dismissed,
however, because of the computer memory and computation required to
compute the bus impedance matrix. Other methods of modeling include a
Taylor series expansion in terms of the variables to be optimized.
Again, this idea was dismissed because of the additional computer
memory and computation required.
A major consideration in the modeling of the power flow equality
constraints is accuracy of modeling the changes in the real power
losses with respect to bus voltages and controllable reactive
injections. In the LP step, it was decided that the cross coupling
between P and v and Q and g could be neglected. However, in the LP
step, the objective was to change reactive power or voltages by
rescheduling reactive power or voltages. Here, the objective is to
move real power losses by' changing controllable~ and v. Therefore,
it may not be appropr;i..ate to neglect the cross coupling of P to v and
55
Q to Q. The importance of this coupling was verified numerically, as
will be shown below. The linearized equality constraints are
presented again here for convenience.
N apij ~Q. OP .. (6-8) c + ---21. ..c~.v j = 0 a Q. . J
avj j=1 J i = 1 , ••• , N
N a Qij A Q j + aQij + AQC. 0 (6-9) L: .,6.Vj = j=1 a oj avj l.
These equations represent the Newton power flow equations and are
subject to limits on bus voltages and controllable reactive power.
The GRG solution technique, as will be described later, requires the
gradient vectors of the real power losses with respect to bus
voltages, bus phase angles, and controllable reactive injections.
Given the model defined above, these gradients are computed as:
0 p _ NBR . ( . ) L - r-- 2gkv. v . sin Q • - Q • ~ L- l.J l. J 0 i k=1
o PL -·-of 0P1 ag
=
af af a~ a~
a .9. a.9. a~ o!
t-1 OPL
a~
OPL ·a!
(6-10)
(6-11)
(6-12) ..
56
In equation (6-11), the variables of interest are the
sensitivities of losses to controllable reactive power generation,
Computer simulations performed showed that to accurately
compute these sensitivities the full Jacobian, as expressed in
equation (6-12), is required. Examples of the errors in losses
sensitivities that may occur are shown in Tables 6-1 and 6-2. These
examples are based on the IEEE 30 bus network. The tables show a
comparison of the real and reactive power loss sensitivities computed
from the full Jacobian versus the fast decoupled power flow Jacobian
with off diagonals neglected. The resulting errors, due to the fast
decoupled method, are shown. As can be seen, the reactive power loss
sensitivities are not close. Real power loss sensitivities are more
accurate but may not be acceptable. Therefore, approximations to the
Jacobian do not achieve the acceptable level of accuracy. This data
substantiates the need to model the full Taylor series expression for
the equality constraints (6-8) and (6-9). The off diagonal terms d_g_ d~
and &.f cannot be approximated. Moreover, in the GRG phase angles,~ ;}V
must also be modeled in addition to v and Q • In the GRG, both the LP - --c
basis matrix B and nonbasis matrix N must be constructed so that:
The basis variables are bus voltages or controllable reactive
power that are not at a limit. In addition, the bus phase angles are
basic except for the reference bus angle which is always nonbasic and
57
TABLE 6-1
ERROR ANALYSIS REACTIVE SENSITIVIES TO REAL POWER LOSSES
Fast Decoupled Newton Full Percent
Bus Jacobian Jacobian Error Error
1 .0009 .0018 .0009 100.0 2 .o .o .o .o 3 .o .o .o .o 4 .0469 .02965 .01725 58.2 5 .o .o .o .o 6 .0066 .00376 .00284 75.5 7 .o .o .o .o 8 .0093 .00564 .00366 64.9 9 .0068 .0052 .0016 30.7
10 .0085 .00623 .00227 36.4 11 .0224 .01174 .01066 90.8 12 .0234 .01235 .01105 89.5 13 .0188 .01025 .00855 83.4 14 .0141 .00953 .00457 47.9 15 .0236 • 01319 .0104 78.9 16 .0330 .01991 .01309 65.7 17 .0451 .02295 .02215 96.5 18 .0022 .0015 .0007 46.7 19 .0233 .01677 .00653 38.9 20 -.0087 -.00628 .00242 38.5 21 .0045 .00234 .00216 92.3 22 .0135 .00906 .0044 49.0 23 .0124 • 01197 .00043 3.5 24 .0.33 .00241 .00089 36.9 25 -.0015 .00013 .00002 15.4 26 .0158 .00858 .0073 85.0 27 .0021 .00079 .00131 73.4 28 .0260 .01466 .0113 76.9 29 .0034 .0026 .0008 30.8 30 .o .oooo .o .o
58
TABLE 6-2
ERROR ANALYSIS REAL POWER SENSITIVITIES TO REAL POWER LOSSES
Fast Percent Bus Decoupled Newton Error Error
1 .037 .037 .o .o 2 .055 .056 .001 1. 7 3 .045 .044 .001 2.3 4 .074 .097 .023 20.6 5 .061 .063 .002 3.2 6 .063 .066 .003 4.5 7 .056 .058 .002 3.4 8 .058 .060 .002 3.3 9 .053 .056 .003 5.4
10 .057 .061 .004 6.6 11 .068 .078 .010 12.8 12 .070 .080 .010 12.5 13 .067 .075 .008 10.7 14 .062 .068 .006 8.8 15 .066 .076 .010 13 .1 16 .080 .096 .016 16.7 17 .093 .117 .024 20.5 18 .055 .056 .001 3.4 19 .067 .077 .010 13.1 20 .021 .018 .003 16. 7 21 .059 .061 .002 3.2 22 .062 .067 .005 7.5 23 .062 .066 .004 6.0 24 .046 .047 .001 2.1 25 .045 .044 .001 2.2 26 .060 .066 .006 9.1 27 .054 .056 .002 3.6 28 .070 .081 .011 13.5 29 .056 .057 .001 1. 75 30 .o .o .o .o
59
is maintained at the reference phase angle. The basis matrix, B, and
the nonbasis matrix, N, are constructed as defined in equations (6-8)
and (6-9).
The Lagrangian function for optimization is:
where x_ is the dual vector and matrix A is the equality constraint
matrix. f is the scalar objective function of real power losses. The
optimal solution is:
dL = ~f t + x. B = 0
d.!B J.!B
dL = df t + x. N = 0
J.!NB J.!NB
dL -A r~B J-JX. .!NB
b = 0 -The GRG technique is used to solve these equations and is
summarized below.
(1) Construct the nonbasis and basis matrices Band N from the linearized constraint equations (6-8) and (6-9).
(2) Compute sensitivities from equations (6-10) through (6-12).
df = J PL -- ---d~ d ~
t
60
In the computation of these sensitivities, only the controllable reactive injections, Q , are of interest. c
(3) Define:
()f
CB l ci.!B c = =
_£NB df J.!NB
(4) Solve for dual vector x_ t = -_£B
(5) Compute the reduced gradient
D.!NB = __lJ: &.!NB
= ..£NB + x_tN
(6) If a variable xNB is near a limit, and 6XNB is moving in
the wrong direction, zero Ll xNB"
(7) Compute induced changes in basis variables
(8) Determine maximum step size, smax' without violating a limit
( 9)
( 10)
s = max N
max L(x1 _ i=1 1
< x + s !::! xi ~ XH ) i
Determine the optimal step size, s t using one dimensional search technique. A simplgp quadratic fit may be used. That is, the loss function P1 is evaluated at three points and an optimal step size is computed.
Update variables
61
(11) If controllable basis variable is moved to a limit exchange nonbasis controllable variable for a corres-ponding basis voltage, or visa versa, then return to Step (1). If no basis limits occur, return to solve another power flow.
Steps (1) and (2) of the algorithm are required to construct basis
and nonbasis variables and to compute the corresponding matrices B and
N. In step (2) the gradients or sensitivities are computed as
required by the extended simplex (or reduced gradient) method. The
cost function c is then constructed in step (3) and the dual vector z is computed in step (4). The reduced gradient can, at that time, be
computed. Step (6) is required so that a nonbasis variable will not
violate a limit. Since a nonbasis variable may exist at any feasible
point, then the nonbasis changes are made zero if a violation of a
limit is eminent. The next step (7) is needed such that the changes
to the basis variables induced by the nonbasis variables are known.
In step (8) the changes in basis and nonbasis variables are used to
compute maximum step size distance. This maximum distance is
determined such that neither a basis or nonbasis variable will violate
a limit. If the move is limited by a basis variable, then an exchange
with a nonbasis variable is necessary. This exchange process is
consistent with the simplex method. If the limit is on a non basis
variable, then no exchange of variables is necessary.
In step (9), the optimum move distance is computed. A quadratic
fit to the loss equation may be used. The real power loss equation
was evaluated as follows:
62
NBR [ *2 PL = L gk + v*2 - 2 cos(Qt - Q~)vt v~J
k=1 vi J 1 J 1 J
where vt = vt + s* av i 1 1
Qt = Qt + s* AQi 1 1
s is the step size and Av and ~Q are the corresponding changes of
h~ = [6XB' A~NB]t. It is important to observe that no re-solution of
the power flow equations was necessary after each step size
evaluation • Moreover, the change in the controllable reactive power
..9.c does not enter directly into the step size evaluation. The result
of step (9) is an optimal step size which is used in combination with
~to update the x in step (10). In step (11) the basis exchange is
done if required. In this algorithm, only bus voltages at generators,
.!c, are allowed to be exchanged with the corresponding reactive
generation, Q • -"C
This simplified the overall exchange considerably
and is consistent with modeling of generator reactive power controls.
However, many types of rules could be constructed for determining
which variables should be exchanged. These rules ·will be valid as
long as the exchange is done such that the objective function will
decrease.
For some networks, it is expected that a number of generator unit
megavars will be at a limit, especially the small units which may be
effective in reducing real power losses. Because of this, many basis
exchanges may occur. The nonbasis variables are taken as the
generators terminal voltages upon entering the optimization step and
63
the corresponding reactive power generator outputs are basic. Once
the reactive generator outputs are at a limit, the exchange will occur
with the generator terminal voltages. With this in mind, the
algorithm was designed to iterate within these eleven steps before
performing another power flow and corresponding relinearization of
power flow equations. That is, using the phase angles, bus voltages
and reactive generator outputs are recomputed in step 10, the matrices
Band N are reconstructed in step 1, and the algorithm cycles through
another step. This procedure is done only if a basis variable is
moved to a limit and a basis exchange is required. If a basis
variable is not moved to a limit, then a local optimum has been
achieved so that the power flow equations must be solved and
re-linearized.
The results of tests on several networks are given in Chapter 7.
After testing on several networks, the stopping criterion was chosen
based on the step size calculation. A small optimal step size in
combination with numerically small values for the reduced gradient
resulted in no significant improvement of the objective and,
therefore, the step size was monitored to evaluate whether to declare
convergence. In the networks tested, monitoring of convergence was
not a significant problem.
6.4 Summary of the Algorithm
An algorithm was developed that uses the simplex method of Linear
Programming as method for solution. A solution summary is shown in
Figure 6-1. Feasibility is obtained in an efficient manner where N
Yes
Perform Basis
Exchan e
Yes
64
"-~ ----1( Start )
FDPF
Construct GRG
Basis
Compute Loss
Sensitivities
Compute Dual
Limit Non-basis
Moves
!Jompute nduced s Changes
Determine Maximum
Step Size
Determine Optimum
Step Size
Update x
No
Perform Modified
LP
Construct LP
Basis
Figure 6-1 - Algorithm Summary
65
equality constraints and N + NC upper and lower bounds are required.
The simplex method was modified so that the nonbasis variable does not
have to be at either an upper or lower bound and, thus, a minimum
amount of reactive control adjustment is needed to eliminate an
infeasibility. After feasibility is obtained, the algorithm will
decrease the objective function. The generalized reduced gradient was
shown to offer many advantages in this stage. The generalized reduced
gradient method is an extension of the simplex method and, as shown,
provides a logical way to move towards the optimum while still
maintaining feasibility. Finally, the full set of power flow
equations are used such that a good linearization is maintained in the
constraint set. It is expected that iterations within the reduced
gradient step may be performed without having to resolve the power
flow equations.
6.5 Computer Implementation
The voltage control algorithm, as developed in this chapter, was
programmed and tested on several networks. The code development was
not done totally from scratch. A standard fast decoupled power flow
algorithm was used to solve the equality constraints, and the Linear
Programming codes, as published by Land and Powell [45], were used to
obtain feasibility. The computer routine GRG step was developed from
scratch, except for the matrix factorization and solution routines,
which were done using codes published by J.K. Reid [50].
Implementation of the GRG follows exactly that of the 11 steps
outlined in Section 6.3.
CHAPTER 7
DISCUSSION OF RESULTS
A computer code was developed and tested on nine networks. These
networks consist of the IEEE 14, 30, 57, and 118 bus networks and a
five bus network published by Stagg [51], a five bus network published
by Stevenson [52], and a three bus network from Elgerd [53]. Also,
the computer code was tested on a 23 bus network published by Biggs
and Laughton [39]. The algorithm performed well on all networks and
showed excellent convergence. For each network studied, a detailed
summary of input data and results are given in Appendices 1 - 8. In
this section a summary of solution data will be discussed and overall
conclusions as to the algorithms performance will be presented.
Tables 7-1 to 7-6 describe the results of testing the IEEE 30
and IEEE 118 bus networks. These tests were to confirm the
feasibility solution of the algorithm. The tests indicate that the
algorithm will eliminate reactive power overloads on units by
rescheduling voltages and reactive power generation. For the IEEE 30
bus network unit 2 was beyond the high reactive power limit by 1.2
megavars. The LP performs one dual simplex pivot. Unit 2 was
rescheduled at its upper limit of 50 megavars and unit 1 had its
output decreased by • 7 megavars. Table 7-7 shows the initial and
final solutions of the algorithm. For those units scheduled at their
limit the 'bus type is modified to one to indicate the unit is no
longer a voltage control bus. Table 7-3 shows the violations existing
66
Network
IEEE 30 Bus Modified
IEEE 118 Bus
67
TABLE 7-1
FEASIBILITY TEST RESULTS
Desired Unit No. Volts
2 1.055
10 .962 16 .963 17 .983 34 -958 43 .99 48 .965 46 1.01
.Amount of Var Infeasibility
-1.2
-6.5 -3-4
-42.1 -2.8
-12.4 -10.0 25.0
LP Control Actions
o Unit 2 volts to 1.0546
o Unit 1 vars decreased .7 vars
0 Rescheduled desired volts at units 18, 17, 46, 43, 10' 48' 16' and 34
68
TABLE 7-2
30 BUS IEEE MODIFIED
INITIAL POWER FLOW SOLUTION
---- GENERATOR SUMMARY ----
NO. BUS NAME VOLTS MW MVAR QMAX QMIN TYPE
1 1 GLEN LYN 1.0600 98.8 -5.5 100.0 -100.0 3 2 2 CLAYTOR 1.0550 80.0 51.2 50.0 -40.0 2 3 5 FIELDALE 1. 0100 50.0 8.7 40.0 -40.0 2 4 8 REUS ENS 1.0100 20.0 13.9 40.0 -1 o.o 2 5 11 ROANOKE 1 .0820 20.0 14.5 24.0 -6.0 2 6 13 HANCOCK 1.0710 20.0 6.9 24.0 -6.0 2
30 BUS IEEE FINAL POWER FLOW SOLUTION
---- GENERATOR SUMMARY ----
OLD NEW NO. BUS NAME VOLTS MW MVAR MVAR QMAX QMIN TYPE
1 1 GLEN LYN 1.0600 98.8 -5.5 -4.8 100.0 -100.0 3 2 2 CLAYTOR 1.0546 80.0 51.2 50.0 50.0 -40.0 1 3 5 FIELDALE 1. 0100 50.0 8.7 8.9 40.0 -40.0 2 4 8 REUSENS 1.0100 20.0 13.9 14.1 40.0 -10.0 2 5 11 ROANOKE 1.0820 20.0 14.5 14.5 24.0 -6.0 2 6 13 HANCOCK 1.0710 20.0 6.9 6.9 24.0 -6.o 2
LP SOLUTION
1 • Moved generator 2 to nonbasis at 50.0 MVARS.
2. Moved generator 2 terminal volts in basis at 1.0546.
69
TABLE 7-3
118 BUS IEEE VIOLATION SUMMARY
Unit Bus Mvar Limit Violation
10 20 -14.5 -8.0 -6.53
16 33 -17-4 -14.0 -3-359
17 35 -50.1 -8.0 -42.057
34 74 -8.8 -6.0 -2.8
43 92 -15.4 -3.0 -12.43
46 103 75.4 40.0 +35-38
48 105 -18.0 -8.0 -10.03
70
TABLE 7-4
118 BUS IEEE LP SUMMARY
MOVED INTO THE BASIS MOVED OUT OF THE BASIS LP Old New Old New
Pivot Gen Bus Volts Volts Gen Bus Vars Vars
1 18 37 • 98 -978 18 37 3.3 -8.0
2 9 19 .97 .934 17 35 -50.1 -8.0
3 46 103 1. 01 .997 46 103 75.4 40.0
4 40 89 1.005 .999 43 92 -15.4 -3.0
5 8 16 . 97 .967 10 20 -14.5 -8.0
6 47 104 .971 .9692 48 105 -14.0 -8.0
7 14 28 .968 .9661 16 33 -17.4 -14.0
8 31 70 .984 .9813 34 74 -8.8 -6.0
71
TABLE 7-5
118 BUS IEEE INITIAL POWER FLOW SOLUTION
---- GENERATOR SUMMARY ----
NO. BUS NAME VOLTS MW MVAR QMAX QMIN TYPE
1 1 BUS-69 1.0300 514.3 -111. 9 1000.0 -1000.0 3 2 2 BUS-1 0.9550 o.o -4.9 15.0 -5.0 2 3 5 BUS-4 0.9980 -9.0 -64.6 300.0 -300.0 2 4 7 BUS-6 0.9900 o.o 8.9 50.0 -13.0 2 5 9 BUS-8 1.0150 -28.0 42.4 300.0 -300.0 2 6 11 BUS-10 1.0500 450.0 -51.1 200.0 -147.0 2 7 13 BUS-12 0.9900 85.0 87.8 120.0 -35.0 2 8 16 BUS-15 0.9700 o.o 6.5 30.0 -10.0 2 9 19 BUS-18 0.9730 o.o 28.9 50.0 -16.0 2
10 20 BUS-19 0.9620 o.o -14.5 24.0 -8.0 2 11 25 BUS-24 0.9920 -13.0 -10.1 300.0 -300.0 2 12 26 BUS-25 1. 0500 220.0 50.1 140.0 -47.0 2 13 27 BUS-26 1.0150 314.0 6.2 1000.0 -1000.0 ·2 14 28 BUS-27 0.9680 -9.0 4.2 300.0 -300.0 2 15 32 BUS-31 0.9670 7.0 33.0 300.0 -300.0 2 16 33 BUS-32 0.9630 o.o -17.4 42.0 -14.0 2 17 35 BUS-34 0.9840 o.o -50.1 24.0 -8.0 2 18 37 BUS-36 0.9800 o.o 3.3 24.0 -8.0 2 19 41 BUS-40 0.9700 -46.o 25.1 300.0 -300.0 2 20 43 BUS-42 0.9850 -59.0 41.5 300.0 -300.0 2 21 47 BUS-46 1.0050 19.0 -4.1 100.0 -100.0 2 22 50 BUS-49 1 .0250 204.0 120.2 210.0 -85.0 2 23 55 BUS-54 0.9550 48.0 4.0 300.0 -300.0 2 24 56 BUS-55 0.9520 o.o 4.7 23.0 -8.0 2 25 57 BUS-56 0.9540 o.o -2.1 15.0 -8.0 2 26 60 BUS-59 0.9850 155.0 80.2 180.0 -60.0 2 27 62 BUS-61 0.9950 160.0 -41.8 300.0 -100.0 2 28 63 BUS-62 0.9980 o.o 1. 3 20.0 -20.0 2 29 66 BUS-65 1.0050 391.0 77.8 200.0 -67.0 2 30 67 BUS-66 1 .0500 392.0 -2.2 200.0 -67.0 2 31 70 BUS-70 0.9840 o.o 13.4 32.0 -10.0 2 32 72 BUS-72 0.9800 -12.0 -5.2 100.0 -100.0 2 33 73 BUS-73 0.9910 -6.o 12 .1 1 oo.o -100.0 2 34 74 BUS-74 0.9580 o.o -8.8 9.0 -6.0 2 35 76 BUS-76 0.9430 o.o 4.2 23.0 -8.0 2 36 77 BUS-77 1 .0060 o.o 16.3 70.0 -20.0 2 37 80 BUS-80 1.0400 477.0 111.8 280.0 -165.0 2 38 85 BUS-85 0.9850 o.o -6.1 23.0 -8.0 2 39 87 BUS-87 1. 0150 4.0 11.0 1000.0 -100.0 2 40 89 BUS-89 1 .0050 607.0 -5.9 300.0 -210.0 2 41 90 BUS-90 0.9850 -85.0 59.3 300.0 -300.0 2
72
TABLE 7-5 (Continued)
118 BUS IEEE INITIAL POWER FLOW SOLUTION
---- GENERATOR SUMMARY ----
NO. BUS NAME VOLTS MW MVAR QMAX QMIN TYPE --42 91 BUS-91 0.9800 -1 o.o -13.1 100.0 -100.0 2 43 92 BUS-92 0.9900 o.o -15.4 9.0 -3.0 2 44 99 BUS-99 1.0100 -42.0 -17.5 100.0 -1 oo.o 2 45 100 BUS-100 1 .0170 252.0 94.0 155.0 -50.0 2 46 103 BUS-103 1. 0100 40.0 75.4 40.0 -15.0 2 47 104 BUS-104 0.9710 o.o 2.4 23.0 -8.0 2 48 105 BUS-105 0.9650 o.o -18.0 23.0 -8.0 2 49 107 BUS-107 0.9520 -22.0 6.6 200.0 -200.0 2 50 110 BUS-110 0.9730 o.o 0.4 23.0 -8.0 2 51 111 BUS-111 0.9800 36.0 -1.8 1000.0 -100.0 2 52 112 BUS-112 0.9750 -43.0 41.4 1000.0 -100.0 2 53 113 BUS-113 0.9930 -6.0 7.6 200.0 -100.0 2 54 116 . BUS-116 1.0050 184.0 57.0 1000.0 -1000. 0 2
73
TABLE 7-6
118 BUS FINAL SOLUTION
---- GENERATOR SUMMARY ----
OLD NEW NO. BUS NAME VOLTS MW MVAR MVAR QMAX QMIN TYPE -- -
1 1 BUS-69 1 .0300 514.3 -111.9 -109.5 1000.0 -1000.0 3 2 2 BUS-1 0.9550 o.o -4-9 -4-9 15.0 -5.0 2 3 5 BUS-4 0.9980 -9.0 -64.6 -64.3 300.0 -300.0 2 4 7 BUS-6 0.9900 o.o 8.9 8.9 50.0 -13.0 2 5 9 BUS-8 1 • 0150 -28.0 42.4 43.7 300.0 -300.0 2 6 11 BUS-10 1.0500 450.0 -51.1 -51.1 200.0 -147.0 2 7 13 BUS-12 0.9900 85.0 87.8 89.6 120.0 -35.0 2 8 16 BUS-15 0.9671 o.o 6.5 -6.1 30.0 -1 o.o 2 9 19 BUS-18 0.9730 o.o 28.9 30.3 50.0 -16.0 2
10 20 BUS-19 0.9620 o.o -14.5 -8.0 24.0 -8.0 1 11 25 BUS-24 0.9920 -13.0 -10.1 -9.5 300.0 -300.0 2 12 26 BUS-25 1.0500 220.0 50.1 51.2 140.0 -47.0 2 13 27 BUS-26 1. 0150 314.0 6.2 7.0 1000.0 -1000.0 2 14 28 BUS-27 0.9661 -9.0 4.2 -1.0 300.0 -300.0 2 15 32 BUS-31 0.9670 7.0 33.0 34.2 300.0 -300.0 2 16 33 BUS-32 0.9630 o.o -17-4 -14.0 42.0 -14.0 1 17 35 BUS-34 0.9840 o.o -50.1 -8.0 24.0 -8.0 1 18 37 BUS-36 0.9767 o.o 3.3 -8.0 24.0 -8.0 1 19 41 BUS-40 0.9350 -46.0 25.1 -42.0 300.0 -300.0 2 20 43 BUS-42 0.9850 -59.0 41.5 76.0 300.0 -300.0 2 21 47 BUS-46 1 .0050 19.0 -4.1 -4.1 100.0 -1 oo.o 2 22 50 BUS-49 1.0250 204.0 120.2 120.2 21 o.o -85.0 2 23 55 BUS-54 0.9550 48.0 4.0 4.0 300.0 -300.0 2 24 56 BUS-55 0.9520 o.o 4.7 4.7 23.0 -8.0 2 25 57 BUS-56 0 .9540 o.o -2.1 -2.1 15.0 -8.0 2 26 60 BUS-59 0.9850 155.0 80.2 80.2 180.0 -60.0 2 27 62 BUS-61 0.9950 160.0 -41.8 -41.8 300.0 -100.0 2 28 63 BUS-62 0.9980 o.o 1.3 1.3 20.0 -20.0 2 29 66 BUS-65 1.0050 391.0 77.8 79.5 200.0 -67.0 2 30 67 BUS-66 1.0500 392.0 -2.2 -2.2 200.0 -67.0 2 31 70 BUS-70 0.9813 o.o 13.4 4.1 32.0 -10.0 2 32 72 BUS-72 0.9800 -12.0 -5.2 -5.2 100.0 -100.0 2 33 73 BUS-73 0.9910 -6.0 12.1 15 .1 100.0 -100.0 2 34 74 BUS-74 0.9580 o.o -8.8 -6.o 9.0 -6.0 1 35 76 BUS-76 0.9430 o.o 4.2 4.6 23.0 -8.0 2 36 77 BUS-77 1 .'0060 o.o 16.3 16.5 70.0 -20.0 2 37 . 80 BUS-80 1. 0400 477.0 111.8 111.8 280.0 -165.0 2 38 85 BUS-85 0.9850 o.o -6.1 -0.6 23.0 -8.0 2 39 87 BUS-87 1. 0150 4.0 11.0 11.0 1000.0 -100.0 2 40 89 BUS-89 1.0000 607.0 -5-9 -31-4 300.0 -21 o.o 2 41 90 BUS-90 0.9850 -85.0 59.3 66.4 300.0 -300.0 2 42 91 BUS-91 0.9800 -1 o.o -13.1 -13.1 100.0 -100.0 2 43 92 BUS-92 0.9900 o.o -15.4 -3.0 9.0 -3.0 1
74
TABLE 7-6 (Continued)
118 BUS FINAL SOLUTION
---- GENERATOR SUMMARY ----
OLD NEW NO. BUS NAME VOLTS MW MVAR MVAR QMAX QMIN. TYPE -- -- -44 99 BUS-99 1. 0100 -42.0 -17.5 -17.5 100.0 -100.0 2 45 100 BUS-100 1.0170 252.0 94.0 113. 7 155.0 -50.0 2 46 103 BUS-103 0.9994 40.0 75.4 40.0 40.0 -15.0 1 47 104 BUS-104 0.9692 o.o 2.4 2.2 23.0 -8.0 2 48 105 BUS-105 0.9650 o.o -18.0 -8.0 23.0 -8.0 1 49 107 BUS-107 0.9520 -22.0 6.6 6.6 200.0 -200.0 2 50 110 BUS-110 0.9730 o.o 0.4 5.9 23.0 -8.0 2 51 111 BUS-111 0.9800 36.0 -1.8 -1.8 1000.0 -100.0 2 52 112 BUS-112 0.9750 -43.0 41.4 41.4 1000.0 -100.0 2 53 113 BUS-113 0.9930 -6.0 7.6 9.8 200.0 -100.0 2 54 116 BUS-116 1.0050 184.0 57.0 57.0 1000.0 -1000.0 2
75
TABLE 7-7
Solution Summary
Basis Controls No. No. OPF Sub Inverse % Loss At
Network · Units Iterations Iterations Non zeros Reduction Limit
118 Bus 54 3 16 GRG 2024 16 .1 14 unit vars 3 FDPF
30 Bus Mod. 6 1 13 GRG 530 12.4 1 bus volt
30 Bus 6 2 5 GRG 530 8.2 2 bus volt 3 unit var
14 Bus 5 1 7 GRG 187 6.8 None
57 Bus 7 1 5 GRG 1228 12.0 4 unit vars
23 Bus 6 6 6 GRG 354 19.9 4 bus volt 5 FDPF
76
in the 118 IEEE bus network. Seven generators were initially
generating beyond their reactive power limits. Table 7-4 shows the
individual LP pivots necessary to obtain a feasible solution. The
table shows that for each LP pivot the generators that were pivoted
into and out of the basis. Eight LP pivots were required to eliminate
all the reactive power overloads. Table 7-5 and 7-6 show the initial
and final solutions for the 118 IEEE network. As shown, generators
were scheduled at their limit and
appropriately.
their bus type modified
In both networks it is seen that only the generators that are in
violation are scheduled at their reactive power limits and that
non-basis variables are not necessarily scheduled at a limit. Only
one LP solution was required to eliminate infeasibili ties. The
linearized equality constraints used in the LP was a good
respresentation of the power flow equality constraints for
rescheduling unit voltages and reactive power generation.
Table 7-7 summarizes the tests performed for real power loss
minimization on several networks. Several small networks were tested
and test results are further described in the Appendices. In this
section only the larger networks are discussed. The solution summary,
Table 7-7, shows that the number of power flow iterations (FDPF)
required was very moderate. A power flow iteration consists of a
minimum of two power flow mismatch calculations (evaluations of the
equality constraints.) An OPF iteration, as shown in Table 7-7, is
77
defined as a power flow solution and a GRG solution. Sub-iterations
within the power flow and GRG are also shown. A GRG iteration
consists of a basis switch and corresponding update of solution. For
example, the 118 IEEE bus network shows that three OPF iterations were
required and that a total of 14 GRG sub-iterations and three power
flow sub-iterations were required in total. The number power flow
sub-iterations for each network was very small. Also, for each
network the megawatt power loss reduction was significant ranging from
6.8% to 19.9%. These large reductions are unusually high because the
voltage high limit was set at 1.1 per unit. Lower upper limits would
result in lower loss reductions. The high limit of 1.1 is typical of
upper limits other researchers have used [10,43,]. Also, it is seen
from Table 7-7 that not many of the controls were driven to a limit.
Thus, the resulting algorithm was shown to have significant advantages
over other linear programming techniques.
The computer storage requirement for the networks was very modest.
For the 118 bus the basis inverse required 2027 non-zeros, which is
much smaller than 7669 non-zeros reported by Dayal [24] using Wolfe's
quadratic programming technique. Dayal also reported that a maximum
of 15 quadratic programming iterations were required for an economic
dispatch solution to the 118 bus IEEE network. The algorithm
developed here indicates a much greater efficiency in computer usage
and, at the same time, large reductions in real power losses • ..
78
Table 7-8 gives a convergence summary for each network.
Convergence is rapid and as shown most of the loss reduction occurs in
one to three OFF iterations. The convergence table shows that most of
the computation occurs in the GRG step. As will be described in
Chapter 8, this computation may be done efficiently by updating only
the column of that basis inverse that is exchanged. Thus, most of
the computation is in the initial construction and factorization of
the basis inverse.
Tables 7-9 through 7-14 show the reduction of losses each sub-
iteration of the OFF and GRG and tables 7-15 through 7-22 give more
detailed data for each iteration. For each GRG sub-iteration the step
size evaluation is given for each step size evaluation. The value of
the objective function is shown with the optimal step size (OPT) and
the maximum step size (SMAX). For each GRG step if a GRG pivot (or
bus type switch) is needed the following information is given: basis
variables that reaches a limit (XBASIS), the limit of this basis
variable (LIMIT), the bus number of the variable (BUS), and the type
of GRG move. M represents that a generator reactive power was a basis
variable and now has become non-basis variable and V indicates a
voltage was a basis variable and now is to become a nonbasis variable.
NB means that the limit occurred in the non-basis. NL means that a
step size move was made with no limit violation in either basis or
non-basis. Each type M or V requires that the basis inverse be
reconstructed. As can be seen the 118 bus IEEE network was a
79
TABLE 7-8
Convergence Summary
OPF Computation % Loss Basis Network Iteration Required Reduction Exchanges
118 Bus 1 11 GRG 11.4 10 2 3 FDPF 1.0 2
2 GRG 3 1 GRG .5 1
30 Bus Mod. 1 8 GRG 12.4 1
30 Bus 1 1 GRG 5.2 0 2 3 GRG 3.0 3
14 Bus 1 7 GRG 6.8 0
57 Bus 1 4 GRG 12.0 3
23 Bus 1 1 GRG 12.0 0 3 FDPF
2 1 GRG 3.5 0 1 FDPF
3 1 GRG 3.7 0 1 FDPF
4 1 GRG .52 0 5 1 GRG .1 0 6 1 GRG .08 0
80
TABLE 7-9
IEEE 14 BUS TEST RESULTS
OPF Iteration GRG Iteration Losses
0 13.28
1 1 13.075
2 12.92
3 12.79
4 12.66
5 12.54
6 12.44
7 12.37
TABLE 7-10
23 BUS TEST RESULTS
OPF Iteration GRG Iteration Losses
0 47.76
1 1 41.90
2 1 40.2388
3 1 38-7522
4 1 38.20
5 1 38.15
6 1 38.11
7 Converged
81
TABLE 7-11
IEEE 30 BUS MODIFIED TEST RESULTS
OPF Iteration GRG Iteration Losses
0 5.37s
1 1 5.24s7
2 5 .1561
3 5.0645
4 4.917
5 4.s577
6 4.8015
7 4.7s7
8 4.7079
2 Converged
TABLE 7-12
IEEE 30 BUS TEST RESULTS
OPF Iteration GRG Iteration Losses
0 17 .61
1 1 16.69
2 1 16.34
2 16.13
3 16.11
3 Converged
82
TABLE 7-13
IEEE 57 BUS TEST RESULTS
OPF Iteration GRG Iteration Losses
0 1 28.114
1 1 27.89
2 27.22
3 25. 71
4 24.59
2 Converged
83
TABLE 7-14
IEEE 118 BUS TEST RESULTS
OPF Iteration GRG Iteration Losses
0 133.0161
1 1 131.03
2 130.53
3 129.037
4 128.065
5 127 .92
6 127 .1
7 126.3
8 125.1
9 124.18
10 123.44
11 117. 79
2 1 113.22
2 112.083
3 1 111.41
2 111.41
4 Converged
84
TABLE 7-15 118 BUS IEEE GRG SUMMARY
OFF GRG OBJECTIVE FUNC. MAX. OPT. BUS TYPE SWITCH INFO. ITER ITER S=O. S=.5 S=1. STEP STEP XBASIS LIMIT BUS TYPE
------------------ ------------------------1 1 133.0 131.92 131.0 .0129 .0379 70. 70. 36 M
2 131. 0 130.7 130.5 .0058 .0776 23. 23. 50 M
3 130.5 129. 7 129.0 .0221 .0929 23. 23. 47 M
4 129.0 128.5 128.0 .0225 .1642 23. 23. 42 M
5 128.0 127 .9 127 .9 • 0044 .2839 15 • 15. 2 M
6 127.9 127.5 127 .1 • 0261 .2884 -67 • -67. 30 M
7 127 .1 126.8 126.6 .0190 • 5181 32. 32. 31 M
8 126.6 126.4 126.3 • 0112 .4517 15 • 15. 25 M
9 126.3 125.7 125.1 • 0471 .7296 -47 • -47. 12 M
10 125.1 124.1 124.1 .0417 .7217 23. 23. 38 M
11 124.1 123.7 123.4 .0329 .2165 -8. -8. 24 M
12 123.4 120.2 117 .1 .2944 2.414
2 1 117 .8 117 .o 116.4 .0696 .2748 -8. -8. 18 M
2 116.4 114. 7 113.2 .2189 -7798 20. 20. 28 M
3 113.2 112.5 112.1 .0815 .1132 NB
3 1 112.0 111.6 111.4 .0420 .0900 1 .1 1 .1 30 v
2 111.4 111.4 111.4 .0001 .3219 1 .1 1 .1 30 v
85
TABLE 7-16 14 BUS IEEE GRG SUMMARY
OPF GRG OBJECTIVE FUNC. MAX. OPT. BUS TYPE SWITCH INFO. ITER ITER S=O. S=.5 S=1. STEP STEP XBASIS LIMIT BUS TYPE
------------------ ------------------------1 1 13.28 13.07 13.26 .1289 .066 NL
2 1 13.07 12. 95 12.93 .0741 .0629 NL
3 1 12.92 12.79 12.91 .1374 .0696 NL
4 1 12. 79 12.69 12.67 .0750 .0653 NL
5 1 12.67 12.57 12.55 .0828 .0695 NL
6 1 12.55 12.46 12.44 .0743 .0661 NL
7 1 12.44 12.37 12.45 .2623 .1251 NL
TABLE 7-17 23 BUS GRG SUMMARY
OPF GRG OBJECTIVE FUNC. MAX. OPT. BUS TYPE SWITCH INFO. ITER ITER S=O. S=.5 S=1. STEP STEP XBASIS LIMIT BUS TYPE
------------------ ------------------------1 1 47.76 43.33 40.17 .1869 .3740 NL
2 1 41. 91 40.95 40.16 .0941 .2983 NL
3 1 40.24 39.09 38.30 .3849 • 7121 NL
4 1 38.45 38.30 38.20 .1350 .2134 NL
5 1 38.21 38.18 38.16 .0823 .2826 NL
86
TABLE 7-18 30 BUS IEEE GRG SUMMARY
OFF GRG OBJECTIVE FUNC. MAX. OPT. BUS TYPE SWITCH INFO. ITER ITER S=O. S=.5 S=1. STEP STEP XBASIS LIMIT BUS TYPE
------------------ ------------------------1 1 17 .61 17 .06 16.69 .1287 .2278 NB
2 1 16.69 16.49 16.34 .1259 -3378 40. 40. 4 M
2 16.34 16.19 16.13 .2714 .3009 40. 40. 3 M
3 16.13 16.12 16.11 .2126 .9470 NB
3 1 16 .11 17.62 19.50 31.05 -54-86
TABLE 7-19 5 BUS STAGG GRG SUMMARY
OFF GRG OBJECTIVE FUNC. MAX. OPT. BUS TYPE SWITCH INFO. ITER ITER S=O. S=.5 S=1. STEP STEP XBASIS LIMIT BUS TYPE
------------------ ---- ------------------------1 1 4.588 4.606 4.95 .3702 .0819 NL
2 1 4.555 4.523 4.58 .1590 .0686 NL
3 1 4.522 4.544 4.88 .3504 .0753 NL
4 1 4.493 4.462 4.52 .1741 .0748 NL
5 1 4.462 4.475 4.76 .3044 .0688 NL
6 1 4.434 4.405 4.45 .1936 .0830 NL
87
TABLE 7-20
3 BUS STAGG GRG SUMMARY
OPF GRG OBJECTIVE FUNC. MAX. OPT. BUS TYPE SWITCH INFO. ITER ITER S=O. S=.5 S=1. STEP STEP XBASIS LIMIT BUS TYPE
------------------ ------------------------1 1 2.66 1.70 2.67 .1205 .060 NL
2 1 1. 703 1.674 1.661.330 1.238 NL
3 1 1.667 1.631 1.66 .1213 .0627 NL
4 1 1 .631 1.624 1 .61 .333 1 .22 NB
5 1 1.618 1.615 1.61 .0649 .1014 NB
TABLE 7-21 30 BUS IEEE MOD. GRG SUMMARY
OPF GRG OBJECTIVE FUNC. MAX. OPT. BUS TYPE SWITCH INFO. ITER ITER S=O. S=.5 S=1. STEP STEP XBASIS LIMIT BUS TYPE
------------------ ------------------------1 1 5.38 5.26 5.26 .1839 .1375 NL
2 1 5.25 5 .15 5.29 .7868 .3602 NL
3 1 5.12 5.08 5.07 .1647 .1414 NL
4 1 5.06 4.99 5.02 .5509 .3323 NL
5 1 4.99 4.93 4.92 .1516 .1475 NL
6 1 4.92 4.87 4.86 .2868 .3061 NL
88
TABLE 7-22 57 BUS IEEE GRG SUMMARY
OPF GRG OBJECTIVE FUNC. MAX. OPT. BUS TYPE SWITCH INFO. ITER ITER S=O. S=.5 S=1. STEP STEP XBASIS LIMIT BUS TYPE
------------------ ------------------------1 1 28.11 27.99 27.89 • 0033 .0353 9. 9 • 6 M
2 27.89 27.53 27.26 .0168 .0393 5. 5. 2 M
3 27.26 27.24 27.22 .002 .1053 -8. -8. 4 M
4 27.22 26.25 25. 71 .0886 .1218 60. 60. 3 M
5 25.71 25.10 24.50 .0903 11 .6 NB
2 1 24.59 24-58 24-58 .0005 -3.33
89
difficult network in this regard. Many basis switches were performed
but the algorithm continued toward a local optimum.
The last Table 7-23 shows how accurate the linearization was for
each test case. For each FDPF iteration the maximum megawatt and
megavar mismatch is shown. That is, if no mismatch exists, the
equality constraints are valid. Only the 23 bus case had significant
power flow mismatches. As a conclusion, it could be expected that the
linearized Jacobian is an accurate representation of the power flow
equality constraints and that for most networks it may not be
necessary to re-solve the power flow equality equations after each GRG
step.
90
TABLE 7-23
Linearization Summary
GRG OPF No. of FDPF Megawatt Mega var
Network Iteration Sub Iterations Iteration Mismatch Mismatch
57 Bus 1 5 1 o.o 1.0
118 Bus 1 12 1 o.o o.o 2 3 1 o.o o.o 3 2 1 o.o o.o
23 Bus 1 1 1 23.0 -1.0 2 1. 0 o.o
1 1 1 2.0 o.o 3 1 1 1.0 o.o 4 1 1 o.o o.o 5 1 1 o.o o.o
30 Bus Mod. 1 1 1 o.o o.o 2 1 1 o.o o.o 3 1 1 o.o o.o 4 1 1 o.o o.o 5 1 1 o.o o.o 6 1 1 o.o o.o
14 Bus 1 1 1 o.o o.o 2 1 1 o.o o.o 3 1 1 o.o o.o 4 1 1 o.o o.o 5 1 1 o.o o.o 6 1 1 o.o o.o 7 1 1 o.o o.o
30 Bus 1 1 1 o.o o.o 2 3 1 o.o o.o 3 1 1 o.o o.o
3 Bus 1 1 1 o.o o.o 2 1 1 o.o o.o 3 1 1 o.o o.o 4 1 1 o.o o.o 5 1 1 o.o o.o
CHAPTER 8
IMPLEMENTATION CONSIDERATIONS
Voltage Control is of interest to many utilities currently
implementing Energy Control Centers. Many control centers have either
purchased such voltage control software or have specified such a
function in a power control computer system procurement. As a result
of this interest, many questions have been identified. These
questions pertain to reliability, accuracy, execution speed, and
integration of voltage control with other control center functions.
This chapter shall address such issues in an attempt to solve some of
these 'practical problems associated with voltage control.
8.1 Integration of Voltage Control Into An Energy Control Center
An overview of the voltage control algorithm as it would be
implemented into an Energy Control Center is shown in Figure 8-1. As
can be seen, the algorithm is an integral part of the network analysis
function and, therefore, does not control the power system in real-
time. Real-time control is done by a combination of SCADA and AGC.
The production cost program supplements SCADA and AGC and accumulates
the hourly production costs.
The major interface between real-time control functions and
network analysis functions is the telemetered data that is processed
by SCADA that .is used by the State Estimator. The State Estimator
provides estimated voltages and power flows. Another major interface
is the penalty factors that are computed by the network analysis
91
SCAD A AGC
PRODUCTION COSTING
REAL-TIME CONTROL
,,, ~~ TOPOLOGY and STATE ESTIMATOR ~\/
EXTERNAL ---·
MODEL
Figure 8-1 - Energy Control Center Overview
VOLTAGE CONTROL
SECURITY ANALYSIS
PENALTY FACTOR
1..q N
93
function and are available to AGC. These penalty factors are based on
the State Estimator and External Model results. The State Estimator
provides the solution of a portion of the electrical power network
that can be estimated from telemetered data and the External Model
provides a solution of the network that has no telemetry and cannot be
solved by the State Estimation. The voltage control algorithm
provides a recommended voltage and reactive power generation setting
based on the current solutions provided by the State Estimator and
External Model programs. Security analysis is used to assess the
overall security of the network.
Within the network analysis functions, the State Estimator,
External Model, and Penalty Factor may execute anywhere on a period of
a few minutes to fifteen minutes, and Security Analysis may execute on
a period of every fifteen minutes to one half hour. The periodicity
of the voltage control program may be as much as five minutes.
Important questions to be answered are what execution period is
necessary, how often should voltage controls be adjusted and what is
the associated impact on production costs. The choice of execution
periodicity does have a significant impact on computer system loading
and, therefore, does impact on control system costs and operation.
Thus, an important question to be asked is how often are voltage
controls needed and what are the overall impacts on production costs.
94
For some utilities it may be difficult to accurately know the bus
voltage limits and generator reactive power limits. Bus voltage
limits are determined by planning departments of the utility. These
limits are designed such that coordination with distribution system is
maintained and transmission equipment are not stressed at high
voltages. Reactive power limits are determined from the units
capability curves and are related to the real power generation. These
reactive power limits may be affected by unit temperatures, hydrogen
pressures, and exciter field current. It is, therefore, hard to know
these limits exactly. In actual practice, since the limits are
difficult to predict, the plant operations personnel typically operate
the generators in a conservative manner. Because of this, overall
optimum voltage coordination may not be possible. Thus, it is
important to know how much the performance of a voltage control
function would be affected by these limits.
Based on the questions discussed above, the following
considerations will be addressed in this chapter:
0 Impact on voltage control periodicity. must voltage control be executed?
How frequently
o How accurate do the limits have to be and what impact do they have on the voltage control program performance?
In addition to the above items, computer simulation results will
be discussed showing how voltage control impacts AGC. These
simulations will show that a reduction in losses will result in
smaller generation to be dispatched to meet a given level of demand.
95
Finally, modifications to the voltage control algorithm will be
discussed to accommodate voltage control in an on-line computer system
environment.
8.2 Algorithm Modifications For On-Line Operation
In this research, the voltage control algorithm consists of a
power flow, linearization, and optmization steps that are iterated
until overall convergence was achieved. Depending on how accurate the
linearization is, the overall optimization may not require re-solution
of the power flow equations. In Chapter 7 it was shown that for the
systems tested, a good linearization was obtained. In a control
center environment, the voltage control algorithm will execute in a
tracking mode so that the state estimated solution will serve as a
basis from which the voltage control algorithm can use a starting
point. Because of this, it is expected that re-solution of the power
flow may not be necessary. As a result, GRG iterations may be
performed within the GRG step only, and subsequent power flow
solutions may not be necessary.
follows:
initial solution
GRG
The resulting algorithm is as
no
96
In order to increase computation efficiency more, the GRG basis
matrix B can be updated only for the column in which basis switching
occurs. This may be implemented by a sparse variant [ 50 J of the
Bartels-Goloub algorithm. Thus, the final solution algorithm required
is the following:
construct B-1 --~
perform GRG step in total
update ~91. yes i of B
stop
It should be understood that if a good linearized model is not
maintained, some oscillation will occur between the controls required
by the voltage control algorithm, the power system response, and
subsequent executions of voltage control. With this procedure, one
iteration of the voltage control algorithm is as follows:
1. Solve for loss sensitivities, by direct solution of linearized power, this equation:
df df dp d~
B = ~f ()f J,S. ay
2. Solve for the dual vector by direct solution.
df Bx_ = -_£, c = (j :!_
97
3. Compute reduced gradient.
T p.~NB = .£. + X. N
4. Compute changes in non-basis variables.
5. Determine optimum step size and update variables.
6. If a basis S,!T+tch has occurred, update the appropriate column of B •
Three direct solutions and one partial factorization are required
per iteration of the GRG using this method. These on-line
modifications will be tested as part of the Energy Control Center
simulations as discussed in the rest of the chapter.
8.3 EMS Simulation
So that the voltage control program could be tested as though it
were part of an Energy Control Center, a simulator of the energy
control center was constructed as shown in Figure 8-2. The simulator
was constructed to closely resemble an Energy Control Center except
that the penalty factor calculation, contingency analysis, and state
estimation programs were not considered. The simulator used a
schedule of system demands for each time period simulated and then
computed the bus loads, losses, generator dispatch, and the power
system response to voltage control. The load model used fixed load
distribution factors, and for each system demand level the load model
computed individual bus loads. A power flow algorithm was used to
Hourly Demand
1--1-LOAD ECONOMIC
MODEL DISPATCH
Estimated Loss
POWER FLOW
Bus Unit Megawatt I L
Loads Generation
POWER SYSTEM
-1
__ _J
~
~
-,
Noise
l Telemetered . Data _
SCAD A . VOLTAGE
SIMULATION CONTROL ~
Unit Volts and Vars
PRODUCTION OUTPUT -, COSTING SUMMARY
........_
Figure 8-2 - Power System - Control Center Simulation
-'° CX>
99
simulate power system response. From the output of the power flow,
real power losses were computed. The summation of the total demand
and real power losses were then input to the Economic Dispatch
algorithm as the genera ti on to be dispatched ( GTBD) • The Economic
Dispatch then determined the optimum real power settings of the units
given the generation to be dispatched. The Economic Dispatch and
power flow algorithm were iterated until total generation was equal to
demand plus losses.
The voltage control algorithm used the simulated power system
response as telemetered data input and the previous optimum voltage
solution to determine the optimum settings for the hour. It then
transmitted optimum generator terminal voltage and reactive power
generation settings back to the power system for control. Thus, the
simulator demonstrated on-line automatic voltage control. Various
options were also included in the simulation, such as modifying the
frequency of control, modifying time duration of control, and
modification of voltage and reactive power limits.
8.4 Simulation Results
The simulator was tested on several networks over a twenty-four
hour simulation period. The voltage algorithm performed well for all
networks tested and, in addition, was shown to be efficient in terms
of computation.
When executed in a tracking mode, where the previous optimum
solution was used as a starting point for the next solution, the
100
algorithm required typically two GRG iterations. In all simulations,
the GRG was not iterated with a power flow but was executed directly
from the output of the SCADA function. That is, the algorithm
modifications as discussed in Section 8.2 were tested and performed
well. The algorithm was shown to converge without iterating with a
power flow solution and the modifications to update only one column of
the basis matrix at a time were successful.
For each hour of the simulation, voltage optimization required, on
an average, two control cycles with the power simulator function to
achieve convergence. The power system simulator and voltage control
required two iterations before an voltage control setting was
established with all generator dispatched at their optimum real power
settings.
Table 8-1 is a summary of test results to demonstrate how voltage
control affects production costs. The Economic Dispatch function for
this case assumed the fuel cost to be one dollar per megawatt for all
generators. The generation that must be dispatched was numerically
equal to the fuel costs. The daily production without voltage control
was $6811 and with voltage control was $6797. Thus, a $14 a day
production cost savings was achieved since the generation to be
dispatched was reduced by this amount. The corresponding losses have
been plotted on Figure 8-3. The curves are plotted for the conditions
of no voltage control and with voltage control. The figure shows that
the voltage control program resulted in significant loss reduction
throughout the day.
101
TABLE 8-1 SIMULATOR TEST CASE
EQUAL COST CURVES IEEE 30 BUS MOD.
V=1.1
WITHOUT CONTROL WITH CONTROL HOUR DEMAND LOSS GTBD LOSS GTBD -- --
1 210 2.0 212 1.4 211 2 199 1.8 201 1.3 201 3 193 1. 7 194 1.2 194 4 193 1. 7 194 1.2 194 5 199 1. 7 201 1.3 201 6 203 1. 7 204 1.3 204 7 262 2.7 265 2.2 265 8 280 3 .1 283 2.6 283 9 370 4.3 344 3.3 343
10 336 4.3 340 3.7 340 11 350 4.6 355 4.0 354 12 336 4.4 340 3.7 340 13 329 4.2 333 3.6 333 14 336 4.3 340 3.7 340 15 325 4.1 330 3.6 329 16 309 3.7 311 3 .1 311 17 311 3.7 315 3 .1 315 18 294 3.4 297 2.9 295 19 287 3.2 290 2.6 290 20 297 3.4 301 2.9 300 21 305 3.6 308 3.0 303 22 301 3.6 305 3.0 304 23 293 3.5 301 2.9 300 24 238 2.4 240 1.8 240
$6,811.1 $6,797.2
'\
MW Losses
14
13
12
11
10
9
8
7
6
5
4
3
2
1 -
1 2 3
102
No voltage control
Figure 8-3 - MW Losses - 30 Bus IEEE Modified
103
Table 8-2 shows the production costs savings for each hour of the
day for various execution periodicities of voltage control. For
example, if voltage control is executed every hour, the cost savings
would be $114 per day. As the periodicity is decreased the daily cost
savings decreased. The average production cost is at a maximum of $7
per megawatt and, in today's power systems, $30 per megawatt is more
typical so that cost savings is on the order of $500 per day.
Reduction of the periodicity did not impact the performance
significantly. This is an important consideration in implementation
since the dispatcher would not have to continually reset controls
every hour. Also, since the program can be executed on a slower
periodicity, the overall computational requirement is significantly
reduced.
Tables 8-3 through 8-6 shows the hourly production cost savings
for various units and test networks. All networks showed significant
cost savings and showed little sensi ti vi ty to voltage limits. As
voltage limits were increased, the production costs were not
dramatically reduced. Table 8-7 shows the effect of restricting
reactive power generation limits. As can be seen, the limits do not
impact the production cost dramatically.
In conclusion, the voltage control algorithm performed well on the
networks tested. As part of an energy control center, it was shown to
reduce production costs with a significant daily cost savings. The
research demonstrated how the accuracy of limits affected performance
104
TABLE 8-2 PRODUCTION COST SAVINGS
30 BUS IEEE VH = 1.1
HOUR DEMAND BASE COST 3 HOURS 6 HOURS 24 HOURS
1 210 1005 1 1 1 2 199 955 1 1 1 3 192 923 2 2 2 4 192 923 2 2 2 5 199 955 2 1 1 6 203 972 2 2 2 7 262 1268 5 5 2 8 280 1360 3 3 2 9 339 1695 3 3 0
10 336 1675 6 4 0 11 350 1752 5 4 0 12 336 1675 4 4 0 13 329 1634 6 6 0 14 336 1675 4 5 0 15 325 1614 5 5 0 16 308 1514 5 5 0 17 311 1533 5 5 0 18 294 1436 4 4 0 19 287 1397 5 5 0 20 297 1454 3 3 0 21 304 1494 4 4 0 22 301 1474 5 4 0 23 297 1455 4 4 0 24 297 1143 0 0 0 -
$85 $82 $13
105
TABLE 8-3
5 BUS STAGG PRODUCTION COST SAVINGS($)
HOURLY DOLLAR SAVINGS HOUR DEMAND BASE COST VHIGH 1.1 VHIGH 1.09 VHIGH 1.08
------ --------- --------- ---------- ----------1 120 610 1 1 1 2 114 580 2 2 2 3 110 560 2 2 1 4 110 560 2 2 1 5 114 580 2 2 2 6 116 769 3 2 2 7 150 769 3 3 2 8 160 825 3 3 2 9 194 1025 6 5 4
10 192 1013 5 5 4 11 200 1062 6 6 5 12 192 1013 5 5 4 13 188 989 5 5 4 14 192 1013 5 5 4 15 186 977 6 5 4 16 176 918 5 5 4 17 178 929 5 4 4 18 168 872 5 5 5 19 164 849 5 4 4 20 170 883 5 5 4 21 174 906 5 5 4 22 172 895 5 5 4 23 170 883 5 5 4 24 136 695 4 4 3
--- --- ---DAILY TOTAL 99 94 79
106
TABLE 8-4
14 BUS IEEE PRODUCTION COST SAVINGS($)
HOURLY DOLLAR SAVINGS HOUR DEMAND BASE COST VHIGH 1.1 VHIGH 1.09 VHIGH 1.08
------ --------- --------- ---------- ----------1 180 880 0 0 0 2 171 838 0 0 0 3 165 811 1 1 1 4 165 811 1 1 1 5 171 838 1 1 1 6 174 852 1 1 1 7 225 1100 2 2 2 8 240 1176 1 1 1 9 291 1453 4 4 2
10 288 1436 4 4 3 11 300 1504 5 4 3 12 288 1436 5 4 3 13 282 1403 5 4 3 14 288 1436 5 4 3 15 279 1386 5 3 3 16 264 1305 5 4 4 17 267 1320 3 2 2 18 252 1240 4 3 2 19 246 1208 3 3 2 20 255 1256 4 3 3 21 261 1288 4 2 2 22 258 1272 4 3 2 23 255 1256 3 3 2 24 204 995 2 2 1
--- --- ---DAILY TOTAL 72 59 47
107
TABLE 8-5
30 BUS IEEE MOD. PRODUCTION COST SAVINGS($)
HOURLY DOLLAR SAVINGS HOUR DEMAND BASE COST VHIGH 1.1 VHIGH 1.09 VHIGH 1.08
------ --------- --------- ---------- ----------1 210 1005 1 1 1 2 199 955 2 2 2 3 192 923 3 2 2 4 192 923 2 2 2 5 199 955 2 2 2 6 203 972 2 2 2 7 262 1268 5 5 4 8 280 1360 5 4 3 9 339 1695 7 6 5
10 336 1675 6 4 3 11 350 1752 9 5 6 12 336 1675 8 6 5 13 329 1634 7 6 6 14 336 1675 8 6 5 15 325 1614 7 6 5 16 308 1514 6 6 5 17 311 1533 4 4 3 18 294 1436 6 5 4 19 287 1397 5 5 3 20 297 1454 4 3 3 21 304 1494 6 5 4 22 301 1474 5 4 4 23 297 1143 5 4 4 24 297 1455 5 4 4 --- --- --
DAILY TOTAL 114 98 84
108
TABLE 8-6
57 BUS IEEE MOD. PRODUCTION COST SAVINGS($)
HOURLY DOLLAR SAVINGS HOUR DEMAND BASE COST VHIGH 1.1 VHIGH 1.09 VHIGH 1.08
------ --------- --------- ---------- ----------1 840 4848 7 4 4 2 798 4512 5 3 3 3 770 4295 4 2 2 4 770 4295 3 0 0 5 799 4511 5 3 4 6 812 4621 2 4 4 7 1050 6713 11 9 7 8 1120 7407 14 11 8 9 1358 10045 25 19 14
10 1349 9879 27 35 15 11 1400 10556 39 41 37 12 1344 9879 35 35 32 13 1316 9551 34 36 30 14 1344 9878 40 36 30 15 1302 9386 39 32 28 16 1232 8596 31 29 24 17 1246 8748 21 26 21 18 1176 7990 27 25 21 19 1148 7696 25 23 19 20 1190 8139 28 26 22 21 1218 8438 26 24 19 22 1204 8289 28 26 21 23 1190 8139 28 26 22 24 952 5802 16 13 11
--- --DAILY TOTAL 525 488 398
109
TABLE 8-7
30 BUS IEEE MOD. PRODUCTION COST SAVINGS($) VHIGH = 1.1
HOURLY DOLLAR SAVINGS HOUR DEMAND BASE COST DERATE .9 DERATE 8 DERATE .7
------ --------- --------- ---------- ----------1 210 1005 2 2 2 2 199 955 2 2 2 3 192 923 3 2 3 4 192 923I 2 3 3 5 199 955 2 2 3 6 203 972 2 3 3 7 262 1268 5 5 4 8 280 1360 5 5 6 9 339 1695 7 5 2
10 336 1675 7 4 3 11 350 1752 1 -2 -6 12 336 1675 7 6 3 13 329 1634 7 6 4 14 336 1675 7 5 3 15 325 1614 7 6 5 16 308 1514 6 7 5 17 311 1533 4 5 5 18 294 1436 5 5 5 19 287 1397 4 4 5 20 297 1454 4 4 4 21 304 1494 5 5 5 22 301 1474 5 4 4 23 297 1143 5 4 4 24 297 1455 5 5 .2.
DAILY TOTAL $106 $104 $83
110
for several test networks with the result that the algorithm was not
overly sensitive to these limits. Further, it was shown that with
regard to the periodicity of control actions, the dispatcher may have
a great deal of flexibility and that an hour-to-hour adjustment of
controls may not be needed to obtain good performance.
8.5 Dispatcher ~an-Machine Interface
It does not matter how well a power control algorithm has been
developed, designed, and programmed if in the final step to
implementation it is not accepted and used by the "computer system
users". The algorithm could then be judged unsuccessful in an
engineering sense.
control industry.
This is particularly true in the power system
Many algorithms are intended to be used by
dispatchers but fall short of the mark because the algorithm has not
been properly presented to the dispatchers. Presentation in an energy
control environment is defined as the CRT Man-Machine Interface (MMI)
that the dispatcher uses to interface to the computer system. After
all theoretical development is done and all programming tasks are
complete, the result is a series of CRT displays available to the
dispatcher. In this section, a proposed MMI is presented that will
accommodate ease of use for the dispatcher, as well as the features
necessary to take advantage of the voltage control algorithm.
Table 8-8 is an execution control display that the dispatcher
would use to execute voltage control routines and to retrieve data or
print results. Fach major block on the display represents an
111
TABLE 8-8
STUDY VOLTAGE CONTROL
EXECUTION CONTROL
[] INITIALIIZE SAVE CASE
[] INITIALIZE EXECUTION DATA
ENABLE VAR CONTROL > <
ENABLE TAP CONTROL > <
[] INITIALIZE VOLTAGE CONTROLS
[] INITIALIZE PLANNED EQUIPMENT OUTAGES
[] RESCHEDULE BUS LOADS
[] RESCHEDULE MEGAWATT GENERATION
[] EXECUTE VOLTAGE CONTROL
[] REQUEST OUTPUT SUMMARY
SAVE CASE NO > <
REAL TIME > <
112
execution request that is needed by the dispatcher to interface with
voltage control. The save case option shown on the display allows the
dispatcher to request an initial starting point of execution from a
previous study case or from the current state estimated (real-time)
solution that is available. The dispatcher would enter either a save
case number in the save case area or a designator "R" in the real-time
data entry field representing initialization to real-time conditions.
After these entries are made, the dispatcher may then request the
voltage control data files be initialized by placing the CRT cursor in
the initialize save case box. Once the algorithm's data input files
are initialized, the dispatcher must specify which options the program
executes. He can request unit megavars as controls or transformer
taps as controls or both by entering designators in the data entry
fields shown.
A next step in an execution sequence is the initialization of
voltage control data. This step is needed because the dispatcher may
desire to set selected voltage controls at certain values, to modify
limits, or to modify the status of selected voltage controls. These
data items are entered on other displays (Tables 8-9 to 8-11) and are
initialized into the voltage control files once the operator has
selected initial voltage controls.
If the dispatcher performs a network study of conditions in the
future, he may want to consider the impact of planned transmission
outages. In this case it is important to initialize planned outages
113
TABLE 8-9
VOLTAGE CONTROL DISPLAY MENU
[] CONTROL SUMMARY [ J EVENT SUMMARY
[ J POWER PLANT SUMMARY [] POWER PLANT LIMITS
[] VOLTAGE CONTROL [ J TRANSFORMER CONTROL
[ J CAPACITOR CONTROL [ J CONSTRAINT SUMMARY
[ J - VIOLATION SUMMARY [ J VOLTAGE LIMIT DISPLAY
[ J STATION MENU [ J PLANNED OUTAGE LIST
114
TABLE 8-10
VOLTAGE MONITOR
CONTROL SUMMARY
ACTUAL MEGA VAR MEGA VAR GENERATION GEN. & TRANS.
----------- ------------- -------LOSSES xxxxx xxxxxx xxxxxx
COST PER HOUR xxxxx xxxxxx xxxxxx SAVINGS PER HOUR xxxxx xxxxxx xxxxxx ACC. SAVINGS xxxxx xxxxxx xxxxxx NO. CONTROLS xxxxx xxxxxx xxxxxx NO. CONTROL xxxxx xxxxxx xxxxxx
OFF COST
TABLE 8-11
VOLTAGE MONITOR
EVENT SUMMARY
STATION EQUIPMENT EVENT TIME OF EVENT ----------- ----------- --------------------- -------------xxxxxxxxx xxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxx xxxxxxxxx xxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxx xxxxxxxxx xxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxx xxxxxxxxx xxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxx xxxxxxxxx xxxxxxx xxxxxxxxxxxxxxxxxxx xxxxxxx
115
so that the status of equipment will be considered in the voltage
control study.
The next two execution options shown are the rescheduling of bus
loads and rescheduling of megawatt generation. For future studies,
the dispatcher may specify specific loading conditions or enter
specify bus loads in areas of the network. The reschedule bus loads
option is designed to allow the dispatcher to initialize this bus load
data into the voltage control data input. Megawatt generation must be
rescheduled to meet various loading schedules if load changes have
been requested. This execution option is then included to schedule
generator megawatts an economic dispatch function to meet current
loading conditions.
Finally, after all the above steps are complete, the voltage
control program may be executed to determine new reactive power and
voltage controls. The man-machine interface should be designed to
allow the dispatcher the option of executing any one of the steps.
For example, if a save case is requested, the operator may modify the
save case by disconnecting a transmission line and then requesting
execution of the voltage control program directly.
The option to have printed output is another request the
dispatcher can make using the execution control display. ·
Table 8-8 is a display menu the dispatcher may use to select a
particular display for review. By placing a CRT cursor on a block
corresponding to a display, the dispatcher may review the display.
116
Table 8-9 and 8-10 are displays the dispatcher should continually
monitor. The control summary, Table 8-8, shows three columns of data
- actual, optimum if reactive power controls are used only, and
optimum if megavars and transformers were used. Each data i tern is
required so that the dispatcher may judge how well he is following the
optimal voltage controls. Further, for this display the dispatcher
may judge whether control is needed and when it is important to
utilize transformer controls.
The event summary, Table 8-11, shows for each equipment
significant events such as overloads or tripping of equipment
off-line. This summary allows the dispatcher to review past events or
current events in the power system that may require action on his
part. Tables 8-12 and 8-13 are power plant CRT displays that are
needed both by the dispatcher and power plant operator. The
dispatcher must rely on the power plant operator to implement controls
and, therefore, displays at both the power plant and control center
are needed. The power plant operator is responsible for maintaining
reactive power limits at the power plants. The power plant summary
shows the actual and desired unit setting and how much lost savings
results if the unit is operator off-cost. With this display,
performance of the power plant operator and dispatcher may be reviewed
at any time. The power plant limit display is necessary so that the
power plant operator may alter reactive power limits. Any time these
limits are changed, the power plant operator should enter a reason
code so that all changes are documented by a reason.
117
TABLE 8-12
VOLTAGE MONITOR
POWER PLANT SUMMARY
TYPE STATION EQUIPMENT CONTROL DESIRED ACTUAL OFF COST
--------- ----------- ------- ------- ------ --------xxxxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxx xxxxxxx xxxxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxx xxxxxxx xxxxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxx xxxxxxx xxxxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxx xxxxxxx xxxxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxx xxxxxxx
TABLE 8-13
VOLTAGE MONITOR
POWER PLANT LIMITS
NORMAL ACTUAL REASON STATION EQUIPMENT HI LOW HI LOW CODE
----------- ----------- ---- ----- ------xxxxxxxxx xxxxxxx xxxxx xxxxx >XXXX<>XXXXX< >XXXXX<
xxxxxxxxx xxxxxxx xxxxx xxxxx >XXXX<>XXXXX< >XXXXX<
xxxxxxxxx xxxxxxx xxxxx xxxxx >XXXX<>XXXXX< >XXXXX<
xxxxxxxxx xxxxxxx xxxxx xxxxx >XXXX<>XXXXX< >XXXXX<
xxxxxxxxx xxxxxxx xxxxx xxxxx >XXXX<>XXXXX< >XXXXX<
118
The voltage control and voltage monitor displays (Tables 8-14 and
8-15) allow the dispatcher to override the voltage control outputs.
For example, a voltage control may be fixed rather than changed by the
algorithm. Further, the operator may disable a particular voltage
from being considered as a control by specifying a status on the
control display. For transformers control can be implemented as a
high, low, or normal voltage setting, as shown in Table 8-15. Both
displays show the actual voltage, the desired voltage, and the
deviation from desired. Al so, the sensitivity of the particular
voltage to the objective function is shown so that the importance of
the particular voltage control is known.
Table 8-16 shows the capacitor control display. For capacitors it
is expected that the status would be entered on this display and would
be taken as fixed by the voltage control program. The dispatcher
could use the voltage sensitivity shown to determine if it is
desireable to switch the capacitor on.
The constraint summary, Table 8-17, is another display the
dispatcher would monitor quite frequently. It is an indicator of how
secure the power system is in terms of voltage controls. The
violation summary, Table 8-18, gives more detail on the constraints
that are violated. For each constraint the amount of overload and
severity of overload are shown.
119
TABLE 8-14
VOLTAGE MONITOR
VOLTAGE CONTROL DISPLAY
VOLT VOLT VOLT VOLT STATION EQUIPMENT STATUS DESIRED ACTUAL DEVIA. SENS.
--------- ----------- ------ ------- ------ ------xxxxxxxxx xxxxxxxxx >XXXXX< >XXXXXX< xxxxx xxxxx xxxxx xxxxxxxxx xxxxxxxxx >XXXXX< >XXXXXX< xxxxx xxxxx xxxxx xxxxxxxxx xxxxxxxxx >XXXXX< >XXXXXX< xxxxx xxxxx xxxxx xxxxxxxxx xxxxxxxxx >XXXXX< >XXXXXX< xxxxx xxxxx xxxxx
TABLE 8-15
VOLTAGE MONITOR
TRANSFORMER CONTROL DISPLAY
( HI , LO , NO ) VOLT VOLT STATION EQUIPMENT STATUS DESIRED ACTUAL DEVIA. SENS.
---------- ----------- ------ ------- ------ ------xxxxxxxxx xxxxxxxxx >XXXXX< >XXXXXX< xxxxx xxxxx xxxxx xxxxxxxxx xxxxxxxxx >XXXXX< >XXXXXX< xxxxx xxxxx xxxxx xxxxxxxxx xxxxxxxxx >XXXXX< >XXXXXX< xxxxx xxxxx xxxxx xxxxxxxxx xxxxxxxxx >XXXXX< >XXXXXX< xxxxx xxxxx xxxxx
120
TABLE 8-16
VOLTAGE MONITOR
CAPACITOR CONTROL DISPLAY
VOLT MVAR VOLT STATION EQUIPMENT STATUS ACTUAL ACTUAL SENS.
----------- ----------- ------ ------- ------ ------. xxxxxxxxx xxxxxxxxx >XXXXX< xxxxxx xxxxx xxxxx xxxxxxxxx xxxxxxxxx >XXXXX< xxxxxx xxxxx xxxxx xxxxxxxxx xxxxxxxxx >XXXXX< xxxxxx xxxxx xxxxx xxxxxxxxx xxxxxxxxx >XXXXX< xxxxxx xxxxx xxxxx xxxxxxxxx xxxxxxxxx >XXXXX< xxxxxx xxxxx xxxxx
TABLE 8-17
VOLTAGE MONITOR
CONSTRAINT SUMMARY
MEGA VAR BUS VOLT MVA LIMITS LIMITS FLOWS
----------- ------------- -------NO OF VIOL. xxxxx xxxxxx xxxxxx SYSTEM SEVERITY xxxxx xxxxxx xxxxxx MOST SEVERE xxxxx xxxxxx xxxxxx NO. CONTROLS xxxxx xxxxxx xxxxxx
REQUIRED
121
TABLE 8-18
VOLTAGE MONITOR
VIOLATION SUMMARY
LIMIT STATION EQUIPMENT TYPE ACTUAL HI LOW DEVIA SEVR.
--------- ----------- ------ ------- -------- ------xxxxxxxxx xxxxxxxxx xxxxx xxxxx xxxxx xx xx xxxxx xxxxx xxxxxxxxx xxxxxxxxx xxxxx xxxxx xxxxx xxxx xxxxx xxxxx xxxxxxxxx xxxxxxxxx xxxxx xxxxx xxxxx xx xx xxxxx xxxxx xxxxxxxxx xxxxxxxxx xxxxx xxxxx xxxxx xxxx xxxxx xxxxx xxxxxxxxx xxxxxxxxx xxxxx xxxxx xxxxx xx xx xxxxx xxxxx
..
122
Finally, Table 8-19, is a display that the dispatcher may enter
voltage high and low limits. For some power systems, it may be
important that individual high and low limits be maintained for each
bus in the system.
8.6 Extension To Other Controls
Other controls that are available to the operator are the control
of capacitor banks, transformers, and phase shifters. Control of
capacitor banks is done via supervisory control switching request and
transformer control is done either using a voltage setpoint or a
raise/lower through a supervisory control request.
Aldrich [ 1 J has shown transformer control to be effective in
reducing losses but not as effective as generator reactive power.
Discussions with dispatchers indicate that transformer taps are used
to reschedule voltages and to reduce MVA flows near the transformer,
but large scale control of taps to minimize losses may be impractical
to implement.
In the algorithm developed, transformer controls may be readily
implemented if needed. Hobson [32] showed how transformer tap
position changes may be expressed as a function of the changes in
reactive injections at the adjoining busses. It is expected that such
an incremental linearization model could be used within the algorithm
developed. Both voltage setpoints or taps for transformer control may
be used as control variables.
123
TABLE 8-19
VOLTAGE LIMIT DISPLAY
LIMIT STATION EQUIPMENT ACTUAL HI LOW
--------- ----------- ------ ------------xxxxx xxxxx xxxx >XXX< >XXX<
xxxxx xxxxx xx xx >XXX< >XXX<
xxxxx xxxxx xxxx >XXX< >XXX<
xxxxx xxxxx xx xx >XXX< >XXX<
xxxxx xxxxx xx xx >XXX< >XXX<
124
It is recommended that capacitors be prescheduled and considered
fixed during the optimization. Preselection may be done by reviewing
the bus voltage sensitivity to losses at busses where capacitors are
located. Since it is not desirable to switch capacitors often from an
equipment maintenance point of view, it is not recommended to switch
capacitors automatically as part of an on-line voltage control
algorithm. Only in the case where overloads could not be relieved,
either by generation or transformer taps, would capacitors be switched
and in this case, a priority list for switching could be constructed
based on loss sensitivities.
The algorithm presented in this paper is not envisioned to be
extended to model MVA line flow constraints. In most Energy Control
Centers, MVA line overloads are controlled by real power generation;
thus, a security dispatch of real power is more appropriate than
reactive power in relieving MVA overloads. A notable exception to
this case in PJM. There, the objective is to unload transmission line
by rescheduling reactive power and models incorporating MVA flow
constraints are needed. The algorithm developed here could be
extended to model those MVA constraints, as suggested by Hobson [32].
The algorithm developed could be extended to incorporate phase
shifters as controls. Since both the real and reactive power balance
equations have been incorporated as constraints, .and phase angles hav.e.
been included as variables, the necessary coupling between real and
megavar variables can be accommodated. In the feasibility step,
125
however, real and reactive power equations must be included. It is
envisioned that the phase shifter angles be entered as nonbasis
variable and be retained as such throughout the solution.
A major extension of the algorithm is the modelling of inequality
constraints in the GRG step. The algorithm developed can accommodate
simple upper and lower bounds on variables but is not yet designed for
general inequality constraints. The algorithm developed will model
inequality constraints in the LP step but not during optimization. To
extend the GRG step to model inequalities, it is suggested that slack
variables are incorporated as in LP. The slack variables transform
the inequalities to equality constraints. As in LP, the GRG step must
determine the maximum change in nonbasis variables so inequality
constraints are not violated.
CHAPTER 9
CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK
An efficient and effective algorithm for voltage control has been
developed and tested. Further, important considerations have been
addressed for the practical implementation of voltage control in an
Energy Control Center.
As part of the research, it was decided to use approximate
programming techniques to perform the optimization. Both quadratic
programming and a combination of LP and GRG techniques were evaluated
and the latter technique was chosen for development. Modification of
the basic LP and GRG techniques were developed to accommodate the
voltage optimization problem. Also, linearization of the nonlinear
equality power flow constraints were implemented so that the most
efficient implementaion of the LP and GRG steps were achieved. The
composite algorithm was shown to be efficient in execution and
resulted in significant reduction in system overloads. The composite
algorithm should prove to be a considerable improvement over
algorithms that are available to implement in today's Energy Control
Center.
The major suggestions for future research algorithmic development
are twofold. First, a more detailed comparison with quadratic
programming should be done. This comparison should include testing of
both algorithms on a number of networks, as was don·e in this research.
126
127
Second, the composite algorithm should be extended to model other
controls and to model functional inequality constraints.
Finally, research in voltage control should also be performed.
Investigation of the interaction of the penalty factors and voltage
control in an Energy Control Center should be performed. This could
be done by simply extending the voltage control algorithm to also
model megawatt controls. An appropriate objective function would be
to minimize fuel costs. Using this approach, the fuel cost reductions
could be studied using various combinations of controls.
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128
129
13. A. M. Sasson, F. Viloria, F. Aboytes, "Optimal Load Flow Solution Using the Hessian Matrix", IEEE Trans. PAS, Vol. PAS-92, pp. 31-41, 1973.
14. W. R. Barcelo, W. W. Lennon and H. R. Koen, "Optimization of the Real-Time Dispatch with Constraints for Secure Operation of Bulk Power Systems," IEEE Trans. PAS, May/ June, 1977, pp. 741-757·
15. D. G. Luenberger, Introduction to Linear and Non-Linear Programming, Addison-Wesley Publishing Co., 1965.
16. J. L. Bala and A. Thanckachalam, "An Improved Second Order Method for Optimal Load Flow" , Submitted at the IEEE PES Winter Meeting, New York, NY, January, 1978, F78 003-6.
17. R. L. Sullivan, "Controlling Generator MVAR Loading - Using a Static Optimization Technique", IEEE Trans. PAS, May/June, 1972, pp. 906-910.
18. P. A. Chamerel and.A. J. Germond, "An Efficient Constrained Power Flow Technique Based on Active-Reactive Decoupling and Use of Linear Programming", PICA 1981, Philadelphia, PA.
19. A. N. Sasson, et. al., "Combined Use of the Powell and Fletcher-Powell Nonlinear Programming Methods for Optimal Load Flows," IEEE Trans. PAS, Vol. PAS-88, pp. 1530-1537, Oct. 1969.
20. A. N. Sasson, "Nonlinear Programming Solutions for Load-Flow, Minimum-Loss, and Economic Dispatching Problems," IEEE Trans. PAS, Vol. 88, pp. 399-409, Apr, 1969.
21. G. F. Reid and L. Hasdorff, "Economic Dispatch Using Quadratic Programming," IEEE Trans. PAS, Vol. PAS-92, pp. 2015-2023, Nov. /Dec. , 1973.
22. H. Nicholson and M. J. H. Sterling, "Optimum Dispatch of Active and Reactive Generation by Quadratic Programming," IEEE Trans. PAS, Vol. PAS-92, pp. 644-654, Mar/Apr, 1973.
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24. G. Dayal, L. L Grigsby and L. Hasdorff, "Quadratic Programming for Optimal Active and Reactive Power Dispatch Using Special Techniques to Reduce Storage Requirements", A 76 388-9, IEEE PES Summer Meeting, Portland, July 1976.
130
25. Auriel Mordecai, Nonlinear Programming: Analyses and Methods, Prentice-Hall, 1976.
26. O. Alsac and B. Stott, "Optimal Load Flow With Steady-State Security," presented at IEEE PES Summer Meet, Vancouver, B. C., Canada, 1973, Paper T73484-3· Also in IEEE Trans. Power App. Syst., Vol. PAS-93, pp. 745-751, May/June, 1974.
27. 1972-1976 Annual Research Reports from the Electrical Engineering Department - Virginia Polytechnic Institute and State University - Energy Research Group.
28. C. H. Jolissaint, N. V. Arvanitidis, and D. G. Luenberger, "Decomposition of Real and Reactive Power Flows: A Method Suited for On-Line Applications," IEEE Trans. PAS, Vol PAS-91, PP• 661-670, Mar/Apr, 1972.
29. C. M. Shen and M. A. Laughton, "Power System Load Scheduling With Security Constraints Using Dual Linear Programming," Pree. Inst. Elec. Eng., Vol. 117, no. 11, pp. 2117-2127, Nov. 1970.
30. E. Hobson, "Electric Power System Security Control Calculations Using Linear Programming," Ph.D. thesis, University of Waterloo, Ontario, 1979.
31. M. A. El-Shibini and M. B. Dayeh, "Reactive Power Optimization Using Modified Linear Programming Approach," C 75 024-5, IEEE PES Winter Meeting, 1975.
32. E. Hobson, Linear 1979.
"Network Constrained Reactive Power Control Using Programming," IEEE PES Winter Meeting, New York,
33. M. A. Pai and S. R. Paranjothi, "Optimal Power Flow with Security Constraints Using Successive Linear Programming," A 75 455-6, IEEE PES Summer Meeting, San Francisco, 1975.
34. M. Innorta, P. Marannino and M. Mocenigo, "Active and Reactive Power Scheduling with Security and Voltage Constraints," Proc. 5th PSCC, Cambridge, U.K., Sept. 1975·
35. B. Stott and J. L. Marinho, "Linear Programming for Power-System Network Security Applications," IEEE Trans. PAS, Vol. PAS-97, pp. 837-848, May/JUI'le 1979.
36. B. Stott, J. L. Marinho, and O. Alsac, "Review of Linear Programming Applied to Power System Rescheduling," presented at the 11th PICA Conference, Cleveland, Ohio, May 1979.
131
37. R. Lugtu, "Security Constrained Dispatch," IEEE PES Summer Meeting, Los Angeles, CA, July 16-21, 1978.
38. J. A. Bubenko and D. V. Sjelvgren, "Decomposition Technique in a Security Related Optimal Power Flow", Proc. 5th PSCC, Cambridge, U.K., Sept. 1975.
39. M. C. Biggs and M.A. Laughton, "Optimal Electric Power Scheduling: A Large Nonlinear Programming Test Problem Solved by Recursive Quadratic Programming", Mathemetical Programming 13 (1977), pp. 167-182.
40. S. N. Talukdar and T. C. -Giras, "A Fast and Robust Variable Metric Method for Optimum Power Flows," PICA, 1981, Philadelphia, PA.
41. David Himmelblau, Applied Nonlinear Programming, McGraw-Hill Book Company, 1972.
42. J. Peschon, H. W. Dommel, W. Powell, "Optimum Power Flow for Systems Controls," IEEE Trans. PAS, Vol. May/June 1972.
and D. W. Bree, Jr., with Area Interchange PAS-91 , pp. 898-905,
43. D. E. Gill, W. Murray, and M. W. Wright, Practical Optimization, Academic Press, 1981.
44. P. Wolfe, "Methods of Nonlinear Programming" in Nonlinear Programming, J. A. Badie (Ed.) , North-Holland Publishing Co., Amsterdam, 1967.
45. A. Land and S. Powell, Fortran Codes for Mathematical Quadratic and Discrete, (John Wiley and
46. Beale, E. M., "On Quadratic Programming," Naval Research Logistics Quarterly, Vol. 6, pp. 227-243, 1959.
47. P. Wolfe, "Methods of Nonlinear Programming", pp 76-77 in R. L. Graves and P. Wolfe (eds), Recent Advances in Mathematical Programming, McGraw-Hill Company, New York, 1963.
48. B. Stott and O. Alsac, "Fast Decoupled Load Flow," IEEE PES/Summer Meeting and EHV/UHV Conference, Vancouver, B.C., Canada, July 15-20, 1973, Paper T 74 463-7.
49. J. R. Neuenswander, Modern Power Systems, International Textbook Company, 1977.
132
50. J. K. Reid, "Fortran Subroutines for Handling Sparse Linear Programming Bases," A.E.R.E. - R.8269, Computer Science and Systems Division, Harwell, England.
51. G. W. Stagg and A. H. El-Abaid, Computer Methods in Power System Analysis, McGraw Hill, 1968.
52. W. D. Stevenson, Elements of Power System Analysis, McGraw Hill, 1975.
53. O. L. Elgerd, Electric Energy Systems Theory: An Introduction, McGraw Hill, 1972.
54. K.R.C. Mamandur, "Emergency Adjustments to VAR Control Variables to Alleviate Overvoltages, Undervoltages, and Generator VAR Limit Volations," presented at IEEE PES Summer Meeting, Portland, Oregon, July, 1981, 81 SM447-2.
55. A. N. Sasson et al., "A Comparison of Power Systems Static Optimization Techniques," in Proc. 7th PICA Conf. (Boston, Mass., 1971), pp. 329-337.
56. P. B. Henser and B. J. Cory, "Solution of the Minimum Loss and Economic Dispatch Problems Including Real and Imaginary Transformer Tap Ratio," in Proc. 7th PICA Conf. (Boston, Mass., 1971), pp. 228-233.
57. R. Billinton and S. S. Sachdeva, "Real and Reactive Power Optimization by Suboptimum Techniques," IEEE Trans. PAS, Vol. PAS-92, pp. 950-956, May/June, 1973.
58. H. H. Happ, "Optimal Power Dispatch," presented at IEEE PES Summer Meet. , Vancouver, B. C. , Canada, 1973. Paper T73460-3. Also in IEEE Trans. PAS, Vol. PAS-93, pp. 820-829, May/June, 1974.
59. R. B. Gungor, N. F. Tsang, and B. Webb, Jr., "A Technique for Optimizing Real and Reactive Power Schedules," IEEE Trans. PAS, Vol. PAS-90, pp. 1781-1790, July/Aug., 1971.
60. R. L. Sullivan, "System Parameter Dispatching for Generator MVAR Control," presented at IEEE PES Summer Meet., Vancouver, B. C., Canada, 1973. Paper C73459-5.
61. "Controlling Generator MVAR Loading Using a Static Optimization Technique," IEEE Trans. PAS, Vol. PAS-91, pp. 906-910 May/ June, 1972.
133
62. D. W. Wells, "Method for the Secure Loading of a Power System," Proc. Inst. Elec. Eng. Vol. 115, no. 8, pp. 1190-1194, Aug, 1968.
63. L. S. Lasdon, Optimization Theory for Large Systems, London: The Macmillan Co., 1970.
64. H. H. Happ, "Optimal Power Dispatch - A Comprehensive Survey", IEEE Trans. PAS, Vol. PAS-96, No. 3, May/June, 1977.
65. R. R. Shaul ts, "A Simplified Economic Dispatch Algorithm Using Decoupled Network Models', IEEE PES Summer Meeting, Mexico City, Mexico, July 17-22, 1977, Paper A 77 738-8.
66. J. Peschon, W. F. Tinney, D.S. Piercy, O. J. Tveit, and M. Cuenod, "Optimum Control of Reactive Power Flow," IEEE Trans. PAS, Vol. PAS-87 (October, 1968).
67. R. B. Gaugot, N. F. Tsang, B. Webb, Jr., "A Technique for Optimizing Real and Reactive Power Schedules", IEEE Trans. PAS, Vol. PAS-90, July/August, 1971, pp. 1784-1790.
68. T. W. Kay, R. W. Squer, R. P. Shultz, and R. A. Smith, "EHV and UHV Loadabili ty Dependence on VAR Supply Capability", to be presented at the IEEE Winter Power Meeting, 1982.
69. Hadley, G. (1962) Linear Programming, Addison-Wesley, Massachu-setts
70. F. D. Galiana and M. Banaker, "Approximation Formulae for Dependent Load Flow Variables" , IEEE Winter Power Meeting, New York, NY, February, 1980. F80 200-6.
71. H. H. Happ, "Optimal Power Dispatch", Systems Engineering for Power: Status and Prospects, U.S. Dept. of Commerce Publication (NTIS) Conf., 750867.
72. Leon Cooper and David Steinberg, Methods and Applications of Linear Programming, W. B. Saunders Company, 1974.
73. J. H. Wilkinson and C. Reinsch, Handbook for Automatic Computation Volume II, Springer-Verlag, 1971.
74. P. Wolfe, Notices Am. Math Soc. 9(4):308 (1962)
134
75. S. A. Arafeh, R. E. Kilmer, and J. H. Rumbaugh, "Closed-Loop Computer Control of a System of Radial Load Busses, Using Transformers, Capacitors and Reactors", IEEE Trans. PAS , Nov/Dec, 1977, PP• 1731-1740.
76. T. S. Dillon, "Rescheduling, Constrained Participation Factors and Parameter Sensitivities, in the Optimal Power Flow Problem", IEEE Trans. PAS, May 1981, pp. 2628-2634·
77. R. Shaul ts and D. T. Sun, "Optimal Power Fl ow Based Upon P-Q Decomposition", PICA 1982, Philadelphia, PA.
78. R. C. Burchett, et al., "Developments in Optimal Power Flow," PICA, 1981, Philadelphia, PA.
79. F. W. Wu, et al., "A Two-Stage Approach to Solving Large-Scale Optimal Power Flows", PICA, 1979.
80. F. I. Denney, "An Updated List of Current Operational Problems," IEEE Winter Power Meeting paper F77 110-0, Feb., 1977.
APPENDIX 1
FIVE BUS STAGG RESULTS
135,
NO
1 1 2 1 3 2 4 2 5 2 6 3 7 4
136
-------------- BEGIN OPTIMAL POWER FLOW --------------
-----5 BUS STAGG NETWORK----
MAX OPF ITERATIONS 6 MAX PF ITERATIONS 10 MVAR LIMITING CONTROL 1 MODERATE ITER FOR MVAR CNTL 2
HIGH VOLTAGE LIMIT 1.10 LOW VOLTAGE LIMIT .85
NUMBER OF BUSSES 5 NUMBER OF BRANCHES 7 NUMBER OF GENERATORS 2 NUMBER OF TRANSFORMERS 0
-----5 BUS STAGG INITIAL GENERATOR DATA -------
NO BUS NAME VOLTS MW QMAX QMIN
1 2
FROM BUS
NORTH NORTH SOUTH SOUTH SOUTH LAKE MAIN
1 NORTH 2 SOUTH
1.0600 1.0475
------5 BUS STAGG
o.o 500.0 -500.0 40.0 50.0 -40.0
BRANCH FLOW DATA ------
TO BUS TYPE RESISTANCE INDUCTANCE
2 SOUTH 0 .0200 .0600 3 LAKE 0 .0800 .2400 3 LAKE 0 .0600 .1800. 4 MAIN 0 .0600 .1800 5 ELM 0 .0400 .1200 4 MAIN 0 .0100 .0300 5 ELM 0 .0800 .2400
CHARG/TAP
.0600
.0500
.0400
.0400
.0300
.0200
.0500
137
5 BUS STAGG INITIAL POWER FLOW SOLUTION
----- GENERATOR SUMMARY -------
NO BUS NAME VOLTS MW MVAR QMAX QMIN TYPE
1 2
BUS
1 2 3 4 5
NO
1 2 3 4 5 6 7
1 NORTH 2 SOUTH
1 • 0600 129. 6 1.0475 40.0
-7.4 500.0 -500.0 3 29.9 50.0 -40.0 2
5 BUS STAGG INITIAL POWER FLOW SOLUTION
----- BUS SUMMARY -------
NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
NORTH 1.0600 o.oo o.o o.o 3 SOUTH 1.0475 -2.80 20.0 10.0 2 LAKE 1.0242 -5.00 45.0 15.0 0 MAIN 1.0236 -5.33 40.0 5.0 0 ELM 1 .0180 -6.15 60.0 1 o.o 0
5 BUS STAGG INITIAL POWER FLOW SOLUTION
------ BRANCH FLOW SUMMARY ------
FROM BUS TO BUS TYPE MW MVAR MVA
1 NORTH 2 SOUTH 0 88.6 -8.6 89.0 1 NORTH 3 LAKE 0 40.7 1 .1 40.8 2 SOUTH 3 LAKE 0 24.8 3.5 25.1 2 SOUTH 4 MAIN 0 28.1 2.9 28.2 2 SOUTH 5 ELM 0 54.9 7.3 55.4 3 LAKE 4 MAIN 0 18.8 -5.2 19.5 4 MAIN 5 ELM 0 6.3 -2.3 6.7
138
-------5 BUS STAGG FINAL POWER FLOW SOLUTION ----
----- GENERATOR SUMMARY -------
NO BUS NAME VOLTS MW MVAR QMAX QMIN TYPE
1 2
BUS
1 2 3 4 5
NO
1 2 3 4 5 6 7
1 NORTH 2 SOUTH
1.0790 129.4 1.0673 40.0
-9.0 500.0 -500.0 3 29.7 50.0 -40.0 2
------5 BUS STAGG FINAL POWER FLOW SOLUTION ----
----- BUS SUMMARY -------
NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
NORTH 1.0790 o.oo o.o o.o 3 SOUTH 1.0673 -2.70 20.0 1 o.o 2 LAKE 1.0446 -4.82 45.0 15.0 0 MAIN 1 .0441 -5 .14 40.0 5.0 0 ELM 1.0386 -5.92 60.0 1 o.o 0
-------5 BUS STAGG FINAL POWER FLOW SOLUTION ----
------ BRANCH FLOW SUMMARY ------
FROM BUS TO BUS TYPE MW MVAR MVA
1 NORTH 2 SOUTH 0 88.3 -9.6 88.8 1 NORTH 3 LAKE 0 40.7 .7 40.7 2 SOUTH 3 LAKE 0 24.8 3.3 25.0 2 SOUTH 4 MAIN 0 28.0 2.7 28.2 2 SOUTH 5 ELM 0 54.9 7.0 55.3 3 LAKE 4 MAIN 0 18.8 -5.3 19.5 4 MAIN 5 ELM 0 6.3 -2.4· 6.7
139
5 BUS STAGG CONVERGENCE SUMMARY
OPF BUS REAL REACT. REACT. ITER BUS TYPE VOLT ANG. DELXB DELXB DELXNB
------ -------- -------- --------- ---------1 1 0 1.024 -.087 .026 .027 .ooo 2 0 1 .018 -.107 .009 .032 .ooo 3 2 1.047 -.049 -1.888 .032 .003 4 0 1.024 -.093 .021 .028 .ooo 5 3 1.060 o.ooo 1.868 .ooo .088
2 1 0 1.026 -.085 .072 -.016 .ooo 2 0 1.019 -.105 .093 -.017 .000 3 2 1.048 -.046 2.207 -.029 .096 4 0 1.025 -.091 .078 -.016 .ooo 5 3 1 .067 .ooo -2.323 .000 .ooo
3 1 0 1.031 -.086 .023 .027 .ooo 2 0 1.025 -.106 .006 .032 .ooo 3 2 1.054 -.048 -1.988 .033 -.001 4 0 1 .031 -.092 .018 .029 .ooo 5 3 1.067 0.000 1.970 .ooo .088
4 1 0 1.033 -.084 .068 -.014 .ooo 2 0 1.026 -.103 .086 -.015 .000 3 2 1.054 . -.046 1.984 -.026 .089 4 0 1.032 -.090 .073 -.014 .ooo 5 3 1.073 0.000 -2.088 .ooo -.004
5 1 0 1.038 -.085 .020 .028 .ooo 2 0 1.032 -.105 .001 .033 .ooo 3 2 1.061 -.048 -2.136 .034 -.005 4 0 1.037 -.091 .015 .029 .ooo 5 3 1.073 0.000 2.120 .ooo .089
6 1 0 1.039 -.083 .063 -.011 .ooo 2 0 1.032 -.102 .079 -.012 .ooo 3 2 1 .060 -.045 1. 768 -.022 -.001 4 0 1.038 -.089 .067 -.012 .ooo 5 3 1.079 0.000 -1 • 861 .ooo -.000
140
5 BUS STAGG SOLUTION SUMMARY
ITER BUS VOLT MVAR ------ ------
1 1 1.0672 7.9 2 1.0477 14.4
2 1 1. 0666 -8.8 2 1.0543 30.3
3 1 1.0732 7.0 2 1.0542 14.7
4 1 1.0729 -9-4 2 1.0609 30.3
5 1 1.0790 6.2 2 1.0605 14.9
6 1 1.0790 -10.1 2 1.0673 30.5
APPENDIX 2
FIVE BUS STEVENSON RESULTS
142
-------------- BEGIN OPTIMAL POWER FLOW --------------
MAX OPF ITERATIONS 6 MAX PF ITERATIONS 10 MVAR LIMITING CONTROL 1 MODERATE ITER FOR MVAR CNTL 2
HIGH VOLTAGE LIMIT 1.10 LOW VOLTAGE LIMIT .85
NUMBER OF BUSSES 5 NUMBER OF BRANCHES 6 NUMBER OF GENERATORS 2 NUMBER OF TRANSFORMERS 0
-----5 BUS STEV. INITIAL GENERATOR DATA -------
NO BUS NAME VOLTS MW QMAX QMIN
1 1 WHITE 1.0200 65.1 60.0 o.o 2 2 GREEN 1.0400 100.0 60.0 o.o
------ BRANCH FLOW SUMMARY ------
NO FROM BUS TO BUS TYPE RESISTANCE INDUCTANCE CHARG/TAP
1 1 WHITE 3 RED 0 .0100 .4000 0.0000 2 1 WHITE 4 BLUE 0 .1500 .6000 0.0000 3 1 WHITE 5 YELLOW 0 .0500 .2000 0.0000 4 2 GREEN 3 RED 0 .0500 .2000 0.0000 5 2 GREEN 5 YELLOW 0 .0500 .2000 0.0000 6 3 RED 4 BLUE 0 .1000 .4000 0.0000
143
5 BUS STEV. INITIAL POWER FLOW SOLUTION
----- GENERATOR SUMMARY -------
NO BUS NAME VOLTS MW MVAR QMAX QMIN TYPE
1 2
BUS
1 2 3 4 5
NO
1 2 3 4 5 6
1 WHITE 2 GREEN
5 BUS STEV.
1.0200 64.6 1.0400 100.0
35.4 45.0
60.0 60.0
INITIAL POWER FLOW SOLUTION
----- BUS SUMMARY -------
NAME VOLTS ANGLE MW LOAD MVAR LOAD
WHITE 1.0200 o.oo o.o o.o GREEN 1.0400 1 .86 o.o o.o RED .9601 -4.22 60.0 30.0 BLUE .9270 -8.12 40.0 10.0 YELLOW .9931 -2.15 60.0 20.0
5 BUS STEV. INITIAL POWER FLOW SOLUTION
------ BRANCH FLOW SUMMARY ------
FROM BUS TO BUS TYPE MW
1 WHITE 3 RED 0 18.4 1 WHITE 4 BLUE 0 25.0 1 WHITE 5 YELLOW 0 21.2 2 GREEN 3 RED 0 60.2 2 GREEN 5 YELLOW 0 40.0 3 RED 4 BLUE 0 16.2
o.o 3 o.o 2
TYPE
3 2 0 0 0
MVAR MVA
15.5 24.0 11.1 27.4 8.8 22.9
29.3 66.9 15.6 42.9 4.4 16.8
144
-------5 BUS STEV. FINAL POWER FLOW SOLUTION----
----- GENERATOR SUMMARY -------
NO BUS NAME VOLTS MW MVAR QMAX QMIN TYPE
1 1 WHITE 1.1000 63.9 43.8 60.0 o.o 3 2 2 GREEN 1.0998 100.0 33.5 60.0 o.o 2
-------5 BUS STEV. FINAL POWER FLOW SOLUTION ----
----- BUS SUMMARY -------
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
1 WHITE 1.1000 o.oo o.o o.o 3 2 GREEN 1.0998 2.04 o.o o.o 2 3 RED 1.0344 -3.40 60.0 30.0 0 4 BLUE 1 .0094 -6.73 40.0 1 o.o 0 5 YELLOW 1.0659 -1.62 60.0 20.0 0
-------5 BUS STEV. FINAL POWER FLOW SOLUTION ----
------ BRANCH FLOW SUMMARY ------
NO FROM BUS TO BUS TYPE MW MVAR MVA
1 1 WHITE 3 RED 0 17 .3 18.1 25.1 2 1 WHITE 4 BLUE 0 24.6 11. 7 27.3 3 1 WHITE 5 YELLOW 0 20.1 14.0 24.5 4 2 GREEN 3 RED 0 59.9 23.6 64.3 5 2 GREEN 5 YELLOW 0 40.0 9.9 41.2 6 3 RED 4 BLUE 0 15.9 2.9 16.2
145
5 BUS STEV. CONVERGENCE SUMMARY
OPF BUS REAL REACT. REACT. ITER BUS TYPE VOLT ANG. DELXB DELXB DELXNB
------ -------- -------- --------- ---------1 1 0 .928 -.156 .076 .032 .ooo
2 0 .993 -.059 .052 .025 .ooo 3 0 .962 -.098 .043 .027 .ooo 4 2 1.040 -.009 -.616 .038 -.010 5 3 1.020 o.ooo .564 .000 .110
2 1 0 .961 -.142 .036 .004 .ooo 2 0 1.016 -.048 .045 -.007 .ooo 3 0 .981 -.086 .056 -.002 .ooo 4 2 1.036 .007 .416 -.025 .088 5 3 1.069 .ooo -.484 .000 .ooo
3 1 0 .981 -.139 .060 .023 .ooo 2 0 1.042 -.052 .041 .018 .ooo 3 0 1.013 -.087 .033 .019 .ooo 4 2 1 .084 -.006 -.550 .028 -.010 5 3 1.069 0.000 .516 .ooo .089
4 1 0 1 .002 - .131 .022 .002 .ooo 2 0 1.056 -.045 .028 -.005 .ooo 3 0 1.024 -.081 .035 -.001 .ooo 4 2 1.081 .004 .287 -.014 .055 5 3 1.100 o.ooo -.317 .000 .000
5 BUS STEV. SOLUTION SUMMARY
ITER BUS VOLT MVAR ------ ------
1 1 1.0687 57.7 2 1.0358 19.7
2 1 1.0685 31 .6 2 1.0844 43.9
3 1 1.1000 51.5 2 1.0811 24.9
4 1 1.1000 39.8 2 1.1000 36.4
APPENDIX 3
THREE BUS ELGERD RESULTS
146
NO
147
-------------- BEGIN OPTIMAL POWER FLOW --------------
-----3 BUS ELGERD NETWORK-----
MAX OPF ITERATIONS 6 MAX PF ITERATIONS 10 MVAR LIMITING CONTROL 1 MODERATE ITER FOR MVAR CNTL 2
HIGH VOLTAGE LIMIT 1.10 LOW VOLTAGE LIMIT .85
NUMBER OF BUSSES 3 NUMBER OF BRANCHES 3 NUMBER OF GENERATORS 2 NUMBER OF TRANSFORMERS 0
-----3 BUS ELGERD INITIAL GENERATOR DATA -------
NO BUS NAME VOLTS MW QMAX QMIN
1 2
3 BUS1 2 BUS2
1 .0500 1.0974
o.o 200.0 -200.0 60.0 200.0 -200.0
------3 BUS ELGERD BRANCH DATA FLOW SUMMARY ------
FROM BUS TO BUS TYPE RESISTANCE INDUCTANCE CHARG/TAP
1 1 BUS3 2 2 BUS2 3 1 BUS3
2 BUS2 3 BUS1 3 BUS1
0 0 0
.0210
.0210
.0210
.0872
.0872
.0872
.0452
.0452
.0452
-----3 BUS ELGERD GENERATOR SUMMARY -------
NO BUS NAME
1 2
3 BUS1 2 BUS2
VOLTS MW MVAR QMAX QMIN TYPE
1.0500 262.7 39.5 200.0 -200.0 3 1.0974 60.0 116.2 200.0 -200.0 2
148
3 BUS ELGERD INITIAL POWER FLOW SOLUTION
----- BUS SUMMARY -------
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
1 BUS3 1 .0374 -2.76 120.0 60.0 0 2 BUS2 1.0974 -.70 o.o o.o 2 3 BUS1 1.0500 o.oo 200.0 100.0 3
3 BUS ELGERD INITIAL POWER FLOW SOLUTION
------ BRANCH FLOW SUMMARY ------
NO FROM BUS TO BUS TYPE MW MVAR MVA
1 1 BUS3 2 BUS2 0 -60.5 -58.4 84.1 2 2 BUS2 3 BUS1 0 -1.7 57.4 57.5 3 1 BUS3 3 BUS1 0 -60.0 -1.5 60.1
-------3 BUS ELGERD FINAL POWER FLOW SOLUTION ----
----- GENERATOR SUMMARY -------
NO BUS NAME VOLTS MW MVAR QMAX QMIN TYPE
1 2
3 BUS1 2 BUS2
1.1000 261.6 125.7 200.0 -200.0 3 1.0998 60.0 25.0 200.0 -200.0 2
149
-------3 BUS ELGERD FINAL POWER FLOW SOLUTION ----
----- BUS SUMMARY -------
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
NO
1 2 3
1 BUS3 2 BUS2 3 BUS1
1 .0647 1.0998 1.1000
-2.28 -.01 o.oo
-------3 BUS ELGERD
120.0 o.o
200.0
FINAL
60.0 o.o
100.0
0 2 3
POWER FLOW SOLUTION ----
------ BRANCH FLOW SUMMARY ------
FROM BUS TO BUS TYPE MW MVAR MVA
1 BUS3 2 BUS2 0 -59°9 -29.9 66.9 2 BUS2 3 BUS1 0 -.3 -2.9 3.0 1 BUS3 3 BUS1 0 -60.2 -30.1 67.3
150
3 BUS ELGERD CONVERGENCE SUMMARY
OPF BUS REAL REACT. REACT. ITER BUS TYPE VOLT ANG. DELXB DELXB DELXNB
------ -------- -------- --------- ---------1 1 0 1.037 -.048 .017 .107 .ooo
2 2 1.097 -.012 -17.021 .202 -.383 3 3 1.050 .ooo 14.891 .ooo .415
2 1 0 1.038 -.042 .016 .ooo .ooo 2 2 1.074 -.000 .146 -.002 .019 3 3 1.075 .ooo -.154 .ooo .012
3 1 0 1.058 -.041 .014 .019 .ooo 2 2 1.098 -.002 -3.109 .035 -.061 3 3 1.089 .ooo 3.028 .ooo .088
4 1 0 1.059 -.040 .014 .ooo .ooo 2 2 1.094 -.000 .145 -.002 .017 3 3 1.095 .000 -.152 .ooo .010
5 1 0 1.064 -.040 .013 .005 .ooo 2 2 1.100 -.001 -.691 .008 -.004 3 3 1.098 .ooo .681 .ooo .029
3 BUS ELGERD SOLUTION SUMMARY
ITER BUS VOLT MVAR ------ ------1 3 1.0749 128.9
2 1.0744 14.0
2 3 1.0894 107.7 2 1.0982 43.2
3 3 1.0949 127.7 2 1.0944 22.8
4 3 1.0981 121.2 2 1.1000 29.5
5 3 1.1000 126.0 2 1.0998 24.6
APPENDIX 4
14 BUS IEEE RESULTS
151
152
-------------- BEGIN OPTIMAL POWER FLOW --------------
14 BUS IEEE NETWORK
MAX OPF ITERATIONS 6 MAX PF ITERATIONS 10 MVAR LIMITING CONTROL 1 MODERATE ITER FOR MVAR CNTL 2
HIGH VOLTAGE LIMIT 1.10 LOW VOLTAGE LIMIT .85
NUMBER OF BUSSES 14 NUMBER OF BRANCHES 21 NUMBER OF GENERATORS 5 NUMBER OF TRANSFORMERS 3
-----IEEE 14 BUS INITIAL GENERATOR DATA -------
NO BUS NAME VOLTS MW QMAX QMIN
1 1 KANAWHA 1.0600 o.o 100.0 -20.0 2 2 TURNER 1.0450 40.0 50.0 -40.0 3 3 LOGAN 1. 0100 o.o 40.0 o.o 4 6 CLINCH R 1.0700 o.o 24.0 -6.0 5 8 TAZEWELL 1.0900 o.o 24.0 -6.0
153
------14 BUS IEEE BRANCH DATA SUMMARY ------
NO FROM BUS TO BUS TYPE RESISTANCE INDUCTANCE CHARG/TAP
1 1 KANAWHA 2 TURNER 0 .0194 .0592 .0528 2 1 KANAWHA 5 BEAVER C 0 .0540 .2230 .0492 3 2 TURNER 3 LOGAN 0 .0470 .1980 .0438 4 2 TURNER 4 SPRIGG 0 .0581 .1763 .0374 5 2 TURNER 5 BEAVER C 0 .0569 .1739 .0340 6 3 LOGAN 4 SPRIGG 0 .0670 .1710 .0346 7 4 SPRIGG 5 BEAVER C 0 .0133 .0421 .0128 8 4 SPRIGG 7 SALTVILL 4 0.0000 .2091 .9780 9 4 SPRIGG 9 GLEN LYN 1 0.0000 .5562 .9690
10 5 BEAVER C 6 CLINCH R 1 0.0000 .2520 .9320 11 6 CLINCH R 11 DOYLE 0 .0950 .1989 0.0000 12 6 CLINCH R 12 HOMER 0 .1229 .2558 0.0000 13 6 CLINCH R 13 MOSES 0 .0662 .1303 0.0000 14 7 SALTVILL 8 TAZEWELL 0 0.0000 .1761 0.0000 15 7 SALTVILL 9 GLEN LYN 0 0.0000 .1100 0.0000 16 9 GLEN LYN 10 AIRPORT 0 .0318 .0845 0.0000 17 9 GLEN LYN 14 KINCAID 0 .1271 .2704 0.0000 18 9 GLEN LYN 14 KINCAID 0 .1271 .2704 0.0000 19 10 AIRPORT 11 DOYLE 0 .0821 .1921 0.0000 20 12 HOMER 13 MOSES 0 .2209 .1999 0.0000 21 13 MOSES 14 KINCAID 0 .1709 .3480 0.0000
-----IEEE 14 BUS GENERATOR SUMMARY -------
NO BUS NAME VOLTS MW MVAR QMAX QMIN TYPE
1 1 KANAWHA 1 .0600 232.3 -17.4 100.0 -20.0 3 2 2 TURNER 1.0450 40.0 40.8 50.0 -40.0 2 3 3 LOGAN 1. 0100 o.o 22.4 40.0 o.o 2 4 6 CLINCH R 1.0700 o.o 6.7 24.0 -6.0 2 5 8 TAZEWELL 1.0900 o.o 14.0 24.0 -6.0 2
154
IEEE 14 BUS INITIAL POWER FLOW SOLUTION
----- BUS SUMMARY -------
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
1 KANAWHA 1.0600 o.oo o.o o.o 3 2 TURNER 1.0450 -4.98 21.7 12. 7 2 3 LOGAN 1. 0100 -12. 71 94.2 19.0 2 4 SPRIGG 1.0202 -10.36 47.8 -3.9 0 5 BEAVER C 1 • 0214 -8.78 7.6 1. 6 0 6 CLINCH R 1.0700 -14.01 11.2 7.5 2 7 SALTVILL 1.0673 -13.48 o.o o.o 0 8 TAZEWELL 1.0900 -13.48 o.o o.o 2 9 GLEN LYN 1 .0670 -15.09 29.5 -14.6 0
10 AIRPORT 1.0602 -15 .19 9.0 5.8 0 11 DOYLE 1 • 0612 -14.74 3.5 1.8 0 12 HOMER 1.0565 -14.83 6.1 1.6 0 13 MOSES 1.0539 -14.91 13.5 5.8 0 14 KINCAID 1.0523 -15.67 14.9 5.0 0
IEEE 14 BUS INITIAL POWER FLOW SOLUTION
------ BRANCH FLOW SUMMARY ------
NO FROM BUS TO BUS TYPE MW MVAR MVA
1 1 KANAWHA 2 TURNER 0 156. 7 -20.4 158.0 2 1 KANAWHA 5 BEAVER C 0 75.5 3.0 75.5 3 2 TURNER 3 LOGAN 0 73.2 3.6 73.2 4 2 TURNER 4 SPRIGG 0 56.3 -3.2 56.4 5 2 TURNER 5 BEAVER C 0 41.3 .1 41.3 6 3 LOGAN 4 SPRIGG 0 -23.3 1.9 23.4 7 4 SPRIGG 5 BEAVER C 0 -62.6 17 .2 64.9 8 4 SPRIGG 7 SALTVILL 4 29.0 -11.2 31.1 9 4 SPRIGG 9 GLEN LYN 1 16.7 -2.0 16.8
10 5 BEAVER C 6 CLINCH R 1 42.4 13.2 44.4 11 6 CLINCH R 11 DOYLE 0 7.8 1.0 7.8 12 6 CLINCH R 12 HOMER 0 7.4 2.2 7.7 13 6 CLINCH R 13 MOSES 0 16.3 5 .1 17 .1 14 7 SALTVILL 8 TAZEWELL 0 o.o -13.8 13.8 15 7 SALTVILL 9 . GLEN LYN 0 29.1 .7 29.1 16 9 GLEN LYN 10 AIRPORT 0 4.9 6.9 8.4 17 9 GLEN LYN 14 KINCAID 0 5.7 3.2 6.5 18 9 GLEN LYN 14 KINCAID 0 5.7 3.2 6.5 19 10 AIRPORT 11 DOYLE 0 -4.1 1.2 4.3 20 12 HOMER 13 MOSES 0 1 .1 .2 1 .1 21 13 MOSES 14 KINCAID 0 3.6 -1.2 3.8
155
-------IEEE 14 BUS FINAL POWER FLOW SOLUTION ----
----- GENERATOR SUMMARY -------
NO BUS NAME VOLTS MW MVAR QMAX QMIN TYPE
1 1 KANAWHA 1.0968 231.4 -1.5 100.0 -20.0 3 2 2 TURNER 1.0764 40.0 43.8 50.0 -40.0 2 3 3 LOGAN 1. 0296 o.o 12.7 40.0 o.o 2 4 6 CLINCH R 1.0843 o.o .8 24.0 -6.0 2 5 8 TAZEWELL 1.0892 o.o 5.9 24.0 -6.0 2
-------IEEE 14 BUS FINAL POWER FLOW SOLUTION ----
----- BUS SUMMARY -------
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
1 KANAWHA 1.0968 o.oo o.o o.o 3 2 TURNER 1.0764 -4.54 21.7 12.7 2 3 LOGAN 1. 0296 -11.70 94.2 19.0 2 4 SPRIGG 1.0439 -9.51 47.8 -3.9 0 5 BEAVER C 1.0470 -8.04 7.6 1.6 0 6 CLINCH R 1.0843 -13.06 11.2 7.5 2 7 SALTVILL 1 .0797 -12.53 o.o o.o 0 8 TAZEWELL 1.0892 -12.53 o.o o.o 2 9 GLEN LYN 1 • 0815 -14.10 29.5 -14.6 0
10 AIRPORT 1.0747 -14.20 9.0 5.8 0 11 DOYLE 1.0757 -13.77 3.5 1.8 0 12 HOMER 1.0710 -13.86 6.1 1.6 0 13 MOSES 1.0684 -13.94 13.5 5.8 0 14 KINCAID 1.0669 -14.67 14.9 5.0 0
..
156
-------IEEE 14 BUS FINAL POWER FLOW SOLUTION ----
------ BRANCH FLOW SUMMARY ------
NO FROM BUS TO BUS TYPE MW MVAR MVA
1 1 KANAWHA 2 TURNER 0 155.8 -1 o.o 156.1 2 1 KANAWHA 5 BEAVER C 0 74.8 8.5 75.3 3 2 TURNER 3 LOGAN 0 72.7 10.0 73.4 4 2 TURNER 4 SPRIGG 0 56.4 1.5 56.4 5 2 TURNER 5 BEAVER C 0 41.5 3.9 41.6 6 3 LOGAN 4 SPRIGG 0 -23-5 -.7 23.5 7 4 SPRIGG 5 BEAVER C 0 -62.6 12.4 63.8 8 4 SPRIGG 7 SALTVILL 4 29.0 -5.5 29.6 9 4 SPRIGG 9 GLEN LYN 1 16.8 - .1 16.8
10 5 BEAVER C 6 CLINCH R 1 42.3 19.3 46.5 11 6 CLINCH R 11 DOYLE 0 7.7 1.0 7.8 12 6 CLINCH R 12 HOMER 0 7.3 2.2 7.7 13 6 CLINCH R 13 MOSES 0 16.3 5.0 17 .o 14 7 SALTVILL 8 TAZEWELL 0 .o -5.8 5.8 15 7 SALTVILL 9 GLEN LYN 0 29.1 -1.3 29.1 16 9 GLEN LYN 10 AIRPORT 0 4.9 6.9 8.4 17 9 GLEN LYN 14 KINCAID 0 5.7 3.2 6.5 18 9 GLEN LYN 14 KINCAID 0 5.7 3.2 6.5 19 10 AIRPORT 11 DOYLE 0 -4.1 1.2 4.2 20 12 HOMER 13 MOSES 0 1 .1 .2 1 .1 21 13 MOSES 14 KINCAID 0 3.5 -1.3 3.8
157
14 BUS IEEE UNIT VOLTAGES CONVERGENCE
ITER LOSS UNIT 1 UNIT 2 UNIT 3 UNIT 4 UNIT 5 ------- ------- ------- ------- -------
0 13.28 1 .06 1.045 1.01 1.070 1.09
1 13.07 1.0763 1.0432 1 .0114 1.0723 1.0897
2 12.92 1 .0763 1.057 1. 0123 1.0743 1 .0895
3 12.79 1.0883 1.0551 1.0176 1.0767 1.0893
4 12.66 1.0874 1 .0672 1.0200 1.079 1.0892
5 12.55 1.098 1.065 1.0265 1 .0818 1.0892
6 12.44 1 .0968 1.0764 1 .0296 1.0848 1 .0892
14 BUS IEEE UNIT MVARS CONVERGENCE
ITER LOSS UNIT 1 UNIT 2 UNIT 3 UNIT 4 UNIT 5 ------- ------- ------- ------- -------
0 13.28 -17.39 40.77 22.42 6.68 14.05
1 13.07 21.07 1.21 23.68 6.48 13.18
2 12.92 - 6.15 41.45 13-84 4.07 11 .31
3 12. 79 23.45 6.85 19.18 4.07 10.32
4 12.66 - 2.96 42.91 12.41 2 .15 8.51
5 12.55 24.24 9.45 18. 71 2.41 7.58
6 12.44 -1.46 43.71 12. 75 .78 5.86
158
14 BUS IEEE UNIT REDUCED GRADIENT CONVERGENCE
ITER LOSS UNIT 1 UNIT 2 UNIT 3 UNIT 4 UNIT 5 ------- ------- ------- ------- -------
1 13.28 .247 -.027 .021 .034 -.004
2 13.07 .ooo .218 .015 .030 -.004
3 12.92 .173 -.027 .076 .037 -.003
4 12.79 -.013 .185 .044 .034 -.002
5 12.66 .152 -.031 .0860 .040 -.001
6 12.55 -.017 .171 .048 .037 .ooo
APPENDIX 5
30 BUS IEEE MODIFIED RESULTS
159
160
-------------- BEGIN OPTIMAL POWER FLOW --------------
30 BUS IEEE MODIFIED NETWORK
MAX OPF ITERATIONS . 6 MAX PF ITERATIONS 10 MVAR LIMITING CONTROL 1 MODERATE ITER FOR MVAR CNTL 2
HIGH VOLTAGE LIMIT 1.10 LOW VOLTAGE LIMIT • 85
NUMBER OF BUSSES 30 NUMBER OF BRANCHES 41 NUMBER OF GENERATORS 6 NUMBER OF TRANSFORMERS 4
-----IEEE 30 BUS MOD INITIAL GENERATOR DATA -------
NO BUS NAME VOLTS MW QMAX QMIN
1 1 GLEN LYN 1.0600 o.o 500.0 -500.0 2 2 CLAYTOR 1 .0550 80.0 50.0 -40.0 3 5 FIELDALE 1.0100 50.0 40.0 -40.0 4 8 REUS ENS 1. 0100 20.0 40.0 -1 o.o 5 11 ROANOKE 1.0820 20.0 24.0 -6.0 6 13 HANCOCK 1.0710 20.0 24.0 -6.o
161
------30 BUS IEEE MOD. BRANCH FLOW SUMMARY ------
NO FROM BUS TO BUS TYPE RESISTANCE INDUCTANCE CHARG/TAP
1 1 GLEN LYN 2 CLAYTOR 0 .0192 .0575 .0528 2 1 GLEN LYN 3 KUM IS 0 .0452 .1852 .0408 3 2 CLAYTOR 4 HANCOCK 0 .0570 .1737 .0368 4 3 KUMIS 4 HANCOCK 0 .0132 .0379 .0084 5 2 CLAYTOR 5 FIELDALE 0 .0472 .1983 .0418 6 2 CLAYTOR 6 ROANOKE 0 .0581 .1763 .0374 7 4 HANCOCK 6 ROANOKE 0 .0119 .0414 .0090 8 5 FIELDALE 7 BLAINE 0 .0460 .1160 .0204 9 6 ROANOKE 7 BLAINE 0 .0267 .0820 .0170
10 6 ROANOKE 8 REUSENS 0 .0120 .0420 .0090 11 6 ROANOKE 9 ROANOKE 1 0.0000 .2080 -9780 12 6 ROANOKE 10 ROANOKE 1 0.0000 .5560 .9690 13 9 ROANOKE 11 ROANOKE 0 0.0000 .2080 0.0000 14 9 ROANOKE 10 ROANOKE 0 0.0000 .1100 0.0000 15 4 HANCOCK 12 HANCOCK 1 0.0000 .2560 .9320 16 12 HANCOCK 13 HANCOCK 0 0.0000 .1400 0.0000 17 12 HANCOCK 14 LOAD14 0 .1231 .2559 0.0000 18 12 HANCOCK 15 LOAD15 0 .0662 .1304 0.0000 19 12 HANCOCK 16 LOAD16 0 .0945 .1987 0.0000 20 14 LOAD14 15 LOAD15 0 .2210 .1997 0.0000 21 16 LOAD16 17 LOAD17 0 .0824 .1923 0.0000 22 15 LOAD15 18 LOAD18 0 .1073 .2185 0.0000 23 18 LOAD18 19 LOAD19 0 .0639 .1292 0.0000 24 19 LOAD19 20 LOAD20 0 .0340 .0680 0.0000 25 10 ROANOKE 20 LOAD20 0 .0936 .2090 0.0000 26 10 ROANOKE 17 LOAD17 0 .0324 .0845 0.0000 27 10 ROANOKE 21 LOAD21 0 .0348 .0749 0.0000 28 10 ROANOKE 22 JUNCTN22 0 .0727 .1499 0.0000 29 21 LOAD21 22 JUNCTN22 0 .0116 .0236 0.0000 30 15 LOAD15 23 LOAD23 0 .1000 .2020 0.0000 31 22 JUNCTN22 24 LOAD24CA 0 .1150 .1790 0.0000 32 23 LOAD23 24 LOAD24CA 0 .1320 .2700 0.0000 33 24 LOAD24CA 25 JUNCTN25 0 .1885 .3292 0.0000 34 25 JUNCTN25 26 LOAD26 0 .2544 .3800 0.0000 35 25 JUNCTN25 27 CLO VERDA 0 .1093 .2087 0.0000 36 28 CLOVERDA 27 CLOVERDA 1 0.0000 .3960 .9680 37 27 CLOVER DA 29 LOAD29 0 .2198 .4153 0.0000 38 27 CLO VERDA 30 LOAD30 0 .3202 .6027 0.0000 39 29 LOAD29 30 LOAD30 0 .2399 .4533 0.0000 40 8 REUSENS 28 CLOVERDA 0 .0636 .2000 .0428 41 6 ROANOKE 28 CLO VERDA 0 .0169 .0599 .0130
162
IEEE 30 BUS MOD INITIAL POWER FLOW SOLUTION
----- GENERATOR SUMMARY -------
NO BUS NAME VOLTS MW MVAR QMAX QMIN TYPE
1 2 3 4 5 6
1 GLEN LYN 2 CLAYTOR 5 FIELDALE 8 REUSENS
11 ROANOKE 13 HANCOCK
1 .0600 1.0546 1.0100 1.0100 1 .0820 1.0710
98.8 80.0 50.0 20.0 20.0 20.0
-4.8 50.0 8.9
14.0 14.5 6.9
500.0 50.0 40.0 40.0 24.0 24.0
-500.0 -40.0 -40.0 -10.0 -6.0 -6.0
IEEE 30 BUS MOD INITIAL POWER FLOW SOLUTION
BUS NAME
1 GLEN LYN 2 CLAYTOR 3 KUMIS 4 HANCOCK 5 FIELDALE 6 ROANOKE 7 BLAINE 8 REUSENS 9 ROANOKE
10 ROANOKE 11 ROANOKE 12 HANCOCK 13 HANCOCK 14 LOAD14 15 LOAD15 16 LOAD16 17 LOAD17 18 LOAD18 19 LOAD19 20 LOAD20 21 LOAD21 22 JUNCTN22 23 LOAD23 24 LOAD24CA 25 JUNCTN25
----- BUS SUMMARY -------
VOLTS ANGLE MW LOAD MVAR LOAD TYPE
1.0600 1 .0546 1.0288 1. 0213 1.0100 1 .0164 1.0058 1. 0100 1.0549 1 .0500 1.0820 1.0623 1.0710 1 .0476 1.0428 1 .0500 1.0447 1 .0332 1.0307 1 .0347 1.0377 1.0382 1.0321 1 .0262 1.0209
o.oo -1.83 -3-71 -4-42 -6.27 -5.20 -6.17 -5-40 -6.59 -8.46 -4-51 -7 .61 -6.21 -8.51 -8.63 -8.23 -8.59 -9.26 -9-44 -9.25 -8.92 -8.91 -9.11 -9-41 -9-38
o.o 21. 7 2.4 7.6
94.2 o.o
22.8 30.0 o.o 5.8 o.o
11.2 o.o 6.2 8.2 3.5 9.0 3.2 9.5 2.2
17 .5 o.o 3.2 8.7 o.o
o.o 12. 7 1.2 1.6
19.0 o.o
10.9 30.0 o.o 2.0 o.o 7.5 o.o 1.6 2.5 1.8 5.8
.9 3.4
.7 11.2 o.o 1.6 6.7 o.o
3 1 0 0 2 0 0 2 0 0 2 0 2 0 0 0 0 0 0 0 0 0 0 0 0
3 1 2 2 2 2
163
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
26 LOAD26 1 .0032 -9.80 3.5 2.3 0 27 CLO VERDA 1. 0261 -9.11 o.o o.o 0 28 CLOVERDA 1. 0116 -5.65 o.o o.o 0 29 LOAD29 1.0063 -10.34 2.4 .9 0 30 LOAD30 .9948 -11.22 10.6 1.9 0
IEEE 30 BUS MOD INITIAL POWER FLOW SOLUTION
------ BRANCH FLOW SUMMARY ------
NO FROM BUS TO BUS TYPE MW MVAR MVA
1 1 GLEN LYN 2 CLAYTOR 0 59.0 -11.8 60.2 2 1 GLEN LYN 3 KUM IS 0 40.4 6.9 41.0 3 2 CLAYTOR 4 HANCOCK 0 31 .5 8.5 32.6 4 3 KUM IS 4 HANCOCK 0 37.0 7.4 37.7 5 2 CLAYTOR 5 FIELDALE 0 45.1 12.3 46.7 6 2 CLAYTOR 6 ROANOKE 0 39.4 8.9 40.4 7 4 HANCOCK 6 ROANOKE 0 34.9 1.8 34.9 8 5 FIELDALE 7 BLAINE 0 -.o 2.6 2.6 9 6 ROANOKE 7 BLAINE 0 22.9 4.9 23.5
10 6 ROANOKE 8 REUSENS 0 11.8 11.6 16.6 11 6 ROANOKE 9 ROANOKE 1 12.8 -7.7 14.9 12 6 ROANOKE 10 ROANOKE 1 11.2 .1 11.2 13 9 ROANOKE 11 ROANOKE 0 -19.9 -13.4 24.0 14 9 ROANOKE 10 ROANOKE 0 32.8 5.3 33.2 15 4 HANCOCK 12 HANCOCK 1 25.3 15.0 29.4 16 12 HANCOCK 13 HANCOCK 0 -19.9 -6.3 20.9 17 12 HANCOCK 14 LOAD14 0 8.0 2.3 8.3 18 12 HANCOCK 15 LOAD15 0 18.5 6.6 19. 7 19 12 HANCOCK 16 LOAD16 0 7.5 3.1 8.1 20 14 LOAD14 15 LOAD15 0 1.8 .6 1.9 21 16 LOAD16 17 LOAD17 0 4.1 1.1 4.2 22 15 LOAD15 18 LOAD18 0 6.1 1.6 6.3 23 18 LOAD18 19 LOAD19 0 2.9 .6 3.0 24 19 LOAD19 20 LOAD20 0 -6.6 -2.8 7.2 25 10 ROANOKE 20 LOAD20 0 s.9 3.7 9.6 26 10 ROANOKE 17 LOAD17 0 4.8 4.7 6.7 27 10 ROANOKE 21 LOAD21 0 16.2 9.7 18.9 28 1.0 ROANOKE 22 JUNCTN22 0 7.9 4.4 9.1 29 21 LOAD21 22 JUNCTN22 0 -1 .1 -1. 7 2.0 30 15 LOAD15 23 LOAD23 0 .. 5.s 2.7 6.4 31 22 JUNCTN22 24 LOAD24CA 0 6.8 2.6 7.3
164
NO FROM BUS TO BUS TYPE MW MVAR MVA
32 23 LOAD23 24 LOAD24CA 0 2.5 1. 0 2.7 33 24 LOAD24CA 25 JUNCTN25 0 .6 1.3 1.5 34 25 JUNCTN25 26 LOAD26 0 3.6 2.4 4.3 35 25 JUNCTN25 27 CLO VERDA 0 -2.9 -1.0 3.1 36 28 CLOVERDA 27 CLOVERDA 1 16.3 5.5 17.2 37 27 CLOVER DA 29 LOAD29 0 6.2 1.7 6.4 38 27 CLOVERDA 30 LOAD30 0 7 .1 1. 7 7.3 39 29 LOAD29 30 LOAD30 0 3.7 .6 3.8 40 8 REUS ENS 28 CLOVERDA 0 1.8 -3.5 4.0 41 6 ROANOKE 28 CLO VERDA 0 14.5 3.5 14.9
-------IEEE 30 BUS MOD FINAL POWER FLOW SOLUTION ----
----- GENERATOR SUMMARY -------
NO BUS NAME VOLTS MW MVAR QMAX QMIN TYPE
1 1 GLEN LYN 1.0846 98.3 -20.6 500.0 -500.0 3 2 2 CLAYTOR 1. 0854 80.0 50.0 50.0 -40.0 1 3 5 FIELDALE 1.0493 50.0 11.9 40.0 -40.0 2 4 8 REUS ENS 1 .0582 20.0 38.8 40.0 -10.0 2 5 11 ROANOKE 1.0920 20.0 5.4 24.0 -6.0 2 6 13 HANCOCK 1 .0843 20.0 -1. 7 24.0 -6.0 2
165
-------IEEE 30 BUS MOD FINAL POWER FLOW SOLUTION ----
----- BUS SUMMARY -------
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
1 GLEN LYN 1.0846 o.oo o.o o.o 3 2 CLAYTOR 1.0854 -1 .81 21.7 12. 7 1 3 KUM IS 1 • 0619 -3°59 2.4 1.2 0 4 HANCOCK 1. 0561 -4.28 7.6 1.6 0 5 FIELDALE 1.0493 -6.05 94.2 19.0 2 6 ROANOKE 1.0559 -5.08 o.o o.o 0 7 BLAINE 1 .0456 -5.97 22.8 10.9 0 8 REUSENS 1.0582 -5.40 30.0 30.0 2 9 ROANOKE 1 .0825 -6.37 o.o o.o 0
10 ROANOKE 1.0799 -8.13 5.8 2.0 0 11 ROANOKE 1.0920 -4-36 .o.o o.o 2 12 HANCOCK 1.0869 -7.24 11.2 7.5 0 13 HANCOCK 1.0843 -5.88 o.o o.o 2 14 LOAD14 1.0734 -8.10 6.2 1.6 0 15 LOAD15 1.0695 -8.24 8.2 2.5 0 16 LOAD16 1.0769 -7.87 3.5 1.8 0 17 LOAD17 1 .0739 -8.24 9.0 5.8 0 18 LOAD18 1 .0614 -8.85 3.2 .9 0 19 LOAD19 1 .0597 -9.04 9.5 3.4 0 20 LOAD20 1.0640 -8.86 2.2 .7 0 21 LOAD21 1 .0682 -8.57 17 .5 11.2 0 22 JUNCTN22 1.0688 -8.56 o.o o.o 0 23 LOAD23 1 • 0612 -8.72 3.2 1.6 0 24 LOAD24CA 1.0582 -9.03 8.7 6.7 0 25 JUNCTN25 1.0574 -9.04 o.o o.o 0 26 LOAD26 1.0404 -9.43 3.5 2.3 0 27 CLOVERDA 1 .0653 -8.81 o.o o.o 0 28 CLO VERDA 1.0528 -5.52 o.o o.o 0 29 LOAD29 1.0462 -9.95 2.4 .9 0 30 LOAD30 1.0352 -10.76 1 o.6 1.9 0
166
-------IEEE 30 BUS MOD FINAL POWER FLOW SOLUTION ----
------ BRANCH FLOW SUMMARY ------
NO FROM BUS TO BUS TYPE MW MVAR MVA
1 1 GLEN LYN 2 CLAYTOR 0 58.2 -22.9 62.5 2 1 GLEN LYN 3 KUM IS 0 40.1 2.3 40.1 3 2 CLAYTOR 4 HANCOCK 0 31.2 6.5 31.9 4 3 KUM IS 4 HANCOCK 0 36.9 3.3 37.0 5 2 CLAYTOR 5 FIELDALE 0 45.0 8.1 45.7 6 2 CLAYTOR 6 ROANOKE 0 39.1 4.1 39.3 7 4 HANCOCK 6 ROANOKE 0 35.1 -9.8 36.4 8 5 FIELDALE 7 BLAINE 0 -.0 2.2 2.2 9 6 ROANOKE 7 BLAINE 0 22.9 5.0 23.4
10 6 ROANOKE 8 REUS ENS 0 12.0 -9.8 15.5 11 6 ROANOKE 9 ROANOKE 1 12.7 -1.3 12. 7 12 6 ROANOKE 10 ROANOKE 1 11.3 2.2 11.5 13 9 ROANOKE 11 ROANOKE 0 -19.9 -4.6 20.5 14 9 ROANOKE 10 ROANOKE 0 32.7 3.0 32.8 15 4 HANCOCK 12 HANCOCK 1 24.8 21.1 32.6 16 12 HANCOCK 13 HANCOCK 0 -19.9 2.2 20.0 17 12 HANCOCK 14 LOAD14 0 7.8 2.0 8.1 18 12 HANCOCK 15 LOAD15 0 18.3 5.3 19.1 19 12 HANCOCK 16 LOAD16 0 7.4 1.9 7.7 20 14 LOAD14 15 LOAD15 0 1. 7 .3 1. 7 21 16 LOAD16 17 LOAD17 0 4.0 .o 4.0 22 15 LOAD15 18 LOAD18 0 6.1 1.0 6.1 23 18 LOAD18 19 LOAD19 0 2.8 .o 2.8 24 19 LOAD19 20 LOAD20 0 -6.7 -3·4 7.5 25 10 ROANOKE 20 LOAD20 0 8.9 4.3 9.9 26 10 ROANOKE 17 LOAD17 0 4.9 5.8 7.6 27 10 ROANOKE 21 LOAD21 0 16.0 9.4 18.6 28 10 ROANOKE 22 JUNCTN22 0 7.8 4.2 8.9 29 21 LOAD21 22 JUNCTN22 0 -1.3 -2.0 2.4 30 15 LOAD15 23 LOAD23 0 5.5 1. 7 5.8 31 22 JUNCTN22 24 LOAD24CA 0 6.6 2.1 6.9 32 23 LOAD23 24 LOAD24CA 0 2.3 .o 2.3 33 24 LOAD24CA 25 JUNCTN25 0 .2 .2 .2 34 25 JUNCTN25 26 LOAD26 0 3.6 2.4 4.3 35 25 JUNCTN25 27 CLOVERDA 0 -3.4 -2.2 4.0 36 28 CLO VERDA 27 CLOVER DA 1 16.8 6.6 18.0 37 27 CLO VERDA 29 LOAD29 0 6.2
167
30 BUS IEEE MOD. UNIT VOLTAGES CONVERGENCE
ITER LOSS UNIT 1 UNIT 2 UNIT 3 UNIT 4 UNIT 5 UNIT 6 ------- ------- ------- ------- ------- -------
0 5.379 1 .06 1.055 1.01 1 .01 1 .082 1 .071
1 5.249 1.0526 1.0541 1.0167 1.0262 1.0832 1.0717
2 5.156 1.0743 1.0706 1 .0320 1.0283 1.0863 1. 0735
3 5.065 1.0703 1.0713 1.0346 1.0438 1.0877 1.0771
4 4.992 1.0885 1. 0851 1.048 1 .048 1 .0908 1.0825
5 4.917 1.0855 1.0862 1.0502 1.0591 1.0923 1.0847
30 BUS IEEE MOD. UNIT MVARS CONVERGENCE
ITER LOSS UNIT 1 UNIT 2 UNIT 3 UNIT 4 UNIT 5 UNIT 6 ------- ------- ------- ------- ------- -------
0 5.379 - 4.82 50.0 8.88 14.01 14.45 6.88
1 5.249 -23.27 50.0 11.07 33.58 12.17 4. 71
2 5.156 - 8.27 50.0 13.54 17.84 10.65 2.60
3 5.065 -21.46 50.0 11.53 36.64 8.41 .68
4 4.992 - 8.82· 50.0 14.79 21.6 7.26 -.29
5 4.917 -20.53 50.0 11 .95 38.98 5 .18 -1.89
168
30 BUS IEEE MOD. UNIT REDUCED GRADIENT CONVERGENCE
ITER LOSS UNIT 1 UNIT 2 UNIT 3 UNIT 4 UNIT 5 UNIT 6 ------- ------- ------- ------- ------- -------
1 5.29 .009 .010 .042 .006 .ooo .060
2 5.16 .010 .012 .019 .110 .ooo -.028
3 5.06 .009 .016 .040 .003 .ooo .054
4 4.99 .010 .015 .015 .097 .ooo -.021
5 4.92 • 010 .019 .0370 .ooo .ooo .051
APPENDIX 6
30 BUS IEE RESULTS
169
170
-------------- BEGIN OPTIMAL POWER FLOW --------------
30 BUS IEE NETWORK
MAX OPF ITERATIONS 6 MAX PF ITERATIONS 10 MVAR LIMITING CONTROL 1 MODERATE ITER FOR MVAR CNTL 2
HIGH VOLTAGE LIMIT 1.10 LOW VOLTAGE LIMIT • 85
NUMBER OF BUSSES 30 NUMBER OF BRANCHES 45 NUMBER OF GENERATORS 6 NUMBER OF TRANSFORMERS 5
-----30 BUS IEEE INITIAL GENERATOR DATA -------
NO BUS NAME VOLTS MW QMAX QMIN
1 1 GLEN LYN 1.0600 o.o o.o o.o 2 2 CLAYTOR 1.0450 40.0 50.0 -40.0 3 5 FIELDALE 1.0100 o.o 40.0 -40.0 4 8 REUSENS 1.0100 o.o 40.0 -10.0 5 11 ROANOKE 1.0820 o.o 24.0 -6.0 6 13 HANCOCK 1.0710 o.o 24.0 -6.0
-----30 BUS IEEE GENERATOR SUMMARY -------
NO BUS NAME VOLTS MW MVAR QMAX QMIN TYPE
1 1 GLEN LYN 1 .0600 261.0 -15.9 o.o o.o 3 2 2 CLAYTOR 1.0427 40.0 50.0 50.0 -40.0 1 3 5 FIELDALE 1.0100 o.o 37 .1 40.0 -40.0 2 4 8 REUSENS 1.0100 o.o 37.4 40.0 -10.0 2 5 11 ROANOKE 1 .0820 o.o 16.2 24.0 -6.0 2 6 13 HANCOCK 1.0710 o.o 1 o.6 24.0 -6.0 2
171
------30 BUS IEEE BRANCH DATA SUMMARY ------
NO FROM BUS TO BUS TYPE RESISTANCE INDUCTANCE CHARG/TAP
1 1 GLEN LYN 2 CLAYTOR 0 .0192 .0575 .0528 2 1 GLEN LYN 3 KUM IS 0 .0452 .1852 .0408 3 2 CLAYTOR 4 HANCOCK 0 .0570 .1737 .0368 4 3 KUMIS 4 HANCOCK 0 .0132 .0379 .0084 5 2 CLAYTOR 5 FIELDALE 0 .0472 .1983 .0418 6 2 CLAYTOR 6 ROANOKE 0 .0581 .1763 .0374 7 4 HANCOCK 6 ROANOKE 0 .0119 .0414 .0090 8 5 FIELDALE 7 BLAINE 0 .0460 .1160 .0204 9 6 ROANOKE 7 BLAINE 0 .0267 .0820 .0170
10 6 ROANOKE 8 REUSENS 0 .0120 .0420 .0090 11 6 ROANOKE 9 ROANOKE 1 0.0000 .2080 .9780 12 6 ROANOKE 10 ROANOKE 1 0.0000 .5560 .9690 13 9 ROANOKE 11 ROANOKE 0 0.0000 .2080 0.0000 14 9 ROANOKE 10 ROANOKE 0 0.0000 .1100 0.0000 15 4 HANCOCK 12 HANCOCK 1 0.0000 .2560 .9320 16 12 HANCOCK 13 HANCOCK 0 0.0000 .1400 0.0000 17 12 HANCOCK 14 LOAD14 0 .1231 .2559 0.0000 18 12 HANCOCK 15 LOAD15 0 .0662 .1304 0.0000 19 12 HANCOCK 16 LOAD16 0 .0945 .1987 0.0000 20 14 LOAD14 15 LOAD15 0 .2210 .1997 0.0000 21 16 LOAD16 17 LOAD17 0 .0824 .1923 0.0000 22 15 LOAD15 18 LOAD18 0 .1073 .2185 0.0000 23 18 LOAD18 19 LOAD19 0 .0639 .1292 0.0000 24 19 LOAD19 20 LOAD20 0 .0340 .0680 0.0000 25 10 ROANOKE 20 LOAD20 0 .0936 .2090 0.0000 26 10 ROANOKE 17 LOAD17 0 .0324 .0845 0.0000 27 10 ROANOKE 21 LOAD21 0 .0348 .0749 0.0000 28 10 ROANOKE 22 JUNCTN22 0 .0727 .1499 0.0000 29 21 LOAD21 22 JUNCTN22 0 .0232 .0472 0.0000 30 21 LOAD21 22 JUNCTN22 0 .0232 .0472 0.0000 31 15 LOAD15 23 LOAD23 0 .1000 .2020 0.0000 32 22 JUNCTN22 24 LOAD24CA 0 .1150 .1790 0.0000 33 23 LOAD23 24 LOAD24CA 0 .1320 .2700 0.0000 34 24 LOAD24CA 25 JUNCTN25 0 .1885 .3292 0.0000 35 25 JUNCTN25 26 LOAD26 0 .2544 .3800 0.0000 36 25 JUNCTN25 27 CLOVERDA 0 .1093 .2087 0.0000 37 27 CLOVER DA 28 CLO VERDA 1 0.0000 .7920 1.0330 38 27 CLO VERDA 28 CLOVERDA 1 0.0000 .7920 1 .0330 39 27 CLO VERDA 29 LOAD29 0 .2198 .4153 0.0000 40 27 CLOVERDA 30 LOAD30 0 .3202 .6027 0.0000 41 29 LOAD29 30 LOAD30 0 .2399 .4533 0.0000 42 8 REUS ENS 28 CLOVERDA 0 .0636 .2000 .0428 43 6 ROANOKE 28 CLO VERDA 0 .0507 .1797 .0043 44 6 ROANOKE 28 CLOVERDA 0 .0507 .1797 .0043 45 6 ROANOKE 28 CLO VERDA 0 .0507 .1797 .0043
172
30 BUS IEEE INITIAL POWER FLOW SOLUTION
----- BUS SUMMARY -------
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
1 GLEN LYN 1 .0600 o.oo o.o o.o 3 2 CLAYTOR 1.0427 -5.49 21.7 12. 7 1 3 KUM IS 1 .0205 -7.99 2.4 1.2 0 4 HANCOCK 1 .0116 -9.64 7.6 1.6 0 5 FIELD ALE 1. 0100 -14.39 94.2 19.0 2 6 ROANOKE 1.0101 -11.37 o.o o.o 0 7 BLAINE 1 .0023 -13.14 22.8 10.9 0 8 REUS ENS 1.0100 -12.12 30.0 30.0 2 9 ROANOKE 1 .0508 -14.44 o.o o.o 0
10 ROANOKE 1.0449 -16.04 5.8 2.0 0 11 ROANOKE 1 .0820 -14.44 o.o o.o 2 12 HANCOCK 1 .0571 -15.30 11.2 7.5 0 13 HANCOCK 1.0710 -15.30 o.o o.o 2 14 LOAD14 1.0423 -16.20 6.2 1.6 0 15 LOAD15 1 .0376 -16.29 8.2 2.5 0 16 LOAD16 1.0448 -15.89 3.5 1.8 0 17 LOAD17 1 .0396 -16.20 9.0 5.8 0 18 LOAD18 1. 0281 -16.90 3.2 .9 0 19 LOAD19 1.0255 -17.07 9.5 3.4 0 20 LOAD20 1.0296 -16.87 2.2 .7 0 21 LOAD21 1 .0326 -16.49 17.5 11.2 0 22 JUNCTN22 1.0331 -16.48 o.o o.o 0 23 LOAD23 1.0270 -16.69 3.2 1 .6 0 24 LOAD24CA 1.0212 -16.87 8.7 6.7 0 25 JUNCTN25 1 • 0163 -16. 51 o.o o.o 0 26 LOAD26 .9985 -16.93 3.5 2.3 0 27 CLOVERDA 1.0218 -16.02 o.o o.o 0 28 CLO VERDA 1.0067 -11.98 o.o o.o 0 29 LOAD29 1. 0018 -17.26 2.4 .9 0 30 LOAD30 .9903 -18.14 1 o.6 1.9 0
173
30 BUS IEEE INITIAL POWER FLOW SOLUTION
------ BRANCH FLOW SUMMARY ------
NO FROM BUS TO BUS TYPE MW MVAR MVA
1 1 GLEN LYN 2 CLAYTOR 0 177.8 -21.6 179.1 2 1 GLEN LYN 3 KUM IS 0 83.1 5.7 83.3 3 2 CLAYTOR 4 HANCOCK 0 45.7 3.3 45.8 4 3 KUMIS 4 HANCOCK 0 77.9 -2.7 77.9 5 2 CLAYTOR 5 FIELD ALE 0 83.0 1.5 83.0 6 2 CLAYTOR 6 ROANOKE 0 61.8 -.0 61.8 7 4 HANCOCK 6 ROANOKE 0 69.9 -15.8 71.7 8 5 FIELDALE 7 BLAINE 0 -14.1 11.5 18.2 9 6 ROANOKE 7 BLAINE 0 37.4 -2.8 37.5
10 6 ROANOKE 8 REUSENS 0 29.5 -8.3 30.6 11 6 ROANOKE 9 ROANOKE 1 27.9 -8.1 29.1 12 6 ROANOKE 10 ROANOKE 1 16.0 .2 16.0 13 9 ROANOKE 11 ROANOKE 0 o.o -15.8 15.8 14 9 ROANOKE 10 ROANOKE 0 27.9 6.0 28.6 15 4 HANCOCK 12 HANCOCK 1 44.2 14.2 46.4 16 12 HANCOCK 13 HANCOCK 0 o.o -10.5 10.5 17 12 HANCOCK 14 LOAD14 0 7.9 2.4 8.2 18 12 HANCOCK 15 LOAD15 0 17 .9 6.8 19.2 19 12 HANCOCK 16 LOAD16 0 7.2 3.2 7.9 20 14 LOAD14 15 LOAD15 0 1.6 .7 1. 7 21 16 LOAD16 17 LOAD17 0 3.6 1.3 3.9 22 15 LOAD15 18 LOAD18 0 6.0 1.6 6.2 23 18 LOAD18 19 LOAD19 0 2.7 .7 2.8 24 19 LOAD19 20 LOAD20 0 -6.8 -2.8 7.3 25 10 ROANOKE 20 LOAD20 0 9.1 3.7 9.8 26 10 ROANOKE 17 LOAD17 0 5.4 4.5 7.0 27 10 ROANOKE 21 LOAD21 0 15.9 9.9 18. 7 28 10 ROANOKE 22 JUNCTN22 0 7.7 4.5 8.9 29 21 LOAD21 22 JUNCTN22 0 -.8 -.7 1.1 30 21 LOAD21 22 JUNCTN22 0 -.8 -.7 1 .1 31 15 LOAD15 23 LOAD23 0 5.1 2.9 5.9 32 22 JUNCTN22 24 LOAD24CA 0 6.0 3.0 6.7 33 23 LOAD23 24 LOAD24CA 0 1.9 1.3 2.3 34 24 LOAD24CA 25 JUNCTN25 0 -.9 2.0 2.2 35 25 JUNCTN25 26 LOAD26 0 3.6 2.4 4.3 36 25 JUNCTN25 27 CLO VERDA 0 -4·4 -.4 4.4 37 27 CLO VERDA 28 CLOVERDA 1 -8.9 -1.9 9.1 38 27 CLOVER DA 28 CLO VERDA 1 -8.9 -1.9 9.1 39 27 CLO VERDA 29 LOAD29 0 6.2 1. 7 6.4 40 27 CLO VERDA 30 LOAD30 0 7 .1 1. 7 7.3 41 29 LOAD29 30 LOAD30 0 3.7 .6 3.8 42 8 REUSENS 28 CLO VERDA 0 -.6 -.3 .7 43 6 ROANOKE 28 CLOVERDA 0 6.1 -.0 6.1 44 6 ROANOKE 28 CLO VERDA 0 6.1 -.o 6.1 45 6 ROANOKE 28 CLOVERDA 0 6.1 -.o . 6.1
174
-------30 BUS IEEE FINAL POWER FLOW SOLUTION ----
----- GENERATOR SUMMARY -------
NO BUS NAME VOLTS MW MVAR QMAX QMIN TYPE
1 1 GLEN LYN 1.1000 259.5 -17.7 a.a o.o 3 2 2 CLAYTOR 1.0841 40.0 50.0 50.0 -40.0 1 3 5 FIELDALE 1.0555 o.o 40.0 40.0 -40.0 1 4 8 REUSENS 1. 0517 o.o 40.0 40.0 -10.0 1 5 11 ROANOKE 1.1000 o.o 8.7 24.0 -6.o 2 6 13 HANCOCK 1.0969 o.o 4.8 24.0 -6.o 2
-------30 BUS IEEE FINAL POWER FLOW SOLUTION ----
----- BUS SUMMARY -------
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
1 GLEN LYN 1.1000 o.oo o.o o.o 3 2 CLAYTOR 1 .0841 -5.05 21. 7 12. 7 1 3 KUMIS 1.0610 -7.31 2.4 1.2 0 4 HANCOCK 1.0520 -8.82 7.6 1 .6 0 5 FIELDALE 1.0555 -13.25 94.2 19.0 1 6 ROANOKE 1 • 0510 -10.42 o.o o.o 0 7 BLAINE 1.0454 -12.07 22.8 10.9 0 8 REUSENS 1.0517 -11.12 30.0 30.0 1 9 ROANOKE 1.0836 -13.27 o.o o.o 0
10 ROANOKE 1.0807 -14.76 5.8 2.0 0 11 ROANOKE 1.1000 -13.27 o.o o.o 2 12 HANCOCK 1.0908 -14.05 11.2 7.5 0 13 HANCOCK 1.0969 -14.05 o.o o.o 2 14 LOAD14 1.0768 -14.89 6.2 1.6 0 15 LOAD15 1.0726 -14.99 8.2 2.5 0 16 LOAD16 1.0796 -14.61 3.5 1.8 0 17 LOAD17 1.0753 -14.91 9.0 5.s 0 18 LOAD18 1 .0637 -15.56 3.2 .9 0 19 LOAD19 1.0615 -15.72 9.5 3.4 0 20 LOAD20 1 .06.56 -15.53 2.2 .7 0 21 LOAD21 1.0690 -15.19 17.5 11.2 0 22 JUNCTN22 1.0695 -15.17 o.o o.o 0 23 LOAD23 1.0633 -15.37 3.2 1.6 0 24 LOAD24CA 1.0590 -15.55 8.7 6.7 0 25 JUNCTN25 1.0564 -15.22 o.o o.o 0 26 LOAD26 1.0394 -15.61 3.5 2.3 0 27 CLO VERDA 1.0630 -14.77 o.o o.o 0
175
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
28 CLOVERDA 1.0477 -10.99 29 LOAD29 1.0440 -15.91 30 LOAD30 1.0329 -16.72
-------30 BUS IEEE FINAL
o.o 2.4
10.6
o.o .9
1.9
0 0 0
POWER FLOW SOLUTION ----
------ BRANCH FLOW SUMMARY ------
NO FROM BUS TO BUS TYPE MW MVAR
1 1 GLEN LYN 2 CLAYTOR 0 175.8 -23.5 2 1 GLEN LYN 3 KUM IS 0 82.2 5.8 3 2 CLAYTOR 4 HANCOCK 0 45.4 4.4 4 3 KUM IS 4 HANCOCK 0 77.4 -1.2 5 2 CLAYTOR 5 FIELDALE 0 82.8 -.6 6 2 CLAYTOR 6 ROANOKE 0 61.5 .8 7 4 HANCOCK 6 ROANOKE 0 69.8 -16.9 8 5 FIELDALE 7 BLAINE 0 -13.8 13. 7 9 6 ROANOKE 7 BLAINE 0 37 .1 -5.4
10 6 ROANOKE 8 REUSENS 0 29.5 -1o.7 11 6 ROANOKE 9 ROANOKE 1 27.8 -3.9 12 6 ROANOKE 10 ROANOKE 1 16.0 1.4 13 9 ROANOKE 11 ROANOKE 0 .o -8.6 14 9 ROANOKE 10 ROANOKE 0 27.8 3.2 15 4 HANCOCK 12 HANCOCK 1 43.9 18. 7 16 12 HANCOCK 13 HANCOCK 0 .o -4.7 17 12 HANCOCK 14 LOAD14 0 7.8 2.3 18 12 HANCOCK 15 LOAD15 0 17 .8 6.3 19 12 HANCOCK 16 LOAD16 0 7 .1 2.8 20 14 LOAD14 15 LOAD15 0 1.5 .5 21 16 LOAD16 17 LOAD17 0 3.6 .9 22 15 LOAD15 18 LOAD18 0 5.9 1.5 23 18 LOAD18 19 LOAD19 0 2.7 .5 24 19 LOAD19 20 LOAD20 0 -6.8 -2.9 25 10 ROANOKE 20 LOAD20 0 9.1 3.8 26 10 ROANOKE 17 LOAD17 0 5.4 4.9 27 10 ROANOKE 21 LOAD21 0 15.8 9.6 28 10 ROANOKE 22 JUNCTN22 0 7.6 4.4 29 21 LOAD21 22 JUNCTN22 0 -.9 -.8 30 21 LOAD21 22 JUNCTN22 0 .-.9 -.8 31 15 L0AD15 23 LOAD23 0 5.0 2.5 32 22 JUNCTN22 24 LOAD24CA 0 5.8 2.6 33 23 LOAD23 24 LOAD24CA 0 1.8 .8
MVA
177.3 82.4 45.6 77.4 82.8 61.5 71.8 19.5 37.5 31.4 28.1 16.0 8.6
28.0 47.7 4.7 8.1
18.9 7.6 1.6 3.7 6.1 2.7 7.4 9.9 7.3
18.5 8.8 1.2 1.2 5.6 6.4 2.0
176
NO FROM BUS TO BUS TYPE MW MVAR MVA
34 24 LOAD24CA 25 JUNCTN25 0 -1 .1 1.5 1.9 35 25 JUNCTN25 26 LOAD26 0 3.5 2.4 4.3 36 25 JUNCTN25 27 CLOVERDA 0 -4.7 -.9 4.8 37 27 CLO VERDA 28 CLOVER DA 1 -9.0 -2.1 9.2 38 27 CLOVERDA 28 CLOVERDA 1 -9.0 -2.1 9.2 39 27 CLOVER DA 29 LOAD29 0 6.2 1. 7 6.4 40 27 CLOVERDA 30 LOAD30 0 7 .1 1. 7 7.3 41 29 LOAD29 30 LOAD30 0 3.7 .6 3.8 42 8 REUS ENS 28 CLOVERDA 0 -.5 -.1 .5 43 6 ROANOKE 28 CLO VERDA 0 6.2 -.1 6.2 44 6 ROANOKE 28 CLOVERDA 0 6.2 -.1 6.2 45 6 ROANOKE 28 CLO VERDA 0 6.2 -.1 6.2
177
30 BUS IEEE UNIT VOLTAGES CONVERGENCE
ITER LOSS UNIT 1 UNIT 2 UNIT 3 UNIT 4 UNIT 5 UNIT 6 ------- ------- ------- ------- ------- -------
0 17.6 1 .06 1.043 1. 01 1.01 1.082 1 .071
1 16.69 1.100 1.069 1.0084 1.0184 1.0844 1.0741
2 16 .11 1.100 1 .0841 1 .0555 1 .0517 1.100 1.096
30 BUS IEEE UNIT MVARS CONVERGENCE
ITER LOSS UNIT 1 UNIT 2 UNIT 3 UNIT 4 UNIT 5 UNIT 6 ------- ------- ------- ------- ------- -------
0 17.6 -15.89 50.0 37 .11 37.45 16.24 1 o.6
1 16.69 21.27 50.0 14.76 25.81 12.92 5.90
2 16 .11 -17. 7 50.0 40.0 40.0 8.7 4.s
30 BUS IEEE UNIT REDUCED GRADIENT CONVERGENCE
ITER LOSS UNIT 1 UNIT 2 UNIT 3 UNIT 4 UNIT 5 UNIT 6 ------- ------- ------- ------- ------- -------
1 16.69 .018 .024 -.012 .067 .000 .311
2 16.11 .018 .025 .119 .143 .ooo .ooo
3 16 .11 .028 .041 .113 .ooo .000 .ooo
4 16.11 .027 .041 -.001 -.001 .ooo .ooo
5 16.11 .ooo .ooo -.001 .ooo .000 .ooo
..
178
57 BUS IEEE UNIT VOLTAGES CONVERGENCE
ITER LOSS UNIT 1 UNIT 2 UNIT 3 UNIT 4 UNIT 5 UNIT 6 UNIT 7
1. 01 .985 .98 1.005 • 980 1 .015 0 28 .11 1. 04
1 16.69 10745 1.063 1.050 1.0288 1.0463 1.0155 1.0227
57 BUS IEEE UNIT MVARS CONVERGENCE
ITER LOSS UNIT 1 UNIT 2 UNIT 3 UNIT 4 UNIT 5 UNIT 6 UNIT 1
0 28.11 129.72 -.75 -1.96 -.11
1 16.69 69.8 50.0 60.00 -8.00
62.42
67.00
3.75 129.9
46.5
APPENDIX 7
23 BUS RESULTS
179
180
-------------- BEGIN OPTIMAL POWER FLOW --------------
23 BUS NETWORK
MAX OPF ITERATIONS 6 MAX PF ITERATIONS 10 MVAR LIMITING CONTROL 1 MODERATE ITER FOR MVAR CNTL 2
HIGH VOLTAGE LIMIT 1.10 LOW VOLTAGE LIMIT .85
NUMBER OF BUSSES 23 NUMBER OF BRANCHES 30 NUMBER OF GENERATORS 6 NUMBER OF TRANSFORMERS 4
-----23 BUS INITIAL GENERATOR DATA -------
NO BUS NAME VOLTS MW QMAX QMIN
1 1 1.0000 45.0 240.0 -30.0 2 2 1.0000 102.0 240.0 -30.0 3 11 1.0000 211.0 300.0 -40.0 4 14 1.0000 500.4 660.0 -120.0 5 20 1.0000 803.0 970.0 -155.0 6 23 1.0000 1028.0 1270.0 -205.0
181
------23 BUS BRANCH DATA SUMMARY ------
NO FROM BUS TO BUS TYPE RESISTANCE INDUCTANCE CHARG/TAP
1 1 3 0 .0242 .0540 .0118 2 1 4 0 .0309 .0693 .0151 3 2 5 0 .0404 .0888 .0197 4 8 5 0 .0325 .0709 .0157 5 2 7 0 .0615 .1620 .0342 6 3 6 0 .0576 .1520 .0320 7 4 9 0 .0266 .0700 .0148 8 9 7 0 .0229 .0504 .0112 9 8 6 0 .0446 .1003 .0218
10 11 10 0 .0233 .0514 .0456 11 8 10 0 .0597 .1315 .0291 12 9 10 0 .0597 .1315 .0291 13 13 14 0 .0043 .0351 .2373 14 14 12 0 .0043 .0351 .2373 15 15 12 0 .0038 .0307 .2078 16 18 15 0 .0035 .0288 .1951 17 23 13 0 .0089 .0720 -4871 18 16 17 0 .0010 .0080 .0543 19 17 18 0 .0021 .0167 .1133 20 19 18 0 .0016 .0127 .0862 21 20 19 0 .0045 .0362 .2451 22 22 18 0 .0024 .0192 .1298 23 20 21 0 .0019 .0156 .1056 24 21 22 0 .0014 .0114 .0770 25 23 16 0 .0020 .0164 .1109 26 12 8 1 .0023 .0839 1.0000 27 13 8 1 .0023 .0839 1.0000 28 12 9 1 .0019 .1300 1 .oooo 29 13 9 1 .0023 .0839 1.0000 30 1 2 0 .0025 .2000 0.0000
182
23 BUS INITIAL POWER FLOW SOLUTION
----- GENERATOR SUMMARY -------
NO BUS NAME VOLTS MW MVAR QMAX QMIN TYPE
1 1 1.0000 253.4 23.5 240.0 -30.0 3 2 2 1.0000 102.0 61.0 240.0 -30.0 2 3 11 1.0000 211.0 92.9 300.0 -40.0 2 4 14 1.0000 500.4 200.8 660.0 -120.0 2 5 20 1.0000 803.0 190.5 970.0 -155.0 2 6 23 1.0000 1028. 179.5 1270.0 -205.p 2
23 BUS INITIAL POWER FLOW SOLUTION
----- BUS SUMMARY -------
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
1 1.0000 o.oo 64.0 16.0 3 2 1.0000 -5.38 101.0 25.0 2 3 .9912 -1.20 o.o o.o 0 4 .9653 -4.10 47.0 12.0 0 5 .9749 -6.03 51.0 13.0 0 6 .9694 -4.61 41.0 1 o.o 0 7 .9565 -7.33 48.0 12.0 0 8 .9804 -4.61 1.0 o.o 0 9 .9589 -6.56 150.0 38.0 0
10 .9489 -9.91 177.0 44.0 0 11 1.0000 -8.25 130.0 32.0 2 12 .9767 -3.94 60.0 o.o 0 13 .9868 -.30 -40.0 o.o 0 14 1.0000 -2.03 480.0 120.0 2 15 .9570 -3.64 201.0 50.0 0 16 .9652 4 .31 132.0 33.0 0 17 .9552 1.86 344.0 86.0 0 18 .9576 .19 104.0 26.0 0 19 .9538 .12 376.0 94.0 0 20 1..0000 7.93 -100.0 -25.0 2 21 .9734 3.19 375.0 94.0 0 22 .9720 2.14 -21.0 -52.0 0 23 1.0000 10.39 129.0 32.0 2
183
23 BUS INITIAL POWER FLOW SOLUTION
------ BRANCH FLOW SUMMARY ------
NO FROM BUS TO BUS TYPE MW MVAR MVA
1 1 3 0 38.3 -1.1 38.3 2 1 4 0 103 .1 7.0 103.4 3 2 5 0 21. 0 17 .8 27.5 4 8 5 0 30.7 -6.8 31.4 5 2 7 0 26.6 15.4 30.7 6 3 6 0 37.9 -.6 38.0 7 4 9 0 52.8 -10.8 53.9 8 9 7 0 22.1 -5.8 22.9 9 8 6 0 4.0 7.9 8.9
10 11 10 0 81.9 60.9 102.0 11 8 10 0 64.2 -4.0 64.3 12 9 10 0 36.7 -9°5 37.9 13 13 14 0 79.2 -57.2 97.7 14 14 12 0 99.3 44.0 108.6 15 15 12 0 7.9 -71.7 72.1 16 18 15 0 210.6 -25.7 212.1 17 23 13 0 255.7 -13.8 256.0 18 16 17 0 502.4 66.5 506.8 19 17 18 0 155.7 -36.2 159.8 20 19 18 0 -11.5 -30.6 32.7 21 20 19 0 371.2 93.6 382.8 22 22 18 0 171.8 48.1 178.4 23 20 21 0 531.8 121.8 545.6 24 21 22 0 151 .1 -8.5 151.4 25 23 16 0 643.3' 161.3 663.2 26 12 8 1 13.3 -4°7 14.1 27 13 8 1 86.8 8.3 87.2 28 12 9 1 33.2 13.6 35.9 29 13 9 1 123.9 36.1 129.0 30 1 2 0 46.9 1.6 46.9
184
-------23 BUS FINAL POWER FLOW SOLUTION ----
----- GENERATOR SUMMARY -------
NO BUS NAME VOLTS MW MVAR QMAX QMIN TYPE
1 1 1.1000 243.7 27.4 240.0 -30.0 3 2 2 1.0939 102.0 51.9 240.0 -30.0 2 3 11 1.0738 211.0 67.3 300.0 -40.0 2 4 14 1.1000 500.4 186.9 660.0 -120.0 2 5 20 1.1000 803.0 153.8 970.0 -155.0 2 6 23 1.1000 1028. 143.5 1270.0 -205.0 2
-------23 BUS FINAL POWER FLOW SOLUTION ----
----- BUS -SUMMARY -------
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
1 1.1000 o.oo 64.0 16.0 3 2 1.0939 -4.15 101.0 25.0 2 3 1.0922 -.94 o.o o.o 0 4 1.0682 -3.22 47.0 12.0 0 5 1.0736 -4.66 51.0 13.0 0 6 1. 0717 -3.58 41.0 10.0 0 7 1.0580 -5.73 48.0 12.0 0 8 1.0802 -3.45 1.0 o.o 0 9 1.0609 -5.08 150.0 38.0 0
10 1.0386 -7.52 177.0 44.0 0 11 1.0738 -5.83 130.0 32.0 2 12 1.0803 -2.84 60.0 o.o 0 13 1.0891 .12 -40.0 o.o 0 14 1.1000 -1.28 480.0 120.0 2 15 1.0644 -2.54 201.0 50.0 0 16 1.0711 3.97 132.0 33.0 0 17 1.0625 1.98 344.0 86.0 0 18 1.0650 • 61 104.0 26.0 0 19 1.0615 .57 376.'o 94.0 0 20 1.1000 6.98 -100.0 -25.0 2 21 1.0779 3.07 375.0 94.0 0 22 1.0773 2.21 -21.0 -52.0. 0 23 1.1000 8.97 129.0 32.0 2
185
-------23 BUS FINAL POWER FLOW SOLUTION ----
------ BRANCH FLOW SUMMARY ------
NO FROM BUS TO BUS TYPE MW MVAR MVA
1 1 3 0 36.4 -.9 36.4 2 1 4 0 99.3 8.0 99.6 3 2 5 0 19.1 15.2 24.4 4 8 5 0 32.3 -5.3 32.7 5 2 7 0 25.3 12.9 28.4 6 3 6 0 36.2 -.1 36.2 7 4 9 0 49.9 -7.8 50.5 8 9 7 0 23.2 -4.8 23.7 9 8 6 0 5.5 5.5 7.7
10 11 10 0 80.8 35.3 88.2 11 8 10 0 63.8 5.7 64.0 12 9 10 0 36.6 .5 36.6 13 13 14 0 78.1 -56.3 96.3 14 14 12 0 98.2 36.7 104.8 15 15 12 0 12.1 -68.3 69.4 16 18 15 0 214.6 -28.8 216.5 17 23 13 0 256.5 -24.8 257.7 18 16 17 0 503.3 57.4 506.6 19 17 18 0 157 .1 -40.3 162.2 20 19 18 0 -10.0 -32.8 34.3 21 20 19 0 371.5 76.1 379.2 22 22 18 0 172.6 42.0 177.7 23 20 21 0 531.5 102.7 541.4 24 21 22 0 151.9 -16.6 152.8 25 23 16 0 642.5 136.3 656.8 26 12 8 1 15.0 -.3 15.0 27 13 8 1 87.7 11.9 88.5 28 12 9 1 34.8 16.3 38.4 29 13 9 1 125.9 38.9 131.8 30 1 2 0 43.6 4.4 43.8
186
23 BUS UNIT VOLTAGES CONVERGENCE
ITER LOSS UNIT 1 UNIT 2 UNIT 3 UNIT 4 UNIT 5 UNIT 6 ------- ------- ------- ------- ------- -------
0 47.75 1. 00 1 .oo 1.00 1 .oo 1 .oo 1.00
1 41.91 1.0416 1.0069 .9677 1.0331 1.0705 1.100
2 40.24 1. 0414 1 .0186 .9955 1.0494 1.1000 1.100
3 3e.45 1.0728 1.0567 1.1000 1.1000 1.1000 1.100
4 38.24 1.1000 1.0816 1 .0592 1.0957 1.1000 1.100
5 38.16 1.100 1.0845 1.0635 1.1000 1.1000 1.100
6 38.11 1.1000 1.0935 1.0735 1.1000 1 .1000 1.100
7 38.11 1.1000 . 1.0935 1.0735 1.1000 1.1000 1.100
23 BUS UNIT MVARS CONVERGENCE
ITER LOSS UNIT 1 UNIT 2 UNIT 3 UNIT 4 UNIT 5 UNIT 6 ------- ------- ------- ------- ------- -------
0 47.75 23.51 60.96 92.86 200.77 190.48 179.51
1 41 .91 51.19 17 .23 37.82 120. 91 148.98 301.48
2 40.24 31.21 20.42 48.72 132.35 194.53 230.96
3 38.45 18.968 37.767 61.440 208.80 157. 01 151. 00
4 38.24 40.320 38.17 59.52 185 .15 157.70 151.89
5 38.16 35.90 39.89 60.94 194.17 154.88 145.96
6 38.11 27.430 51.89 67.28 186.88 153.80 143.47
7 38.11 27.58 51. 71 67.09 189.00 153.8 143.52
187
23 BUS UNIT REDUCED GRADIENT CONVERGENCE
ITER LOSS UNIT 1 UNIT 2 UNIT 3 UNIT 4 UNIT 5 UNIT 6 ------- ------- ------- ------- ------- -------
1 41. 91 -.173 .037 .177 .377 .535 .223
2 40.24 .295 .124 .174 .313 .ooo -.002
3 38.45 .160 .141 .131 .ooo .ooo .103
4 38.24 .016 .065 -.032 .ooo .ooo .141
5 38.16 .052 .035 .0520 .ooo .ooo .ooo
APPENDIX 8
57 BUS IEEE RESULTS
188
189
-------------- BEGIN OPTIMAL POWER FLOW --------------
50 BUS IEEE NETWORK
MAX OPF ITERATIONS 6 MAX PF ITERATIONS 10 MVAR LIMITING CONTROL 1 MODERATE ITER FOR MVAR CNTL 2
HIGH VOLTAGE LIMIT 1.10 LOW VOLTAGE LIMIT .85
NUMBER OF BUSSES 57 NUMBER OF BRANCHES 80 NUMBER OF GENERATORS 7 NUMBER OF TRANSFORMERS 17
----- IEEE 57 BUS INITIAL GENERATOR DATA -------
NO BUS NAME VOLTS MW QMAX QMIN
1 1 AMOS 1.0400 o.o 200.0 -200.0 2 2 BAKER 1. 0100 o.o 50.0 -17 .o 3 3 CRAWFORD .9850 40.0 60.0 -1 o.o 4 6 ELLS .9800 o.o 25.0 -8.0 5 8 GRANGE 1.0050 450.0 200.0 -140.0 6 9 HOMER .9800 o.o 9.0 -3.0 7 12 LOESCHER 1.0150 31 o.o 155.0 -50.0
------57 BUS IEEE BRANCH DATA SUMMARY ------
NO FROM BUS TO BUS TYPE RESISTANCE INDUCTANCE CHARG/TAP
1 1 AMOS 2 BAKER 0 .0083 .0280 .1290 2 1 AMOS 15 OAKDALE 0 .0178 .0910 .0988 3 1 AMOS 16 ANDY 0 .0454 .2060 .0546 4 1 AMOS 17 ANDY 0 .0238 . .1080 .0286 5 2 BAKER 3 CRAWFORD 0 .0298 .0850 .0818 6 3 CRAWFORD 4 DOYLE 0 .0112 .0366 .0380 7 3 CRAWFORD 15 OAKDALE 0 .0162 .0530 .0544 8 4 DOYLE 5 DAWSON 0 .0625 .1320 .0258 9 4 DOYLE 6 ELLS 0 .0430 .1480 .0348
190
NO FROM BUS TO BUS TYPE RESISTANCE INDUCTANCE CHARG/TAP
10 4 DOYLE 18 DOYLE 1 0.0000 .5550 .9700 11 4 DOYLE 18 DOYLE 1 0.0000 .4300 .9780 12 5 DAWSON 6 ELLS 0 .0302 .0641 .0124 13 6 ELLS 7 FARLIE 0 .0200 .1020 .0276 14 6 ELLS 8 GRANGE 0 .0339 .1730 .0470 15 7 FARLIE 8 GRANGE 0 .0139 .0712 .0194 16 7 FARLIE 29 FARLIE 1 0.0000 .0648 .9670 17 8 GRANGE 9 HOMER 0 .0099 .0505 .0548 18 9 HOMER 10 JENKINS 0 .0369 .1679 .0440 19 9 HOMER 11 KINCAID 0 .0258 .0848 .0218 20 9 HOMER 12 LOESCHER 0 .0648 .2950 .0772 21 9 HOMER 13 MOSES 0 .0481 .1580 .0406 22 9 HOMER 55 HOMER 1 0.0000 .1205 .9400 23 10 JENKINS 12 LOESCHER 0 .0277 .1262 .0328 24 10 JENKINS 51 JENKINS 1 0.0000 .0712 .9300 25 11 KINCAID 13 MOSES 0 .0223 .0732 .0188 26 11 KINCAID 41 KINCAID 1 0.0000 .7490 .9550 27 11 KINCAID 43 KINCAID 1 0.0000 .1530 .9580 28 12 LOESCHER 13 MOSES 0 .0178 .0580 .0604 29 12 LOESCHER 16 ANDY 0 .0180 .0813 .0216 30 12 LOESCHER 17 ANDY 0 .0397 .1790 .0476 31 13 MOSES 14 NESTLE 0 .0132 .0434 .0110 32 13 MOSES 15 OAKDALE 0 .0269 .0869 .0230 33 13 MOSES 49 MOSES 1 0.0000 .1910 .8950 34 14 NESTLE 15 OAKDALE 0 .0171 .0547 .0148 35 14 NESTLE 46 NESTLE 1 0.0000 .0735 .9000 36 15 OAKDALE 45 OAKDALE 1 0.0000 .1042 .9550 37 18 DOYLE 19 RICHTER 0 .4610 .6850 0.0000 38 19 RICHTER 20 RICHTER 0 .2830 .4340 0.0000 39 20 RICHTER 21 RICHTER 1 0.0000 .7767 1.0430 40 21 RICHTER 22 BEAVER2 0 .0736 .1170 0.0000 41 22 BEAVER2 23 BEAVER1 0 .0099 .0152 0.0000 42 22 BEAVER2 38 STANTON 0 .0192 .0295 0.0000 43 23 BEAVER1 24 POOL 0 .1660 .2560 .0084 44 24 POOL 25 PO OLA 1 0.0000 1.1820 1 .0000 45 24 POOL 25 PO OLA 1 0.0000 1 .2300 1.0000 46 24 POOL 26 POOL 1 0.0000 .0473 1.0430 47 25 PO OLA 30 CHESTER 0 .1350 .2020 0.0000 48 26 POOL 27 HAMEL 0 .1650 .2540 0.0000 49 27 HAMEL 28 WYN COTE 0 .0618 .0954 0.0000 50 28 WYN COTE 29 FARLIE 0 .0418 .0587 0.0000 51 29 FARLIE 52 N VEXLEY 0 .1442 .1870 0.0000 52 30 CHESTER . 31 HANOVER 0 .3260 .4970 0.0000 53 31 HANOVER 32 UXBRIDGE 0 .5070 .7550 0.0000 54 32 UXBRIDGE 33 LUXBRIDG 0 .0392 .0360 0.0000 55 32 UXBRIDGE 34 UXBRIDGE 1 0.0000 .9530 .9750 56 34 UXBRIDGE 35 COPLEY M 0 .0520 .0780 .0032 57 35 COPLEY M 36 COPLEY 0 .0430 .0537 .0016
191
NO FROM BUS TO BUS TYPE RESISTANCE INDUCTANCE CHARG/TAP
58 36 COPLEY 37 N COPLEY 0 .0290 .0366 0.0000 59 37 N COPLEY 38 STANTON 0 .0651 .1009 .0020 60 36 COPLEY 40 TAUNTON 0 .0300 .0466 0.0000 61 37 N COPLEY 39 W TAUNTO 0 .0239 .0379 0.0000 62 38 STANTON 44 S OAKDAL 0 .0289 .0585 .0020 63 38 STANTON 48 AIRPORT1 0 .0312 .0482 0.0000 64 38 STANTON 49 MOSES 0 .1150 .1770 .0060 65 39 W TAUNTO 57 W TAUNTO 1 0.0000 1.3550 .9800 66 40 TAUNTON 56 W TAUNTO 1 0.0000 1.1950 .9580 67 41 KINCAID 42 AIRPORT3 0 .2070 .3520 0.0000 68 41 KINCAID 43 KINCAID 0 0.0000 .4120 0.0000 69 44 S OAKDAL 45 OAKDALE 0 .0624 .1242 .0040 70 46 NESTLE 47 AIRPORT2 0 .0230 .0680 .0032 71 47 AIRPORT2 48 AIRPORT1 0 .0182 .0233 0.0000 72 48 AIRPORT1 49 MOSES 0 .0834 .1290 .0048 73 49 MOSES 50 MANX 0 .0801 .1280 0.0000 74 50 MANX 51 JENKINS 0 .1386 .2200 0.0000 75 52 N VEXLEY 53 VEXLEY 0 .0762 .0984 0.0000 76 53 VEXLEY 54 VEXLEY S 0 .1878 .2320 0.0000 77 54 VEXLEY S 55 HOMER 0 .1732 .2265 0.0000 78 56 W TAUNTO 41 KINCAID 0 .5530 .5490 0.0000 79 56 W TAUNTO 42 AIRPORT3 0 .2125 .3540 0.0000 80 57 W TAUNTO 56 W TAUNTO 0 .1740 .2600 0.0000
IEEE 57 BUS INITIAL POWER FLOW SOLUTION
----- GENERATOR SUMMARY -------
NO BUS NAME VOLTS MW MVAR QMAX QMIN TYPE
1 1 AMOS 1.0400 478.9 129.4 200.0 -200.0 3 2 2 BAKER 1.0100 o.o -.7 50.0 -17 .o 2 3 3 CRAWFORD .9850 40.0 -2.4 60.0 -10.0 2 4 6 ELLS .9800 o.o .4 25.0 -8.0 2 5 8 GRANGE 1. 0050 450.0 62.9 200.0 -140.0 2 6 9 HOMER .9800 o.o 3.3 9.0 -3.0 2 7 12 LOESCHER 1. 0150 310.0 129.6 155.0 -50.0 2
..
192
IEEE 57 BUS INITIAL POWER FLOW SOLUTION
----- BUS SUMMARY -------
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
1 AMOS 1.0400 o.oo 55.0 17 .o 3 2 BAKER 1.0100 -1.19 3.0 88.0 2 3 CRAWFORD .9850 -5.98 41.0 21.0 2 4 DOYLE .9817 -7.33 o.o o.o 0 5 DAWSON .9767 -8.53 13.0 4.0 0 6 ELLS .9800 -8.66 75.0 2.0 2 7 FARLIE .9837 -7.60 o.o o.o 0 8 GRANGE 1.0050 -4.48 150.0 22.0 2 9 HOMER .9800 -9.59 121.0 26.0 2
10 JENKINS .9861 -11.46 5.0 2.0 0 11 KINCAID .9736 -10.20 o.o o.o 0 12 LOESCHER 1.0150 -10.48 377.0 24.0 2 13 MOSES .9784 -9.81 18.0 2.3 0 14 NESTLE .9694 -9.35 10.5 5.3 0 15 OAKDALE .9875 -7 .19 22.0 5.0 0 16 ANDY 1.0134 -8.86 43.0 3.0 0 17 ANDY 1 .0175 -5.40 42.0 8.0 0 18 DOYLE 1.0094 -11.54 27.2 9.8 0 19 RICHTER 1.0046 -13.36 3.3 .6 0 20 RICHTER 1.0141 -13.80 2.3 1.0 0 21 RICHTER .9993 -12.85 o.o o.o 0 22 BEAVER2 1.0050 -12.86 o.o o.o 0 23 BEAVER1 1.0034 -12.92 6.3 2.1 0 24 POOL .9917 -13.26 o.o o.o 0 25 PO OLA .9662 -18.43 6.3 3.2 0 26 POOL .9521 -12.93 o.o o.o 0 27 HAMEL .9781 -11.50 9.3 .5 0 28 WYNCOTE .9945 -10.48 4.6 2.3 0 29 FARLIE 1.0090 -9.78 17.0 2.6 0 30 CHESTER .9426 -18.97 3.6 1.8 0 31 HANOVER .9074 -19.59 5.8 2.9 0 32 UXBRIDGE .9110 -18.52 1.6 .8 0 33 LUXBRIDG .9085 -18.56 3.8 1.9 0 34 UXBRIDGE .9586 -14.22 o.o o.o 0 35 COPLEY M .9647 -13.96 6.0 3.0 0 36 COPLEY .9737 -13.67 o.o o.o 0 37 N COPLEY .9825 -13.47 o.o o.o 0 38 STANTON 1.0095 -12.74 14.0 7.0 0 39 W TAUNTO .9804 -13.52 o.o o.o 0
193
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
40 TAUNTON .9707 -13.69 o.o o.o 0 41 KINCAID .9952 -14.10 6.3 3.0 0 42 AIRPORT3 .9649 -15.55 7 .1 4.4 0 43 KINCAID 1.0090 -11.36 2.0 1.0 0 44 S OAKDAL 1. 0140 -11.86 12.0 1.8 0 45 OAKDALE 1.0345 -9.28 o.o o.o 0 46 NESTLE 1 .0583 -11.13 o.o o.o 0 47 AIRPORT2 1.0311 -12.53 29.7 11.6 0 48 AIRPORT1 1.0249 -12.62 o.o o.o 0 49 MOSES 1.0345 -12.96 18.0 8.5 0 50 MANX 1.0221 -13.43 21.0 10.5 0 51 JENKINS 1 .0519 -12.55 18.0 5.3 0 52 N VEXLEY .9787 -11.51 4.9 2.2 0 53 VEXLEY .9693 -12.26 20.0 1 o.o 0 54 VEXLEY S .9954 -11.72 4.1 1.4 0 55 HOMER 1.0307 -10.81 6.8 3.4 0 56 W TAUNTO .9667 -16.09 7.6 2.2 0 57 W TAUNTO .9629 -16.61 6.7 2.0 0
194
IEEE 57 BUS INITIAL POWER FLOW SOLUTION
------ BRANCH FLOW SUMMARY ------
NO FROM BUS TO BUS TYPE MW MVAR MVA
1 1 AMOS 2 BAKER 0 102.0 75.0 126.6 2 1 AMOS 15 OAKDALE 0 149.0 34.4 152.9 3 1 AMOS 16 ANDY 0 79.3 -.9 79.3 4 1 AMOS 17 ANDY 0 93.4 3.9 93.5 5 2 BAKER 3 CRAWFORD 0 97.6 -4.6 97.7 6 3 CRAWFORD 4 DOYLE 0 59.6 -10.4 60.5 7 3 CRAWFORD 15 OAKDALE 0 34.3 -17.3 38.5 8 4 DOYLE 5 DAWSON 0 14.0 -4.0 14.6 9 4 DOYLE 6 ELLS 0 14.3 -4°5 15.0
10 4 DOYLE 18 DOYLE 1 13.5 1.0 13.6 11 4 DOYLE 18 DOYLE 1 17 .3 -.7 17 .3 12 5 DAWSON 6 ELLS 0 .8 -5.9 6.0 13 6 ELLS 7 FARLIE 0 -17 .5 -1.3 17.5 14 6 ELLS 8 GRANGE 0 -42.3 -6.6 42.9 15 7 FARLIE 8 GRANGE 0 -77°9 -13.1 79.0 16 7 FARLIE 29 FARLIE 1 60.3 14.1 62.0 17 8 GRANGE 9 HOMER 0 178.2 19.8 179.3 18 9 HOMER 10 JENKINS 0 17.2 -9.1 19.5 19 9 HOMER 11 KINCAID 0 13.0 2.5 13.2 20 9 HOMER 12 LOESCHER 0 2.6 -15.9 16.1 21 9 HOMER 13 MOSES 0 2.4 -1. 7 2.9 22 9 HOMER 55 HOMER 1 19.0 10.5 21.7 23 10 JENKINS 12 LOESCHER 0 -17.7 -20.2 26.9 24 10 JENKINS 51 JENKINS 1 29.8 12.8 32.4 25 11 KINCAID 13 MOSES 0 -9.9 -4°3 10.8 26 11 KINCAID 41 KINCAID 1 9.2 3.6 9.9 27 11 KINCAID 43 KINCAID 1 13.6 5.0 14.5 28 12 LOESCHER 13 MOSES 0 -.4 61.2 61.2 29 12 LOESCHER 16 ANDY 0 -33°5 8.8 34.6 30 12 LOESCHER 17 ANDY 0 -48.5 9.2 49.4 31 13 MOSES 14 NESTLE 0 -10.3 22.9 25.1 32 13 MOSES 15 OAKDALE 0 -48.9 5.0 49.2 33 13 MOSES 49 MOSES 1 32.6 34.5 47.4 34 14 NESTLE 15 OAKDALE 0 -69.0 -9°9 69.7 35 14 NESTLE 46 NESTLE 1 48.2 28.3 55.9 36 15 OAKDALE 45 OAKDALE 1 37.5 .2 37.6 37 18 DOYLE 19 RICHTER 0 3.6 -1.6 4.0 38 19 RICHTER 20 RICHTER 0 .2 -2.4 2.4 39 20 RICHTER 21 RICHTER 1 -2.1 -3°4 3.9
195
NO FROM BUS TO BUS TYPE MW MVAR MVA
40 21 RICHTER 22 BEAVER2 0 -2.1 -3.5 4.1 41 22 BEAVER2 23 BEAVER1 0 1 o.o 3.9 1o.7 42 22 BEAVER2 38 STANTON 0 -12.1 -7·4 14.2 43 23 BEAVER1 24 POOL 0 3.7 1.8 4.1 44 24 POOL 25 PO OLA 1 7.3 2.5 7.7 45 24 POOL 25 PO OLA 1 7.0 2.4 7.4 46 24 POOL 26 POOL 1 -10.8 -2.4 11.0 47 25 PO OLA 30 CHESTER 0 8.1 5.9 1 o.o 48 26 POOL 27 HAMEL 0 -10.9 -2.6 11.2 49 27 HAMEL 28 WYN COTE 0 -20.4 -3.5 20.7 50 28 WYN COTE 29 FARLIE 0 -25.3 -6.3 26.1 51 29 FARLIE 52 N VEXLEY 0 18.0 2.7 18.2 52 30 CHESTER 31 HANOVER 0 4.4 3.8 5.8 53 31 HANOVER 32 UXBRIDGE 0 -1.6 .7 1. 7 54 32 UXBRIDGE 33 LUXBRIDG 0 3.9 2.0 4.4 55 32 UXBRIDGE 34 UXBRIDGE 1 -7.0 -2.1 7.4 56 34 UXBRIDGE 35 COPLEY M 0 -7.2 -2.8 7.7 57 35 COPLEY M 36 COPLEY 0 -13.3 -5.7 14.5 58 36 COPLEY 37 N COPLEY 0 -16.9 -9.9 19.6 59 37 N COPLEY 38 STANTON 0 -20.8 -12.9 24.4 60 36 COPLEY 40 TAUNTON 0 3.5 4.1 5.4 61 37 N COPLEY 39 W TAUNTO 0 3.9 2.9 4.8 62 38 STANTON 44 S OAKDAL 0 -24.5 4.4 24.9 63 38 STANTON 48 AIRPORT1 0 -17 .8 -20.8 27.4 64 38 STANTON 49 MOSES 0 -4.9 -11.4 12.4 65 39 W TAUNTO 57 W TAUNTO 1 3.8 2.9 4.8 66 40 TAUNTON 56 W TAUNTO 1 3.4 4.0 5.3 67 41 KINCAID 42 AIRPORT3 0 8.9 3.4 9.6 68 41 KINCAID 43 KINCAID 0 -11.6 -3.0 12.0 69 44 S OAKDAL 45 OAKDALE 0 -36.7 2.4 36.8 70 46 NESTLE 47 AIRPORT2 0 48.2 26.3 54.9 71 47 AIRPORT2 48 AIRPORT1 0 18.0 13.5 22.5 72 48 AIRPORT1 49 MOSES 0 -.1 -7.8 7.8 73 49 MOSES 50 MANX 0 9.4 4.1 10.3 74 50 MANX 51 JENKINS 0 -11.6 -6.5 13.3 75 52 N VEXLEY 53 VEX LEY 0 12.5 -.3 12.5 76 53 VEXLEY 54 VEXLEY S 0 -7.7 -4.6 9.0 77 54 VEXLEY S 55 HOMER 0 -12.0 -6.3 ·13.5 78 56 W TAUNTO 41 KINCAID 0 -5.5 .6 5.5 79 56 W TAUNTO 42 AIRPORT3 0 -1.6 1.4 2.1 80 57 W TAUNTO 56 W TAUNTO 0 -2.9 .6 2.9
196
-------IEEE 57 BUS FINAL POWER FLOW SOLUTION ----
----- GENERATOR SUMMARY -------
NO BUS NAME VOLTS MW MVAR QMAX QMIN TYPE
1 1 AMOS 1.0707 475.6 71.1 200.0 -200.0 3 2 2 BAKER 1.0591 o.o 50.0 50.0 -17.0 1 3 3 CRAWFORD 1.0458 40.0 60.0 60.0 -1 o.o 1 4 6 ELLS 1.0258 o.o -8.0 25.0 -8.0 1 5 8 GRANGE 1.0443 450.0 70.8 200.0 -140.0 2 6 9 HOMER 1.0125 o.o 9.0 9.0 -3.0 1 7 12 LOESCHER 1. 0178 310.0 42.6 155.0 -50.0 2
-------IEEE 57 BUS FINAL POWER FLOW SOLUTION ----
----- BUS SUMMARY -------
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
1 AMOS 1.0707 o.oo 55.0 17 .o 3 2 BAKER 1.0591 -1.41 3.0 88.0 1 3 CRAWFORD 1. 0458 -6.00 41.0 21.0 1 4 DOYLE 1.0385 -7.15 o.o o.o 0 5 DAWSON 1.0266 -8.10 13.0 4.0 0 6 ELLS 1.0258 -8.15 75.0 2.0 1 7 FARLIE 1.0260 -7.16 o.o o.o 0 8 GRANGE 1.0443 -4.25 150.0 22.0 2 9 HOMER 1. 0125 -8.92 121 .o 26.0 1
10 JENKINS 1.0045 -10.49 5.0 2.0 0 11 KINCAID 1 .0040 -9.46 o.o o.o 0 12 LOESCHER 1.0178 -9.40 377.0 24.0 2 13 MOSES 1. 0054 -9.06 18.0 2.3 0 14 NESTLE 1.0028 -8.71 10.5 5.3 0 15 OAKDALE 1.0280 -6.80 22.0 5.0 0 16 ANDY 1.0248 -8.02 43.0 3.0 0 1T ANDY 1.0387 -4.88 42.0 8.0 0 18 DOYLE 1.0692 -10.93 27.2 9.8 0 19 RICHTER 1.0593 -12.46 3.3 .6 0 20 RICHTER 1.0648 -12.80 2.3 1.0 0 21 RICHTER 1. 0411 -12.02 o.o o.o 0 22 BEAVER2 1.0455 -12.01 o.o o.o 0 23 BEAVER1 1.0442 -12.07 6.3 2.1 0 24 POOL 1.0367 -12.41 o.o o.o 0 25 PO OLA 1 .0144 -17 .15 6.3 3.2 0
197
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
26 POOL .9954 -12.10 o.o o.o 0 27 HAMEL 1.0222 -10.77 9.3 .5 0 28 WYNCOTE 1.0386 -9.83 4.6 2.3 0 29 FARLIE 1.0528 -9.19 17 .o 2.6 0 30 CHESTER .9916 -17.63 3.6 1.8 0 31 HANOVER .9570 -18.15 5.8 2.9 0 32 UXBRIDGE .9577 -17.13 1. 6 .8 0 33 LUXBRIDG .9554 -17.17 3.8 1.9 0 34 UXBRIDGE 1. 0018 -13.26 o.o o.o 0 35 COPLEY M 1.0071 -13.01 6.0 3.0 0 36 COPLEY 1 • 0153 -12.74 o.o o.o 0 37 N COPLEY 1.023.5 -12.55 o.o o.o 0 38 STANTON 1 .0491 -11.89 14.0 7.0 0 39 W TAUNTO 1.0214 -12.59 o.o o.o 0 40 TAUNTON 1 .0122 -12.75 o.o o.o 0 41 KINCAID 1.0305 -13.09 6.3 3.0 0 42 AIRPORT3 1.0030 -14.48 7 .1 4.4 0 43 KINCAID 1.0418 -10.55 2.0 1.0 0 44 S OAKDAL 1.0544 -11.09 12.0 1.8 0 45 OAKDALE 1.0757 -8.75 o.o o.o 0 46 NESTLE 1.0957 -10.38 o.o o.o 0 47 AIRPORT2 1.0691 -11.68 29.7 11.6 0 48 AIRPORT1 1.0630 -11. 77 o.o o.o 0 49 MOSES 1.0691 -12.03 18.0 8.5 0 50 MANX 1.0526 -12.42 21 .o 10.5 0 51 JENKINS 1.0740 -11.50 18.0 5.3 0 52 N VEXLEY 1 .0235 -10.78 4.9 2.2 0 53 VEX LEY 1 .0141 -11.47 20.0 1 o.o 0 54 VEXLEY S 1.0362 -10.93 4.1 1.4 0 55 HOMER 1.0672 -10.04 6.8 3.4 0 56 W TAUNTO 1 .0059 -15.02 7.6 2.2 0 57 W TAUNTO 1.0031 -15.51 6.7 2.0 0
198
-------IEEE 57 BUS FINAL POWER FLOW SOLUTION ----
------ BRANCH FLOW SUMMARY ------
NO FROM BUS TO BUS TYPE MW MVAR MVA
1 1 AMOS 2 BAKER 0 103.8 7.4 104.1 2 1 AMOS 15 OAKDALE 0 149.1 23.9 151.0 3 1 AMOS 16 ANDY 0 76.9 9.0 77.5 4 1 AMOS 17 ANDY 0 91 .1 13.8 92.1 5 2 BAKER 3 CRAWFORD 0 99.4 -18.7 101.2 6 3 CRAWFORD 4 DOYLE 0 60.2 .8 60.2 7 3 CRAWFORD 15 OAKDALE 0 35.9 21.3 41.7 8 4 DOYLE 5 DAWSON 0 14.7 1.2 14.8 9 4 DOYLE 6 ELLS 0 14.0 3.1 14.4
10 4 DOYLE 18 DOYLE 1 13.6 .7 13.6 11 4 DOYLE 18 DOYLE 1 17.4 -1. 2 17 .5 12 5 DAWSON 6 ELLS 0 1.5 -.1 1.5 13 6 ELLS 7 FARLIE 0 -17 .2 1.8 17.3 14 6 ELLS 8 GRANGE 0 -42.4 -3.7 42.5 15 7 FARLIE 8 GRANGE 0 -78.3 - -10.2 78.9 16 7 FARLIE 29 FARLIE 1 61.0 14.6 62.7 17 8 GRANGE 9 HOMER 0 178.1 34.8 181.5 18 9 HOMER 10 JENKINS 0 16.9 -.9 16.9 19 9 HOMER 11 KINCAID 0 13.2 5 .1 14.2 20 9 HOMER 12 LOESCHER 0 2.4 -6.3 6.7 21 9 HOMER 13 MOSES 0 2.7 1.7 3.2 22 9 HOMER 55 HOMER 1 18.6 9.1 20.7 23 10 JENKINS 12 LOESCHER 0 -16.9 -8.3 18.8 24 10 JENKINS 51 JENKINS 1 28.6 9.6 30.2 25 11 KINCAID 13 MOSES 0 -9.5 -.o 9.5 26 11 KINCAID 41 KINCAID 1 9.1 3.2 9.7 27 11 KINCAID 43 KINCAID 1 13.5 4.4 14.2 28 12 LOESCHER 13 MOSES 0 -3.6 19.6 20.0 29 12 LOESCHER 16 ANDY 0 -31.4 -2.6 31.5 30 12 LOESCHER 17 ANDY 0 -46.5 -2.2 46.5 31 13 MOSES 14 NESTLE 0 -11.2 9.0 14.3 32 13 MOSES 15 OAKDALE 0 -49.8 -11.0 51.0 33 13 MOSES 49 MOSES 1 32.5 32.8 46.2 34 14 NESTLE 15 OAKDALE 0 -70.0 -24.0 74.0 35 14 NESTLE 46 NESTLE 1 48.3 28.7 56.2 36 15 OAKDALE 45 OAKDALE 1 37.7 1.4 37.7 37 18 DOYLE 19 RICHTER 0 3.8 -.9 3.9 38 19 RICHTER 20 RICHTER 0 .4 -1.6 1. 7 39 20 RICHTER 21 RICHTER 1 -1. 9 -2.6 3.2
199
NO FROM BUS TO BUS TYPE MW MVAR MVA
40 21 RICHTER 22 BEAVER2 0 -1 .9 -2.7 3.3 41 22 BEAVER2 23 BEAVER1 0 9.5 2.7 9.9 42 22 BEAVER2 38 STANTON 0 -11.4 -5.4 12.6 43 23 BEAVER1 24 POOL 0 3.2 .6 3.2 44 24 POOL 25 PO OLA 1 7.4 2.3 7.7 45 24 POOL 25 PO OLA 1 7 .1 2.2 7.4 46 24 POOL 26 POOL 1 -11.3 -3.0 11. 7 47 25 POOLA 30 CHESTER 0 8.1 6.0 10.1 48 26 POOL 27 HAMEL 0 -11.3 -3 .1 11. 7 49 27 HAMEL 28 WYNCOTE 0 -20.8 -3.9 21.2 50 28 WYN COTE 29 FARLIE 0 -25.7 -6.6 26.5 51 29 FARLIE 52 N VEXLEY 0 18.1 2.7 18.3 52 30 CHESTER 31 HANOVER 0 4.4 4.0 6.0 53 31 HANOVER 32 UXBRIDGE 0 -1.5 1.0 1.8 54 32 UXBRIDGE 33 LUXBRIDG 0 3.8 1.9 4.3 55 32 UXBRIDGE 34 UXBRIDGE 1 -7.0 -1.8 7.2 56 34 UXBRIDGE 35 COPLEY M 0 -7.0 -2.3 7.4 57 35 COPLEY M 36 COPLEY 0 -13.1 -5.0 14.0 58 36 COPLEY 37 N COPLEY 0 -16.8 -9.5 19.3 59 37 N COPLEY 38 STANTON 0 -20.5 -12.8 24.2 60 36 COPLEY 40 TAUNTON 0 3.5 4.6 5.8 61 37 N COPLEY 39 W TAUNTO 0 4.0 3 .1 5.0 62 38 STANTON 44 S OAKDAL 0 -24.7 2.9 24.8 63 38 STANTON 48 AIRPORT1 0 -17 .2 -19.1 25.7 64 38 STANTON 49 MOSES 0 -4.3 -9.3 10.3 65 39 W TAUNTO 57 W TAUNTO 1 3.9 3.1 5.0 66 40 TAUNTON 56 W TAUNTO 1 3.5 4.5 5.7 67 41 KINCAID 42 AIRPORT3 0 8.9 2.9 9.3 68 41 KINCAID 43 KINCAID 0 -11.5 -2.6 11.8 69 44 S OAKDAL 45 OAKDALE 0 -36.8 .9 36.8 70 46 NESTLE 47 AIRPORT2 0 48.3 26.8 55.3 71 47 AIRPORT2 48 AIRPORT1 0 18.0 13.9 22.7 72 48 AIRPORT1 49 MOSES 0 .5 -5.6 5.6 73 49 MOSES 50 MANX 0 1 o.6 7.2 12.8 74 50 MANX 51 JENKINS 0 -10.5 -3.5 11.1 75 52 N VEXLEY 53 VEXLEY 0 12.8 -.o 12.8 76 53 VEXLEY 54 VEXLEY S 0 -7.3 -3.7 8.2 77 54 VEXLEY S 55 HOMER 0 -11.6 -5°3 12.7 78 56 W TAUNTO 41 KINCAID 0 -5.4 1.0 5.4 79 56 W TAUNTO 42 AIRPORT3 0 -1.6 1.8 2.4 80 57 W TAUNTO 56 W TAUNTO 0 -2.8 .8 2.9
200
57 BUS IEEE UNIT VOLTAGES CONVERGENCE
ITER LOSS UNIT 1 UNIT 2 UNIT 3 UNIT 4 UNIT 5 UNIT 6 UNIT 7
0 28 .11 1. 04 1. 01 .985 .98 1.005 • 980 1. 015
1 16.69 1.0745 1.063 1.050 1.0288 1.0463 1.0155 1.0227
57 BUS IEEE UNIT MVARS CONVERGENCE
ITER LOSS UNIT 1 UNIT 2 UNIT 3 UNIT 4 UNIT 5 UNIT 6 UNIT 7
0 28.11 129.72 - .75
1 16.69 50.0
-1.96 -·.11
60.00 -8.00
62.42
67.00
3.75 129.9
9.0
APPENDIX 9
118 BUS IEEE RESULTS
201
202
11.8 BUS IEEE INITIAL POWER FLOW SOLUTION
----- GENERATOR SUMMARY -------
NO BUS NAME VOLTS MW MVAR QMAX QMIN TYPE
1 1 BUS-69 1 .0300 513.8 -112.6 200.0 -50.0 3 2 2 BUS-1 .9550 o.o -3.0 15.0 -5.0 2 3 5 BUS-4 .9980 -9.0 -12.6 300.0 -300.0 2 4 7 BUS-6 .9900 o.o 16.4 50.0 -13.0 2 5 9 BUS-8 1 • 0150 -28.0 57.0 300.0 -300.0 2 6 11 BUS-10 1.0500 450.0 -51.0 200.0 -147.0 2 7 13 BUS-12 .9900 85.0 91 .8 120.0 -35.0 2 8 16 BUS-15 .9700 o.o 4.4 30.0 -10.0 2 9 19 BUS-18 .9730 o.o 26.3 50.0 -16.0 2
10 20 BUS-19 .9634 o.o -8.0 24.0 -8.0 1 11 25 BUS-24 .9920 -13.0 -10.7 300.0 -300.0 2 12 26 BUS-25 1.0500 220.0 51.3 140.0 -47.0 2 13 27 BUS-26 1. 0150 314.0 4.5 1000.0 100*00 2 14 28 BUS-27 .9680 -9.0 2.3 300.0 -300.0 2 15 32 BUS-31 .9670 7.0 31 .9 300.0 -300.0 2 16 33 BUS-32 .9639 o.o -14.0 42.0 -14.0 1 17 35 BUS-34 .9851 o.o -8.0 24.0 -8.0 1 18 37 BUS-36 .9800 o.o 3.4 24.0 -8.0 2 19 41 BUS-40 .9700 -46.0 27.9 300.0 -300.0 2 20 43 BUS-42 .9850 -59.0 41.5 300.0 -300.0 2 21 47 BUS-46 1 .0050 19.0 -4.2 100.0 -100.0 2 22 50 BUS-49 1.0250 204.0 120.0 210.0 -85.0 2 23 55 BUS-54 .9550 48.0 3.9 300.0 -300.0 2 24 56 BUS-55 .9520 o.o 4.7 23.0 -8.0 2 25 57 BUS-56 .9540 o.o -2.3 15.0 -8.0 2 26 60 BUS-59 .9850 155.0 80.9 180.0 -60.0 2 27 62 BUS-61 .9950 160.0 -41.9 300.0 -100.0 2 28 63 BUS-62 .9980 o.o 1.3 20.0 -20.0 2 29 66 BUS-65 1. 0050 391.0 77.9 200.0 -67.0 2 30 67 BUS-66 1.0500 392.0 -2.2 200.0 -67.0 2 31 70 BUS-70 .9840 o.o 11. 7 32.0 -10.0 2 32 72 BUS-72 .9800 -12.0 -5.2 100.0 -100.0 2 33 73 BUS-73 .9910 -6.0 11.9 100.0 -100.0 2 34 74 BUS-74 .9595 o.o -6.0 9.0 -6.0 1 35 76 BUS-76 .9430 o.o 3.4 23.0 -8.0 2 36 77 BUS-77 1.0060 o.o 15.7 70.0 -20.0 2 )7 80 BUS-80 1 .0400 477.0 112.0 280.0 -165.0 2 38 85 BUS-85 .9850 o.o -6.4 23.0 -8.0 2 39 87 BUS-87 1 • 0150 4.0 11.0 1000.0 -100.0 2 40 89 BUS-89 1.0050 607.0 -12.5 300.0 -210.0 2
203
NO BUS NAME VOLTS MW MVAR QMAX QMIN TYPE
41 90 BUS-90 .9850 -85.0 59.3 300.0 -300.0 2 42 91 BUS-91 .9800 -10.0 -15.0 100.0 -100.0 2 43 92 BUS-92 .9925 o.o -3.0 9.0 -3.0 1 44 99 BUS-99 1.0100 -42.0 -17.5 100.0 -100.0 2 45 100 BUS-100 1 .0170 252.0 108.5 155.0 -50.0 2 46 103 BUS-103 1.0007 40.0 40.0 40.0 -15.0 1 47 104 BUS-104 .9710 o.o 5.4. 23.0 -8.0 2 48 105 BUS-105 .9659 o.o -8.0 23.0 -8.0 1 49 107 BUS-107 .9520 -22.0 5.7 200.0 -200.0 2 50 110 BUS-110 .9730 o.o 5.0 23.0 -8.0 2 51 111 BUS-111 .9800 36.0 -1.8 1000.0 -100.0 2 52 112 BUS-112 .9750 -43.0 41.5 1000.0 -100.0 2 53 113 BUS-113 .9930 -6.0 7.5 200.0 -100.0 2 54 116 BUS-116 1.0050 18*40 56.4 1000.0 100*00 2
118 BUS IEEE INITIAL POWER FLOW SOLUTION
----- BUS SUMMARY -------
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
1 BUS-69 1.0300 o.oo o.o o.o 3 2 BUS-1 .9550 -19.74 51.0 27.0 2 3 BUS-2 .9714 -19.21 20.0 9.0 0 4 BUS-3 .9676 -18.86 39.0 10.0 0 5 BUS-4 .9980 -15 .12 30.0 12.0 2 6 BUS-5 1 .0017 -14.67 o.o 0.0 0 7 BUS-6 .9900 -17.42 52.0 22.0 2 8 BUS-7 .9893 -17.87 19.0 2.0 0 9 BUS-8 1.0150 -9.46 o.o o.o 2
10 BUS-9 1.0429 -2.21 o.o o.o 0 11 BUS-10 1.0500 5.37 o.o o.o 2 12 BUS-11 .9850 -17. 71 70.0 23.0 0 13 BUS-12 .9900 -18.24 47.0 1 o.o 2 14 BUS-13 .9682 -19.12 34.0 16.0 0 15 BUS-14 .9836 -18.99 14.0 1.0 0 16 BUS-15 .9700 -19.37 90.0 30.0 2 17 BUS-16 .9838 -18.58 25.0 1 o.o 0 18 BUS-17 .9947 -16.86 11.0 3.0 0 19 BUS-18 .9730 -19.08 60.0 34.0 2 20 BUS-19 .9634 -19.57 45.0 25.0 1 21 BUS-20 .9585 -18.74 18.0 3.0 0 22 BUS-21 .9594 -17.20 14.0 8.0 0
204
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
23 BUS-22 .9707 -14.69 1 o.o 5.0 0 24 BUS-23 1.0010 -9.86 7.0 3.0 0 25 BUS-24 .9920 -10.10 o.o o.o 2 26 BUS-25 1.0500 -2.74 o.o o.o 2 27 BUS-26 1. 0150 - • 71 o.o o.o 2 28 BUS-27 .9680 -15.34 62.0 13.0 2 29 BUS-28 .9616 -17.05 17.0 7.0 0 30 BUS-29 .9632 -18.04 24.0 4.0 0 31 BUS-30 .9872 -11.41 o.o o.o 0 32 BUS-31 .9670 -17.92 43.0 27.0 2 33 BUS-32 .9639 -15. 91 59.0 23.0 1 34 BUS-33 .9710 -20.00 23.0 9.0 0 35 BUS-34 .9851 -19.36 59.0 26.0 1 36 BUS-35 .9805 -19.80 33.0 9.0 0 37 BUS-36 .9800 -19.80 31.0 17.0 2 38 BUS-37 .9909 -18.90 o.o o.o 0 39 BUS-38 .9646 -13.14 o.o o.o 0 40 BUS-39 .9701 -22.19 27.0 11.0 0 41 BUS-40 .9700 -23.22 20.0 23.0 2 42 BUS-41 .9668 -23.62 37.0 1 o.o 0 43 BUS-42 .9850 -21. 91 37.0 23.0 2 44 BUS-43 .9777 -19.25 18.0 7.0 0 45 BUS-44 .9844 -16.53 16.0 8.0 0 46 BUS-45 .9862 -14.62 53.0 22.0 0 47 BUS-46 1.0050 -11.76 28.0 10.0 2 48 BUS-47 1.0163 -9-48 34.0 o.o 0 49 BUS-48 1 .0206 -10.29 20.0 11.0 0 50 BUS-49 1.0250 -9.28 87.0 30.0 2 51 BUS-50 1 .0011 -11.32 17.0 4.0 0 52 BUS-51 .9669 -13.94 17 .o 8.0 0 53 BUS-52 .9568 -14.90 18.0 5.0 0 54 BUS-53 .9460 -15.88 23.0 11.0 0 55 BUS-54 .9550 -14.97 113.0 32.0 2 56 BUS-55 .9520 -15.26 63.0 22.0 2 57 BUS-56 .9540 -15.07 84.0 18.0 2 58 BUS-57 .9706 -13.86 12.0 3.0 0 59 BUS-58 .9590 -14.72 12.0 3.0 0 60 BUS-5.9 .9850 -10.88 277.0 113.0 2 61 BUS-60 .9931 -6.99 78.0 3.0 0 62 BUS-61 .9950 -6.09 o.o o.o 2 63 BUS-62 -9980 -6.71 77.0 14.0 2 64 BUS-63 .9701 -7.29 o.o o.o 0 65 BUS-64 .9844. -5.56 o.o o.o 0 66 BUS-65 1.0050 -2.44 o.o o.o 2 67 BUS-66 1 .0500 -2.66 39.0 18.0 2
205
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
68 BUS-67 1 .0197 -5.30 28.0 7.0 0 69 BUS-68 1.0030 -2.51 o.o o.o 0 70 BUS-70 .9840 -7 .21 66.0 20.0 2 71 BUS-71 .9858 -7.36 o.o o.o 0 72 BUS-72 .9800 -11.40 o.o o.o 2 73 BUS-73 .9910 -7.58 o.o o.o 2 74 BUS-74 .9595 -8.29 68.0 27.0 1 75 BUS-75 .9693 -7.08 47.0 11.0 0 76 BUS-76 .9430 -8.19 68.0 36.0 2 77 BUS-77 1.0060 -3.27 61.0 28.0 2 78 BUS-78 1. 0034 -3.57 71 .o 26.0 0 79 BUS-79 1.0092 -3.27 39.0 32.0 0 80 BUS-80 1.0400 -1.02 130.0 26.0 2 81 BUS-81 .9981 -1.98 o.o o.o 0 82 BUS-82 .9889 -2.75 54.0 27.0 0 83 BUS-83 .9848 -1.56 20.0 1 o.o 0 84 BUS-84 • 9801 .97 11.0 7.0 0 85 BUS-85 .9850 2.53 24.0 15.0 2 86 BUS-86 .9867 1 .16 21.0 10.0 0 87 BUS-87 1. 0150 1.42 o.o o.o 2 88 BUS-88 .9875 5.66 48.0 1 o.o 0 89 BUS-89 1.0050 9.72 o.o o.o 2 90 BUS-90 .9850 3.31 78.0 42.0 2 91 BUS-91 .9800 3.33 o.o o.o 2 92 BUS-92 .9925 3.83 65.0 1 o.o 1 93 BUS-93 .9871 .82 12.0 7.0 0 94 BUS-94 .9907 -1.33 30.0 16.0 0 95 BUS-95 .9810 -2.31 42.0 31.0 0 96 BUS-96 .9928 -2.47 38.0 15.0 0 97 BUS-97 1.0114 -2.10 15.0 9.0 0 98 BUS-98 1.0235 -2.57 34.0 8.0 0 99 BUS-99 1.0100 -2.94 o.o o.o 2
100 BUS-100 1 .0170 -1.94 37.0 18.0 2 101 BUS-101 .9926 -.37 22.0 15.0 0 102 BUS-102 .9912 2.33 5.0 3.0 0 103 BUS-103 1.0007 -5.54 23.0 16.0 1 104 BUS-104 .9710 -8.28 38.0 25.0 2 105 BUS-105 .9659 -9.40 31.0 26.0 1 106 BUS-106 .9618 -9.65 43.0 16.0 0 107 BUS-107 .9520 -12.44 28.0 12.0 2 108 BUS-108 .9668 -10.60 2.0 1.0 0 109 BUS-109 .9675 -11.04 8.0 3.0 0 110 BUS-110 .9730 -11.88 39.0 30.0 2 111 BUS-111 .9800 -10.24 o.o o.o 2 112 BUS-112 .9750 -14.98 25.0 13.0 2
206
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
113 BUS-113 .9930 -16.88 o.o o.o 2 114 BUS-114 .9606 -16.23 8.0 3.0 0 115 BUS-115 .9605 -16.24 22.0 7.0 0 116 BUS-116 1.0050 -2.95 o.o o.o 2 117 BUS-117 .973s -19.78 20.0 8.0 0 118 BUS-118 .9505 -8.06 33.0 15.0 0
207
118 BUS IEEE INITIAL POWER FLOW SOLUTION
------ BRANCH FLOW SUMMARY ------
NO FROM BUS TO BUS TYPE MW MVAR MVA
1 2 BUS-1 3 BUS-2 0 -12.2 -13.1 17.9 2 2 BUS-1 4 BUS-3 0 -38.8 -16.9 42.3 3 5 BUS-4 6 BUS-5 0 -103.8 -24.3 106.6 4 4 BUS-3 6 BUS-5 0 -68.4 -14.2 69.9 5 6 BUS-5 7 BUS-6 0 89.0 3.5 89.1 6 7 BUS-6 8 BUS-7 0 36.1 -4·9 36.4 7 9 BUS-8 10 BUS-9 0 -440.6 -89.7 449.7 8 6 BUS-5 9 BUS-8 1 -340.6 -88.3 351.8 9 10 BUS-9 11 BUS-10 0 -445·3 -24.4 445.9
10 5 BUS-4 12 BUS-11 0 64.8 -.3 64.8 11 6 BUS-5 12 BUS-11 0 77.9 2.5 77.9 12 12 BUS-11 13 BUS-12 0 34.9 -35.6 49.9 13 3 BUS-2 13 BUS-12 0 -32.3 -20.0 38.0 14 4 BUS-3 13 BUS-12 0 -9.6 -12.5 15.8 15 8 BUS-7 13 BUS-12 0 17.0 -6.6 18.3 16 12 BUS-11 14 BUS-13 0 35.7 11.2 37.4 17 13 BUS-12 15 BUS-14 0 19.0 2.4 19.1 18 14 BUS-13 16 BUS-15 0 1.4 -4.0 4.3 19 15 BUS-14 16 BUS-15 0 4.9 2.9 5.7 20 13 BUS-12 17 BUS-16 0 8.3 4.2 9.3 21 16 BUS-15 18 BUS-17 0 -103.2 -23.7 105.9 22 17 BUS-16 18 BUS-17 0 -16.7 -3.8 17 .1 23 18 BUS-17 19 BUS-18 0 80.3 24.0 83.8 24 19 BUS-18 20 BUS-19 0 19.4 14.0 23.9 25 20 BUS-19 21 BUS-20 0 -10.0 4.9 11.1 26 16 BUS-15 20 BUS-19 0 12.0 12.0 17 .o 27 21 BUS-20 22 BUS-21 0 -28.0 4.5 28.4 28 22 BUS-21 23 BUS-22 0 -42.2 -2.3 42.3 29 23 BUS-22 24 BUS-23 0 -52.6 -6.9 53.1 30 24 BUS-23 25 BUS-24 0 14.4 6.6 15.8 31 24 BUS-23 26 BUS-25 0 -166.5 -23.1 168.1 32 26 BUS-25 27 BUS-26 1 -94.7 -15.1 95.9 33 26 BUS-25 28 BUS-27 0 143.8 30.1 146.9 34 28 BUS-27 29 BUS-28 0 32.7 -.6 32.7 35 29 BUS-28 30 BUS-29 0 15.5 -6.5 16.8 36 18 BUS-17 31 BUS-30 1 -230.9 -66.9 240.4 37 9 BUS-8 31 BUS-30 0 72.0 24.6 76.1 38 27 BUS-26 31 BUS-30 0 219.3 -14.0 219.8 39 18 BUS-17 32 BUS-31 0 15.3 11.1 18.9 40 30 BUS-29 32 BUS-31 0 -8.6 -8.6 12.1 41 24 BUS-23 33 BUS-32 0 91.4 5.8 91.6
208
NO FROM BUS TO BUS TYPE MW MVAR MVA
42 32 BUS-31 33 BUS-32 0 -29.4 11.4 31.5 43 28 BUS-27 33 BUS-32 0 12. 7 .5 12.8 44 16 BUS-15 34 BUS-33 0 7.4 -4.5 8.7 45 20 BUS-19 35 BUS-34 0 -3.6 -10.3 10.9 46 36 BUS-35 37 BUS-36 0 .6 4.5 4.5 47 36 BUS-35 38 BUS-37 0 -33.6 -13.5 36.2 48 34 BUS-33 38 BUS-37 0 -15.6 -10.6 18.9 49 35 BUS-34 37 BUS-36 0 30.5 8.6 31.7 50 35 BUS-34 38 BUS-37 0 -93.1 -35.8 99.8 51 38 BUS-37 39 BUS-38 1 -239.2 -80.9 252.6 52 38 BUS-37 40 BUS-39 0 53.6 3.4 53.7 53 38 BUS-37 41 BUS-40 0 42.7 -3.2 42.8 54 31 BUS-30 39 BUS-38 0 56.4 16.7 58.8 55 40 BUS-39 41 BUS-40 0 25.6 -8.1 26.9 56 41 BUS-40 42 BUS-41 0 14.2 1.6 14.3 57 41 BUS-40 43 BUS-42 0 -13.1 -6.0 14.4 58 42 BUS-41 43 BUS-42 0 -22.8 -1.4 24.0 59 44 BUS-43 45 BUS-44 0 -18.0 -.7 18.0 60 35 BUS-34 44 BUS-43 0 -.o 2.3 2.3 61 45 BUS-44 46 BUS-45 0 -34.2 6.0 34.8 62 46 BUS-45 47 BUS-46 0 -37 .1 -3.4 37.2 63 47 BUS-46 48 BUS-47 0 -31.6 -.4 31.6 64 47 BUS-46 49 BUS-48 0 -15.0 -5.8 16.0 65 48 BUS-47 50 BUS-49 0 -9.2 -12.2 15.3 66 43 BUS-42 50 BUS-49 0 -66.1 5.8 66.4 67 43 BUS-42 50 BUS-49 0 -66.1 5.8 66.4 68 46 BUS-45 50 BUS-49 0 -50.5 -1.8 50.5 69 49 BUS-48 50 BUS-49 0 -35.1 3.3 35.3 70 50 BUS-49 51 BUS-50 0 53.7 13.4 55.4 71 50 BUS-49 52 BUS-51 0 66.7 20.4 69.8 72 52 BUS-51 53 BUS-52 0 28.6 6.2 29.3 73 53 BUS-52 54 BUS-53 0 10.4 2.0 1 o.6 74 54 BUS-53 55 BUS-54 0 -12.7 -5.6 13.8 75 50 BUS-49 55 BUS-54 0 37.9 13.1 40.0 76 50 BUS-49 55 BUS-54 0 37.8 11.2 39.5 77 55 BUS-54 56 BUS-55 0 7 .1 1.5 7.2 78 55 BUS-54 57 BUS-56 0 18.6 4.3 19.1 79 56 BUS-55 57 BUS-56 0 -21.5 -5.8 22.3 80 57 BUS-56 58 BUS-57 0 -23.1 -9.1 24.8 81 51 BUS-50 58 BUS-57 0 36.0 9.1 37 .1 82 57 BUS-56 59 BUS-58 0 -6.7 -3.7 7.7 83 52 BUS-51 59 BUS-58 0 18.9 3.1 19.1 84 55 BUS-54 60 BUS-59 0 -30.3 -7.5 31.2 85 57 BUS-56 60 BUS-59 0 -27.9 -4.2 28.2 86 57 BUS-56 60 BUS-59 0 -29.2 -3.9 29.5
209
NO FROM BUS TO BUS TYPE MW MVAR MVA
87 56 BUS-55 60 BUS-59 0 -34·4 -8.3 35.4 88 60 BUS-59 61 BUS-60 0 -44·5 3.9 44.7 89 60 BUS-59 62 BUS-61 0 -53.0 5.4 53.3 90 61 BUS-60 62 BUS-61 0 -113.2 8.7 113.5 91 61 BUS-60 63 BUS-62 0 -10.0 -7 .1 12.3 92 62 BUS-61 63 BUS-62 0 25.7 -13.9 29.2 93 60 BUS-59 64 BUS-63 1 -148.9 -53.8 158.3 94 64 BUS-63 65 BUS-64 0 -148.9 -64.6 162.3 95 62 BUS-61 65 BUS-64 1 -33.2 -14.8 36.3 96 39 BUS-38 66 BUS-65 0 -183.1 -54.4 191.0 97 65 BUS-64 66 BUS-65 0 -182.6 -64.5 193.6 98 50 BUS-49 67 BUS-66 0 -133.8 4.8 133.9 99 50 BUS-49 67 BUS-66 0 -133.8 4.8 133.9
100 63 BUS-62 67 BUS-66 0 -37 .1 -17 .3 41.0 101 63 BUS-62 68 BUS-67 0 -24.3 -14.4 28.2 102 66 BUS-65 67 BUS-66 1 11.8 72.3 73.2 103 67 BUS-66 68 BUS-67 0 53.1 19.3 56.5 104 66 BUS-65 69 BUS-68 0 9.3 -20.7 22.8 105 48 BUS-47 1 BUS-69 0 -56.8 13.7 58.4 106 50 BUS-49 1 BUS-69 0 -47°3 12.7 49.0 107 69 BUS-68 1 BUS-69 1 -131.0 126.9 182.4 108 1 BUS-69 70 BUS-70 0 104.6 11.9 105.3 109 25 BUS-24 70 BUS-70 0 -10.8 -.1 10.8 110 70 BUS-70 71 BUS-71 0 5.7 -6.8 8.9 111 25 BUS-24 72 BUS-72 0 12.1 .8 12.1 112 71 BUS-71 75 BUS-75 0 -.4 7.0 7.0 113 71 BUS-71 73 BUS-73 0 6.0 -13.0 14.3 114 70 BUS-70 74 BUS-74 0 17.4 11.5 20.9 115 70 BUS-70 75 BUS-75 0 1.4 8.1 8.2 116 1 BUS-69 75 BUS-75 0 108.0 15.0 109.0 117 74 BUS-74 75 BUS-75 0 -50.8 -8.0 51.4 118 76 BUS-76 77 BUS-77 0 -60.8 -21.2 64.4 119 1 BUS-69 77 BUS-77 0 60.8 2.0 60.9 120 75 BUS-75 77 BUS-77 0 -34·3 -8.7 35.4 121 77 BUS-77 78 BUS-78 0 45.3 6.6 45.7 122 78 BUS-78 79 BUS-79 0 -25.8 -18.3 31.7 123 77 BUS-77 80 BUS-80 0 -96.8 -37.3 103.8 124 77 BUS-77 80 BUS-80 0 -44·5 -20.5 49.0 125 79 BUS-79 80 BUS-80 0 -64.9 -29.5 71.3 126 69 BUS-68 81 BUS-81 0 -43.8 -12.0 45.4 127 80 BUS-80 81 BUS-81 1 43.8 -65.9 79.2 128 77 BUS-77 82 BUS-82 0 -3.2 17 .2 17 .4 129 82 BUS-82 83 BUS-83 0 -47.1 24.2 53.0 130 83 BUS-83 84 BUS-84 0 -24.7 14.2 28.5 131 83 BUS-83 85 BUS-85 0 -42-7 12.3 44.4 132 84 BUS-84 85 BUS-85 0 -36.3 9.5 37.5 133 85 BUS-85 86 BUS-86 0 17.2 -7-4 18. 7
210
NO FROM BUS TO BUS TYPE MW MVAR MVA
134 86 BUS-86 87 BUS-87 0 -3.9 -15.1 15.6 135 85 BUS-85 88 BUS-88 0 -50.3 7.6 50.9 136 85 BUS-85 89 BUS-89 0 -71.2 .7 71. 2 137 88 BUS-88 89 BUS-89 0 -98.9 -2.5 98.9 138 89 BUS-89 90 BUS-90 0 58.2 -4.7 58.4 139 89 BUS-89 90 BUS-90 0 11o.7 -5·4 110.9 140 90 BUS-90 91 BUS-91 0 1.3 4.5 4.6 141 89 BUS-89 92 BUS-92 0 201.7 -7.0 201.9 142 89 BUS-89 92 BUS-92 0 63.6 -6.6 63.9 143 91 BUS-91 92 BUS-92 0 -8.8 -8.5 12.2 144 92 BUS-92 93 BUS-93 0 57.8 -10.7 58.7 145 92 BUS-92 94 BUS-94 0 52.3 -14.2 54.2 146 93 BUS-93 94 BUS-94 0 44.8 -18.5 48.5 147 94 BUS-94 95 BUS-95 0 41.0 9.3 42.0 148 80 BUS-80 96 BUS-96 0 19.0 20.8 28.1 149 82 BUS-82 96 BUS-96 0 -10.2 -6.7 12.2 150 94 BUS-94 96 BUS-96 0 19.9 -9.5 22.1 151 80 BUS-80 97 BUS-97 0 26.4 25.5 36.7 152 80 BUS-80 98 BUS-98 0 28.9 8.3 30.1 153 80 BUS-80 99 BUS-99 0 19.5 8.2 21.1 154 92 BUS-92 100 BUS-100 0 31.5 -17 .2 35.9 155 94 BUS-94 100 BUS-100 0 4.3 -49.1 49.3 156 95 BUS-95 96 BUS-96 0 -1 .3 -21.4 21.4 157 96 BUS-96 97 BUS-97 0 -11 .1 -19.9 22.8 158 98 BUS-98 100 BUS-100 0 -5.3 2.4 5.9 159 99 BUS-99 100 BUS-100 0 -22.7 -4.6 23.2 160 100 BUS-100 101 BUS-101 0 -16.7 22.0 27.6 161 92 BUS-92 102 BUS-102 0 44.7 -7.7 45.3 162 101 BUS-101 102 BUS-102 0 -39.0 9.3 40.1 163 100 BUS-100 103 BUS-103 0 121.1 -4.3 121.2 164 100 BUS-100 104 BUS-104 0 56.4 10.6 57.4 165 103 BUS-103 104 BUS-104 0 32.3 7.9 33.3 166 103 BUS-103 105 BUS-105 0 42.9 6.6 43.4 167 100 BUS-100 106 BUS-106 0 60.6 9.1 61.3 168 104 BUS-104 105 BUS-105 0 48.7 .2 48.7 169 105 BUS-105 106 BUS-106 0 8.8 4.3 9.8 170 105 BUS-105 107 BUS-107 0 26.7 -1.9 26.8 171 105 BUS-105 108 . BUS-108 0 24.1 -10.8 26.4 172 106 BUS-106 107 BUS-107 0 24.0 -3.4 24.2 173 108 BUS-108 109 BUS-109 0 21.8 -10.3 24.1 174 103 BUS-103 110 BUS-110 0 60.6 3.2 60.6 175 109 BUS-109 110 BUS-110 0 13.7 -12.8 18.8 176 110 BUS-110 111 BUS-111 0 -35.7 1.0 35.7 177 110 BUS-110 112 BUS-112 0 69.5 -30.6 75.9 178 18 BUS-17 113 BUS-113 0 2.6 4.5 5.2 179 33 BUS-32 113 BUS-113 0 3.5 -17.2 17 .6
211
NO FROM BUS TO BUS TYPE MW MVAR MVA
180 33 BUS-32 114 BUS-114 0 9.2 2.4 9.5 181 28 BUS-27 115 BUS-115 0 20.9 4.4 21.3 182 114 BUS-114 115 BUS-115 0 1.2 .9 1.5 183 69 BUS-68 116 BUS-116 0 184.1 -11.4 197 .5 184 13 BUS-12 117 BUS-117 0 20.2 5.2 20.8 185 75 BUS-75 118 BUS-118 0 40.5 25.4 47.9 186 76 BUS-76 118 BUS-118 0 -7.2 -11.5 13.5
212
-------------- BEGIN OPTIMAL POWER FLOW --------------
118 IEEE BUS NETWORK
MAX OPF ITERATIONS 6 MAX PF ITERATIONS 10 MVAR LIMITING CONTROL 1 MODERATE ITER FOR MVAR CNTL 2
HIGH VOLTAGE LIMIT 1.10 LOW VOLTAGE LIMIT .85
NUMBER OF BUSS ES 118 NUMBER OF BRANCHES 186 NUMBER OF GENERATORS 54 NUMBER OF TRANSFORMERS 9
-----118 BUS IEEE INITIAL GENERATOR DATA -------
NO BUS NAME VOLTS MW QMAX QMIN
1 1 BUS-69 1.0300 516.4 200.0 -50.0 2 2 BUS-1 .9550 o.o 15.0 -5.0 3 5 BUS-4 -9980 -9.0 300.0 -300.0 4 7 BUS-6 .9900 o.o 50.0 -13.0 5 9 BUS-8 1.0150 -28.0 300.0 -300.0 6 11 BUS-10 1 .0500 450.0 200.0 -147.0 7 13 BUS-12 .9900 85.0 120.0 -35.0 8 16 BUS-15 .9700 o.o 30.0 -10.0 9 19 BUS-18 .9730 o.o 50.0 -16.0
10 20 BUS-19 .9620 o.o 24.0 -8.0 11 25 BUS-24 .9920 -13.0 300.0 -300.0 12 26 BUS-25 1 .0500 220.0 140.0 -47.0 13 27 BUS-26 1. 0150 314.0 1000.0 100*00 14 28 BUS-27 .9680 -9.0 300.0 -300.0 15 32 BUS-31 .9670 7.0 300.0 -300.0 16 33 BUS-32 .9630 o.o 42.0 -14.0 17 35 BUS-34 .9840 o.o 24.0 -8.0 18 37 BUS-36 .9800 o.o 24.0 -8.0 19 41 BUS-40 .9700 -46.o 300.0 -300.0 20 43 BUS-42 .9850 -59.0 300.0 -300.0 21 47 BUS-46 1.0050 19.0 100.0 -100.0 22 50 BUS-49 1.0250 204.0 210.0 -85.0 23 55 BUS-54 .9550 48.0 300.0 -300.0
213
NO BUS NAME VOLTS MW QMAX QMIN
24 56 BUS-55 .9520 o.o 23.a -8.a 25 57 BUS-56 .954a a.a 15.a -8.a 26 6a BUS-59 .985a 155.a 18a.o -6a.a 27 62 BUS-61 .995a 16a.a 3aa.a -1ao.a 28 63 BUS-62 .9980 o.a 20.a -2a.a 29 66 BUS-65 1.ao5a 391.a 2oa.a -67.0 3a 67 BUS-66 1. a5aa 392.a 2aa.a -67.a 31 70 BUS-7a .984a a.o 32.0 -1a.a 32 72 BUS-72 .9800 -12.a 1ao.o -1oa.a 33 73 BUS-73 .991a -6.0 100.a -1aa.a 34 74 BUS-74 .958a o.o 9.0 -6.a 35 76 BUS-76 .9430 o.a 23.0 -8.a 36 77 BUS-77 1. oa6o a.a 70.0 -20.0 37 80 BUS-8a 1.04oa 477.0 28a.o -165.a 38 85 BUS-85 .9850 o.a 23.0 -8.a 39 87 BUS-87 1. a150 4.0 1000.0 -1ao.o 4a 89 BUS-89 1.oa5a 607.a 3aa.o -210.a 41 9a BUS-9a .9850 -85.0 3ao.o -3aa.a 42 91 BUS-91 .9800 -1 a.a 100.0 -100.a
-43 92 BUS-92 .99ao a.o 9.0 -3.0 44 99 BUS-99 1 .01 oa -42.a 100.0 -100.0 45 100 BUS-100 1.0170 252.0 155.a -50.0 46 103 BUS-103 1. 01 oa 40.0 40.0 -15.0 47 104 BUS-1a4 .9710 o.o 23.a -8.0 48 105 BUS-1 a5 .965a o.o 23.0 -8.0 49 107 BUS-1a7 .9520 -22.0 200.0 -200.a 50 110 BUS-11a .9730 o.o 23.0 -8.0 51 111 BUS-111 .9800 36.0 1000.0 -100.0 52 112 BUS-112 .9750 -43.0 1ooa.o -1oa.o 53 113 BUS-113 .9930 -6.0 200.0 -100.a 54 116 BUS-116 1.0050 -184.a 1ooa.o 1oa*Oa
214
------ 118 BUS IEEE INITIAL BRANCH FLOW SUMMARY ------
NO FROM BUS TO BUS TYPE CONDUCT SUSCEPT CHARG/TAP
1 2 BUS-1 3 BUS-2 0 .0303 .0999 .0254 2 2 BUS-1 4 BUS-3 0 .0129 .0424 .0108 3 5 BUS-4 6 BUS-5 0 .0017 .0080 .0021 4 4 BUS-3 6 BUS-5 0 .0241 .1080 .0284 5 6 BUS-5 7 BUS-6 0 .0119 .0540 .0143 6 7 BUS-6 8 BUS-7 0 .0046 .0208 .0055 7 9 BUS-8 10 BUS-9 0 .0024 .0305 1.1620 8 6 BUS-5 9 BUS-8 1 0.0000 .0267 1. 0150 9 10 BUS-9 11 BUS-10 0 .0026 .0322 1 .2300
10 5 BUS-4 12 BUS-11 0 .0209 .0688 .0175 11 6 BUS-5 12 BUS-11 0 .0203 .0682 .0174 12 12 BUS-11 13 BUS-12 0 .0060 .0196 .0050 13 3 BUS-2 13 BUS-12 0 .0187 .0616 .0157 14 4 BUS-3 13 BUS-12 0 .0484 .1600 .0406 15 8 BUS-7 13 BUS-12 0 .0086 .0340 .0087 16 12 BUS-11 14 BUS-13 0 .0223 .0731 .0188 17 13 BUS-12 15 BUS-14 0 .0215 .0707 .0182 18 14 BUS-13 16 BUS-15 0 .0744 .2444 .0627 19 15 BUS-14 16 BUS-15 0 .0595 .1950 .0502 20 13 BUS-12 17 BUS-16 0 .0212 .0834 .0214 21 16 BUS-15 18 BUS-17 0 .0132 .0437 .0444 22 17 BUS-16 18 BUS-17 0 .0454 .1801 .0466 23 18 BUS-17 19 BUS-18 0 .0123 .0505 .0130 24 19 BUS-18 20 BUS-19 0 .0112 .0493 .0114 25 20 BUS-19 21 BUS-20 0 .0252 .1170 .0298 26 16 BUS-15 20 BUS-19 0 .0120 .0394 .0101 27 21 BUS-20 22 BUS-21 0 .0183 .0849 .0216 28 22 BUS-21 23 BUS-22 0 .0209 .0970 .0246 29 23 BUS-22 24 BUS-23 0 .0342 .1590 .0404 30 24 BUS-23 25 BUS-24 0 .0315 .0492 .0498 31 24 BUS-23 26 BUS-25 0 .0156 .0800 .0864 32 26 BUS-25 27 BUS-26 1 0.0000 .0382 1.0410 33 26 BUS-25 28 BUS-27 0 .0318 .1630 .1764 34 28 BUS-27 29 BUS-28 0 .0191 .0855 .0216 35 29 BUS-28 30 BUS-29 0 .0237 .0943 .0238 36 18 BUS-17 31 BUS-30 1 0.0000 .0388 1. 0410 37 9 BUS-8 31 BUS-30 0 .0043 .0504 .5140 38 27 BUS-26 31 BUS-30 0 .0080 .0860 .9080 39 18 BUS-17 32 BUS-31 0 .0474 .1563 .0399 40 30 BUS-29 32 BUS-31 0 .0108 .0331 .0083 41 24 BUS-23 33 BUS-32 0 .0317 .1153 .1173 42 32 BUS-31 33 BUS-32 0 .0298 .0985 .0251
215
NO FROM BUS TO BUS TYPE CONDUCT SUSCEPT CHARG/TAP
43 28 BUS-27 33 BUS-32 0 .0229 .0755 .0193 44 16 BUS-15 34 BUS-33 0 .0380 .1244 .0319 45 20 BUS-19 35 BUS-34 0 .0752 .2470 .0632 46 36 BUS-35 37 BUS-36 0 .0022 .0102 .0027 47 36 BUS-35 38 BUS-37 0 .0110 .0497 .0132 48 34 BUS-33 38 BUS-37 0 .0415 .1420 .0366 49 35 BUS-34 37 BUS-36 0 .0087 .0268 .0057 50 35 BUS-34 38 BUS-37 0 .0026 .0094 .0098 51 38 BUS-37 39 BUS-38 1 0.0000 .0375 1.0690 52 38 BUS-37 40 BUS-39 0 .0321 .1060 .0270 53 38 BUS-37 41 BUS-40 0 .0593 .1680 .0420 54 31 BUS-30 39 BUS-38 0 .0046 .0540 .4220 55 40 BUS-39 41 BUS-40 0 .0184 .0605 .0155 56 41 BUS-40 42 BUS-41 0 .0145 .0487 .0122 57 41 BUS-40 43 BUS-42 0 .0555 .1830 .0466 58 42 BUS-41 43 BUS-42 0 .0410 .1350 .0344 59 44 BUS-43 45 BUS-44 0 .0608 .2454 .0607 60 35 BUS-34 44 BUS-43 0 .0413 .1681 .0423 61 45 BUS-44 46 BUS-45 0 .0224 .0901 .0224 62 46· BUS-45 47 BUS-46 0 .0400 .1356 .0332 63 47 BUS-46 48 BUS-47 0 .0380 .1270 .0316 64 47 BUS-46 49 BUS-48 0 .0601 .1890 .0472 65 48 BUS-47 50 BUS-49 0 .0191 .0625 .0160 66 43 BUS-42 50 BUS-49 0 .0715 .3230 .0860 67 43 BUS-42 50 BUS-49 0 .0715 .3230 .0860 68 46 BUS-45 50 BUS-49 0 .0684 .1860 .0444 69 49 BUS-48 50 BUS-49 0 .0179 .0505 .0126 70 50 BUS-49 51 BUS-50 0 .0267 .0752 .0187 71 50 BUS-49 52 BUS-51 0 .0486 .1370 .0342 72 52 BUS-51 53 BUS-52 0 .0203 .0588 .0140 73 53 BUS-52 54 BUS-53 0 .0405 .1635 .0406 74 54 BUS-53 55 BUS-54 0 .0263 .1220 .0310 75 50 BUS-49 55 BUS-54 0 .0730 .2890 .0738 76 50 BUS-49 55 BUS-54 0 .0869 .2910 .0730 77 55 BUS-54 56 BUS-55 0 .0169 .0707 .0202 78 55 BUS-54 57 BUS-56 0 .0027 .0095 .0073 79 56 BUS-55 57 BUS-56 0 .0049 .0151 .0037 80 57 BUS-56 58 BUS-57 0 .0343 .0966 .0242 81 51 BUS-50 58 BUS-57 0 .0474 .1340 .0332 82 57 BUS-56 59 BUS-58 0 .0343 .0966' .0242 83 52 BUS-51 59 BUS-58 0 .0255 .0719 .0179 84 55 BUS-54 60 BUS-59 0 .0503 .2293 .0598 85 57 BUS-56 60 BUS-59 0 .0825 .2510 .0569 86 57 BUS-56 60 BUS-59 0 .0803 .2390 .0536 87 56 BUS-55 60 BUS-59 0 .0474 .2158 .0565 88 60 BUS-59 61 BUS-60 0 .0317 .1450 .0376 89 60 BUS-59 62 BUS-61 0 .0328 .1500 .0388 90 61 BUS-60 62 BUS-61 0 .0026 .0135 .0146
216
NO FROM BUS TO BUS TYPE CONDUCT SUSCEPT CHARG/TAP
91 61 BUS-60 63 BUS-62 0 .0123 .0561 .0147 92 62 BUS-61 63 BUS-62 0 .0082 .0376 .0098 93 60 BUS-59 64 BUS-63 1 0.0000 .0386 1.0410 94 64 BUS-63 65 BUS-64 0 .0017 .0200 .2160 95 62 BUS-61 65 BUS-64 1 0.0000 .0268 1. 0150 96 39 BUS-38 66 BUS-65 0 .0090 .0986 1.0460 97 65 BUS-64 66 BUS-65 0 .0027 .0302 .3800 98 50 BUS-49 67 BUS-66 0 .0180 .0919 .0248 99 50 BUS-49 67 BUS-66 0 .0180 .0919 .0248
100 63 BUS-62 67 BUS-66 0 .04s2 .2180 .0578 101 63 BUS-62 68 BUS-67 0 .0258 .1170 .0310 102 66 BUS-65 67 BUS-66 1 0.0000 .0370 .9350 103 67 BUS-66 68 BUS-67 0 .0224 .1015 .0268 104 66 BUS-65 69 BUS-68 0 .0014 .0160 .6380 105 48 BUS-47 1 BUS-69 0 .0844 .2778 .0709 106 50 BUS-49 1 BUS-69 0 .0985 .3240 .0828 107 69 BUS-68 1 BUS-69 1 0.0000 .0370 .9350 108 1 BUS-69 70 BUS-70 0 .0300 .1270 .1320 109 25 BUS-24 70 BUS-70 0 .1022 .4115 .1020 110 70 BUS-70 71 BUS-71 0 .0088 .0355 .0088 111 25 BUS-24 72 BUS-72 0 .0488 .1960 .0488 112 71 BUS-71 75 BUS-75 0 .0446 .1800 .0444 113 71 BUS-71 73 BUS-73 0 .0087 .0454 .0118 114 70 BUS-70 74 BUS-74 0 .0401 .1323 .0337 115 70 BUS-70 75 BUS-75 0 .0428 .1410 .0360 116 1 BUS-69 75 BUS-75 0 .0405 .1220 .1240 117 74 BUS-74 75 BUS-75 0 .0123 .0406 .0103 118 76 BUS-76 77 BUS-77 0 .0444 .14so .0368 119 1 BUS-69 77 BUS-77 0 .0309 .1010 .1038 120 75 BUS-75 77 BUS-77 0 .0601 .1999 .0498 121 77 BUS-77 78 BUS-78 0 .0038 .0124 .0126 122 78 BUS-78 79 BUS-79 0 .0055 .0244 .0065 123 77 BUS-77 80 BUS-80 0 .0170 .0485 .0472 124 77 BUS-77 80 BUS-80 0 .0294 .1050 .0228 125 79 BUS-79 80 BUS-80 0 .0156 .0704 .0187 126 69 BUS-68 81 BUS-81 0 .0018 .0202 .8080 127 80 BUS-80 81 BUS-81 1 0.0000 .6370 1.0690 128 77 BUS-77 82 BUS-82 0 .0298 .0853 .0817 129 82 BUS-82 83 BUS-83 0 .0112 .0366 .0380 130 83 BUS-83 84 BUS-84 0 .0625 .1320 .0358 131 83 BUS-83 85 BUS-85 0 .0430 .14so .0348 132 84 BUS-84 85 BUS-85 0 .0302 .0641 .0123 133 85 BUS-85 Sp BUS-86 0 .0350 .1230 .0276 134 86 BUS-86 87 BUS-87 0 .0283 .2074 .0445 135 85 BUS-85 88 BUS-88 0 .0200 .1020 .0276 136 85 BUS-85 89 BUS-89 0 .0239 .1730 .0470 137 88 BUS-88 89 BUS-89 0 .0139 .0712 .0193
217
NO FROM BUS TO BUS TYPE CONDUCT SUSCEPT CHARG/TAP
138 89 BUS-89 90 BUS-90 0 .0518 .1880 .0528 139 89 BUS-89 90 BUS-90 0 .0238 .0997 .1060 140 90 BUS-90 91 BUS-91 0 .0254 .0836 .0214 141 89 BUS-89 92 BUS-92 0 .0099 .0505 .0548 142 89 BUS-89 92 BUS-92 0 .0393 .1581 .0414 143 91 BUS-91 92 BUS-92 0 .0387 .1272 .0327 144 92 BUS-92 93 BUS-93 0 .0258 .0848 .0218 145 92 BUS-92 94 BUS-94 0 .0481 .1580 .0406 146 93 BUS-93 94 BUS-94 0 .0223 .0732 .0188 147 94 BUS-94 95 BUS-95 0 .0132 .0434 .0111 148 80 BUS-80 96 BUS-96 0 .0356 .1820 .0494 149 82 BUS-82 96 BUS-96 0 .0162 .0530 .0544 150 94 BUS-94 96 BUS-96 0 .0269 .0869 .0230 151 80 BUS-80 97 BUS-97 0 .0183 .0934 .• 0254 152 80 BUS-80 98 BUS-98 0 .0238 .1080 .0286 153 80 BUS-80 99 BUS-99 0 .0454 .2060 .0546 154 92 BUS-92 100 BUS-100 0 .0648 .2950 .0772 155 94 BUS-94 100 BUS-100 0 .0178 .0580 .0604 156 95 BUS-95 96 BUS-96 0 .0171 .0547 .0147 157 96 BUS-96 97 BUS-97 0 .0173 .0885 .0240 158 98 BUS-98 100 BUS-100 0 .0397 .1790 .0476 159 99 BUS-99 100 BUS-100 0 .0180 .0813 .0216 160 100 BUS-100 101 BUS-101 0 .0277 .1262 .0328 161 92 BUS-92 102 BUS-102 0 .0123 .0559 .0146 162 101 BUS-101 102 BUS-102 0 .0246 .1120 .0294 163 100 BUS-100 103 BUS-103 0 .0160 .0525 .0536 164 100 BUS-100 104 BUS-104 0 .0451 .2040 .0541 165 103 BUS-103 104 BUS-104 0 .0466 .1584 .0407 166 103 BUS-103 105 BUS-105 0 .0535 .1625 .0408 167 100 BUS-100 106 BUS-106 0 .0605 .2290 .0620 168 104 BUS-104 105 BUS-105 0 .0099 .0378 .0099 169 105 BUS-105 106 BUS-106 0 .0140 .0547 .0143 170 105 BUS-105 107 BUS-107 0 .0530 .1830 .0472 171 105 BUS-105 108 BUS-108 0 .0261 .0703 .0184 172 106 BUS-106 107 BUS-107 0 .0530 .1830 .0472 173 108 BUS-108 109 BUS-109 0 .0105 .0288 .0076 174 103 BUS-103 110 BUS-110 0 .0391 .1813 .0461 175 109 BUS-109 110 BUS-110 0 .0278 .0762 .0202 176 110 BUS-110 111 BUS-111 0 .0220 .0755 .0200 177 110 BUS-110 112 BUS-112 0 .0247 .0640 .0620 178 18 BUS-17 113 BUS-113 0 .0091 .0301 .0077 179 33 BUS-32 113 BUS-113 0 .0615 .2030 .0518 180 33 BUS-32 114 BUS-114 0 .0135 .0612 .0163 181 28 BUS-27 115 BUS-115 0 .0164 .0741 .0197 182 114 BUS-114 115 BUS-115 0 .0023 .0104 .0028 183 69 BUS-68 116 BUS-116 0 .0003 .0041 .1640 184 13 BUS-12 117 BUS-117 0 .0329 .1400 .0358 185 75 BUS-75 118 BUS-118 0 .0145 .0481 .0120 186 76 BUS-76 118 BUS-118 0 .0164 .0544 .0136 PROGRAM IS DONE
218
118 BUS IEEE FINAL POWER FLOW SOLUTION
----- GENERATOR SUMMARY -------
NO BUS NAME VOLTS MW MVAR QMAX QMIN TYPE
1 1 BUS-69 1.0993 492.4 -84.3 200.0 -50.0 3 2 2 BUS-1 1.0268 o.o 15.0 15.0 -5.0 1 3 5 BUS-4 1. 0587 -9.0 -35.0 300.0 -300.0 2 4 7 BUS-6 1.0465 o.o 6.7 50.0 -13.0 2 5 9 BUS-8 1.0638 -28.0 -34-7 300.0 -300.0 2 6 11 BUS-10 1.0947 450.0 -72.2 200.0 -147.0 2 7 13 BUS-12 1.0446 85.0 37.3 120.0 -35.0 2 8 16 BUS-15 1.0445 o.o 23.6 30.0 -1 o.o 2 9 19 BUS-18 1.0446 o.o 27.3 50.0 -16.o 2
10 20 BUS-19 1.0382 o.o -8.0 24.0 -8.0 1 11 25 BUS-24 1.0547 -13.0 -6.4 300.0 -300.0 2 12 26 BUS-25 1.0978 220.0 -47-0 140.0 -47.0 1 13 27 BUS-26 1 • 0816 314.0 62.7 1000.0 -1000.0 2 14 28 BUS-27 1.0476 -9.0 34.5 300.0 -300.0 2 15 32 BUS-31 1 .0363 7.0 22.5 300.0 -300.0 2 16 33 BUS-32 1.0369 o.o -14.0 42.0 -14.0 1 17 35 BUS-34 1.0709 o.o -8.0 24.0 -8.0 1 18 37 BUS-36 1.0648 o.o -8.0 24.0 -8.0 1 19 41 BUS-40 1.0389 -46.0 19.2 300.0 -300.0 2 20 43 BUS-42 1.0381 -59.0 18.0 300.0 -300.0 2 21 47 BUS-46 1. 0580 19.0 -18.6 100.0 -100.0 2 22 50 BUS-49 1.0801 204.0 61.1 21 o.o -85.0 2 23 55 BUS-54 1. 0460 48.0 42.6 300.0 -300.0 2 24 56 BUS-55 1.0409 o.o -8.0 23.0 -8.0 1 25 57 BUS-56 1.0441 o.o 15.0 15.0 -8.0 1 26 60 BUS-59 1.0686 155.0 83.4 180.0 -60.0 2 27 62 BUS-61 1. 0752 160.0 -10.3 300.0 -100.0 2 28 63 BUS-62 1.0769 o.o 20.0 20.0 -20.0 1 29 66 BUS-65 1. 0593 391.0 49.1 200.0 -67.0 2 30 67 BUS-66 1.1000 392.0 -67.0 200.0 -67.0 1 31 70 BUS-70 1 .0636 o.o 32.0 32.0 -10.0 1 32 72 BUS-72 1.0247 -12.0 -15.2 100.0 -100.0 2 33 73 BUS-73 1.0567. -6.0 -7.6 100.0 -100.0 2 34 74 BUS-74 1.0422 o.o -6.0 9.0 -6.0 1 35 76 BUS-76 1. 0355 o.o 23.0 23.0 -8.0 1 36 77 BUS-77 1.0768 o.o 70.0 70.0 -20.0 1 37 80 BUS-80 1.0889 477.0 -19.8 280.0 -165.0 2 38 85 BUS-85 1.0806 o.o 23.0 23.0 . -8.0 1 39 87 BUS-87 1.0506 4.0 -8.3 1000.0 -100.0 2
219
NO BUS NAME VOLTS MW MVAR QMAX QMIN TYPE
40 89 BUS-89 1 .0983 607.0 -0.6 300.0 -210.0 2 41 90 BUS-90 1.0797 -85.0 47 .1 300.0 -300.0 2 42 91 BUS-91 1. 0816 -1 o.o 4.1 100.0 -100.0 2 43 92 BUS-92 1.0795 o.o -3.0 9.0 -3.0 1 44 99 BUS-99 1. 0691 -42.0 -13.8 100.0 -100.0 2 45 100 BUS-100 1.0758 252.0 59.0 155.0 -50.0 2 46 103 BUS-103 1.0609 40.0 40.0 40.0 -15.0 1 47 104 BUS-104 1.0403 o.o 23.0 23.0 -8.0 1 48 105 BUS-105 1.0320 o.o -8.0 23.0 -8.0 1 49 107 BUS-107 1.0184 -22.0 3.6 200.0 -200.0 2 50 110 BUS-110 1.0196 o.o 23.0 23.0 -8.0 1 51 111 BUS-111 1.0217 36.0 -8.2 1000.0 -100.0 2 52 112 BUS-112 1.0002 -43.0 7.0 1000.0 -100.0 2 53 113 BUS-113 1.0526 -6.0 -24.8 200.0 -100.0 2 54 116 BUS-116 1.0528 -184.0 -10.4 1000.0 -1000.0 2
220
118 BUS IEEE FINAL POWER FLOW SOLUTION
----- BUS SUMMARY -------
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
1 BUS-69 1.0993 o.oo o.o o.o 3 2 BUS-1 1.0268 -16.88 51.0 27.0 1 3 BUS-2 1.0329 -16.27 20.0 9.0 0 4 BUS-3 1.0349 -16.03 39.0 1 o.o 0 5 BUS-4 1.0587 -12.67 30.0 12.0 2 6 BUS-5 1.0642 -12.30 o.o o.o 0 7 BUS-6 1.0465 -14.63 52.0 22.0 2 8 BUS-7 1.0452 -15.02 19.0 2.0 0 9 BUS-8 1. 0638 -7.64 o.o o.o 2
10 BUS-9 1.0917 -1.04 o.o o.o 0 11 BUS-10 1.0947 5.92 o.o o.o 2 12 BUS-11 1.0434 -14.91 70.0 23.0 0 13 BUS-12 1.0446 -15. 31 47.0 1 o.o 2 14 BUS-13 1.0315 -16.20 34.0 16.0 0 15 BUS-14 1.0437 -16.05 14.0 1.0 0 16 BUS-15 1.0445 -16.55 90.0 30.0 2 17 BUS-16 1.0429 -15.68 25.0 1 o.o 0 18 BUS-17 1.0616 -14.29 11.0 3.0 0 19 BUS-18 1. 0446 -16.26 60.0 34.0 2 20 BUS-19 1.0382 -16. 71 45.0 25.0 1 21 BUS-20 1. 0321 -16.00 18.0 3.0 0 22 BUS-21 1.0314 -14.67 14.0 8.0 0 23 BUS-22 1. 0397 -12.49 10.0 5.0 0 24 BUS-23 1.0630 -8.22 7.0 3.0 0 25 BUS-24 1.0547 -8.44 o.o o.o 2 26 BUS-25 1.0978 -1.78 o.o o.o 1 27 BUS-26 1 .0816 -0.02 o.o o.o 2 28 BUS-27 1.0476 -13.15 62.0 13.0 2 29 BUS-28 1.0377 -14.57 17 .o 7.0 0 30 BUS-29 1.0345 -15.36 24.0 4.0 0 31 BUS-30 1. 0511 -9.52 o.o o.o 0 32 BUS-31 1.0363 -15.23 43.0 27.0 2 33 BUS-32 1.0369 -13.55 59.0 23.0 1 34 BUS-33 1.0521 -17 .15 23.0 9.0 0 35 BUS-34 1. 0709 -16.64 59.0 26.0 1 36 BUS-35 1.0658 -16.99 33.0 9.0 0 37 BUS-36 1.0648 -16. 99 31.0 17 .o 1 38 BUS-37 1.0772 -16.26 o.o o.o 0 39 BUS-38 1.0367 -11.25 o.o o.o 0
221
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
40 BUS-39 1. 0461 -18.95 27.0 11.0 0 41 BUS-40 1.0389 -19. 77 20.0 23.0 2 42 BUS-41 1. 0321 -20.11 37.0 1 o.o 0 43 BUS-42 1.0381 -18.56 37.0 23.0 2 44 BUS-43 1. 0578 -16.55 18.0 7.0 0 45 BUS-44 1.0529 -14 .18 16.0 8.0 0 46 BUS-45 1.0487 -12.47 53.0 22.0 0 47 BUS-46 1.0580 -9.82 28.0 1 o.o 2 48 BUS-47 1. 0735 -7.90 34.0 o.o 0 49 BUS-48 1 .0761 -8.57 20.0 11.0 0 50 BUS-49 1 .0801 -7.67 87.0 30.0 2 51 BUS-50 1.0655 -9.57 17 .o 4.0 0 52 BUS-51 1.0437 -11. 96 17.0 8.0 0 53 BUS-52 1.0370 -12.62 18.0 5.0 0 54 BUS-53 1. 0336 -13.70 23.0 11.0 0 55 BUS-54 1.0480 -12.97 113.0 32.0 2 56 BUS-55 1.0409 -13.17 63.0 22.0 1 57 BUS-56 1.0441 -13.04 84.0 18.0 1 58 BUS-57 1. 0504 -11. 02 12.0 3.0 0 59 BUS-58 1.0418 -12.68 12.0 3.0 0 60 BUS-59 1.0686 -9.35 277.0 113.0 2 61 BUS-60 1.0735 -5.96 78.0 3.0 0 62 BUS-61 1. 0752 -5.19 o.o o.o 2 63 BUS-62 1.0769 -5.71 77.0 14.0 1 64 BUS-63 1.0439 -6.24 o.o o.o 0 65 BUS-64 1.0546 -4.71 o.o o.o 0 66 BUS-65 1. 0593 -1.84 o.o o.o 2 67 BUS-66 1.1000 -1.86 39.0 18.0 1 68 BUS-67 1.0837 -4°35 28.0 7.0 0 69 BUS-68 1.0534 -1.94 o.o o.o 0 70 BUS-70 1.0636 -6.27 66.o 20.0 1 71 BUS-71 1.0601 -6.33 o.o o.o 0 72 BUS-72 1.0247 -9.36 o.o o.o 2 73 BUS-73 1.0567 -6.44 o.o o.o 2 74 BUS-74 1.0422 -7.21 68.0 27.0 1 75 BUS-75 1.0507 -6.18 47.0 11.0 0 76 BUS-76 1. 0355 -7.24 68.0 36.0 1 77 BUS-77 1.0768 -2.76 61.0 28.0 1 78 BUS-78 1.0723 -2.98 71.0 26.0 0 79 BUS-79 1.0736 -2.66 39.0 32.0 0 80 BUS-80 1.0889 -0.45 130.0 26.0 2 81 BUS-81 1.0470 -1.43 o.o o.o 0 82 BUS-82 1.0666 -2.25 54.0 27.0 0 83 BUS-83 1.0689 -1.28 20.0 10.0 0 84 BUS-84 1.0726 .79 11.0 7.0 0 85 BUS-85 1.0806 2.07 24.0 15.0 1 86 BUS-86 1. 0611 1. 24 21. 0 1o.0 0
222
BUS NAME VOLTS ANGLE MW LOAD MVAR LOAD TYPE
87 BUS-87 1 .0506 1. 75 o.o o.o 2 88 BUS-88 1.0830 4.74 48.o 1 o.o 0 89 BUS-89 1.0983 8.18 o.o o.o 2 90 BUS-90 1.0797 2.84 78.0 42.0 2 91 BUS-91 1 .0816 2.73 o.o o.o 2 92 BUS-92 1.0795 3.33 65.0 1 o.o 1 93 BUS-93 1.0664 .87 12.0 7.0 0 94 BUS-94 1.0621 -0.89 30.0 16.0 0 95 BUS-95 1 .0525 -1.74 42.0 31.0 0 96 BUS-96 1.0625 -1.88 38.0 15.0 0 97 BUS-97 1.0714 -1.49 15.0 9.0 0 98 BUS-98 1.0768 -1.85 34.0 8.0 0 99 BUS-99 1. 0691 -2.19 o.o o.o 2
100 BUS-100 1.0758 -1.26 37.0 18.0 2 101 BUS-101 1 • 0651 -0.05 22.0 15.0 0 102 BUS-102 1.0737 2.13 5.0 3.0 0 103 BUS-103 1.0609 -4-43 23.0 16.0 1 104 BUS-104 1._0403 -6.98 38.0 25.0 1 105 BUS-105 1.0320 -7 .91 31.0 26.0 1 106 BUS-106 1.0278 -8.12 43.0 16.0 0 107 BUS-107 1.0184 -10.55 28.0 12.0 2 108 BUS-108 1.0253 -8.85 2.0 1.0 0 109 BUS-109 1.0228 -9.20 8.0 3.0 0 110 BUS-110 1.0196 -9.82 39.0 30.0 1 111 BUS-111 1. 0217 -8.24 o.o o.o 2 112 BUS-112 1.0002 -12.22 25.0 13.0 2 113 BUS-113 1 .0526 -14.19 o.o o.o 2 114 BUS-114 1.0369 -13.87 8.0 3.0 0 115 BUS-115 1.0372 -13.88 22.0 7.0 0 116 BUS-116 1.0528 -2.36 o.o o.o 2 117 BUS-117 1.0296 -16.70 20.0 8.0 0 118 BUS-118 1.0377 -7.06 33.0 15.0 0
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