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The use of a genetic algorithm and fuzzy logic to estimate load allocation for adaptive voltage control in an electrical power distribution network

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Title:
The use of a genetic algorithm and fuzzy logic to estimate load allocation for adaptive voltage control in an electrical power distribution network
Creator:
Micek, Robert Joseph
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Language:
English
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xi, 71 leaves : ; 28 cm

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Subjects / Keywords:
Electric power distribution -- Mathematical models ( lcsh )
Electric power-plants -- Mathematical models ( lcsh )
Voltage regulators ( lcsh )
Electric power distribution -- Mathematical models ( fast )
Electric power-plants -- Mathematical models ( fast )
Voltage regulators ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references (leaf 71.).
General Note:
Department of Electrical Engineering
Statement of Responsibility:
by Robert Joseph Micek.

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University of Colorado Denver
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Auraria Library
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Resource Identifier:
66464103 ( OCLC )
ocm66464103
Classification:
LD1193.E54 2005m M52 ( lcc )

Full Text
THE USE OF A GENETIC ALGORITHM AND FUZZY LOGIC TO ESTIMATE
LOAD ALLOCATION FOR ADAPTIVE VOLTAGE CONTROL IN AN
ELECTRICAL POWER DISTRIBUTION NETWORK
by
Robert Joseph Micek
B.S.E.E, University of Nebraska at Lincoln, 1978
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Electrical Engineering
2005


2006 by Robert Joseph Micek
All rights reserved.


This thesis for the Master of Science
degree by
Robert Joseph Micek
has been approved
by
Miloje Radenkovic
Jeffrey Selman


Micek, Robert J (M.S., Electrical Engineering)
The Use of a Genetic Algorithm and Fuzzy Logic to Estimate Load Allocation for
Adaptive Voltage Control in an Electrical Power Distribution Network
Thesis directed by Professor Miloje Radenkovic
ABSTRACT
Voltage regulation is a critical function in the supply and distribution of electricity.
End use equipment operates efficiently within a narrow band of voltage limits. This
thesis examines one approach for voltage control which can adapt to changing load
conditions in a medium voltage distribution network. The voltage control seeks to
obtain as narrow a regulation range as possible centered at the rated nominal
voltage.
Knowledge of the voltage at each customer location is necessary for the voltage
control optimization. An estimate of these voltages is possible if a load distribution
estimate can be made for the network on a real time basis. This thesis creates a
genetic algorithm with fuzzy logic to estimate the load distribution. An unbalanced
load flow solution is derived to provide the fast convergence needed by the genetic
algorithm as part of its fitness evaluation. Results of a test case with the proposed
IV


algorithm as part of its fitness evaluation. Results of a test case with the proposed
approach are compared to a benchmark practice. The benchmark practice maintains
a constant voltage at the substation bus which is designed to meet industry standard
voltage limits at the customer premises.
This abstract accurately represents the content of the candidates thesis. I
recommend its publication.
Signed
Miloje Radenkovic
v


DEDICATION
I dedicate this thesis to my parents for their love, support and faith, to my daughter
for her dedication and example, and to my son for his sense of humor.


ACKNOWLEDGEMENT
I wish to thank my advisor, Joe Beaini for his knowledge and help, as well as Dr.
Radenkovic and Mr. Selman for their assistance. I also wish to thank Professors
Ren Su and P.K. Sen for affording me the opportunity to complete this program.


CONTENTS
Figures....................................................................x
Tables....................................................................xi
Chapter
1. Introduction............................................................1
1.1 Purpose of Distribution Voltage Regulation.............................1
1.2 Purpose of Thesis......................................................2
1.3 Industry Standard Voltage Regulation Practices.........................3
2. Status of Current Research..............................................6
3. Scope of Thesis.........................................................8
4. Physical Network Model.................................................9
4.1 By Phase Representation................................................9
5. Genetic Algorithm.....................................................13
5.1 Genetic Algorithm Search..............................................13
5.2 Chromosome Representation............................................14
5.3 Problem Constraints..................................................14
5.4 Fuzzy Logic Operator.................................................15
viii
5.5 Fitness Value
16


5.6 Load Flow Program.................................................17
6. Voltage Regulation Optimization....................................28
7. Tests..............................................................32
7.1 Model.............................................................32
7.2 Fuzzy Logic Operator..............................................34
7.3 Genetic Algorithm.................................................36
7.4 Benchmark Comparison..............................................39
8. Results............................................................41
8.1 Genetic Algorithm Voltage Variation...............................41
8.2 LTC Set-point Optimization........................................48
9. Conclusions and Discussion.........................................52
Appendix
A. Development of Unbalanced Load Flow Equations.....................55
B. Matlab Genetic Algorithm Settings.................................63
C. Genetic Algorithm Results.........................................65
References............................................................71
ix


FIGURES
Figure
1.1 LTC Control Diagram....................................................5
5.1 Unbalanced Load Flow Model............................................19
6.1 Set-point Optimization Control Diagram Part 1.........................30
6.2 Set-point Optimization Control Diagram Part 2........................31
7.1 Study Network Model...................................................34
7.2 Power Factor Probability Function....................................35
8.1 GA Variation Test Phase A.............................................43
8.2 GA Variation Test Phase B............................................44
8.3 GA Variation Test Phase C............................................45
8.4 Optimized A Phase Voltage............................................49
8.5 Current Practice A Phase Voltage.....................................49
8.6 Optimized B Phase Voltage............................................50
8.7 Current Practice B Phase Voltage.....................................50
8.8 Optimized C Phase Voltage............................................51
8.9 Current Practice C Phase Voltage.....................................51
x


TABLES
Table
1.1 IEEE Std. 141 Recommended Voltage Ranges.....................2
8.1 Genetic Algorithm Power Variation Test.......................46
xi


1. Introduction
1.1 Purpose of Distribution Voltage Regulation
Voltage regulation is a critical function in the supply of electricity. End use
equipment is designed to operate properly within a narrow range of its rated value.
The Institute for Electrical and Electronics Engineers (IEEE) has established
recommended normal operating ranges to meet these requirements. Table 1.1
shows the recommended voltage ranges contained in IEEE Std. 141 [1], The range
is typically between +5% to -8% of nominal under normal operating conditions.
IEEE Std. 141 allows a 3% voltage drop from the customers service entrance to the
end loads. The electric supplier must design the delivery infrastructure to maintain
a range of +5% to -5% to accommodate the customers wiring. This corresponds to
a range of 114 volts to 126 volts for a nominal voltage of 120 volts. The regulation
range is defined as the difference between the highest and lowest steady state
voltages seen by the customer over a given time period. The IEEE Std. 141 limits
given above determine the maximum and minimum limits of the regulation range.
1


Table 1.1 IEEE Std. 141 Recommended Voltage Ranges
Nominal System Voltage Nominal utilization voltage Voltage Range A
3-wire 4-wire Maximum Minimum
Utilization and Service voltage Service voltage Utilization voltage
120/240 115/230 126/252 114/228 110/220
208/120 200 218/126 197/114 191/110
480/277 460 504/291 456/263 440/254
1.2 Purpose of Thesis
It is possible that voltage regulation control which can adapt to changing load
conditions could reduce the regulation range experienced by the majority of
customers since few customers are typically located in very close proximity to the
substation. A more tightly controlled supply to maintain optimal voltages at the
customers delivery point would result in more efficient operation of motor and
lighting loads. This would increase the efficiency of energy usage and delivery.
This thesis will attempt to provide a more accurate load estimate and control
methodology than is currently practiced. The control method should react to
varying load conditions and determine an optimum LTC setpoint to provide nominal
rated voltage to as many customers as possible without violating any IEEE voltage
limits.
2


1.3 Industry Standard Voltage Regulation Practices
Voltage regulation of urban medium voltage distribution circuits is typically
provided by a load tap changer (LTC) located on the load side of the substation
power transformer. The LTC controls are normally set to maintain the substation
medium voltage bus voltage within a specific bandwidth. Figure 1.2 shows a
typical voltage regulator control diagram for an LTC. This type of control is
designed to maintain a minimum customer service voltage level at all times without
excessive operation of the LTC.
This practice does not provide optimum operation of the distribution network in that
customers near the end of the circuit will experience greater swings in their supplied
voltage than is necessary. This is due to the fact that the substation bus voltage is
not adjusted to accommodate for the difference between lightly loaded and heavily
loaded conditions. An excessive regulation range can result in premature customer
equipment failures and unanticipated excursions beyond acceptable voltage supply
limits. For example, voltages above the nominal value will cause heating elements
and incandescent light bulbs to operate at a higher temperature shortening their
expected life spans. According to IEEE Std. 141 [1], a four percent rise in voltage
above the nominal value results in a forty two percent reduction in life for
incandescent bulbs. Voltages below nominal will produce less horsepower from
3


motors which can cause excessive heating if they are too closely matched to their
driven load.
The lower limit of the IEEE recommended voltage range is used by utilities to select
the LTC set-point which allows for the worst case voltage drop under peak loading
conditions when the network is in a normal configuration. An estimate of load
distribution is made on a model of the system network under peak loading to
determine if allowable voltage drop restrictions are exceeded. Some utilities use
historical customer billing data and peak circuit loads to estimate load distribution.
Others apportion the load based on connected transformer capacity.
4


c/1
Figure 1.1 LTC Control Diagram


2. Status of Current Research
A significant amount of research has been accomplished over the last fifteen years
in the area of conservation voltage reduction. Conservation voltage reduction is
most commonly used as a form of demand reduction during peak periods of
electricity usage. Voltages are monitored at the last customer to insure that the
minimum voltage levels are not exceeded. Kirshner [2] reported that
Commonwealth Edison implemented a conservation voltage reduction program with
the installation of 6700 voltage monitors. The monitors were installed in the electric
meter housing and read manually. The voltmeters recorded the highest and lowest
voltages seen by the meter since the last read was performed. More recent research
efforts have concentrated on load estimation based on historical load patterns by
customer class. Fletcher and Saed [3] discussed data model requirements for such a
load estimate in their paper. This thesis attempts to use a genetic algorithm for load
estimation rather than historical load profiles. It also seeks a minimization of the
regulation range for the majority of customers to improve service levels.
Genetic algorithms are a branch of evolutionary computing which have numerous
optimization applications. The most common application in the power industry has
been in the area of network planning. Ramirez-Rosado [4] used a genetic algorithm
6


to meet future load growth through network expansion in the most effective
arrangement. Chen [5] describes a genetic algorithm which decides which
transformer connections should be used along a feeder to minimize load unbalance.
Load estimates were provided by historical benchmark data in both of the
referenced papers.
7


3. Scope of Thesis
Four things are needed to determine an optimum LTC set-point under varying load
conditions. An accurate model of the physical network provides the transfer
function response to varying supply voltages and load demands. A reasonable
estimate of load demands throughout the network provides the driving function to
determine the bus voltage levels. A load flow program is needed to determine the
network losses which are not accounted for in the load estimate and to solve the
nonlinear system equations. Lastly, an optimization method that uses the first three
tools is needed to find the LTC set-point which provides the smallest voltage
deviation from the rated value to as many customers as possible.
The scope of the thesis is to model one branch of the network to the estimated
lowest voltage bus, create a genetic algorithm to estimate the load supplied by each
load bus in the branch, create an unbalanced load flow program for the fitness
functions network solution, and develop a set-point optimization method.
8


4. Physical Network Model
4.1 By Phase Representation
An unbalanced by-phase load flow model and program are used in this thesis
because they provide improved accuracy over a balanced load flow solution. Most
feeders are unbalanced due to the presence of a large number of customers fed by
only one phase. This leads to unbalanced voltages and delivered power at each bus.
A balanced load flow solution would not be able to account for this and its accuracy
would suffer accordingly. Kersting and Phillips [6] determined that a balanced load
flow solution provides poor performance when modeling an unbalanced distribution
network both in the ability to converge to a solution and in the inaccuracy of the
results.
A model of the network needs to identify the source bus, load buses, and the
impedances of the tie lines between each bus. Since an unbalanced phase model is
desired, the effect of the currents in the other phases must be taken into account for
each tie line section.
Current in the other two phases of a line section will produce an inductive coupling
with the current carried in the phase under consideration. This creates an additional
9


voltage drop in the wire. The inductance due to this coupling is given in
Stevensons [7] Equation 3.63 as:
Eq. 4.1.1
2
Lma = *10-7 *ln(l/(D12 *D23 *D31))*(Ib +IC) Henries/meter
where: Lma = the mutual inductance in phase a due to currents lb and Ic
lb, Ic the current in the other two phases
D)2, D23 ,D3i = the distance between conductors 1, 2, and 3
An unbalanced circuit creates a return current that travels through the neutral
conductor as well as the earth in a grounded system. This current also produces an
inductive coupling with the current in the phase under consideration. Carsons
Equations [8] modify the mutual reactance to include the effect of the return current.
The equivalent impedance is given by:
Eq. 4.1. 2
Zm = k,f + jk2f(k3 InD + ln-y %) ohms/mile
where: Zm = the mutual impedance in one phase caused by the other phase
currents
k| = .00158837 ohms/hertz-mile
f = frequency
j = the square root of minus one
k2 = .0020224 ohms/hertz-mile
k3 = 7.67860
D = \jD12 *D23 *D31 where Dxx is in feet
p = resistivity of earth in ohm-meters
10


Two impedances are thus required for each line section. The self impedance, Zs, is
associated with the voltage drop in a wire due only to the current in that wire.
The second impedance, Zm is associated with the voltage drop in a wire due to the
current carried in the wires of the other phases. An equivalent impedance matrix
can then be made such that:
Eq. 4.1. 3
Vda- Zs Zm Zm V
Vdb - Zm Zs Zm lb
1 u > 1 Zm ZmZs .1.-
where: Vda, Vdb, Vdc = the voltage drop in the line for phases a,b,c
This is the industry standard for modeling unbalanced distribution networks
according to Mr. Wayne Carr, the president and founder of Milsoft Utility
Solutions. Milsoft is one of the preeminent providers of load flow and fault
analysis software to the utility industry. Mr. Carr was kind enough to discuss this
subject and provided some of the papers referenced in this study.
Loads can be modeled as constant complex power loads. This is an accurate
portrayal of the loads for the fitness function portion of this thesis since the loads
are modeled with specified source voltages. A constant impedance or constant
current load will have reduced power demands as bus voltages are lowered whereas
11


a constant complex power load will not vary in its demand. Commonwealth Edison
has reported a .52% reduction in energy when the voltage was reduced by 1% [2].
This will cause an error in the LTC set-point optimization part of the thesis. For
example a 2% voltage reduction could result in a 1.04% reduction in demand. This
would result in an error of approximately .06 volts at the last bus if the voltage was
118 volts at that bus. This is an acceptable amount of error for the purposes of the
thesis.
12


5. Genetic Algorithm
5.1 Genetic Algorithm Search
A genetic algorithm is a stochastic search method which mimics an evolutionary
process. It is used to maximize or minimize an objective function. A genetic
representation of possible problem solutions creates the chromosome structure. An
initial population of chromosomes is created to begin the process. An evaluation
method is needed to rank individual chromosomes in terms of their fitness as an
optimum solution. An evaluation of the initial population is done to select the
appropriate individuals to create the next generation. These individuals are called
the parents. Individuals can be passed intact to the next generation or used to create
a new individual which retains portions of the original chromosome. New
individuals called offspring are created by two distinct methods. Crossover selects
genes from two chromosomes and splices them together to create a new
chromosome. Mutation randomly changes some of the genes in one chromosome to
create a new individual. Gen and Cheng [9] state that conventional genetic
algorithms principally use the crossover operator and that performance is heavily
dependent upon it. Offspring are evaluated for fitness and compared to the initial
population. A new generation is created by selecting the best individuals from the
13


offspring and initial population. This process is continued until one of the stopping
criteria is met.
5.2 Chromosome Representation
The chromosome for this problem consists of the predicted real and reactive power
leaving each load bus. The number of genes required in each chromosome would
be 6n where n is the number of buses in a three phase system. A real distribution
branch has buses that do not supply power on every phase. This is because single
phase taps are often made to serve small commercial buildings and residential
subdivisions. These genes do not need to be represented in the chromosome since
there is no real or reactive power supplied to the missing phases. This reduces the
number of genes to 6n-2m, where m is the number of missing phases.
5.3 Problem Constraints
The real and reactive powers leaving a bus will always be a positive number when
there are no capacitor banks or generators connected to the network. This represents
one of the problem constraints. Another set of constraints relate to the conservation
of energy. The real and reactive powers injected into the network from the
substation bus are generally measured quantities. The sum of the genes associated
with a particular phase and type of power must be less than the power injected into
14


the network. In fact, this sum must be less than the injected power by the amount of
the system losses. The problem constraints can be expressed by:
where:
Q-SO
Ps.>L'!.,Pi
z:=1p4ps-p,
Qs >Z".,Q
Z"Q.aQs.-Q
mloss, max
mloss, max
Eq. 5.3.1
Pim = the mth phase real power at bus i
Qim = the mth phase reactive power at bus i
Psm = the mth phase real power injected into the network
Qsm= the mth phase reactive power injected into the network
Pmloss,max = the maximum expected mth phase real power losses
Qmioss.max = the maximum expected mth phase reactive losses
n = the number of buses in the network
5.4 Fuzzy Logic Operator
Loads encountered in practice will have power factors that typically range from .5
lagging to unity. Fuzzy logic can be used to determine the suitability of an
individual chromosome to reflect this expected range of power factors. A fuzzy
logic operator embedded in the genetic algorithm will provide direction to the
stochastic search while allowing unlikely areas of the solution space to be explored.
A probability function must first be established to describe the expected distribution
of values. A penalty function can then be assigned based on the expected
15


probability of any given power factor to determine the fitness of a chromosome in
the genetic algorithm.
5.5 Fitness Value
Gen and Cheng [9] recommend the use of a penalty function to convert a
constrained problem into an unconstrained problem. Penalties are assigned to any
chromosome which does not meet the problem constraints. A penalty determined
by the fuzzy logic operator mentioned in the preceding section can also be included
to direct the search away from unlikely solutions. These penalties constitute the
penalty function. The fitness value determined for each chromosome is the sum of
the penalty function and the fitness function. Weighting factors can be used with
each assigned penalty to assist in the selection process. These factors are chosen
based on the value of each constraint or probability of certain genes. The fitness
function is the objective function for which an optimum solution is desired.
The fitness function in this thesis calculates the voltages at the last bus for each
chromosome and compares them to the specified values. The fitness function
should be minimized to produce the least error in the predicted voltages. The fitness
function is calculated as follows:
Eq. 5.4.1

spec
V V
nb, calc | nb, spec
'V -V
nc, calc nc, spec
)*wv
16


|Vnx ca]c | = the calculated x phase voltage magnitude at the last bus
where:
Vn, sp£C = the specified x phase voltage magnitude at the last bus
Wv = the fitness function weighting factor
5.6 Load Flow Program
A load flow program is needed to calculate the voltages used in the fitness function.
An iterative ladder technique is commonly used in commercial load flow programs.
This type of solution is rather simple and well adapted to a large distribution
network. Kersting [6] provides an excellent description of this method. It involves
the use of an assumed voltage at the last bus and calculation of the corresponding
source voltage. All of the line currents are then adjusted through the use of a
correction factor. The correction factor is the ratio between the specified and
calculated source voltages. The adjusted line currents are then used with the
specified source voltage to calculate the last bus voltage. This process is repeated
until a convergence limit is met.
Another method that has been used for load flow analysis is the Newton-Raphson
method [7]. This is an older method that was used for balanced load flow analysis
of transmission systems. The disadvantage of this method is the determination of a
large matrix called the Jacobian whose size grows exponentially with the number of
buses in the study. This methods advantage lies in its rapid convergence on a
17


solution. It will converge quadratically [10] within certain limits. Rapid
convergence is desirable when many load flow solutions are to be made.
There is a large computational load required to run a load flow analysis for each
possible solution generated by the genetic algorithm. A two to five minute solution
time for the entire process is desirable to make voltage control changes if necessary
when the load is rapidly changing during peak conditions. Accuracy of the load
flow solution is also critical to the process since it is used to determine the fitness
values.
The Newton-Raphson method for a load flow solution is developed from the basic
network equations. Two modifications need to be made to the method outlined in
Stevensons text [7] for an unbalanced load flow solution. The first modification
must change the development of the admittance network to include the impact of
mutual impedances. The second modification must expand the network equations to
include all three phases.
18


Bus la
Bus 2a
11
Figure 5.1 Unbalanced Load Flow Model
Figure 5.1 portrays a-phase of the simple network used to develop the equations for
the admittance network. lLXm is the mth phase current leaving the network at Bus x.
Vmxy is the voltage drop due to the mutual coupling of the current in the other two
phase wires. Zmxy denotes the mutual impedance between phases in line xy. Zsxy is
the self impedance of line xy. These terms define what is known as the primitive
impedance matrix. The primitive impedance matrix is then used to form the
admittance matrix.
19


The network equations for the model of Figure 5.1 are:
EQ. 5.5.1
I.-" "yyy yyyy yy 1 lala 1 1 a 1 b 1 laic 1 \a2a 1 la^ 1 Ia2c 1 Ia3a 1 Ia3b 1 Ia3c VIa~
I,b YYYYYYY YY 11bla 1lblb 1 lblc 1 lb2a 1lb2b 1 lb2c 1 lb3a 1lb3b xlb3c Vlb
I., YYYYY YYYY 1 lcla 11 c lb 1 Iclc 1 lc2a 1 lc2b 1 lc2c 1 lc3a 1 lc3b 1 lc3c V,c
I2a YYYYY YYYY 1 2ala 1 2a I b 1 2alc 1 2a2 a 1 2a2b 1 2a2c 1 2a3a 1 2a3b 1 2a3c V2a
I2b YYYYYYYYY 12bla 1 2blb 12blc 12b2a 1 2b2b 1 2b2c 1 2b3a 1 2b3b 1 2b3c V2b
I2c YYY YYYYYY A 2c!a 12clb x2clc 12c2a 1 2c2b 1 2c2c 12c3a 12c3b 1 2c3c V2c
ha YYY YYY YYY 1 3ala 13alb 1 3a 1 c 1 3a2a 1 3a2b 1 3a3c 1 3a3a 1 3a3b 1 3a3c V3a
13b YYYYYY YYY 1 3bla 1 3blb 13blc 1 3b2a 1 3b2b 1 3b2c 1 3b3a 1 3b3b 1 3b3c V3b
I*. YYY YYY YYY L 1 3cla 1 3clb 1 3clc 1 3c2a 1 3c2b 1 3c2c 1 3c3a 1 3c3b 1 3c3c J _V3C.
where: Iuv = the v phase current injected into the network at bus u
Vxy = the y phase voltage at bus x
Yuvxy the admittance between current Iuv and voltage Vxy
The admittance elements are derived from the primitive impedance matrix.
Applying Kirchoff s current law for phase a at bus 1 yields:
EQ. 5.5.2
Iia =V,aYla +(Vla V2a -(I12b +I12c)Zm12)Ys12 +(Vla V3a -(I13b +I13c)Zm13)Ys13
EQ. 5.5.3
I,. = Vla(Yla + Ys12 + Ys13) V2aYsl2 V3aYs13 -(I12b + I12c)Zm12Ys12
13b + ^13c )Zm,3Ys,3
where: Ysxy=l/Zsxy
Yia = the a phase source impedance at bus 1
20


Ixyz = the z phase current that flows from bus x to bus y
This equation is identical to the one developed in Stevensons text with the
exception of the terms due to the mutual coupling with other phase currents.
Similar equations can be written for the other phase currents at bus 1. The same
application can then be used at buses 2 and 3. These are shown in Appendix A.
The mutual coupling terms need to be expressed in terms of their dependence on the
associated bus voltages rather than line currents. This development is also given in
Appendix A. If we define two variables gxy and hxy as:
EQ. 5.5.4
gxy =ZmXyYSxy
EQ. 5.5.5
hxy =0 + 2gxy)(l-gxy)
Then Equation 5.5.2 can be rewritten as:
EQ.5.5.6
Ii.=Vlt(Ylt+Ys12(l + gI2) + Ys13(l + g13))-V2aYs12(l + g12)-V3>Ys13(l + g13)
-(V, +Vlc)(YsI2g12/h12 +Ysl3g13/h13) + (V2b + V2c)(Ysl2g12/h12)
+ (V3b+V3c)(Ys13g13/h13)
Equation 5.5.6 reduces to the one developed in Stevensons text when there is no
mutual coupling. The admittance matrix elements can now be created. The general
values are shown below.
21


EQ. 5.5.7
Yk^,=Ylm+X-!Ys11(l + g)
11 l*k
where: n = the number of buses in the network
for m p:
V V"_ Vs JLsl
* kmkp ^ki ,
hki
for j k :
Yjmkn, =-Ysjk(l + gjk)
for j k and m p:
Y** =Ys*(fi)
njk
Defining Vkm = |Vkm|zpkm, Vk*m = |Vkm|Z-Pkm Yjmkp = |Yjmkp|Z0jmkp,
calculate the complex power conjugate, S^m = Vk*mIkm = Pkm iQkm
s;a = v;avlaYlala +v*vlbYlalb + v;.vleY1>le +v,;v2aYla2a +v,;v2bYla2b
+v;av2cYla2c +v* v3aYla3a +v,;v3bYla3b +v;av3cYla3e
EQ. 5.5.8
EQ. 5.5.9
EQ. 5.5.10
we can
EQ. 5.5.11
22


EQ. 5.5.12
P =|Vr|Y1>l,|cos8l, + |VjVlb||Y,.lb|CoS(01,lb +Plb -P,a)
+ lvIK|Y,c|cos(ei,lc +pic -Pl,) + |Vl,||V2,||Y,|cos(eM, +p2a-p)
+ |V,.||V2l||YMb|cos(e,ub+plb -Pla) +1Vla|V2c||Yla2c|cos(91a2b +P2t -p)
+ lvuIVJ,|Y,.1,|cos(9J> +p2. -p) + |Vl,||VJb||Y1.Jb|cos(eiub + PJb -P)
+ |V,a|V2t||Y,aJC|coS(0laJI +p,b -p)
In general,
= E1, SJVJNMcos^ +P# -PJ
EQ. 5.5.13
Solving for Qta,
Q,.=Hv1,r|Y,,l,|sin9lb,,-|V1,||V,b||Yjsin(0l,,l + plb-P,,)
-|V,.||Vlt||YIC|sin(eulb +P,b p) |Vla||V2a||Yla2a|sin(0la3a +P2a p,a)
-|Vla||V2b||Yla2b|sin(eia2b + p2b -p,a> -|Vla|V2a|Yla2c|sin(0la2c +P2t -p,a)
-|vJVJa|Yla3a|Sm(0laJ.+pJa-Pla)-|V,a||VJb|YlJsin(0,a3b + pJb-Pla)
-|V,.||VJt||YlaJt|sin(0
la3c + P3c-P.a)
EQ. 5.5.14
In general,
Qkn, = -El, Zj=a !V^ l|Vij |Yknuj |sin(0kmij + Pjj 'PkJ
EQ. 5.5.15
The genetic algorithm generates the estimated real and reactive power by phase
delivered at each load bus. These are used by the load flow program as the
specified complex power exiting the network at each load bus. The Newton-
23


Raphson method uses Equations 5.5.13 and 5..5.15 to calculate the complex power
entering or leaving the network at each bus using assumed values for the voltage at
each bus. The specified source voltages are commonly used for the initial estimate.
The calculated complex power values are then compared to the specified values as
shown below to determine the error.
EQ.5.5.16
AP = P
^rk rk, spec
-P,
k, calc
k Q k, spec Q k, calc
where: Pk, spec the specified real power at load bus k
Qk.spec = the specified reactive power at load bus k
Pk, caic = the calculated real power at load bus k
Qk, caic = the calculated reactive power at load bus k
The Newton-Raphson method uses a Taylor series expansion without the higher
order differential terms as shown below to calculate the needed corrections for the
bus voltages.
Ap.=I>&8%i +
+I>v,5%
where:
AVj = the correction to be made to V,
APj = the correction to be made to p;
EQ. 5.5.17
24


Equation 5.5.17 can be shown in matrix form as:
EQ. 5.5.18
Bus 1 is used as the swing bus and not included in the above matrix. The power is
not specified at the swing bus in order to allow it to vary in accordance with the
system losses. The voltage at the swing bus is held constant while the voltages at
the remainder of the buses are adjusted in accordance with the above equation. The
large matrix of partial derivatives is called the Jacobian. Its size is the chief reason
this method is not used for large networks. Multiplying both sides of Equation
5.5.18 by the inverse of the Jacobian calculates the desired changes in the assumed
bus voltages. The addition of these changes to the assumed values completes the
first iteration. The process is continued until the desired convergence limit is met.
25


The elements of the Jacobian matrix are calculated as follows:
=I +P, -P.) for ij km: EQ. 5.5.19
5% =-v"viYsin k/^SVk = YlmkmCOSlcmm + 2ji=] XjCOS(kmj + Pij Pkm) for ij km: EQ. 5.5.21
8% = VkmYkljCOSfe,^ +P -P,m) EQ. 5.5.22
5Q75P, =S !!*; V>-VsY-cos<6'. +P -P> for ij km: EQ. 5.5.23
6Q74,J=-v"vY-cos(e- +pi -p-> EQ. 5.5.24
EQ. 5.5.25
5Ql7sv =-2V.Y-si- vsyMisMe^+Pj- / u v km ij*km -PkJ
26


for ij km:
6Q
km j
'8Y;
=-VkmYkmijsin(ekmij +P,J -Pkm)
EQ. 5.5.26
27


6. Voltage Regulation Optimization
The goal of this thesis is to attempt to minimize the regulation range for the greatest
number of customers and to center the range at nominal rated voltage. A new
optimized LTC set-point will need to be determined. The new set-point should
provide the lowest root mean squared (rms) value of all the bus voltage deviations
from the nominal rated value. The rms value, AV^ is defined as:
Eq. 6.1. 1
AVm = JX;jVbl Vr)2 /n
where: Vbi = the voltage at the ith customer
Vr = nominal rated voltage value
n = the number of customers
The value AVavg can be defined as:
Eq. 6.1. 2
AV... = A load flow solution can be run with the normal voltage set-point to determine the
base case load bus voltages, the voltage drops from the load bus to each customer,
AV^, and AVavg. The set-point can be changed by AVavg to determine a new set-
point. A new load flow solution can be run with the revised set-point and new
values calculated for AV^, and AVavg. The process can be repeated until the
28


desired convergence limit for AV^ is achieved. Figure 6.1 shows the control
diagram of the optimization method.
29


Figure 6.1 Set-point Optimization Control Diagram Part 1
30


Calculate rms voltage
deviation, V^. Create
new LTC Set-point =
Vs.
Y
_________I________
Export new Set-
point to Voltage
Regulator Control.

Calculate average
voltage deviation,
DVaVg. Calculate new
Vs=Vs-DVavg.
Count=l.
Figure 6.2 Set-point Optimization Control Diagram Part 2


7. Tests
7.1 Model
There were over 2,300 load points supplied by the actual circuit used to develop the
test model. It was not feasible to model thousands of load buses and achieve a
reasonable solution time. This program was developed in Matlab which uses
separate files for each subroutine. This has a much slower execution time than
compiled code. The selected test model uses only the branch to the customer with
the lowest estimated voltage on a heavily loaded feeder. Using the most critical
branch rather than the entire network supplied by one transformer reduced the
number of load buses from thousands to seven. This should provide an accurate
scalable demonstration for the test. This also provides a model of most concern to
meet minimum voltage limits. If there were another branch of significant concern
on a feeder, the process could be repeated by a similar model for that branch using
the bus voltages and estimated delivered power at the bus where the new branch
connects to the initial one. This schema should meet the accuracy and speed
requirements of the problem.
Figure 7.1 shows the model of the branch considered in this study. The model
represents seven load buses along a critical branch of a heavily loaded circuit at a
32


major distribution utility. The loads supplied by each load bus are assumed to be
complex power loads. The self and mutual impedances used in this study came
from a major supplier of load flow analysis software. The inputs to the program
consist of the type of conductor and length for each tie line between the load buses
as well as the voltage and complex power injected at Bus 1. The impedances are
then calculated using the self and mutual impedance values.
Equivalent loads were created to model the voltage ranges seen by the customers
supplied from the load buses. All of the loads were placed at an equivalent load
point of the distribution network beyond the models load bus on a per phase basis.
The equivalent load point was determined by inspection and connected transformer
capacity. The voltage drop from the load bus to the equivalent load point was
determined by a separate load flow program. This drop was then subtracted from
the load bus voltage calculated by the Matlab program to model the voltage at the
equivalent customers location on the primary network. An additional 3 volt drop
was also subtracted to model the voltage drop through the secondary network to the
customers premises.
33


Bus 7 Bus 8
Figure 7.1 Study Network Model
7.2 Fuzzy Logic Operator
A probability function was created to estimate the likelihood of any given power
factor of the load supplied by a network bus for each phase. Figure 7.2 illustrates
the probability function used in the program.
34


Figure 7.2 Power Factor Probability Function
A power factor less than 1 was assigned a probability of zero. Power factors of .7
or greater were given the same probability. A power factor of. 1 was assigned a
probability equal to 5% of that assigned to power factors of .7 or greater. Power
factors between .1 and .7 had a proportional probability assigned to them. The area
under the curve represents the probability of all values which must equal 100%. If
the maximum probability of the higher power factors is defined as x, the area under
the curve can be expressed by:
35


Eq. 7.2.1
AREA = .6 (.05x) + .5 .6 (.95x) + ,3x = 1
Solving for x provides the value of 1.626 as shown in Figure 7.2. The power factor
of the loads at each bus were evaluated by phase. A penalty of. 15 was assessed for
power factors below 1. No penalty was assigned for power factors of .7 or greater.
Power factors from .1 to .7 were penalized based on the probability function. A
penalty of .15 (1 (probability / 1.626)) was assessed for power factors in the .1 to
.7 range.
7.3 Genetic Algorithm
A genetic algorithm is used to estimate the power delivered by each load bus. The
chromosome consists of the real and reactive power per phase supplied by each load
bus. There are seven load buses in the test model. Three of the seven buses did not
have three phase taps associated with them. Two of these buses have single phase
taps and the third bus has two single phase taps. This means there are five missing
phases in the model. The total number of genes required for this model is (6x7)-
(5x2) or 32. Zeros are spliced to the chromosome at the appropriate positions
before the load flow program was run to determine the fitness function.
The genetic algorithm tool used by Matlab only allows an unconstrained problem
solution. All of the genes generated by the genetic algorithm were first converted to
positive numbers before the fitness value is calculated in order to meet the first
36


problem constraint. Due to the limited amount of time to complete this thesis, the
algorithms performance using an unconstrained search with negative values was
not evaluated.
Losses up to 5% were allowed before a penalty factor was assessed to meet the
second set of problem constraints. The penalty factor was ten times the amount by
which the constraint was exceeded for any given phase or type of power. A penalty
factor was also assessed by the fuzzy logic operator as described in the previous
section. These penalty factors were then summed and added to the fitness function
to form the fitness value. This is the value passed back to the genetic algorithm. As
an example, a penalty of .3 was assessed if any constraint was exceeded by 100 kw
or 100 kvars. The same penalty was assessed if two phases on any bus supplied
loads with a power of less than ten percent.
A multiplier of 2.5 was used in the fitness function for any difference between the
calculated and specified voltages at the last load bus. This would result in a fitness
function value of .25 if the difference between any specified voltage and its
calculated value from the chromosome was .1 volts on a 120 volt base. The values
of 100 kw and .1 volts were selected as reasonable limits of resolution for the load
flow program and LTC optimization. The multiplier for the power conservation
penalty factor results in a slightly higher value than the voltage. This was done
37


because these constraints proved much easier for the genetic algorithm to solve than
the voltage limits. Test runs done to meet only the power constraints derived an
acceptable solution in approximately 4 generations on a consistent basis. The
rationale was to put a slightly higher emphasis on the power constraints so that the
algorithm could find a solution that met these constraints before zeroing in on the
fitness function. The power factor penalty was not given as large an emphasis in
order to allow greater flexibility in the search of the solution space.
A number of variances from the default values provided in Matlabs genetic
algorithm tool were assessed in an effort to speed minimization and obtain the
smallest fitness value. The only changes which seemed to offer a significant
improvement on a consistent basis was a varying population length of [100, 50, 20],
and a crossover fraction of 67%. The other default values in the tool were an elite
count of 2 individuals, a gaussian mutation with a shrink and scale of 1, scattered
crossover and a 20% forward migration at an interval of 20 generations. Appendix
B contains the definitions of these variables provided in Matlab. The criteria to
terminate the genetic algorithm were set at a maximum of 150 generations, a fitness
value of zero, and a stall time of 20 generations.
Three iterations of the load flow program were used rather than setting a
convergence limit. Five load flow tests were run with the model wherein the power
38


levels were adjusted from 80% to 120% of an assumed case. This was done to test
the programs convergence. All of the tests converged to less than .1 volts after
three iterations.
Per unit quantities are used in the load flow solution. These quantities divide the
real values for voltage, current, impedance and power by what are known as its base
values. These values were used because they can easily be converted across
transformers. This was not necessary for the study but would be of benefit if the
study were extended through the secondary network. A base voltage of 13.2 kv for
line to line voltage and a base power for three phase power of 10 MVA was
selected. The selection was made to closely match the existing conditions of the
actual circuit. The base values for impedance and current are 17.42 ohms and
437.4 amps. They are derived as shown below:
Eq. 7.2. 2
MVAbase
(KVbase)2
MVAbase
7.4 Benchmark Comparison
The following voltage regulation practice at a major distribution utility was used for
comparison in this test. The substation bus voltage of 7,852 volts line to ground is
39


maintained within a bandwidth of 140 volts. This is the equivalent of a 123.6 volt
set-point and a 2.2 volt bandwidth on a 120 volt base voltage.
Maximum and minimum loading cases were run to estimate the regulation ranges
seen by the customers. The assumed voltage drop from the models load bus to the
customer were proportionally adjusted based on the minimum to peak loading ratio
to account for reduced power flows in the minimum loading case. Values of
AVavg and AVnns were calculated and a new LTC set-point determined. The load
flow program was run with the new set-point and new customer voltages calculated.
The algorithms results were then compared against the utilitys practices to
measure any possible performance improvement.
40


8. Results
8.1 Genetic Algorithm Voltage Variation
Ten runs were completed by the program for the maximum loading case to test the
variations in the search method. The average solution time was one minute forty
seconds per run. This was below the two to five minute solution time previously
stated as the goal. The load estimates derived by the genetic algorithm are shown in
Appendix C. Two of the runs converged to the same solution set. Runs 4 and 7
reached a solution with a fitness value of .0945. The other eight runs had fitness
values that ranged from. 122 to .232. All of the runs had acceptable fitness values.
The complex powers contained in the different solutions varied considerably. Table
8.1 displays the estimated loads and best fitness values from these ten runs of the
genetic algorithm. For example, the estimated a-phase reactive power at Bus 7
ranged from 5 kvars to 417 kvars, a variation of 8,340%! According to Duan [11],
many power problems are NP hard problems which means that an optimum solution
can not be found. That appears to be the case with this problem since there are a
number of solutions which have acceptable fitness values but vary widely from each
other.
41


Genetic algorithms do not always present a practical view of possible solutions
since they are designed to search both the feasible and infeasible regions of the
solution space. This behavior can be seen in Table 8.1 by examining the power
factor of the estimated loads. Runs 1, 6, 8, and 9 all had one bus with a power
factor of less than 40 percent. This is less than the expected value in a real
distribution network where power factors would typically range from 50 to 95
percent. The fuzzy logic operator did well in directing the genetic algorithms
search. However, some excursions beyond expected values does still occur.
The solution sets did define a fairly narrow deviation range when the bus voltages
were compared. Figures 8.1 through 8.3 show the variations for each phase. The
maximum variation of .36 volts occurred on a-phase at Bus 6.
The maximum voltage variations on b and c phases were approximately .3 volts.
A voltage variation of .1 to .2 volts would be desirable for tight voltage control.
The genetic algorithm is able to find solutions which meet the voltage and power
conservation criteria. The fuzzy logic operator did aid in the restriction of the loads
expected power factor range. However, the variations in predicted bus voltages
were too large for accurate voltage optimization.
42


Volts
Phase A
Figure 8.1 GA Variation Test Phase A
H I tHHH


Volts
Phase B
Run 1
Run 2
M Run 3
Run 4
Run 5
Run 6
IRun 7
---Run 8
Run 9
Run 10


Volts
Phase C
U\
124.00
123.00
122.00
121.00
120.00
119.00
118.00
Bus 1 Bus 2 Bus 3 Bus 4 Bus 5 Bus 6 Bus 7 Bus 8
Figure 8.3 GA Variation Test Phase C
II ItHHH


Table 8.1 Genetic Algorithm Power Variation Test
Run Real Power (kw) Reactive Power (kvar) Bus
Phase a Phase b Phase c Phase a Phase b Phase c
1 987 607 561 528 625 101 2
405 1161 529 33 274 122 3
Fitness 0 0 174* 0 0 468* 4
Value = 0 0 83 0 0 84 5
.122 185 240 532 78 69 281 6
286 214 305 149 30 24 7
443 569 0 353 349 0 8
2 669 1441 140 258 264 4 2
761 385 624 46 8 226 3
Fitness 0 0 298 0 0 93 4
Value = 0 0 326 0 0 193 5
.1575 302 218 441 233 452 336 6
369 358 363 5 345 212 7
314 448 0 605 294 0 8
3 924 1014 425 97 310 179 2
425 977 338 135 19 646 3
Fitness 0 0 394 0 0 3 4
Value = 0 0 539 0 0 12 5
.232 757 194 101 627 413 99 6
247 352 417 262 393 151 7
70 301 0 38 213 0 8
4,7 945 1349 229 236 376 52 2
348 449 842 358 116 127 3
Fitness 0 0 316 0 0 124 4
Value = 0 0 297 0 0 311 5
.0945 145 458 119 140 583 28 6
645 204 389 24 181 448 7
250 437 0 368 136 0 8
5 1138 835 738 161 135 139 2
372 776 299 110 583 165 3
Fitness 0 0 384 0 0 187 4
Value = 0 0 82 0 0 84 5
.147 111 748 640 89 37 345 6
534 461 141 417 584 160 7
259 12 0 333 11 0 8
Power factor is between 30 and 40 percent.
46


Table 8.1 (Cont.) Genetic Algorithm Power Variation Test
Run Real Power (kw) Reactive Power (kvar) Bus
Phase a Phase b Phase c Phase a Phase b Phase c
6 708 877 540 94 87 169 2
606 975 315 219 441 325 3
Fitness 0 0 342 0 0 117 4
Value = 0 0 358 0 0 266 5
.147 661 267 656 366 166 195 6
255 236 30 129 365 12 7
122* 442 0 333* 280 0 8
8 655 1047 168 33 295 195 2
571 792 1038 493 127 43 3
Fitness 0 0 202 0 0 17 4
Value = 0 0 241 0 0 99 5
.156 656 264* 161 257 610* 269 6
108 313 470 95 188 428 7
319 456 0 256 129 0 8
9 894 885 544 325 196 258 2
369 878 455 30 268 98 3
Fitness 0 0 241 0 0 98 4
Value = 0 0 125 0 0 278 5
.223 489 274* 252 196 717* 157 6
306 601 575 410 162 180 7
271 183 0 195 46 0 8
10 542 841 172 387 469 29 2
801 926 828 264 36 173 3
Fitness 0 0 208 0 0 229 4
Value = 0 0 160 0 0 133 5
.162 771 222 372 86 405 183 6
38 591 433 9 181 328 7
263 210 0 382 254 0 8
* Power factor is between 30 and 40 percent.
47


8.2 LTC Set-point Optimization
The LTC optimization was developed using a solution from the genetic algorithm
for the minimum and maximum circuit loading. This will indicate the possible
improvement from adaptive voltage control from the benchmark practice were
accurate prediction of the bus voltages possible. These results are shown in Figures
8.4 through 8.9. The graphs illustrate the voltages experienced by customers at the
equivalent load points under minimum and maximum loading conditions. The
number of customers at each equivalent load point is shown along the graphs x-
axis. These figures show a dramatic reduction in the regulation range for all
customers. For example, Figure 8.5 shows that the customers on a-phase would see
a 2 to 3 volt regulation range under the benchmark practice. Figure 8.4 shows that
the range is reduced to .2 to 1.2 volts. In fact, the customers experiencing the
greatest voltage variations achieved an approximately 60% reduction in their
regulation range under optimization compared to the benchmark method.
48


Optimized Phase A
Maximum lead condition
m Mnimm lead condition
Figure 8.4 Optimized A Phase Voltage
Figure 8.5 Current Practice A Phase Voltage
49


Phase B Current Practice
JO
1
12200
121.00
120.00
119.00
naoo
117.00
116.00
115.00
114.00
- Maxi mm load condtion
- Mnirnm load concftion
373 248 194 57
No. of Customers
241
Figure 8.7 Current Practice B Phase Voltage
50


Optimized Phase C
No. of Customers
Maximum load condtion
Mnimm load condition
Figure 8.8 Optimized C Phase Voltage
Phase C Current Practice
-a Maxi mm load condtion
Mrimm load condtion
Figure 8.9 Current Practice C Phase Voltage
51


9. Conclusions and Discussion
The thesis derived a modification of the Newton-Raphson method to analyze
unbalanced networks. The modified Newton Raphson method did meet the
performance requirements of this thesis. It would not be a good candidate for
solution of a large network due to the size of the Jacobian matrix.
The results of the previous chapter demonstrate that adaptive voltage control can
produce a significant improvement in regulation over the benchmark practice. The
ability to adjust the voltage control under changing load conditions can provide a
much tighter control of the voltage experienced by all the customers. The
maintenance of tighter voltage supply limits to the customer would improve the
operation and longevity of customer equipment. Although the genetic algorithm
was not able to estimate loads, it did provide a reasonable estimate of the voltage
profile. Further improvements in accuracy would be necessary before the voltage
profile could be used for adaptive voltage control. Its accuracy does appear
sufficient to validate the results generated from a load estimate derived from
historical usage patterns. An accurate load estimate derived by historical usage
should provide a voltage profile similar to that of the genetic algorithm.
The test did not include single phase lines as part of the model. This could have
been done by the additional simultaneous equations for just the phase under
52


consideration. Practical deployment of an adaptive voltage control would have to
include this capability.
It is noted from Figures 8.4 through 8.9 that the minimum load curves for both the
optimized condition and the current practice are nearly identical. As stated
previously, the current practice maintains the substation bus voltage at a level that
ensures minimum voltage limits are not exceeded during maximum loading
conditions. The minimum load curve may indicate that design criteria such as
circuit loading and service area are limited to provide this profile under more
normal conditions.
The model network did not have any customers close to the substation. The first
load bus was 6000 feet from the substation. An urban area substation supplies a
number of circuits from one transformer. Any one of these circuits would be likely
to have customers supplied from a load bus within a half mile of the substation.
This would tend to provide a greater spread between minimum and maximum load
cases for both the optimized and current practice methods of voltage control.
Figures 8.4 through 8.9 display the voltages obtained if the voltages were
maintained at the LTC set-point. An actual voltage regulator control has an
associated bandwidth to limit the number of LTC operations. A 2.2 volt bandwidth
53


which is used under the current practice means that the spreads shown in the Figures
could be as much as 2.2 volts greater for short periods of time.
The genetic algorithm runs which came to the exact same solution set found local
minimums. All of the solutions with fitness values below .25 are probably equally
acceptable since this fitness value approaches the accuracy of the load flow solution.
54


Appendix A
Development of Unbalanced Load Flow Equations
These equations are developed for the simple model of Figure 5.1. Applying
Kirchoffs current law at bus 1:
Eq. A. 1
I,a =VlaYla +(VU V2a -(II2b +I12c)Zm12)YsI2 +(Vla -V3a -(I13b + II3c)ZmI3)Ys13
= Vla(Yla + Ys12 + Ys13) V2aYs12 V3aYs13 -(I12b + li2c)2m12Ys]2
_ 13b + Il3c)Zm!3Ysi3
Eq. A. 2
I,b = VlbYlb +(Vlb V2b -(I12a + I12c)Zm12)Ys12 +(VIb V3b -d13a + I13c)ZmI3)Ys13
= Vlb(Y,b + Ysl2 +Ys]3)-V2bYs12 -V3bYs13 -d12a +Il2c)Zm12Ys,2
-d,3a+I13c)Zm13Ys13
Eq. A. 3
I,c =VlcYlc +(Vlc V2c -d12a +I12b)Zm12)Ys12 +(Vlc -V3c -d,3a + I13b)Zml3)Ys13
= Vle(Yle + Ys12 +Ys13)-V2cYs12 V3cYs13 -d12a +I12b)Zm12Ys12
'(^13a + ^13b)^mi3Ysi3
Applying Kirchoffs voltage law from bus 1 to bus 2 for phase a:
Eq. A. 4
Y)a = I12aZst2 +di2b^12c)Zm,2 + V2a
Eq. A. 5
%b ^12b^S12 12a *"^]2c)Zmi2 + V2b
55


Eq. A. 6
V,c 112c12 + 12a + ^12b)^mi2 + ^2c
Solving for Ii2a, 112b, and Ii2c,
!i2a =(V,. -V2a)Ys12 -(I12b +II2c)Zm12Ys12
^ 12b = C^lb _^2b)^S12 (I]2a ^12c )^mi2^S12
Ii2c=(Vlc-V2c)YsI2-(II2a+I12b)Zm12Ys12
Combining Equations A.7 and A.8,
Eq. A. 7
Eq. A. 8
Eq. A. 9
Eq. A. 10
^12a "*"^12b = (%a ^2a + ^lb 'V2b)YS,2 12a ^12b )^mi2^S12 2Zm,2Ys,2 I 12c
Eq. A. 11
flu. +Il2b)(l + ZmI2Ys12) = (Vla -V2a + Vlb -V2b)Ys12 -2Zm12Ys12Il2c
^ 12a + ^ 12b
(V.a ~ V2a +Vlb V2b)Ys12 2Zm12Ys12I,2c
l + Zm12Ys12
Combining Equations A.8 and A.9 yields:
Eq. A. 12
^ 12b + ^ 12c
(Vlb-V2b+Vlc-V2c)Ys
2c >
l + Zm12Ysl2
2Zml2Ys12Il2a
Eq. A. 13
56


Combining Equations A.7 and A.9,
Eq. A. 14
^ 12a + I|2c
(Vla-V2a+Vc-V2c)Ys12-2Zm12Ys12I12b
l + Zml2Ysp
Substituting Equation A. 13 into Equation A.7,
I|2a=(VIa-V2a)Ysi2-
Eq. A. 15
Zm12Ys,2((V,b V2b + Vlc V2c)Ys12 -2Zm12Ys12I12a)
l + Zm12Ys12
= (v y2 )Ysi2 Zmi2YSp(V,b -V2b +Vlc V2c) | 2Zm,22YSpI12a
l + Zm12Ys12
l + Zm12Ys12
Combining like terms:
Eq. A. 16
2Zmf2YSp _ Zm,2Ysf2 (Vlb V2b + Vlc V2c)
1l2at1 / tVla V 2a / 1 ^ 12
1 + Zm12Ys12
l + Zml2Ys12
I
12a
(VIa -V2a)Ys12
Zm12YSp(Vlb V2b +Vlc V2c)
1 + Zml2Ys12
2Zmf2Ys,~
l + Zml2Ysl2
(V,a V2a)Ys,2(l + Zm,2Ys,2)-Zm,2Ys^2(V,b V2b +V,C V2c)
l + Zm]2Ys12 -2ZmpYsf2
(V,a ~ V2a)Ys,2(l + Zm,2Ys,2)-Zm,2Ys^2(V,b V2b + V,c V2c)
(1 + 2Zm12Ys12)(l-Zm12Ys12)
Eq. A. 17
57


Similarly, substituting Equation A. 14 into Equation A.8,
Eq. A. 18
_ (V,b ~V2b)Ys12(l + Zm,2Ys12)-Zm,2Ysf2(V,a V2a + V,c V2c)
(1 + 2Zml2 Ys12 )(1 Zm12 Ys12 )
And substituting Equation A. 12 into Equation A.9,
Eq. A. 19
_ (V.c - )Ys.20 + Zml2Ys12) Zm12Ys,22 (V V2a +V,b V2b)
(l + 2Zm12Ys12)(l-Zml2Ys12)
Applying the same method to solve for Ii3a, I^b, and Ii3C yields:
_ (Vla V3a)Ys13(l + Zm,3Ys,3)-Zm,3Ysf3(V,b V3b +Vlc -V3c)
(1 + 2Zm13Ys13)(l -Zml3Ys13)
_ (V,b ~ V3b)Ys,3(l + Zm,3Ys,3)-Zm,3Ysf3(V,a V3a + V,c V3c)
(1 + 2Zm13Ys13)(l Zm13Ys13)
Eq. A. 20
Eq. A. 21
(V ~V3c)Ys,3(l + Zm,3Ys,3)-Zm,3Ysf3(V,a V3a +V,b V3b)
(1 + 2Zm13Ys13)(l Zm13Ys13)
Eq. A. 22
Combining Equations A. 18 and A. 19,
Eq. A. 23
I|2b + I|2c
(V,b -V2b)Ys,2(l + Zm,2Ys,2)-Zm,2Ysf2(V,a -V2a +V,C -V2c)
(1 + 2Zm12Ys12)(l Zm12Ys12)
58


, (V,c-V2c)Ys12(l + Zm12Ys12)-Zm12Ys^(V,a-V2a+V,b-V2b)
(1 + 2Zm12Ys12)(l Zml2Ys12)
_ (V,b -V2b + V,c V2c)Ys,2(l + Zm,2Ys12)-Zm,2Ys^2(V,c V2c +V,b V2b)
(1 + 2Zml2Ysl2 )(1 Zml2 Ys12 )
Zm,2Ys*2(2V,a -2V2a)
(1 + 2Zm12Ys]2)(l Zm12Ysl2)
(Vlb -V2b + V.c V2c)Ysl2 +Zm,2Ys^2(V,b -V2b +V,C -V2c -V,c + V2c -V,b + V2b)
(1 + 2Zm12Ys12)(l Zm12Ys12)
Zm12Ys^2(2V,a 2V2a)
(1 + 2Zml2Ysl2)(l Zm12Ys12)
_ (Vlb V2b + V.c V;c)Ys,2 Zm,2Ysf2(2V,a -2V2a)
(1 + 2Zm12Ysl2)(l Zml2Ysl2)
Combining Equations A. 17 and A. 19 yields:
I)2a ^12c
(v -V2a + V.c -V2c)Ys,2 -Zm,2YSl22(2V,b -2V2b)
(1 + 2Zm12Ys12)(l Zml2Ys,2)
Eq. A. 24
Combining Equations A. 17 and A. 18 yields:
I]2a + ^12b ~
(V,a V2a + V,b V2b)Ys,2 -Zm,2Ysf2(2V,c -2V2c)
(1 + 2Zm12Ys,2)(l Zml2Ys12)
Eq. A. 25
Applying the same method for Ii3a, Ii3b, and Ii3C yields:
^13b + f13c '
(VIb V3b + V]c -V3c)YsI3 Zm13Ys*3(2Vla -2V3a)
(1 + 2Zml3Ysl3)(l Zm13Ysl3)
Eq. A. 26
59


Eq. A. 27
113a + 113c
(V|a ~Vja + V,c -V3c)Ysl3 -Zm,3Ys,23(2V,b -2V3b)
(1 + 2Zm13Ys13)(l Zm13Ys13)
T +I _(V,a-V3a+Vlb-V3b)Ys13-Zm13Ysf3(2Vlc-2V3C)
113a + 113b
(l + 2Zm13Ysl3)(l-Zml3Ysl3)
Eq. A. 28
Substituting Equations A.23 and A.26 into Equation A.l,
I.a VU(YU + Ys]2 + Ys)-V2aYs12 -V3aYs13
Eq. A. 29
- (Zm12Ys]2)
(V.b V2b + V -V2c)Ys,2 -Zm,2Ysf2(2V,a -2V2a)
(1 + 2Zm12Ysl2)(l -Zm12Ysl2)
-(Zm13Ys13)
(V.b V3b + Yc V3c)Ys,3 -Zm,3Ysf3(2Vla -2V3a)
(1 + 2Zm13Ysl3)(l Zm^YSu)
= Vla(Yla+Ys12 +
2Zmf2Ys'2
(l + 2Zm12Ys12)(l -Zm12Ys12)
- +Ys,
2Zmf,Ys3
(l + 2Zm13Ys,3)(l-Ztn13YS|3)
)'V2a(Ysl2 +
2Zm,2Ys,2
(l + 2Zml2Ysl2)(l-Zml2Ys,2)
)
V3a(Ys,3 +
2ZmfjYsJ3
(l + 2Zml3Ys13)(l-Zml3Ys1j)
)
-(Vlb+Vlc)(
Zm12Ys12

Zm,3Ysl3
(1 + 2Zm12Ys12)(l Zm12 Ys12) (1 + 2Zm13Ys13)(l Zm13Ys13)
)
+ (V2b+V2c)(
Zm,2Ys12
(l + 2Zm12Ys12)(l-Zm12Ys12)
)
60


+ (V3b+V3c)(
Zm13Ys13
(1 + 2Zm13Ys13)(l Zml3Ys13)
)
If we define gip and hiP as:
Eq. A. 30
for i p:
Sip =ZmipYsiP
hip=(l + 2ZmipYsip)(l-ZmipYsip)
Then Equation A.29 can be rewritten as:
I.a = Vla(Y,a+Ys12(l +
2gf2
2g ?,
) + Ys]3(1 +
(l + 2gI2)(l-gi2) (l + 2g13)(l-gl3)
Eq. A. 31
))
-V2aYsI2(l +
2g,22
(l + 2g12)(l-g12)
)-V3aYs13(l +
2gf3
(l + 2g13)(l-g13)
)
" (VIb + Vle + (v2b + V2c )(^^) + (V3b + V3c )(^^)
h|2 h13 i2 h13
= Vla(Y]a+Ys12(l + g12-2gl22+2gf2) + Ys13(l + g13-2gf3+2g?3))
- V2aYsl2(l + g12 -2g,22 +2g,22)-V3aYs13(l + g13 -2g23 +2g?i)
"(Vlb+Vlc)(
Ysi2gi2 Ysngi3
)+(v2l + v^X-^M+fv,, + V)(
Ys13g13,
M3
M3
= Vu(Yl,+Ysl2(l + gl2) + Ys(l + g]J))-VYSl!(l + g1I)-V1,Ys1,(l + gIJ)
(v^+v^x^a+^i
^YSl2g.2
r YS13gl3 '
-)+(V2b + V2c + (V3b + v3c x^^M
M2
M2
61


Substituting Equations A.24 and A.27 into Equation A.2,
Eq. A. 32
I,b = Vlb(Yll>+Ysl20+g12) + Ysl,(l + gl)))-V2lYs12(l + gl2)-VYs,J(l + g)
-(V,, + VJ(^^ + ^% + (V2. + V2,)3^) + (V +VJc)3^)
h12 h13 h12 h13
Substituting Equations A.25 and A.28 into Equation A.3,
Eq. A. 33
I= Vlc(Y1,+Ysll(l + g,2) + Ys,J(l + g))-VJ,Ys12(l + g,J)-V*Ys,1(l + g)
-(V,. +Vlb)(^Iil!H-^il)+(V2, +V2l)(^^-) + (VJ, + VJb)3^)
62


Appendix B
Matlab Genetic Algorithm Settings
Population size is a vector of length greater than 1, the algorithm creates multiple
subpopulations. Each entry of the vector specifies the size of a subpopulation.
Initial range specifies lower and upper bounds for the entries of the vectors in the
initial population.
Rank scaling scales the raw scores based on the rank of each individual, rather than
its score which removes the effect of the spread of raw scores.
Stochastic uniform selection chooses parents for the next generation based on their
scaled values from the fitness scaling function. It lays out a line in which each
parent corresponds to a section of the line of length proportional to its expectation.
The algorithm moves along the line in steps of equal size, one step for each parent.
At each step, the algorithm allocates a parent from the section it lands on. The first
step is a uniform random number less than the step size.
Elite count specifies the number of individuals that are guaranteed to survive to the
next generation. It must be less than or equal to Population Size.
Crossover fraction specifies the fraction of the next generation, other than elite
individuals, that are produced by crossover. The remaining individuals, other than
elite individuals, in the next generation are produced by mutation.
Gaussian mutation makes small random changes in the individuals which provides
genetic diversity and enables the GA to search a broader space. It adds a random
number to each vector entry of an individual. This random number is taken from a
Gaussian distribution centered on zero. The variance of this distribution can be
controlled with two parameters. The Scale parameter determines the variance at the
first generation. The Shrink parameter controls how variance shrinks as generations
go by. If the Shrink parameter is 0, the variance is constant. If the Shrink
parameter is 1, the variance shrinks to 0 linearly as the last generation is reached
Scattered crossover combines two individuals, or parents, to form a new
individual, or child, for the next generation. It first creates a random binary vector. It
63


then selects the genes where the vector is a 1 from the first parent, and the genes
where the vector is a 0 from the second parent, and combines the genes to form the
child. For example,
pi = [ab c d e f g h]
p2 = [l 2345 67 8]
random crossover vector = [ 1 1 0 0 1 0 0 0]
child = [ab34e67 8]
Migration is the movement of individuals between subpopulations, which the
algorithm creates if population size has a length greater than 1. Every so often, the
best individuals from one subpopulation replace the worst individuals in another
subpopulation. Forward migration takes place toward the last subpopulation. That
is the nth subpopulation migrates into the (n+l)'th subpopulation. Fraction controls
how many individuals move between subpopulations. Fraction is the fraction of the
smaller of the two subpopulations that moves. If individuals migrate from a
subpopulation of 50 individuals into a population of 100 individuals and fraction is
0.1, 5 individuals (0.1 50) migrate. Individuals that migrate from one
subpopulation to another are copied. They are not removed from the source
subpopulation. Interval controls how many generations pass between migrations.
Stopping criteria determine what causes the algorithm to terminate.
Generations specifies the maximum number of iterations the genetic algorithm
performs.
Time limit specifies the maximum time in seconds the genetic algorithm runs
before stopping.
Fitness limit If the best fitness value is less than or equal to the value of Fitness
limit, the algorithm stops.
Stall generations If there is no improvement in the best fitness value for the
number of generations specified by Stall generations, the algorithm stops.
Stall time limit If there is no improvement in the best fitness value for an interval
of time in seconds specified by Stall time limit, the algorithm stops.
64


Appendix C
Genetic Algorithm Results
Run 1
Best: 0.12219 Mean: 0.41703
Current Best Indvickjal
Run 2
Best: 0.1574S Mean: 0.35704
CLment Best IncividLBl
65


Run 3
Best: 0.2318 Mean: 0.59399
Run 4
Best: 0.094454 Meert 0.48758
66


Run 5
Best: 0.14739 Meert 0.54362
Current Best Indvidual
Run 6
Best: 0.14679 Mean: 0.3393
9
60 80
Generation
Current Best Indivickjal
140
67


Run 7
Best: 0.004454 Mean 0.48758
Run 8
Best: 0.15633 Mean 0.32801
68


Run 9
Best: 0.2232 Mean: 0.47285
Run 10
Best: 0.16169 Meert 0.45526
69


Minimum Loading Optimization Run
Best: 0.0053979 Mean: 0.13903
8r
10 20 30 40 50 60 70 80 90 100
Generation
CLrrcnt Best Individual
Maximum Loading Optimization Run
Best: 0.12232 Mean: 0.63501
70


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71