SENSITIVITY BASED VOLT/VAR CONTROL AND LOSS OPTIMIZATION

By

ANURAG R. KATTI

A THESIS PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2012

c⃝ 2012 ANURAG R. KATTI

2

To my parents

3

ACKNOWLEDGMENTS

I thank all the people who have supported me over the duration of this thesis and

beyond. In particular I would like to thank my parents and my advisor Dr. Pramod

Khargonekar at the Dept. of Electrical and Computer Engineering. I would also like to

thank Dr. Wonsuk ”Daniel” Lee at the Dept. of Agriculture and Biological Engineering,

University of Florida for giving me the opportunity to work on interesting projects, my

friends and colleagues in the Precision Agriculture Laboratory. And last but not the

least, I would like to thank my friends and past roommates Diwakar Raghunathan, Kiran

Tumkur, Niki Nachappa and Ugandhar Reddy and all my friends over the years.

4

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 OVERVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Distributed Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Distribution Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Voltage and Reactive Power Control . . . . . . . . . . . . . . . . . . . . . 18

3 PROBLEM DESCRIPTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 VOLTAGE-VAR CONTROL AND LOSS MINIMIZATION . . . . . . . . . . . . . 32

5.1 Voltage Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.2 Centralized Voltage Control and Loss Optimization . . . . . . . . . . . . . 34

5.2.1 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.2.2 Constraints and Optimization Problem . . . . . . . . . . . . . . . . 39

5.3 Decentralized Voltage Control and Loss Optimization . . . . . . . . . . . . 395.4 Optimization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.4.1 Quadratic Programming . . . . . . . . . . . . . . . . . . . . . . . . 425.4.2 Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . 43

5.5 Results and Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6 GENERATOR SITING AND SIZING . . . . . . . . . . . . . . . . . . . . . . . . 49

6.1 Generator Siting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.2 Generator Sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.3 Impact of DG Penetration on Loss . . . . . . . . . . . . . . . . . . . . . . 55

7 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

APPENDIX: FEEDER CONFIGURATIONS . . . . . . . . . . . . . . . . . . . . . . . 59

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6

LIST OF TABLES

Table page

2-1 Comparison of voltage, current and loss with and without DG . . . . . . . . . . 19

5-1 Comparison of optimization performance for case 1 . . . . . . . . . . . . . . . 45

5-2 Comparison of optimization performance for case 2 . . . . . . . . . . . . . . . 46

5-3 Comparison of optimization performance for case 3 . . . . . . . . . . . . . . . 47

7

LIST OF FIGURES

Figure page

2-1 Schematic of a Power grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2-2 N+1 load feeder with a distributed generator connected at the last node . . . . 19

2-3 Current drawn at different voltages for different load types . . . . . . . . . . . . 20

5-1 Sensitivity of 34 node feeder for four different DG positions . . . . . . . . . . . 33

5-2 Variation of sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5-3 No result condition of the 13 node feeder . . . . . . . . . . . . . . . . . . . . . 47

6-1 Sensitivity of 34 node feeder for six different DG positions . . . . . . . . . . . . 50

6-2 Sensitivity for a decreasing voltage profile . . . . . . . . . . . . . . . . . . . . . 50

6-3 Sensitivity for an increasing voltage profile . . . . . . . . . . . . . . . . . . . . . 51

6-4 Sensitivity of current to DG location . . . . . . . . . . . . . . . . . . . . . . . . 52

6-5 Sensitivity of power loss to DG location . . . . . . . . . . . . . . . . . . . . . . 53

6-6 Variation of losses for different penetration levels and number of DGs . . . . . 55

A-1 Schematic of IEEE 13 node test feeder . . . . . . . . . . . . . . . . . . . . . . 59

A-2 Schematic of IEEE 34 node test feeder . . . . . . . . . . . . . . . . . . . . . . 60

8

Abstract of Thesis Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of the

Requirements for the Degree of Master of Science

SENSITIVITY BASED VOLT/VAR CONTROL AND LOSS OPTIMIZATION

By

ANURAG R. KATTI

May 2012

Chair: Pramod KhargonekarMajor: Electrical and Computer Engineering

The objective of this study is to control voltage at different points in a distribution

grid to within a specified range and minimize power loss. Sensitivity coefficients are

used to determine the reactive power dispatch of distributed generators connected in the

grid. Power loss is also represented in terms of sensitivity coefficients to simultaneously

optimize the voltage profile and power loss. Two variants of the optimization algorithm

are discussed - a centralized control algorithm based on the complete state of the

system and a decentralized algorithm using only the local information.

To study the influence of the location of generators in the grid, the properties of

sensitivity and its variation for different generator locations are studied. Two different

sensitivity coefficients - current and power loss sensitivity with respect to location of

generator are developed from the voltage sensitivity values. The use of these sensitivity

coefficients in siting and sizing of generators are discussed. And finally the influence

of the number of generators and penetration of distributed generation on voltage and

distribution loss are discussed through simulations.

9

CHAPTER 1INTRODUCTION

Distribution systems are the last stage in the delivery of power to the customer.

Power produced by generators is transported through the transmission network at high

voltages to distribution networks and delivered to the customer at the utility voltage

typically around 120V/240V for residential customers and possibly for other types of

customers with larger power requirements. The transfer of power from the substation -

the beginning of the traditional distribution system to the customer causes current to flow

downstream in the distribution grid. Currents in a circuit cause a voltage drop between

nodes and power loss in the conductor joining them. To deliver the maximum power

to the end user, the power lost during the transfer of energy must be minimized. Utility

companies must also ensure the quality of power at the output terminals. One aspect

of power quality deals with the voltage magnitude voltage at the outlets must not vary

more than 5% of the nominal voltage (120V) according to ANSI standards [1].

To ensure quality of power and provide the most efficient transfer of power, the

distribution company has to perform voltage control and loss minimization respectively.

In the traditional distribution grid, control was achieved by adjusting the taps on the

on load tap change transformer and voltage regulators or adjusting the reactive power

compensation of any capacitor banks or other compensation devices. To ensure the

most economical compensation, optimization was necessary. Because of the use of

reactive power (VAR) sources for compensation, the operation is called Volt/VAR control.

The optimization function could be the cost of compensation, the number of tap changes

since the lifetime of tap change transformers is limited, loss, etc.

In recent times however, there’s been a call to upgrade the distribution grid and

make it more smarter and allow it to handle power flow in the opposite directions as

well, among a list of other improvements i.e. allow for generators to be connected at

the distribution level (called distributed generators or DGs) and not just at the high

10

voltage (HV) level. This initiative has come to be known as smart grid and a few such

installations are already in development [2]. Customers who use generators for stand-by

power or cheap, alternative power are embracing the idea of DGs. This has encouraged

planners to envision a grid that can accommodate generators on the consumer

side from which the utilities can purchase excess energy during shortages. Such

improvements would make it unnecessary to buy more power or invest in expensive

generators to satisfy the growing demands.

While distributed generation has many advantages to offer, its effect on the grid is

still being studied, especially at high penetrations. In the current state of low penetration

of DGs in the grid, their effect is negligible. However, it is expected that the penetration

of DGs will increase in the future. Ambitious targets of 30% renewable penetration

in the US grid by 2030 have been made. Although not all of it is in the form of DGs,

they are expected to form a significantly large portion of renewable sources. European

countries already have a significant percentage of their generation produced by DGs

and renewable energy [3] and it is estimated to grow further in the future.

With the growing presence of DGs, studying its effects at high penetrations

becomes necessary because generators will affect the direction of power flow; their

effects need to be considered more carefully when multiple generators are connected at

multiple locations. Integration of new sources leads to problems with control, protection,

islanding and maintenance, to list a few. Multiple new sources embedded into the grid

would only complicate the matter. This study explores one aspect of integration of DGs

called Voltage/VAR Control which aims to control the voltage and power flow in the

distribution grid through VAR compensation using DGs. Another aim of this study is to

minimize loss during power flow in the distribution grid - called distribution losses.

Loss profile can also benefit from the proper placement of DGs on the grid; for

example, a thumb rule is line currents can be reduced by placing power sources close

to load centers. Since loss is directly proportional to the square of magnitude of line

11

current, reducing the line current reduces losses. Similarly, observations can be made

on sizing of DGs in a grid and the effect of higher penetration of DGs on the loss profile.

Sensitivity is a concept associated with the power flow Jacobian and is calculated

by inverting the Jacobian matrix. Sensitivity of a node denotes the change in voltage at

that node for a unit change in power at some node on the grid. If the power change can

be affected by a DG, sensitivity can indicate the amount of power output necessary to

effect a required change.

Based on the concept of sensitivity VVC, loss minimization, siting and sizing of

DGs and the effect of DG penetration on losses are studied. The study has been

organized as follows: chapter 2 discusses the motivation for the study and gives an

overview of distribution networks, distributed generation, the Voltage-VAR control

problem and distribution losses. Chapter 3 formulates the voltage/VAR control problem

mathematically. Chapter 4 lists past studies in Voltage-VAR control, use of distributed

generators for voltage control, distribution loss and optimization and a compensation

technique based on sensitivity. Chapter 5 discusses a modified algorithm based on

sensitivity to incorporate loss optimization in the Volt-VAR control problem. Chapter

6 discusses siting and sizing of DGs and the effect of increasing DG penetration in

distribution networks. Chapter 7 concludes the study with a note on the applications of

DGs, VVC and loss minimization using DGs and future work.

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CHAPTER 2OVERVIEW

The power industry is experimenting with changes in the manner of power delivery

and new avenues of power generation and improvement in delivery are being sought.

The changes bring with them a new set of challenges and problems. This study aims

to tackle a small set of those challenges pertaining to the inclusion of distributed

generation and comment on the effect of increased penetration of these generators in

the grid and their effect on the distribution system losses.

2.1 Motivation

This study aims to accomplish a fourfold objective:

1. Voltage control with reactive power compensation using DGs2. Distribution loss minimization3. Siting and sizing of DGs4. Study the effect of increasing penetration of DGs in the distribution grid

Voltage control is a necessary operation required to be performed by a distribution

company to maintain power quality. Traditionally DGs haven’t been included in the

control operations but with the increasing number of DGs [4], [5], it may soon become

feasible to use them for control operations. DGs, especially the inverter interfaced DGs

have quick response rates and can respond quickly to changing conditions.

Losses are a big concern in electricity transmission and distribution. The U.S.

is among the biggest consumers of electric power but it loses over 260 billion kWh

every year - the highest in the world, despite having an efficient system that loses only

6% [6], [7]. The lost energy translates to a cost of about $20 billion [5], [7]. China,

with comparable power consumption to the U.S. loses less power in transmission and

distribution. European countries have a similarly efficient system. Smaller land area

also limits the losses in these nations. But larger countries like India and Brazil with

much lower consumptions than the U.S. lose nearly a quarter and a sixth of its energy

respectively in transmission and distribution. India is in fact second only to the U.S. in

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the absolute amount of power lost (nearly 220 billion kWh). There is thus a need to

improve efficiency in both transmission and distribution for economic reasons.

Increasing distribution efficiency would also reduce the energy loss in the

transmission system due to the reduction in power transmitted over long lines. DGs

with their local siting offer the possibility of reducing energy transmission over long

ranges. Technological improvements in the field of renewable energy generation also

offers the possibility that further expansion in generation distributed or otherwise and

power consumption can be from clean energy with a smaller carbon footprint.

However a framework for their use and control needs to be developed. This study

is a step in that direction with control and compensation achieved using sensitivity

coefficients. Chapter develops the mathematical formulation of the voltage control

problem but a brief overview of the popular DG technologies, distribution systems and

voltage and loss control in sections 2.2 -2.4.

2.2 Distributed Generation

Distributed generation is a blanket term used to describe small scale power

generators that are connected at the distribution level or on the customer side of

power meter [8]. While there’s no consensus on the power output of DGs, most studies

consider outputs ranging from kilowatts (KW) to a few megawatts (MW) as distributed

generation. DGs have been classified in some studies [4] into micro: up to 5KW; small:

5KW-5MW; medium: 5MW-50MW; and large: 50MW-300MW.

Despite the growing interest and the reducing costs of renewable electricity such

as wind and solar, fossil fuel based generators are still the most economical and

reliable forms of generation and micro turbines are among the cleanest of combustion

based generators and when used as a cogeneration unit it can have efficiencies of

80% and above. Micro turbines burn fuel at high temperature and pressure and the

resulting fumes cause rotation of turbines blades at high speeds. When coupled with an

alternator, this produces electricity. Micro turbines can be designed for a wide variety of

14

fuels such as fuel oil, natural gas, etc. Micro turbines are small in size, clean and can

operate for long periods of time with low maintenance [8].

Fuel cells [8] generate electricity through electrons generated by an electrochemical

reaction. The electrons travel through the electrical circuit connected to the cell

producing direct current. Fuel cells require a constant supply of fuel - for example,

hydrogen to operate. Fuel cells can have an efficiency of over 50% even without CHP

and over 80% with cogeneration.

Unlike fuel based generators, renewables harness the natural sources of energy

which also makes them intermittent sources; for example: solar cells cannot work

efficiently on a cloudy day, a wind turbine cannot generate electricity when there is no

wind and droughts will halt production in a hydroelectric power station.

The most popular forms of renewable energy are solar-thermal power, solar

photo-voltaic cells but the most popular is probably wind energy. Wind energy has

been used to do work for a long time and they’re being used to generate electricity

as well. Wind turbines are designed to intercept the path of the wind which causes

rotation of the turbine blades. They are typically connected to an induction generator to

produce electricity but synchronous generators are in use as well [9]. To produce usable

electricity, steady winds are necessary. Wide open spaces are therefore ideal to set up

wind turbines and wind farms; for example, Midwest USA is well suited for large wind

farms. But some of the strongest winds are observed over the sea and it is estimated

that wind energy is more abundant off-shore than on-shore [10]. Despite having one

of the largest installed wind capacities in the world, USA does not have many off-shore

farms. Off-shore farms are abundant in many European countries where wind power is

already a significant portion of the generated power; example: 20% in Denmark or 10%

in Ireland and Spain [11].

Solar power is utilized in two ways - directly converting to electricity with a

photovoltaic cell or indirectly with a concentrated solar power where the sunlight is

15

focused to a small region using lenses and mirrors to generate steam to rotate turbines.

Photovoltaic cells convert solar energy to electricity using the photovoltaic effect where

a voltage difference is induced across P-N junction by shining radiation (solar radiation)

on one of the surfaces [12]. Photovoltaic cells generate DC voltage and additional

electronics (called inverters) are needed to convert it to AC for interconnection with the

grid. While the cost of solar generator modules is reducing, it is still more expensive to

produce a unit of energy using solar that the more traditional generators.

With the emphasis on revamping the grid into a smart grid which seeks to support

plug and play usage capability for DGs, it can be assumed that DGs are going to

become an integral part of the electric grid because DGs can be used to expand the

power capacity of a distribution system without purchasing additional power or build

expensive, new generator stations. Another advantage of distributed generation is

that the generators are much smaller than the centralized generation resources and

therefore cheaper. Therefore new technological improvements can easily be deployed in

the form of distributed generators.

Distributed generators do not currently have an active role in providing ancillary

services to the grid; they are instead expected to produce power at a constant rate at

a constant power factor. During low voltage situations they are required to ride through

or disconnect from the grid in severe cases. Two reasons [13], [14] for their passive

connection are 1> DGs do not have sufficient generation capability to have a significant

effect and 2> a control algorithm operating in parallel with the utility control operations

might aggravate the situation. However with modern, fast acting, electronic control

systems and communication networks, DGs can be included in a coordinated control

plan to provide voltage and power support. This study is an attempt to devise such a

coordinated control technique.

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2.3 Distribution Networks

Due to economies of scale, generation of power was traditionally done at remote

locations close to the fuel source and away from the consumers. Distribution networks

are delivery systems to bring power from the generators though the transmission grid to

the consumer. The transmission system which begins at the generator and ends at the

distribution substation is a meshed network for increased reliability and power sharing.

But the distribution system (beginning at the distribution substation and ending at the

customer’s premises) is mainly radial i.e. lines starting from the substation rarely form

loops.

Figure 2-1. Schematic of a Power grid. Source: US Department of Energy [15]

Most nodes of a distribution system are rarely connected to more than two other

nodes. The series of branches forming a chain are known as feeder lines and the

feeder(s) connected to the substation bus are known as the main feeder. The others

are known as laterals or sub-feeders. Nodes are any points of interest in the network,

generally points with a connected load, a lateral, a transformer, DG, regulator, etc.

Another difference between distribution and transmission systems is that series

resistance of distribution lines as a fraction of the series reactance (typically referenced

by R/X ratio) is much higher for distribution lines whereas in transmission lines the

reactance is dominant. A consequence of this property is real power can also be

dispatched for voltage regulation whereas in transmission systems, reactive power

produces a bigger voltage change for the same amount of dispatch. Although this

17

study is limited to the conventional reactive power (VAR) support, from a voltage

regulation stand-point, real power dispatch can produce a similar result assuming the

line reactance and resistance are comparable.

A distribution system may also be unbalanced i.e. all three phases of the power

system may not be equally loaded; one or more phases may not even be used. This is

one of the reasons that traditional power flow algorithms used for transmission systems

cannot be used for distribution systems. Due to the unbalanced nature, the high R/X

ratios and the radial nature of the grid, Newton-Raphson type methods may fail to

converge. Therefore other methods better suited for radial distribution system conditions

have been developed. The forward-backward method sweep based on ladder theory

[16] is used in this study for all power flow operations.

The power flow algorithm treats the substation bus as the slack node and the

remaining nodes as PQ nodes. Including PV nodes in the feeder complicates power

flow because keeping a constant voltage at a particular node requires a VVC operation.

Therefore, for simplicity even DGs are considered as PQ nodes with a negative load

value to indicate that they feed power into the network instead of consuming it.

As evidenced by the radial topology, the distribution grid was not originally designed

for a bidirectional flow of power. Although the cables can handle the reverse flow of

current, protection devices such as distance relays assume a unidirectional flow of

current. A bidirectional flow will cause a reduction in line currents which can adversely

affect the detection capacities of the relay. It is also a concern for service personnel

operating on a faulty line - in a radial structure it is easy to de-energize a line by cutting

off the main supply to the line. But with DGs connected, the line may be islanded -

which implies that the line is carrying current from the DGs but not the main supply.

Line voltage regulators operation is also affected since they estimate voltage at a

downstream node based on the current through its line compensation circuit and

a secondary source located downstream disrupts this relation. Therefore a control

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procedure that does not depend heavily on line currents to estimate voltage must be

made available for use with DGs.

2.4 Voltage and Reactive Power Control

The ANSI standard [1] requires the voltages during steady state operation to be as

follows: on a nominal voltage of 120V, the service voltage is allowed a leeway of

Voltage-VAR control or VVC refers to regulating the voltage by feeding or consuming

reactive power as necessary. While real and reactive powers and node voltage and

phase are all intricately linked, there’s a stronger relation between reactive power and

voltage magnitude; between real power and voltage angle. This phenomenon exists

because of the decoupling of real and reactive power that occurs if the line resistance

is much smaller than the reactance and voltage magnitude at all nodes is maintained at

around 1pu. Line impedance is a fixed parameter and has to be chosen during system

design but the second condition is valid when the grid is adequately controlled and

maintained. For the case of transmission lines, line reactive impedance is indeed more

than resistance, but for distribution lines it is not necessarily true. Depending on the

ratio of reactance and resistance of a line both active and reactive power may have

equal effect on the voltage of the grid but by convention, reactive power is chosen for

compensation. In case of reactive power (VAR) compensation, the rule of thumb is:

injecting VAR into the grid increases the voltage while absorbing it reduces the voltage.

Traditionally voltage control has been done using switching circuits, transformers,

line drop compensators, step voltage regulators, load shedding, reactive power

compensation using capacitor banks, etc. With the growing popularity of distributed

generation or disperse generation other avenues of compensation have opened up. This

study is concerned with VVC but one that is based on sensitivity of voltage to reactive

power injections from DGs. The study also explores the possibility of using voltage

sensitivity and VVC to reduce distribution power loss.

19

Reducing losses involves reducing line currents all along the feeder. This can also

be restated as reducing the voltage difference between adjacent nodes. It is easy to

prove that distributing power sources across the feeder reduces the line current and

thereby losses. For example in Figure 2-2, a single DG is connected at the last node of

an N + 1 node feeder. Assuming all the nodes have an equal sized load connected to

it and they draw the same amount of current irrespective of the voltage at the node, the

current at the source bus is N ∗ Iload without the DG. If the DG assumes an equal load,

the current drawn from each source would be N ∗ Iload/2. Table 1 compares the lowest

voltage, maximum current and losses for the case with and without DG. Without any

form of voltage control or compensation, voltage magnitude decreases steadily along

the length of the feeder beginning at the substation (node 0).

Figure 2-2. N+1 load feeder with a distributed generator connected at the last node

With the DG however, the decrease in voltage is lower because the net current from

a single source is smaller than the current drawn from the substation without any DG.

Therefore, the voltage reduces moving from either end of the feeder towards the center.

For simplicity N is taken to be even.

Table 2-1. Comparison of voltage, current and loss with and without DGWithout DG With DG

Lowest voltage V0 − ZlineNIload V0 − ZlineNIload/2

Maximum current NIload1

2NIload

Total power loss 1

6N(N + 1)(2N + 1)I 2loadRline

1

12N(N + 1)(N + 2)I 2loadRline

20

It can be inferred that increasing the number of power sources reduces the

maximum line current. Since loss is proportionate to the square of the line current,

reducing the maximum current magnitude has a huge impact on the total distribution

loss in a feeder. In the example, reduction is by almost a factor of 4. In the example

of Figure 2-2 the type of load used is constant current load - where the current drawn

is independent of the voltage. Power consumed by a load is given by VI*, therefore

if a constant current load is connected at a higher voltage it consumes more power.

The other commonly used types of load are constant power and constant impedance.

Constant power loads draw the same amount of power irrespective of the voltage

but current drawn reduces with increasing voltage. Constant impedance loads have

constant impedance regardless of the voltage but power increases as the square of

voltage and current increases linearly with voltage.

Figure 2-3. Current drawn at different voltages for different load types

If all loads on a grid were of the same type, loss minimization would be a simple

problem. For a constant power load a higher voltage load is preferred therefore letting

the node with the highest voltage to be at 1.05 pu. (maximum allowed voltage according

to ANSI standards) would be sufficient. For constant impedance letting the lowest

voltage be 0.95 pu. would solve the control problem. For constant current loads as

long as the nodal voltages are within allowable limits, no control is necessary. For a

21

homogeneous load type control is simple irrespective of the load sizes but actual loads

are not homogeneous and optimization is required to determine the best configuration

and dispatch. Chapter 3 lists some of the techniques used in previous studies.

22

CHAPTER 3PROBLEM DESCRIPTION

An electrical system is governed by power flow equations which are a result of

Kirchhoff’s current law and Ohm’s law. These equations define the relationship between

the voltage at each node in the grid and the loads or generators connected to them.

Knowing the voltage at each node, it is possible to know the current in all the lines; the

exact power consumed or injected at each node and other metrics such as stability

of the grid etc. The voltages at all nodes (defined by a voltage phasor magnitude and

angle) are known as the state of the system. The set power flow equations can be

represented as

F (x , u) = 0

where x is the state vector and u is the vector of all control variables such as tap position

or voltage regulator, on load tap changing transformer, generator power output, etc. and

the loads at different nodes. F is the relation between x and u defined by Kirchhoff’s

current law and Ohm’s law. For ease of calculation all variables are represented in the

per unit system.

Voltage control according to ANSI standards requires that the utility voltage not vary

more than 5% from the nominal voltage of 1 pu. If x can be separated as

x =

|V |

θ

where |V | is the vector of node voltage magnitudes and θ is the bus angle, then voltage

control implies 0.95 ≤ |Vi | ≤ 1.05 for all nodes i = 1, 2, ...N

if Vi goes out of bounds, ucontrol can be adjusted so that Vi is within limits again.

In

u =

ucontrol

uload

23

ucontrol is a vector of all control variables and uload is the vector of load values. The vector

of control signals, ucontrol , can be chosen in different ways to achieve the required result.

Hence an objective function is necessary to choose best vector based on some criteria.

If tap changing transformers are used, the number of tap changes is often a criterion. If

compensation methods are being used and it costs the utility different rates for different

types of compensation then the most economical dispatch is sought. Line losses are

also often considered for optimization since losses can be controlled by varying the

voltage at the different nodes.

The power lost as heat on a single line between nodes i , j , is given as

Pilossj = Ii

2

j Ri j = (Vi − Vj)2/Ri j

where V = |V | ∗ e j θ and Ri j is the resistance of the line between nodes i and j .

Power systems however, are not single lines but have three phases, which in case of

a distribution network may be unbalanced. Therefore the total loss is calculated as a

product of vectors and matrices as:

Pilossj = Real {VI ∗} = Real

{(Vi − Vj)Zi

∗j−1(Vi − Vj)

∗}If the system is three phase, Vi is a 3× 1 complex vector of voltages of the three phases

of each node and Zi j is a 3× 3 complex matrix.

The total system loss is obtained by adding the loss over all the lines

P losstotal =

∑i ,j

Real{(Vi − Vj)Zi

∗j−1(Vi − Vj)

∗} (3–1)

For two nodes i , j not connected to each other, Z ∗i−1

j is a zero matrix and doesn’t

contribute to the loss. Therefore the general VVC with loss optimization problem can be

written as

24

Minimize P losstotal =

∑i ,j

Real{(Vi − Vj)Z

∗i−1

j Vi − Vj∗} (3–2)

Such that F (x , u) = 0

0.95pu ≤ |Vi | ≤ 1.05pu and

umini ≤ ui ≤ umax

i ∀i = 1, 2, ...M

where ui are the control variables.

In this study the control variables used are the reactive power produced by DGs

connected at different nodes in the distribution grid. Adjusting the power production

may not always be sufficient control mechanism and voltage regulators may need to

be adjusted as well. The tap changing operation is not included in the optimization but

performed if optimization fails to produce a feasible result. The optimization and results

are discussed in depth in chapter 5.

Chapter 4 discusses the past studies done in the field of voltage VAR control, siting

and sizing of reactive power sources and use of voltage sensitivity for control. In addition

to VAR optimization, placement of DGs on the feeder can also be used to adjust losses -

some positions are better suited for loss reduction than others. Similarly the capacity of

the DGs can also be optimized for a better performance. The optimal siting and sizing of

DGs is discussed in chapter 6. Also discussed in the chapter is the effect of increasing

penetration of DGs on distribution losses.

25

CHAPTER 4LITERATURE REVIEW

Voltage and reactive power control (Volt/VAR control or VVC) is an important

task that has been studied many times for both the transmission grid and distribution

grid. While new innovative techniques are being sought for voltage control, the most

commonly used methods are still the time tested ones such as feeder reconfiguration

[17], [18] to minimize line currents. The radial structure of the distribution system also

supports regulation through step voltage regulator with line drop compensator [16].

Injecting reactive power using compensation devices such as static VAR compensators

(SVCs), static synchronous compensators (STATCOMs) and other flexible alternating

current transmission system (FACTS) devices can be used to boost voltage as well

as control the phase angle [19]. The simplest and most commonly used form of

compensation though is capacitor banks which may be located at the substation or

along the feeder line. Traditional control techniques have dealt with optimizing the

positions of the taps in the transformers or controlling the output of the compensation

devices [20] or both while optimizing for economic or other system constraints. Due to

the non-linearity of the control problem, evolutionary algorithms such as particle swarm

optimization [21], [22] or genetic algorithm [23], [24] have been extensively used for

optimization.

Recent technological improvements have made DGs popular as a parallel source of

power for important or sensitive loads. Their capacity to inject excess power into the grid

has made them a viable option for compensation. Although DGs are being connected

to the grid, their involvement in providing ancillary services is negligible. There is still

concern regarding integration and control of new generators in the distribution grid

although extensive literature is available on various aspects of DG integration and

utilization from the various technologies it entails [4], [5], [8], [25], their impact - both

economic [26] and on the voltage profile [27], [28]; and incentives to promote their use

26

[29]; to the concerns and challenges of using DGs [25], [30]. A lot of research work has

also been done on control involving DGs [13], [27], [31], [32] and more importantly sizing

and siting the DGs on the grid [28], [33], [34].

With the current focus on upgrading the electric grid to a smart grid with support for

decentralized control, different methods of decentralized control are being researched.

Multi agent systems (MAS) [35], [36] are among the ideas being explored. The algorithm

developed by Baran and Markabi [37] to determine the optimum reactive power dispatch

of DGs using linear programming, is an example. A similar algorithm that dispatches

both real and reactive is discussed in [38].

Sensitivity of different types have also been used to determine the location of DGs

on the grid [39], [40] as well. But Gozel and Hocaoglu [41] with an intention to avoid

Jacobian and admittance matrix, developed an analytical method to locate and size

DGs on a radial system using a loss sensitivity factor based on the current injection

matrix. The goal was to determine the amount of injection required to reduce the losses

to a minimum. But with a loss function that can be derived from measured voltage

sensitivities, it may be easier to calculate the loss sensitivity coefficients.

Voltage and Reactive Power control using Sensitivity. The algorithm developed

by Markabi and Baran [37] implemented a simple multi-agent distributed VVC algorithm

based on sensitivity coefficients. They used sensitivity coefficients derived from the

power flow Jacobian to determine dispatch using linear programming. However, to

make the algorithm decentralized and independent of the grid architecture, sensitivity

coefficients of nodes without a DG were eliminated through Kron reduction. But the

new coefficients are not measurable quantities since they have been adjusted by Kron

reduction and made system dependent. Nevertheless, the concept of using sensitivity

to determine dispatch is a useful result. Dispatch is can still be calculated using the

measured sensitivity values which are the true instantaneous sensitivity values.

27

A single feeder line with multiple DGs connected along the length of it with the DGs

carrying most of the load was considered in the study. Each DG is assumed to be an

agent with intelligence. The remaining nodes are passive with no intelligence. All the

agents can communicate among themselves to share any necessary information. Each

agent performs three important tasks monitoring, moderating and dispatch.

Monitoring refers to checking the node voltage and verifying that it is always within

the specified range (within 5% of the nominal). When the voltage is no longer within

limits, the agent corresponding to the (most) affected node acts as the moderator.

They note that the voltages of the downstream nodes are usually the most severely

affected and it might be necessary to have a dummy DG with no output connected

at these nodes to monitor the voltage at the these points. It essentially translates to

having a measuring device like a PMU connected at the end node and extending the

communication networks till the end of the feeder. If that may not be possible then some

method of estimating the voltage at the end is necessary for example by assuming a

constant voltage difference between the end of the feeder and its nearest DG unit.

When a nodal voltage violates the operating limits, the closest DG senses it and

communicates with the other DGs and requests for reactive support and receives

their bids. The bids are the maximum support each DG can lend and the sensitivity

coefficient for the particular node. The moderator then decides the optimal dispatch

scheme for DGs. The DGs on receiving the dispatch change the output power to suit

requirements. This is the dispatch mode.

In general, feeding reactive power into the grid increases the voltage while

consuming it reduces the voltage magnitude. This behavior is captured by the sensitivity

coefficients and simple linear programming can calculate the dispatch scheme. The

problem is formulated as

Minimize∑i

�Qi (4–1)

28

Such that �Vk = Vmin − V 0

k and

Qmini ≤ Q0

i + �Qi ≤ Qmaxi

where �Qi is the change in reactive power output of i th DG, V 0k is the current voltage at

node k and Q0i is the current reactive power output of i th DG.

The DG causing the highest sensitivity to the affected node is chosen and it

supports the node to whatever extent it can. If the reactive power support of that DG

does not suffice, the DG with the next highest sensitivity helps and so on until the

voltage excess or deficiency has been compensated and all the voltages are within the

specified limits. If there are n nodes in the distribution system and V1,V2 ...Vn are the

node voltage magnitudes (assuming a balanced feeder, but the theory can easily be

expanded to unbalanced systems) and there are m (m ≤ n) DGs with reactive power

Q1,Q2 ...Qm, the voltage sensitivity is defined as

βi j =∂Vi

∂Qj

∀i = 1, 2, ... n and ∀j = 1, 2 ...m (4–2)

The partial derivatives form the Jacobian matrix used in Newton-Raphson power flow

and it defines the relation between VAR support of the DGs and voltage at the nodes.

The effect of the m DGs on its node voltage can be more accurately expressed by

considering that the net reactive power injection at most nodes is zero. Hence n voltage

equations can be reduced to only m equations by Kron reduction. The coefficients of the

variables obtained thus are the required sensitivities. It is easy to determine the VAR

support from equations (4–3) and (4–4).

If H is the Jacobian matrix, P is the vector of real power injections are each node, Q

is the reactive power injected, x is the vector of voltage magnitudes and θ is the vector of

node voltage angles, then

f =

P

Q

and x =

θ

V

29

�f = H × �x

H =

HPθ HPV

HQθ HQV

(4–3)

HQV is the partial of reactive power with respect to voltage magnitude. Reactive support

can be obtained using real and reactive power decoupling, which is based on the fact

that the voltage magnitude at a node affects the reactive power injected at that node

rather than the real power which is affected by the voltage angle. Hence

�Q/V = HQV�V (4–4)

�Q is divided by the node voltage, V , to linearize the power flow equations. Since �Q is

zero for the load nodes, the rows and columns of HQV can be rearranged to

HQV =

B11 B12

B21 B22

where

B11 = partial derivative of reactive power injection at the load nodes with respect to

voltages at the DG nodes

B12= partial derivative of reactive power injection at the load nodes with respect to

voltages at the DG nodes

B21= partial derivative of reactive power injection at the DG nodes with respect to

voltages at the load nodes

B22= partial derivative of reactive power injection at the DG nodes with respect to

voltages at these nodes. 0

�Q/V

=

B11 B12

B21 B22

�VL

�VDG

(4–5)

�VL is the vector of voltages at the load nodes.

30

Therefore,

�VDG = (B22 − B21B1−1

1 B12)−1�QDG/V or (4–6)

�VDG = β�QDG (4–7)

β is the sensitivity matrix. The elements of β determine the reactive support that each

DG provides. Therefore equation (4–1) changes to

Minimize∑i

�Qi (4–8)

Such that �Vk =

m∑i=1

βk i�Qi = Vmin − V 0

k

0 ≤ Q0

i + �Qi ≤ Qmaxi

Equation (4–8) indicates that the best solution to this problem is to have maximum

dispatch for the DG with the highest sensitivity until generator capacity is reached. If that

is not sufficient, DG with the next largest sensitivity adjusts its output and so forth until

the voltage drop has been compensated. A drawback of this algorithm is that it adjusts

the voltage to bring it within acceptable operating range but only barely. So if voltage

exceeds 1.05pu it is lowered to 1.05pu, if it falls below 0.95pu it is raised to 0.95pu. This

may not be the best voltage profile for a distribution feeder since the load is always

changing; even small variations in load or DG output can cause the voltages to go out

of range again. To maintain it at a voltage slightly higher than acceptable level might

seem like a good option but that only invites the question, ’how high?’. The answer lies

not in raising or lowering the voltage to a fixed level, but to optimize the voltage for other

parameters. Loss is the objective chosen in this study because it not only reduces the

wastage of energy lines but also frees up the line capacity for more useful power flow.

A new set of constraints and optimization function can therefore be devised to

account for these. A few modifications to the algorithm are suggested which employ a

quadratic optimization function rather than a linear one. This is discussed in chapter

31

5. Chapter 5 also discusses voltage sensitivity and its behavior to changes in load and

power generation.

32

CHAPTER 5VOLTAGE-VAR CONTROL AND LOSS MINIMIZATION

The objective of voltage-VAR control can be expanded from just maintaining the

voltage at the end of each feeder node within the specified voltage range to also protect

the voltage against variations in the system conditions such as loading, faults, loss

of power, etc. With access to power generation sources at different locations in the

grid it becomes possible to not only maintain voltages within operable regions but

manipulate the power flow so that other objectives can be achieved. The most important

advantage of using voltage sensitivity for this purpose is it allows for the estimation

of the new state - and subsequently other parameters that are a function of nodal

voltage - without performing load flow analysis. A complex set of nonlinear equations

can be approximated to a linear combination of sensitivity coefficients, saving precious

computation resource and time.

5.1 Voltage Sensitivity

Sensitivity has been defined in chapter 4 (equation (4–5)) as the inverse of the

power flow Jacobian. Voltage sensitivity with respect to change in reactive power

injection at a node (hereby referred to as voltage sensitivity or just sensitivity; this study

deals exclusively with reactive power and voltage change unless otherwise specified) is

an n × n block in the 2n × 2n Jacobian matrix. Sensitivity is an easy way of estimating

the new state of the grid when the DG outputs are changed since it is the observed,

steady state voltage change for a unit change in power production. Figure 5-1 shows

the sensitivity of phase A for all nodes of a 34 node feeder for 1KVAR change in output

power of a DG when connected at four different locations. The missing sensitivity values

are a consequence of an unbalanced feeder - not all nodes utilize all three phases.

Using sensitivity, for some change in power production, the voltage change in the

feeder nodes can be estimated by scaling the sensitivity at these nodes the appropriate

amount. The relation is described by equation (4–7). However, the expression for β,

33

the sensitivity is modified from equation (4–6); instead of using sensitivity of only small

subset of nodes from a reduced set of power flow equations, the sensitivity of all nodes

represented by the columns of HQ−1

V are used. Equation (4–6) exploits the fact that

power injection at non-DG nodes is zero. Hence the sensitivity calculated from only

DG connected nodes does not represent the the measurable change in voltage at the

particular nodes; it is a calculated quantity, although it may still be numerically similar

to the measurable sensitivity. Since the magnitude of voltage change is quite small (of

the order of 10−4p.u./KVAR change in power injection), the mismatch between the

calculated and the observed value may go unnoticed. Patterns and trends [42] similar

to those of the calculated sensitivity of equation (4–6) are also observable for the actual

measured sensitivity coefficients.

Figure 5-1. Sensitivity of 34 node feeder for four different DG positions

For a perfectly decoupled system, the sensitivity parameters are a constant with

HQV given by the admittance matrix. But even for a lossless line it is not possible

to have a perfectly decoupled system and the sensitivity varies with voltage. Over

large voltage ranges sensitivity is non-linear but over smaller ranges - such as in LV

or MV compensation schemes it can be approximated to a linear trend (figure 5-2).

Mathematically this observation can be made by not approximating the nodal voltage to

34

1pu. This is also the expression used in this study. But the most accurate value can be

obtained by inverting the Jacobian

H =

HPθ HPV

HQθ HQV

with no assumptions or approximations. The columns represent the sensitivity

coefficients for all the nodes - except the substation bus - for all possible DG locations.

The sensitivity of the substation bus is 0 due to the control action of the substation

control system. In the Jacobian of dimension 2n × 2n (when number of nodes in the

feeder is n + 1; sensitivity of the substation bus is zero) and the voltage sensitivity to

reactive power is a n × n block in the bigger matrix.

Figure 5-2 is the plot of sensitivity of a particular node with DG fixed at a different

node and the reactive power output of the DG is varied from from -100 KVAR to

+100KVAR in steps of 1 KVAR. Plot A is the result of repeating the process for different

values of real power output. While plot A is for all three phases, plot B displays the

sensitivity for a single phase (phase A) with respect to the node voltage instead of the

reactive power output.

Figure 5-2 is typical of most nodes. It is observed that while sensitivity is not a

constant, it can be approximated as a linear function for most cases. The centralized

voltage control optimization function is derived assuming a constant sensitivity and the

consequences of assuming linearly varying sensitivity are explained.

5.2 Centralized Voltage Control and Loss Optimization

If the sensitivity is fairly constant, a single coefficient can be selected for each

node and phase for control. But due to the varying voltage and loading and injection

conditions, sensitivity also varies. Hence there are fewer approximations. The

centralized control approach uses all the voltage and approximate sensitivity values for

voltage and loss estimation. The decentralized algorithm which is given to a coordinated

35

A Variation with change in reactive dispatch

B Variation with node voltage

Figure 5-2. Variation of sensitivity

control structure based on local information is discussed in section 5.3 but both of them

optimize the same quantity - loss. The decentralized version is an extension of the

centralized version.

36

5.2.1 Objective Function

The expression for distribution loss on a single branch was derived in equation

(3–1) as

Minimize P losstotal =

∑i ,j

Real{(Vi − Vj)Z

∗i−1

j (Vi − Vj)∗}

Such that F (x , u) = 0

0.95pu ≤ |Vi | ≤ 1.05pu and

umini ≤ ui ≤ umax

i ∀i = 1, 2, ...M

where Zi j is the impedance between two nodes i and j . Considering the total loss and

not only the real power loss,

lossi j = (V2 − V1)Z∗−1

12 (V2 − V1)∗

If the voltages after optimization are V ′1 and V ′

2 then

lossi j = (V ′2 − V ′

1)Z∗−1

12 (V ′2 − V ′

1)∗

but V ′ = V + �V and �V can be estimated using voltage sensitivity as

lossi′j = (V2 + �V2 − V1 − �V1)Z

∗−1

12 (V2 + �V2 − V1 − �V1)∗

Simplifying,

lossi′j =(V2 − V1)Z

∗−1

12 (V2 − V1)∗ + (�V2 − �V1)Z

∗−1

12 (�V2 − �V1)∗

+ (V2 − V1)Z∗−1

12 (�V2 − �V1)∗ + (�V2 − �V1)Z

∗−1

12 (V2 − V1)∗

The first term on the right of the equation is lossi j hence

lossi′j − lossi j = (�V2 − �V1)Z

∗−1

12 (�V2 − �V1)∗

+ (V2 − V1)Z∗−1

12 (�V2 − �V1)∗ + (�V2 − �V1)Z

∗−1

12 (V2 − V1)∗ (5–1)

37

Since lossi j denotes the initial (or current) state of loss in the line, it is a constant with

respect to optimization and can be neglected. This corresponds to optimizing for change

in total loss rather than the total loss itself.

Thus the new value of f is f ′

f ′ =∑i ,j

(Vi − Vj)Zi∗−1

j (�Vi − �Vj)∗ + (�Vi − �Vj)Zi

∗−1

j (Vi − Vj)∗

+∑i ,j

(�Vi − �Vj)Zi∗−1

j (�Vi − �Vj)∗ (5–2)

The change in voltage at k th node �Vk , can be represented as

�Vk =∑i

β ik ∗ �Q i

DG

Or, in vector form:

�Vk = �QTDGβ (5–3)

Substituting in equation (5–1) and simplifying,

f ′ =∑i ,j

(Vi − Vj)Zi∗−1

j (βi∗j �QDG) + (�QT

DGβi j)Zi∗−1

j (Vi − Vj)∗

+∑i ,j

(�QTDGβi j)Zi

∗−1

j (βi∗j �QDG)

where βi j = βi − βj is a 3×m matrix containing the sensitivity coefficients of each of the

three phases of a particular node with respect to all the connected DGs. Therefore,

f ′ =∑i ,j

((Vi − Vj)Z∗−1βi

∗j )�QDG + �QDG(βi jZ

∗−1(Vi − Vj)∗)

+∑i ,j

�QDG(βi jZ∗−1βi

∗j )�QDG

This can be further simplified to a quadratic equation in vector form as

f ′ = A�QDG + �QTDGB + �QT

DGC�QDG

38

Considering only the real power loss, the final objective function is

f ′ = P�QDG +�QTDGQ�QDG (5–4)

where P = Real{A+ B} and Q = Real{C}

A second voltage regulator control level exists above this DG dispatch control to

utilize the control options already incorporated in the feeder. Due to the existence of

DGs, tap positions cannot be accurately selected by the line compensator circuit and

downstream voltage has to be communicated to the regulator controller or the tap can

be shifted by one position at a time. For each change in tap settings, the optimization

algorithm for the DGs is executed. The combination of both these operations determines

the ideal settings.

Unlike the compensation algorithm of Markabi and Baran, this method of loss

control is not given to dispersive control; this is a centralized power flow and loss control

algorithm because to develop the objective function all the node voltages are necessary.

An agent based decentralized variant of this algorithm can also be devised; discussed

in section 5.3. However with the increasing incorporation of communication channels

between the agents of a power grid, it becomes possible to locate the control center

at any location, including a node on the feeder line. It is through the communication

network that a dispersed control of the grid can be achieved.

Another important note regarding the centralized algorithm is that �V used in

the derivation is a complex quantity whereas voltage sensitivity in the Jacobian is the

change in voltage magnitude with respect to DG output, which is a real number. Using

angular sensitivity and voltage sensitivity to reactive power and the voltage profile, the

complex �V can be calculated. In the simulations however, the complex voltage change

is measured. By choosing a small enough change in reactive power injection and using

that for sensitivity the magnitude of the complex value and the actual change in voltage

39

magnitude can be made comparable. For the IEEE 13 node and 34 node feeders,

1KVAR was found to serve the purpose.

5.2.2 Constraints and Optimization Problem

The general constraints of the system are the voltage constraints given in equation

(3–1) are:

0.95pu ≤ |Vi |+ |�Vi | ≤ 1.05pu

and capacity constraints of the DGs

Qmini ≤ Qi + �Qi ≤ Qmax

i ∀i . (5–5)

Using the relation (5–2) in equations (5–4) and (5–5)

0.95− |Vi | ≤ |�Vi | = �QTDGβi ≤ 1.05− |Vi | (5–6)

And

Qmini −Qi ≤ �Qi ≤ Qmax

i −Qi (5–7)

The complete optimization problem can, therefore, be defined as

Minimize f ′ = P�QDG + �QTDGQ�QDG (5–8)

Such that: Vmini ≤ �QDG

Tβi ≤ Vmi ax and

Qmini −Qi ≤ �Qi ≤ Qmax

i −Qi

An evolutionary technique called particle swarm optimization (PSO) is used to solve the

optimization problem.

5.3 Decentralized Voltage Control and Loss Optimization

Decentralized voltage control is implemented using data available locally to the

controller. Unlike the case of centralized control where data from all the sensors in

the network are to be collected at an aggregator site before decisions are taken,

decentralized controller reacts to local phenomena. This is particularly useful when

40

communication systems are experiencing problems due to weather or other reasons.

Implementing a system with multiple decision making bodies also makes the system

robust to other contingencies and intentional attacks. In this study however, the myriad

controllers do not act individually, rather each one monitors the local conditions and

coordinates with other such controllers called agents to achieve the required goal. This

is known as a multi agent system.

The control structure is similar to the one described in section 4 with changes in the

information being exchanged, and the optimization algorithm used to decide dispatch.

The agents control DG installations and voltage regulators and perform monitoring (of

voltage it its point of contact), moderation (negotiate with other agents) and dispatch

(execute optimization algorithm and communicate dispatch scheme to other agents)

operations.

Due to the absence of all node voltages, equation (5–8) cannot be used. A

small modification to the objective function of equation (5–8) can be used for local

optimization. The central control scheme generates a new objective function before

optimization to reflect the voltage profile of the feeder which is an operation the control

agent cannot perform. If a reference voltage profile of the feeder could be used instead,

voltage data collection would be unnecessary to determine the objective function. For

example, equation (5–1) determines the change in the loss with respect to the initial

condition - the voltage condition on the entire feeder before optimization. Similarly if

the initial condition can be measured against a fixed reference, a new function does not

have to be calculated before each execution. A base case voltage profile - for example,

when none of the DGs were connected - can be used as a reference,. If the voltages for

this case are represented as V refk , change in loss can be calculated as

lossi′j = (V ref

i +δV refi +�Vi−V ref

j −δV refj −�Vj)Zi

∗−1

j (V refi +�Vi+δV ref

i −V refj −�Vj−δV ref

j )∗

41

where V refi + δV ref

i + �Vi = V ′i , the new voltage to be estimated and V ref

i + δV refi = Vi is

the voltage before control.

lossi′j =

((V ref

i + δV refi − V ref

j − δV refj ) + (�Vi − �Vj)

)× Zi

∗−1

j

((V ref

i + δV refi − V ref

j − δV refj ) + (�Vi − �Vj)

)∗lossi

′j = (V ref

i + δV refi − V ref

j − δV refj )Z ∗−1(V ref

i + δV refi − V ref

j − δV refj )∗

+ (�Vi − �Vj)Z∗−1(V ref

i + δV refi − V ref

j − δV refj )∗

+ (V refi + δV ref

i − V refj − δV ref

j )Z ∗−1(�Vi − �Vj)∗

+ (�Vi − �Vj)Z∗−1(�Vi − �Vj)

∗

The first term of lossi ′j is lossi j , line loss for the current voltage profile. Therefore,

lossi′j − lossi j = (�Vi − �Vj)Z

∗−1(V refi − V ref

j + δVirefj )∗

+ (V refi − V ref

j + δVirefj )Z ∗−1(�Vi − �Vj)

∗

+ (�Vi − �Vj)Z∗−1(�Vi − �Vj)

∗ (5–9)

The first term of lossi ′j can be ignored as it is a constant with respect to the optimization

problem. The remaining three terms form the optimization function. V refk is known since

it is the reference profile; an approximate value for δV refk will be calculated using local

voltage information. The precise value is not necessary since the value required for

estimating the loss is (δV refi − δV ref

j ) the change in voltage difference between two

adjacent nodes. Similar to equations (5–1) to (5–3), by substituting (5–2) in (5–9) and

simplifying,

f =∑i ,j

lossi′j − lossi j = A�QDG + B�QDG + �QT

DGC�QDG (5–10)

Such that: Vmink ≤ �QT

DGβi ≤ Vmaxk and

Qmini −Qi ≤ �Qi ≤ Qmax

i −Qi∀i = 1, 2, ...m

42

Where

A =∑i ,j

((V refi − V ref

j )Zi∗−1

j βi j + (V refi − V ref

j )′Zi∗−1

jTβi j) and

B =∑i ,j

(δVirefj Z ∗−1βi j + δVi

refj

TZ ∗−1Tβi j) and

C =∑i ,j

(βiTj Zi

∗−1

jTβi j)

δVirefj = (δV ref

i − δV refj ) is estimated using the local voltage value. Since a local event

such as a voltage violation triggers the optimization, it is assumed that the voltage at

the specific node is known. The reference voltage at that node is also known. Using

the difference between the two known voltages, some compensation dispatch that will

bridge the difference is calculated i.e. (�QDG) such that (�QDG) ∗ βk = δV refk at errant

node k is calculated. The dispatch scheme does not necessarily have to be feasible

since it is a theoretical value used to estimate δVirefj . It is possible to allocate the entire

dispatch to one generator but it is preferred to use as many generators as possible. With

(�QDG) known, δV refi = (�QDG) ∗ βi can be calculated for all nodes; subsequently, B

can also be calculated.

5.4 Optimization Algorithm

5.4.1 Quadratic Programming

The basic quadratic optimization problem is defined by the objective function G

where x is the vector to be optimized, H is a symmetric matrix and f is a vector:

Minimize G =1

2xTHx + f Tx (5–11)

Such that

Ax ≤ b

Aeqx = beq

lb ≤ x ≤ ub

43

In the loss minimization problem however, the H matrix is not symmetric due to the

complex nature of voltage, sensitivity and line series impedance; also due to the use of

complex conjugates to calculate the H matrix. Therefore H is made symmetric with the

relation

H = (H + HT )/2

However, despite converting the Hessian to a symmetric matrix, the Matlab function

quadprog for quadratic programming failed to produce feasible results.

5.4.2 Particle Swarm Optimization

Due to the failure of constrained quadratic optimization, a swarm based technique

called Particle Swarm Optimization (PSO) [21], deriving inspiration from the flocking

behavior of birds was chosen.

PSO employs a large swarm of agents each of which can save information about

itself and the swarm and also change its course of motion according to this information.

The agents are initiated at random locations in the search space which denotes the

range of values each variable in the optimization problem can take. The variables to

be determined form the ”location” vector of the agent. Each location in the search

space has an associated fitness value which is the value of the objective function for

the particular sequence of variable values. During an iteration of the algorithm, each

agent updates its location value by adjusting it velocity according to equations (5–12)

and (5–13). During this ”exploration” each stores its personal (pbest) and the swarm’s

global (gbest) best values. These indicate the best location where the particular agent

experienced the best objective value function and the best value for the whole swarm

respectively. Each agent has its own personal best but the swarm shares a common

global best location and value. By updating its own location based on pbest and gbest ,

every agent converges to the optimal location.

vi = w ∗ vi + c1 ∗ rand ∗ (pbesti − xi) + c2 ∗ rand ∗ (gbest − xi) (5–12)

44

xi = xi + vi (5–13)

where xi is the position vector of agent i , pbesti is the best location it has been to, gbest

is the best position for the entire swarm. c1 and c2 are constants and w is the inertial

constant.

5.5 Results and Observations

The performance of three algorithms:

1. Dispatch linearly proportional to the sensitivity of the affected node (Linear)2. Centralized compensation using PSO (PSO)3. Decentralized compensation using PSO (Dispersed)

are tested for 3 different cases: case 1> optimizing when reactive output of all DGs is

zero; case 2> all the loads are scaled down to 80% and; case 3> all the loads are scaled

up to 130%.

In the first case, all the DGs are connected to the grid but the DGs are working

at power factor=1. By adjusting the reactive power output of each DG, total system

loss is optimized. While the test feeders assume a static load, real loads are changing

constantly. Assuming that the the loads do not generally change steeply, 20% reduction

in all loads or 30% increase in all values from the base load is considered the worst

case scenario and the performance is measured for both these conditions. An

implication of low sensitivity values is that large dispatches are necessary to be able

to effect a significant change on the voltage. A very optimistic case where the DGs bear

most of the load is considered. But a consequence of this is that power production (both

real and reactive) needs to be stepped down when the loads are scaled down or the

substation will behave like a load and absorb power instead of feeding it to the grid.

Therefore, the DGs are scaled down by a factor of 30%. The DGs are not scaled up

when the load increases though.

The three algorithms for the three cases are tested on the IEEE 13 and 34 node

feeders (Appendix). For both the feeders, the capacitor banks are ignored and all

45

reactive support is assumed by the DGs. While the capacitor can also be modeled as a

DG with discrete output values and zero real power, it is simpler to ignore the capacitors.

Additionally, for the case of the 34 node feeder, the load on node 27 is not scaled since it

is a very large load and it is connected after a step down transformer. Hence voltage is

very sensitive to even minor changes in the size of this load (figure 5-1). It is assumed to

be a constant load.

The tables below tabulate the results for the three cases for the two feeder

configurations. Both Linear and Dispersed varieties optimize the reactive powers only if

a node is violating the voltage limits. PSO variety on the other hand, attempts to reduce

loss in any situation. If node voltage is outside limits, even that is adjusted. Therefore in

Tables 5-1 and 5-2 the gain in loss savings is zero for Linear and Dispersed for 13 node

feeder. In case of the 34 node feeder, because the voltage regulator taps cannot be

calculated accurately, not all the voltages are within acceptable limits, therefore the DG

outputs are varied. A huge gain in loss savings is observed for the 34 node feeder with

the maximum being achieved by the Dispersed method. But the method also required

the changing of taps. Since taps are changed only when DG optimization fails, it can be

inferred that PSO performs better in terms of optimization even if the results are not as

good. For the 13 node feeder, a modest 5% savings was observed.

Table 5-1. Comparison of optimization performance for case 1

Feeder Optimization Optimized Saving Estimation Tap Power Reductiontype Algorithm Loss (KW) (%) Error (KW) changes (KVA)

Linear 0 0 0 0 013 PSO 61.04 5.73 0.03 0 67.45

Dispersed 0 0 0 0 0

Linear 62.06 34.30 1.26 0 82.1534 PSO 55.10 41.67 0.39 0 53.46

Dispersed 47.26 49.98 0.50 3 82.53

The 34 node feeder seems to be more amenable to optimization. Significant

savings can be observed through optimization. Dispersed and PSO appear to be

46

working equally well although PSO gave slightly better results overall for case 2 (table

5-2). Because Linear limits itself to adjusting the voltage to the ANSI standard, its loss

optimization performance is not on par with the others.

Table 5-2. Comparison of optimization performance for case 2

Feeder Optimization Optimized Saving Estimation Tap Power Reductiontype Algorithm Loss (KW) (%) Error (KW) changes (KVA)

Linear 0 0 0 0 013 PSO 40.26 5.80 0.03 0 80.31

Dispersed 0 0 0 0 0

Linear 64.00 22.47 1.22 0 119.734 PSO 48.31 41.47 1.29 0 84.83

Dispersed 49.71 39.79 1.28 0 72.71

In table 5-3, ’NA’ implies that a steady state condition with no voltage violations

was not achieved through optimization. Fig 5-3 depicts the cause for this. It can be

seen in graph of phase C that after optimization, the minimum voltage is just above

0.95pu whereas the maximum voltage is far above 1.05pu. The reactive power dispatch

necessary to reduce the peak voltage will also push the lower voltage below the 0.95

range. In the next iteration, the minimum value is raised but the maximum value also

breaks the upper limit. The cycle repeats and a solution cannot be found. It needs

a concerted effort of the regulators and DGs to solve it. Hence PSO and Dispersed

converged to a solution but Linear did not.

In case of the 34 node feeder, node 5 just before the first regulator experiences a

low voltage which is further aggravated by the increase in load. The DG at the node

does not have sufficient capacity to raise the voltage. The voltage may in fact have been

made worse by the presence of the DG because the current flow from the DG may have

increased the voltage drop in the region. Therefore none of the methods have a positive

influence on the voltage and loss problem.

It can be seen that all the methods had a positive loss saving despite the magnitude

in some cases. Optimization also led to an overall reduction in power consumption in

47

Table 5-3. Comparison of optimization performance for case 3

Feeder Optimization Optimized Saving Estimation Tap Power Reductionnodes Algorithm Loss (KW) (%) Error (KW) changes (KVA)

Linear NA NA NA NA NA13 PSO 108.02 17.49 4.56 5 187.13

Dispersed 108.33 17.26 5.32 5 97.61

Linear NA NA NA NA NA34 PSO NA NA NA NA NA

Dispersed NA NA NA NA NA

A Phase A B Phase B

C Phase C

Figure 5-3. No result condition of the 13 node feeder

48

all cases. Therefore optimization is economical for not only reducing the line losses

but also the total power consumption. Thus if power has an associated generation and

distribution cost, expenses can be reduced with the use of DGs.

It is observed that the loss estimation error is also quite small. With increasing

scaling of the loads though, the error increases due to the approximations involved in the

sensitivity parameter, such as approximating the sensitivity curve to a linear relationship,

the minor influence of increasing reactive dispatch on sensitivity of DGs connected at

other locations, etc. Therefore, it might be useful to continuously monitor voltage and

dispatch reactive power rather than perform optimization when a voltage violation occurs

so that voltage difference and sensitivity variation can be contained.

It was observed in figure 5-2 that sensitivity is not a constant but can be approximated

to a linearly varying quantity. Then the sensitivity can be written as

β = α�Q + β0

where α is the slope of the line. Therefore

V = β�Q = α(�Q)2 + β0�Q

If voltage is a quadratic function, loss would be a quartic function. But the observed

improvement in estimation due to a quartic loss function was not significant enough to

account for a higher order loss function.

49

CHAPTER 6GENERATOR SITING AND SIZING

Chapter 5 discussed optimization of DG output for optimal loss performance where

the DGs were already connected at certain locations on the grid. However, looking at

the voltage sensitivity values, it is apparent that connecting DGs at some positions yield

better sensitivity than other nodes i.e. the same amount of power produces a higher

voltage difference for some DG locations than others. In this chapter sensitivity as

applied to determining the position of the DG and its size are discussed. The effect of

having too many or too few DGs are also considered.

6.1 Generator Siting

The location of the DG on the feeder is an important factor in determining the effect

it has on voltages at other nodes, power flow in the grid and consequently power loss.

For example, DGs connected lower down the feeder causes a greater change in voltage

for the same power than a DG connected upstream to it. Generally, farther away from

the substation a DG is located, higher are the sensitivity coefficients corresponding

to it. Figure 6-1 plots sensitivities for phase A of the 34 node feeder for 6 different DG

locations. Sensitivity is higher for almost all nodes when the DG is located at the last

node of the main feeder. The sixth scenario has the DG located on a lateral branch

rather than the main feeder unlike the first 5 cases.

Even when a node has a DG connected to it, it had a lower sensitivity when

compared to a DG located lower downstream. It should be noted that this relation

is valid for a general decreasing trend in the voltages. For other types of profiles,

the sensitivity patterns are different. Example, figure 6-2 shows the sensitivity for a

uniformly decreasing voltage profile on a 21 node feeder for different DG locations;

figure 6-3 shows sensitivity for an increasing trend. Similarly based on the true voltage

conditions, the sensitivity pattern for any feeder configuration may be calculated. In

50

Figure 6-1. Sensitivity of 34 node feeder for six different DG positions

general the voltage profile on most feeders is of the decreasing type, therefore sensitivity

of becomes more negative moving down the feeder.

Figure 6-2. Sensitivity for a decreasing voltage profile for different DG locations

For voltage control operations, achieving a large voltage change with a small

amount of dispatch is desirable for economic reasons. Based on the sensitivity values

we can infer that to conserve power, it is ideal to locate the DG on the node with the

highest impact on voltage - typically the nodes at the end of feeder lines and laterals.

But for loss optimization, it is not only necessary to use the least amount of power for

51

Figure 6-3. Sensitivity for an increasing voltage profile for different DG locations

compensation, it is also essential that lines carry the least amount of current. Current

and power sensitivity coefficients are useful in comparing these performances. Both

these indices however, can be derived from voltage sensitivity. Power sensitivity is

the total change in power that a unit amount of reactive power generation causes i.e.

equation (5–3) for a single DG and �QDG = 1unit. Comparing the power sensitivity at

different nodes and choosing the node with the smallest numerical value can decide the

best location.

Current sensitivity by definition, is the rate of change of line currents with change

in DG injection power. But current in a line can be represented using the voltage at the

nodes at the end of the line as:

Ii j = (Vi − Vj)/Zi j or

Ii j + �Ii j = (Vi + �Vi − Vj − �Vj)/Zi j

But �Vk = βk × 1, therefore

�Ii j = βi j/Zi j (6–1)

52

By comparing the �Ii j values for all pairs i , j and all DG locations, DG sites can

be identified. Figure 6-4 plots the current sensitivity values for all the nodes for all

possible DG locations on an example 20 node feeder (Appendix). Since the aim is to

reduce loss by reducing current, the DG site should be chosen so that the reduction in

current is maximum. For the first few candidate sites, current sensitivity is nearly flat

downstream of the DG node and drops for nodes upstream to it. For the last few nodes,

current sensitivity drops from around node 10 but increases downstream to it. This is not

preferable. Hence the best candidate is node 10 and the nodes immediately adjacent to

it.

Figure 6-4. Sensitivity of current to DG location

The selection procedure can be made even simpler considering loss sensitivity.

Loss sensitivity has already been defined in equations (5–1) and (5–2). By setting

�Vk = βk ∗ 1unit, loss sensitivity for a DG location can be obtained. Figure 6-5 plots

the variation of loss with different DG locations. Locating it at node 10 reduces loss the

maximum, hence it is the ideal candidate.

If multiple DGs are to be installed then it is better to choose one position at a time.

When another DG location is being cited, all the previously selected DGs are connected

to their respective nodes, then the loss sensitivities are calculated for all the candidate

nodes to decide the new DG location.

53

Figure 6-5. Sensitivity of power loss to DG location

Using power loss sensitivity also accounts for load distribution in the grid and

positioning the DG close to the load center so that current flow is reduced. The example

feeder had most of the load concentrated between node 4 and node 10 of the feeder.

The optimal node, as selected by power loss sensitivity and current sensitivity is the

node at the end of the load concentration, which also implies that of the 7 nodes with

the highest loads, node 10 has the highest voltage sensitivity. Therefore power loss

sensitivity combines the advantage of both voltage and current sensitivities into a single

quantity.

6.2 Generator Sizing

Generator sizing and generator siting are two interdependent operations with the

location of the DG site influencing the amount generation necessary at a particular

position and the size of already installed DGs influencing how many more generators

are necessary and where. Generator sizing can be done in many ways: generation

capacity can be added in multiples of some base quantity as needed through optimization

but usually sizing and siting is performed as a combined operation. Some DG

configurations are considered for installation in the system and optimization is carried

out to determine the best candidates from the available choices and the optimal site for

their installation.

54

In [41] sizing operation was done using loss sensitivity parameters to optimize

the loss characteristics of the grid. An analytical expression for loss was derived from

the current equations and bus incidence matrix. By differentiating this expression and

equating it to zero, the size of the DG was calculated so that net loss in the circuit is

minimum. With the loss expression of equation (5–1) and (5–2), the problem can be

solved as a simple quadratic equation. If the new configuration does not lead to out of

range voltages, the DG is retained. For example, for the 34 node feeder a -80 KVAR DG

was indicated as necessary to minimize loss.

But in line with the aim of compensation during faults and expansion of generation

capability using DGs, the intent is to calculate the minimum size of DG. In the IEEE 34

node feeder, node 27 often suffers from low voltage, reaching down to less than 0.9pu

in some cases. A DG which can provide for such a situation is required. If sensitivity is

assumed to be a constant, the amount of reactive power necessary can be calculated

as QDG = �V /β. The value of QDG might turn out to be too high. A work around for

this is to share the necessary power over multiple DGs. Or some portion of the reactive

power can be relegated to real power. Depending on the grid conditions, the real and

reactive power sensitivity values may be comparable. In such a case it maybe useful to

provide the minimum voltage support using real power injection and protection against

changes using reactive power dispatch. For example: if the lowest voltage is 0.88pu,

sensitivity is 6 × 10−4pu/KVAR and support capability up to 1pu is necessary, then

Q = 0.12/6 × 10−4 = 200KVAR. If the real power sensitivity is equal to the reactive

power sensitivity, for a DG operating at 0.85 power factor, the ratio of reactive and real

power is roughly 0.6. Therefore, P = 125KW and Q = 75KVAR.

If there’s more than one node to account for, a linear programming with inequality

constraints may be set up to simultaneously solve for multiple DG sizes as:

n∑j=1

βiqj Qj + βi

rjRj ≥ Vi ∀i = 1, ...m, j = 1, ... n (6–2)

55

where m is the number of nodes and n are the number of DGs and

Qj/Rj ≤ tan(cos−1 (θj)

)(6–3)

θj is the power factor of the DG.

6.3 Impact of DG Penetration on Loss

To study DG impact, the 34 node feeder was chosen. Up to 10 DGs were free to be

placed. DGs were added one at a time and each one was placed at a node so that the

total loss was a minimum. The size of the DGs was varied depending on the number of

generators. The DG size was fix calculated as Loadtotal/3 ∗ P/100/N where Loadtotal is

the total load on all the 3 phases of the system, P is the penetration in percent and N is

the number of DGs. Penetration was varied in steps of 5% from 0 to 80% and number of

DGs was varied from 1 to 10 in steps of 1. Losses were plotted a function of the number

of DGs and penetration (figure 6-6).

Figure 6-6. Variation of losses for different penetration levels and number of DGs

The observed data suggests that loss reduces with increasing DG penetration but

the reduction with increasing number of DGs is not very apparent. At larger penetration

levels, having more DGs does seem to be helpful but in the case of the 34 node feeder,

10 were not necessary. Similar loss levels were observed even with 5 or 6 DGs. It

was also observed that while multiple DGs were located at the same node at lower

56

penetrations, they were scattered across a few nodes at higher penetration and

DG count; clustering of DGs at the a node indicates that a bigger sized generator is

necessary at that location and the total number of DGs necessary is not really 10 but

the number of unique DG locations. This also suggests that loss may increase if N

unique locations are desired and different size DGs are not preferred. Siting and sizing

therefore plays an important role in loss minimization.

57

CHAPTER 7CONCLUSION

Integration of distributed generation into the grid is a hot research topic and it

promises many benefits. But it also brings with it many problems related to control,

protection, islanding, etc. In this thesis one aspect of that problem, namely distribution

line loss, was tackled using sensitivity coefficients. An algorithm was devised to mitigate

the losses and improve power flow.

It was found that including distributed generation in the distribution grid can help

reduce the line losses by reducing the maximum current flowing in a line. Since loss is

proportional to the square of the current, even small reduction in current can improve the

voltage and loss profile significantly. Inclusion of DGs and proper dispatch of reactive

power also led to a reduction in the total power consumed in the feeder. Therefore DGs

can play an important role in peak shaving operations. It is economically advantageous

to integrate DGs into the grid. DGs also contribute to the capacity expansion of the utility

grid and prevent the purchase of expensive power from the energy market.

However, DGs in large numbers can play havoc with existing control operations

such as voltage regulation. A line voltage regulator estimates the voltage condition at

a downstream node and adjusts its taps to control the voltage at the target node. The

estimation is based on the current flowing through its line drop compensator circuit.

By changing the current flow pattern, DGs cause the regulators to underestimate the

severity of voltage problems and cause inadequate control. Therefore, when DGs are

included in the grid in sufficient quantity it is essential for voltages to be measured rather

than estimated. And it may also become necessary for DGs themselves to participate in

the control operation.

Optimizing loss in the grid requires complete knowledge of the system conditions for

efficient operation. It is better performed as a centralized control technique. However, it

is still possible to obtain satisfactory results through localized control with coordination

58

of the other DGs. Such a technique was implemented using a multi agent control

operation.

Loss optimization can greatly benefit from the proper location of DGs in the

distribution system. From simulations on an example system it was observed that

placing DGs close to the load concentration is beneficial even though voltage sensitivity

would indicate placement near the end of the feeder line because the same amount

of power dispatch can effect a larger change in voltage. Loss sensitivity is a better

parameter to decide the siting of DGs. It has also been found possible to decide the size

of the DG using voltage sensitivity. Simulations on the penetration of DGs in the grid and

to an extent, increasing the number of DGs were found to favor minimization of loss. All

results regarding loss seem to favor the inclusion of distributed generators into the grid.

However some observations during the study do call for further research such as:

the small value of sensitivity. It implies large dispatch for a significant change in voltage,

which further implies large DGs. That may cause the substation to absorb power instead

of supplying it. Therefore, care must be taken during DG sizing to ensure that current

flow is not reversed at the substation bus. A fixed DG size would interfere with this when

the load is varying. Therefore real power output of the DGs must also be varied along

with the reactive power, especially in small systems.

Simulations in this study were implemented on small sized systems with very few

nodes. Simulations on bigger and more complicated networks need to be performed and

studied, particularly the behavior of sensitivity and the effect of penetration and number

of DG on the system. Future studies including the model of the distributed generator

and its response to the the dynamic behavior of the grid are necessary. The response to

increasing penetration of the distributed generators,the varying load conditions and the

ability of the generators to track load changes and control operations without interfering

with each other even at high penetrations need to be studied.

59

APPENDIX: FEEDER CONFIGURATIONS

Standard IEEE test feeders were chosen for all voltage control and power

optimization simulations [43]. The feeders chosen were of 13 nodes and 34 nodes

respectively. The one line diagrams of the two feeders are shown in Figure 7 and 7.

An example 20 node feeder was used to test DG location using the various sensitivity

coefficients.

IEEE 13 Node Feeder. The 13 node feeder is a small, highly loaded, wye-connected,

unbalanced test feeder [44] with capacitor compensation as well as a voltage regulator.

The nodes have been renumbered according to Figure 7 for convenience. For voltage

control simulations, two DGs at nodes 2 and 8 are considered. Both the DGs are of the

same capacity - 500KW and a maximum of 300KVAR. The DGs are considered to have

a constant real power output but the reactive power is varied according to necessity.

Figure A-1. Schematic of IEEE 13 node test feeder. Source: IEEE 13 Node Feeder [43]

60

IEEE 34 Node Feeder. The 34 node feeder is a long, lightly loaded, wye-connected,

unbalanced test feeder [44] based an an actual feeder in Arizona. It has two capacitor

banks and two lne voltage regulators for voltage control. The nodes have been

renumbered according to Figure 7 for convenience. For voltage control simulations,

five DGs are located at nodes 5, 9, 15, 19 and 27. Attempting to size all the DGs

equally resulted in voltge violations so they are sized at either 100 KW or 120 KW and a

maximum of 60KVAR.

Figure A-2. Schematic of IEEE 34 node test feeder. Source: IEEE 34 Node Feeder [43]

Example 20 node Feeder. The 20 node feeder is an overloaded feeder with a

major load concentration at the beginning of the feeder with about 75% of the total

load connected equally to the nodes 4 through 10. It was designed primarily to check

if the load center can be detected by the different sensitivity coefficients. The feeder

is a balanced system with nodes located at uniform distances and the lines having

R = 1,X = 1 and no impedance between the phases. Like the 34 node standard feeder

it operates at 24.9 KV.

61

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BIOGRAPHICAL SKETCH

Anurag R. Katti completed his Bachelor of Engineering in electronics and communication

engineering from Visvesvaraya Technological University, Belgaum in 2008. He worked at

the Indian Institute of Science, Bangalore after graduation on swarm based optimization,

pattern recognition, robotics and control. In 2009 he decided to pursue graduate studies

and he was admitted to the University of Florida for the Master of Science program in

electrical engineering with a focus on control systems. He soon realized his interest in

green energy and power systems. His current research interests include power systems

and control, smart grids, distribution systems, green energy and optimization. This

thesis is the culmination of his work in the field and completes the Master of Science

thesis requirements.

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