Senses and the endocrine system Essay Example | Topics and Well Written Essays - 500 words

Senses and the endocrine system - Essay Example These abnormalities that occur sometimes have no effect on the systems while in other cases the effects are detrimental. One of the common seen abnormalities as people age is the formation of cataracts. Cataracts are a clouding of the lenses in the eye to where light becomes hard to pass through. As a result of the light not being able to penetrate, problems with vision occur. Cataracts can form in a person’s eyes as the result of multiple conditions. Exposure to ultraviolet light can cause proteins to denature, which can lead to the condition. Certain diseases such as diabetes have been known to cause this to occur. Eye trauma and genetic issues can also be accounted for with the development of cataracts (Goldstein, 2010). Cataracts are an abnormality that is forming which affects the visual field of the individual. This is because the lenses in eyes are biologically centered around having a clear lens that is adjusted by the retina to allow light in depending on the light le vel. Even though aging of biological components is inherent as humans age, the formation of cataracts is often caused or started by external or internal determinalistic factors that cannot be accounted for. As a result, these are biological factors, which represent an abnormality to normal human physiology.

Plato Essay Example | Topics and Well Written Essays - 250 words - 1

Plato - Essay Example In an attempt to analyze the life and ideas of Plato, this paper thus purposes to reflect on chapter 4 of the book â€Å"Living Philosophy: A Historical Introduction to philosophical ideas†. Having been born around 428 BCE in Athens, Plato revolutionized the way of thinking both in Greece and the rest of the world. In his idea of rationalism, Plato regarded reasoning as the cornerstone source and test for knowledge. In his rationalism idea, Plato taught that there is existence of certain truths that cannot be changed, and there is a need for intellectual understanding to identify the truths. These truths were thus believed to be the governing forces of humanity and the universe. Drawing an analogy with the sleep to elaborate his viewpoint of Immortality, Morality, and the Soul, he understood immortality of the soul in terms of reincarnation. Just as a person can wake up after sleep, he believed the dead would wake up in another body (Vaughn, P. 52). Among Platos famous dialogues as depicted in the book â€Å"Living Philosophy: A Historical Introduction to philosophical ideas† is his notion of the Republic. He draws most of his inference of the Republic from the works of Socrates thus coming to an argument point that a republic is a society of people who are governed by justice and fulfilment in life in terms of happiness. Thus, members of the republic must all work together socially and economically to ensure every member is justly and fairly treated according to one’s contribution to the

Static Voltage Stability Analysis In Power Systems Engineering Essay

Static Voltage Stability Analysis In Power Systems Engineering Essay Voltage stability, one of the principal aspects of power system stability, has been the main reason for many of major power system blackout incidents over the last few decades. It is acknowledged universally that voltage stability is and will remain a challenge in the 21st century, even likely to increase in importance. Therefore a better understanding of voltage stability in power systems is necessary for power engineers, who might participate in the planning, designing, and operation of modern power systems. This report talks about a relevant engineering thesis project: Static Voltage Stability Analysis in Power Systems, which is carried out for 2 semesters from July 2009 to June 2010. The aim of this thesis project is to conduct a more comprehensive study into the theory of static voltage stability, and investigate a new approach for power flow analysis: 3-dimension P-Q-V curve. First of all, the basic knowledge of static voltage stability is reviewed, and analysis on an elementary power system, radial system, is carried out including power flow study, P-V and Q-V curve analysis. Based on the 2- dimension P-V and Q-V plotting, the relationship of P, Q, and V is studied and a new method for static voltage stability analysis is tried: P-Q-V curve. The second part of this project focuses on the analysis of WSCC three-generator-nine-bus system. Simulation of the system is carried through by means of UWPFLOW and POWERWORLD. Direct power flow method and continuation power flow method are applied and the weakest bus is studied. Last but not least, curves are obtained and results are discussed. Keywords: Static Voltage Stability; Radial System; Power Flow Method; Continuation Power Flow Method; P-V Curve; Q-V Curve; P-Q-V Curve. CONTENTS à ¦Ã¢â‚¬ËœÃ‹Å" à ¨Ã‚ ¦Ã‚  i ABSTRACT ii CONTENTS iv CHAPTER 1 INTRODUCTION 1 CHAPTER 2 POWER SYSTEM VOLTAGE STABILITY 8 CHAPTER 3 STATIC VOLTAGE STABILITY ANALYSIS OF ELEMENTARY POWER SYSTEM 11 CHAPTER 4 STATIC VOLTAGE STABILITY ANALYSIS OF WSCC NINE-BUS SYSTEM 26 CHAPTER 5 CONCLUSION 39 REFERENCES 41 ACKNOWLEDGEMENTS 43 APPENDIX A MATLAB CODES FOR FIGURE 3.8 44 APPENDIX B MATLAB CODES FOR FIGURE 4.2 46 APPENDIX C MATLAB CODES FOR FIGURE 4.3 47 APPENDIX D MATLAB CODES FOR FIGURE 4.4 48 APPENDIX E MATLAB CODES FOR FIGURE 4.5 49 APPENDIX F MATLAB CODES FOR FIGURE 4.6 50 APPENDIX G MATLAB CODES FOR FIGURE 4.7 51 APPENDIX H DATA OF WSCC NINE-BUS SYSTEM 52 CHAPTER 1 INTRODUCTION An Overview of Modern Power System A power system is a network of conductors and devices which allows electrical energy to be transferred from the generating power stations to load centers through transmission network. Since the first electric network in the United States was established at the Pearl Street Station in New York City by Thomas Edison in 1882 [1], power systems have been experiencing more than 100 years development and improvement. Nowadays, modern power system has developed to be a complex interconnected network, which can be subdivided into four parts: Generation Private and publicly owned generators produce the electricity that feeds into high voltage grids. Transmission High voltage transmission grids transport power from generating units at various locations to distribution systems which ultimately supply the load. Distribution Distribution systems deliver the power from local bulk supply points to the consumers service-entrance equipments. Loads Loads of power systems are composed of industrial, commercial, and residential load. Figure 1.1 Modern Power System [2] Power System Stability A power system is said to be stable if it has the property that it retains a state of equilibrium under normal operating conditions and regains an acceptable state of equilibrium after being subjected to a disturbance. Of all the complex phenomena on power system, power system stability is the most intricate to understand and challenging to analyze [3]. Damage to power system stability may cause the system to blackout or collapse as well as other catastrophic incidents, leading to enormous social and economic losses. Classification of Power System Stability Based on the systems different properties, network structures and operation modes, the system instability can behave in many different ways. Accordingly power system stability study is divided mainly into three fields: angle stability, frequency stability and voltage stability. The diagram below shows visually the classification of power system stability. Figure 1.2 Classification of Power System Stability History of Study on Power System Stability Initially, angular stability was firstly paid attention to and studied since power transmission capability had traditionally been limited by either rotor angle (synchronous) stability or by thermal loading capability. And the blackout problems had been associated with transient stability, which were diminished by fast short circuit clearing, powerful excitation systems and varies special stability controls [3]. In other words, nowadays the theory and methods on angular stability are relatively more complete. Meanwhile, study on voltage stability had been quite slow, which mainly attributed to two reasons: Incidents caused by voltage instability or voltage collapse occurred relatively late, not until which did people paid attention to voltage instability problems. Understanding of voltage instability was not so profound as other kinds of instability problems in the early days. Varies of issues arose during the study on voltage stability such as load-based modeling, dynamic behaviors of different components as well as their interaction, and so on. Overview of Power System Voltage Stability Voltage Instability Incidents in the World Power system voltage stability was firstly introduced in 1940s, but failed to draw peoples attention until 1970s, since which voltage instability and collapse had resulted in several major system failures or blackouts throughout the world, as listed below [4, 5, 22]: August 22, 1970, Japan, 30 minutes; September 22, 1970, New York, several hours; September 22, 1977, Jacksonville, Florida, few minutes; December 19, 1978, France, 26 minutes; August 4, 1982, Northern Belgium, 4.5 minutes; September 2, November 26, December 28 30, 1982, Florida, 1-3 minutes; May 21, 1983, Northern California, 2 minutes; December 27, 1983, Sweden, 55 seconds; June 11, 1984, Northeastern USA, several hours; May 17, 1985, South Florida, 4 seconds; April 1986, Winnipeg, Canada Nelson River HVDC links, 1 second; May 20, 1986, England, 5 minutes; November 1986, SE Brazil, Paraguay, 2 seconds; January 12, 1987, Western France, 6-7 minutes; July 20, 1987, Illinoisand India, several hours; July 23, 1987, Tokyo Japan, 20 minutes; August 22, 1987, Western Tennessee, 10 seconds; July 2, 1996, Western System Coordination Council (WSCC), Northern USA; August 1996, Malaysia; August 14, 2003, USA Canada; September 28, 2003, Italy. Progress of Study on Voltage Stability The large numbers of worldwide voltage collapse incidents made it become the focus of worlds attention to study voltage stability of power system. In the 1982s researching list of Electric Power Research Institute (EPRI) in USA, voltage stability was considered as the most significant issue. Over the last thirty years, and especially over about the last twenty years, utility engineers, consultants, and university researchers have intensely studied voltage stability. Hundreds of technical papers have resulted, along with conferences, symposiums, and seminars. Utilities have developed practical analysis techniques, and are now planning and operating power systems to prevent voltage instability for credible disturbances [6]. Importance of Voltage Stability in Future In a foreseeable future, the global fast-growing power consumption will require more intensive use of available transmission facilities, which means an operation of power systems closer to their voltage stability limits. The increased use of existing transmission is made possible, in part, by reactive power compensation [6]. Undoubtedly, voltage stability is and will remain a challenge in the 21st century, even likely to increase in importance. Therefore a better understanding of voltage stability in power systems is necessary for power engineers, who might participate in the planning, designing, and operation of modern power systems. Topic Definition and Scope The topic of this project is Static Voltage Stability Analysis in Power Systems, which mainly focuses on the following: Overview of the phenomena of static voltage stability; Analysis associated with the phenomena; Reasons why voltage collapse happens; Measures to improve static voltage stability. In consideration of restrictions on the simulation, a three-generator-nine-bus case is used throughout the whole project while a typical two-bus (one-generator-one-load) case is used for the P-Q-V curve analysis. Aims and Objectives The main objective of this project is to get a wider and deeper understanding of static voltage stability in power systems, which can be reduced into sub-objectives: To conduct a more comprehensive study into the theory of static voltage stability; To look for reasons why voltage collapse happens; To investigate a new approach for power flow analysis: 3-dimension P-Q-V plotting; To propose proper measures of improving static voltage stability in power systems; To conclude generation direction and load direction for the analyzed power system. CHAPTER 2 POWER SYSTEM VOLTAGE STABILITY Basic Concepts of Voltage Stability IEEE Definitions IEEE [7] provided a formal definition of voltage stability and relative concepts as given below: Voltage Stability: Voltage stability is the ability of a system to maintain voltage so that when load admittance is increased, load power will increase and so that both power and voltage are controllable. Voltage Collapse: Voltage collapse is the process by which voltage instability leads to very low voltage profile in a significant part of the system. Voltage Security: Voltage security is the ability of a system not only to operate stably, but also to remain stable (as for as the maintenance of system voltage is concerned) following any reasonable credible contingency or adverse system change. CIGRE Definitions Nevertheless, the above definitions of voltage stability conditions were not directly compatible with the general IEEE definition for stability concept. Hence new definitions were given in CIGRE report [8], which are as following: Voltage Stability: A power system, at a given operating state and subjected to a given disturbance, is voltage stable if voltages near loads approach post-disturbance equilibrium values. The disturbed state is within the region of the stable post-disturbance equilibrium. Voltage Instability: Voltage instability is the absence of voltage stability, and results in progressive voltage decrease (or increase). Destabilizing control reaching limits, or other control actions (e.g. load connection), however, may establish global stability. Voltage Collapse: Following voltage instability, a power system undergoes voltage collapse if the post-disturbance equilibrium voltages near loads are below acceptable limits. Voltage collapse in the system may be either total (blackout) or partial. Voltage collapse is more complex than simple voltage instability leading to a low-voltage profile in a significant part of the power system. Other Relative Concepts Large-disturbance Voltage Stability: Large-disturbance voltage stability is concerned with a systems ability to control voltages following large disturbances such as system faults, loss of generation, or circuit contingencies. The study period of interest may extend from a few seconds to tens of minutes. Therefore, long-term dynamic simulations are required for analysis. Small-disturbance Voltage Stability: Small-disturbance voltage stability is concerned with a systems ability to control voltages following small perturbations such as incremental changes in system load. For such case, static analysis is effectively used. Relation of Voltage Stability to Rotor Angle Stability Voltage stability and rotor angle (or synchronous) stability are more or less interlinked. Transient voltage stability is often interlinked with transient rotor angle stability, and slower forms of voltage stability are interconnected with small-disturbance rotor angle stability. Voltage Stability is concerned with load areas and load characteristics. For rotor stability, we are often concerned with integrating remote power plants to a large system over long transmission lines. Voltage stability is basically load stability, and rotor angle stability is basically generator stability [6]. In a large interconnected system, voltage collapse of a load is possible without loss of synchronism of any generators. Transient voltage stability is usually closely associated with transient rotor angle stability. Long-term voltage stability is less interlinked with rotor angle stability. We can consider that if voltage collapses at a point in a transmission system remote from loads, it is an angle instability problem. If voltage collapses in a load area, it is possibly mainly a voltage instability problem. CHAPTER 3 STATIC VOLTAGE STABILITY ANALYSIS OF ELEMENTARY POWER SYSTEM Introduction of an Elementary Model: Radial System Simple radial system network is used to develop most of the concepts of the static voltage stability. Once basic concepts are understood, we can represent as much as appropriate in computer simulation, which will be carried out in Chapter 4. Figure 3.1 shows an equivalent circuit of the power system, and a model called radial system is formed to represent such power system, as shown in Figure 3.2. Figure 3.1 Equivalent Circuit of Power System Figure 3.2 Radial System Model The sending-end and receiving-end voltages are assumed to be fixed and can be interpreted as points in large systems where voltages are stiff or secure. The sending end and receiving end are connected by an equivalent reactance. Basic Analysis of Radial System Active Power Transmission Applying the radial system in Figure 3.2, the relations can be easily calculated: Similarly, for the sending end: The familiar equations for and are equal since we assume a lossless system, and maximum power transferred is at a power load angle equal to 90 degree. Note that the 90-degree maximum power angle is nominal, in other words, maximum power occurs at a different angle if we apply transmission losses or resistive shunt loads. And the case with impedance load at the receiving end will be discussed in section 3.2.2. Reactive Power Transmission In the study of the static voltage stability in power system, the transmission of reactive power is especially of interest. Usually we are interested in variable voltage magnitudes. Particularly, we are interested in the reactive power that can be transmitted across a transmission line, or a transformer as the receiving-end voltage sags during a voltage emergency or collapse. Considering the reactive power flow over the transmission line alone, we can write approximate formulas for Equations (3.3) and (3.5) in terms of small angles by using : From Equations (3.6) and (3.7), it can be observed that reactive power transmission depends mainly on voltage magnitudes and flows from the higher voltage to the lower voltage. Such observation, however, cannot be applied in the case of high stress, i.e. high power transfers and angles, where the angle is large enough and no longer approaches 1. This is important as voltage stability problems normally happen during highly stressed conditions. Difficulties with Reactive Power Transmission Reactive Power Transmission Behavior in Different Cases First of all, take an example of the radial system in Figure 3.2, assuming X=0.2 p.u. with varied values of voltage magnitude and angles, i.e. varied loading conditions. Applying Equations (3.3) and (3.5), and can be calculated as listed in the following table: Conditions (p.u.) (p.u.) (degree) (p.u.) (p.u.) Lightly loaded 1.10 1.00 10 0.634 0.416 Moderately loaded 1.05 0.90 20 1.072 0.390 Heavily loaded 1.00 0.80 50 2.429 -0.629 Table 3.1 Reactive Power Transmission in varied conditions From the table, it is clear that at higher loading, transmission lines are more difficult to transfer reactive power and reactive power cannot be transmitted across large power angles (the value of becomes negative in the case with a power angle of 50 degree). Minimizing Transfer of Reactive Power High angles are due to long lines and high real power transfers. It is therefore required to maintain voltage magnitude profiles with voltages of approximately 1 p.u.. Compared with real power transfers, reactive power cannot be transmitted across long distances. It has been observed that the greater distance of the reactive power sources from the reactive demand will lead to: [9] greater voltage gradient on the lines supplying the reactive power greater amount of required reactive power compensation more difficult to control the voltage level Another reason to minimize the transfer of reactive power is minimizing the real and reactive losses. The purpose to reduce real losses is due to economic reasons while minimizing the reactive losses can reduce investment in reactive devices such as shunt capacitors. As we know, the losses across the series impedance of a transmission line are and . For , we have: and Obviously, to minimize losses, we should minimize reactive power transfer and keep voltage high at the same time. Keeping voltage high to minimize reactive losses helps maintain voltage stability. In other words, reactive power should be generated close to the receiving end. Power Flow Analysis In a power system, powers are known rather than currents. Thus power flow analysis is backbone of static voltage stability studies. Power flow analysis, also known as load flow analysis, involves the calculation of power flows and voltages of a transmission network for specified terminals or bus conditions. Bus Classification In solving a power flow problem, a power system is supposed to be operating under balanced conditions and a single-phase model is used. Associated with each bus are four quantities: active power P, reactive power Q, voltage magnitude , and voltage angle. The following types of buses (nodes) are represented, and at each bus two of the above four quantities are specified: Voltage-controlled (P-V) buses: These buses are the generator buses. They are also known as regulated buses or P-V buses. For such kind of buses, the real power P and voltage magnitude are specified, while the reactive power Q and the voltage angle are unknown. Load (P-Q) buses: Load buses are also called P-Q buses as their real power P and reactive power Q are specified. The voltage magnitude and angle are to be determined. Slack (Swing) bus: Such bus is taken as reference of the whole power system. For a slack bus, the voltage magnitude and voltage angle are specified. As the power losses in the system are not known a priori, at least one bus must have unspecified P and Q. Thus the slack bus is the only bus with known voltage. This bus makes up the difference between the scheduled loads and generated power that are caused by the losses in the network [1]. Traditionally while analyzing, the voltage magnitude of slack bus is assumed to be 1 p.u. and the voltage angle is assumed to be 0 degree. Transmission Line Modeling The transmission line is traditionally represented with two types of models: nominal model and nominal T model, as shown in Figure 3.3 and Figure 3.4 where Z is the series impedance and Y is the shunt admittance due to the line charging capacitance. Neither nominal T or nominal à Ã¢â€š ¬ exactly represent the actual line, however, they brings great convenience in the power flow analysis, especially in the application of NEWTON-RAPHSON method, which will be discussed in the coming section. Figure 3.3 Nominal Model Figure 3.4 Nominal T Model NEWTON-RAPHSON Power Flow Method In order to include all the three types of buses (P-V bus, P-Q bus and slack bus as introduced in 3.3.1) at the same case, a 3-bus power system is considered as shown in Figure 3.5, where Bus 1 is the slack bus, i.e. and are specified as . Bus 2 is a voltage-controlled bus, i.e. and are known while and are unknown. Bus 3 is a load bus, i.e. and are known while and are unknown. Figure 3.5 3-bus Power System The network performance equation of such a sample is: where Applying the bus-loading equations: Now NEWTON-RAPHSON Power Flow Method can be approached as: P-V Curve Analysis P-V curve is useful for conceptual analysis of static voltage stability and for study of radial system, where P is the load in an area and V is the voltage at a critical or representative bus. Besides, P can also be the power transferred across a transmission interface or interconnection. Voltage at several busses can be plotted. Consider the radial system as shown in Figure 3.2. The receiving-end active power can be expressed as in the Equation 3.2. Then a P-V cueve can be plotted as in Figure 3.6, which shows relationship between P and V at the receving end for different values of load power factor and the locus of the critical operting point is shown by the dotted line. Nornally, only the operting points above the locus of the critical points represent satisfying operating condition. A sudden reduction in power factor or increase in Q can thus cause the system to change from a stable operating condition to an unsatisfactory and possibly unstable [10]. Figure 3.6 V versus P for different power factors [10] Q-V Curve Analysis Q-V curve is presently the workhorse method of voltage stability analysis at many utilities [6]. Considering the system in Figure 3.2, we can obtain reactive power both at sending end and receiving end, or and by means of Equation (3.5) and Equation (3.3). Then a Q-V cueve can be plotted as in Figure 3.7, which shows relationship between Q and V. The reactive power margin is the MVAr distance from the operating point to either the bottom of the curve, or to a point thaere the voltage squared characteristic if an applied capacitor is tanfent to the V-Q curve [6]. Additionally, the slope of the V-Q curve indicates the stiffness of the bus. Figure 3.7 Typical Q V Curve A New Method for Static Voltage Stability Analysis: P-Q-V Curve Analysis Introduction of MATLAB Software MATLAB is a numerical computing environment and fourth generation programming language. Developed by The MathWorks, MATLAB allows matrix manipulation, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs in other languages [18]. An additional package, Simulink, adds graphical multi-domain simulation. This project greatly benefits from MATLAB to handle 3-dimension curve drawing for P-Q-V curve study, as well as the matrix manipulation associated with power flow analysis, 2-dimension curve plotting for P-V/Q-V curve study in the analysis of WSCC nine-bus system, which will be described in details in CHAPTER 4. P-Q-V Curve In this section, for convenience of forming an ideal voltage source, we assume the angle of the to be zero while the angle of to be degree. Then Equation 3.2 and 3.3 become: Noting that We can eliminate in Equations 3.16 and 3.17, which obtains or Obviously, with specified , and , Equation 3.19 shows relationship of , and . To work out such relationship visually, MATLAB is applied and a P-Q-V curve is obtained as below, where P stands for , Q stands for V stands for and E stands for . Refer to Appendix A for details on MATLAB codes, with the assumsion that E = 1 , X= 0.2 and the values of tan are chosen randomly as [-0.41, -0.2, 0, 0.2, 0.41, 1, 10, 100, 1000]. Figure 3.8 P Q V Curve CHAPTER 4 STATIC VOLTAGE STABILITY ANALYSIS OF WSCC NINE-BUS SYSTEM Introduction of WSCC Nine-bus System WSCC nine-bus system is a typical testing system develped by Western Systems Coordinating Council. It is commonly uesd in jornals and papers for power system studying. Figure 4.1 shows an overview of the WSCC nine-bus system. Refer to Appendix H for parameters of this system. Figure 4.1 Single Line Diagram of WSCC Nine-bus System Introduction of UWPFLOW Software For determining the static voltage stability of the WSCC nine-bus system, UWPFLOW software is used. This software has been developed by University of Waterloo, Canada, and distributed free on the Power Globe. It was written in C and runs under DOS and UNIX enviroments. It has no limitation on the system size other than those imposed by memory limitation in the corresponding enviroment, i.e. RAM and swap space in the UNIX and exrended memory in DOS [16, 20]. UWPFLOW is a research tool that has been designed to calculate local bifurcation characterized by a singularity in the power system Jacobian. This was developed based on power flow method. This software also generates a series of output files that allow further analysis. UWPFLOW reads AC power flow data in WSCC format [11] or IEEE common format [12], DC data in ETMSP format [13], FACTS devices data in s special format described in [14], and steady state load model data in OH format [15]. However in the present study IEEE common format data is used. Additional UN format data is required for bifurcation analysis, such as direction of generation change, direction of load change and maximum genertion limit [10]. The software assumes that one parameter the loading factor, is allowed to change. All steady state system controls remain operational unless otherwise specified by means of the software option. Introduction of POWERWORLD Software POWERWORLD Simulator is an interactive power system simulation package designed to simulate high voltage power system operation on a time frame ranging from several minutes to several days [17]. POWERWORLD provides a linear programming based optimal power flow package Simulator OPF, which ideally suits to do power flow analysis. Whats more, the planning-mode tool Simulator PVQV fulfills the need of Q-V curve drawing. Throughout the project, PowerWorld Simulator will be used to carry out power flow analysis and Q-V curve study of the twelve-bus case. Analysis of WSCC Nine-bus System Direct Method: Repeated Power Flow First of all, the WSCC nine-bus system in Figure 4.1 is built in UWPFLOW software. By running the system and increasing the loading level of step by step, attention will be focused on getting convergence and the maximum loading level. For loading direction, assume all the loads are increased by the same ratio, and only generator at Bus-1 is allowed to dispatch required additional real power. With the load P and Q increased simultaneously with the ratio of 10%, in the same loading direction, the bus voltages in per unit measurement are tabulated in Table 4.1. Couples of data points are collected near the system divergence point. Table 4.1 has shown that the system started to collapse (or diverge) at the point where all loads at the 3 load buses are increased in the same direction till 116%. Note that in Table 4.1, the starting point is denoted as 0% as there is no additional loads added, which is named as basic load. Then we can conclude from Table 4.1 that the maximum loading level for the WSCC nine-bus system is at additional of 116% loading direction on all 3 load buses. Load Increment (%) Bus5 Bus7 Bus9 P (MW) Q (Mvar) V (p.u.) P (MW) Q (Mvar) V (p.u.) P (MW) Q (Mvar) V (p.u.) 0 90 30 1.0129 100 35 1.0162 125 50 1.0261 10 99 33 1.0069 110 38.5 1.0105 137.5 55 0.9886 20 108 36 1.004 120 42 1.0053 150 60 0.981 30 117 39 0.9928 130 45.5 0.999 162.5 65 0.972 40 126 42 0.9846 140 49 0.993 175 70 0.9625 50 135 45 0.9753 150 52.5 0.9862 187.5 75 0.9516 60 144 48 0.9648 160 56 0.979 200 80 0.9394 70 153 51 0.953 170 59.5 0.9711 212.5 85 0.9257 80 162 54 0.9396 180 63 0.9626 225 90 0.9102 90 171 57 0.9242 190 66.5 0.9532 237.5 95 0.8923 100 180 60 0.9061 200 70 0.9428 250 100 0.8714 110 189 63 0.881 210 73.5 0.9239 262.5 105 0.84 112 190.8 63.6 0.8737 212 74.2 0.9167 265 106 0.83 114 192.6 64.2 0.8657 214 74.9 0.9087 267.5 107 0.8191 115 193.5 64.5 0.86 215 75.25 0.9024 268.75 107.5

The Great Gatsby As A Satire Essay -- essays research papers

The Great Gatsby as a Satire   Ã‚  Ã‚  Ã‚  Ã‚  Satire is an implement used by authors to point out a flaw of society or group of people in general. There are different levels of satire that the author can use. For example, the author may employ a type a formal satire known as Juvenalian satire. Here, the writer points out a subject with anger and contempt for it in a bitter fashion. There is also the contrasting form of Juvenalian satire called Horatian satire. Here, the writer points out a subject with a gentleness and jovial tenderness. The second main type of satire is informal. This is the type of satire used in The Great Gatsby. Here, Fitzgerald uses Nick to point out the character's flaws and makes each person the butt of the witticism by what they themselves do. The supposed 'guests'; at all of Gatsby's parties are prime examples of satire in The Great Gatsby. Many people who attended the parties were never even invited. This disregard for propriety illustrates the crassness and thoughtlessness that seemed to run rampant among the rich and famous during the twenties. An example of carelessness is when a large group of people at one of the huge soirees, decides to continue the party in the massive, expensive fountain in Gatsby's lawn. They just jump right in and begin to dance without concern for their health, much less concern for the well being of the fountain. After the galas had died down, most of the participants went home, leaving a monstrous mess...

Computational Efficiency of Polar

Lecture Notes on Monte Carlo Methods Fall Semester, 2005 Courant Institute of Mathematical Sciences, NYU Jonathan Goodman, [email  protected] nyu. edu Chapter 2: Simple Sampling of Gaussians. created August 26, 2005 Generating univariate or multivariate Gaussian random variables is simple and fast. There should be no reason ever to use approximate methods based, for example, on the Central limit theorem. 1 Box Muller It would be nice to get a standard normal from a standard uniform by inverting the distribution function, but there is no closed form formula for this distribution 2 x unction N (x) = P (X < x) = v1 ? e? x /2 dx . The Box Muller method is a 2 brilliant trick to overcome this by producing two independent standard normals from two independent uniforms. It is based on the familiar trick for calculating ? 2 e? x I= /2 dx . This cannot be calculated by â€Å"integration† – the inde? nite integral does not have an algebraic expression in terms of elementary f unctions (exponentials, logs, trig functions). However, ? 2 e? x I2 = ? /2 e? y dx 2 ? /2 ? 2 e? (x dy = +y 2 )/2 dxdy . The last integral can be calculated using polar coordinates x = r cos(? ), y = r sin(? with area element dxdy = rdrd? , so that 2? I2 = r = 0? e? r 2 /2 rdrd? = 2? r = 0? e? r 2 /2 rdr . ? =0 Unlike the original x integral, this r integral is elementary. The substitution s = r2 /2 gives ds = rdr and ? e? s ds = 2? . I 2 = 2? s=0 The Box Muller algorithm is a probabilistic interpretation of this trick. If (X, Y ) is a pair of independent standard normals, then the probability density is a product: 2 2 1 1 ? (x2 +y2 )/2 1 e . f (x, y ) = v e? x /2  · v e? y /2 = 2? 2? 2? 1 Since this density is radially symmetric, it is natural to consider the polar coordinate random variables (R, ? de? ned by 0 ? ? < 2? and X = R cos(? ), and Y = R sin(? ). Clearly ? is uniformly distributed in the interval [0, 2? ] and may be sampled using ? = 2? U1 . Unlike the original dis tribution function N (x), there is a simple expression for the R distribution function: 2? r G(R) = P (R ? r) = r =0 ?=0 r 1 ? r 2 /2 e rdrd? = 2? e? r 2 /2 rdr . r =0 The same change of variable r 2 /2 = s, r dr = ds (so that r = r when s = r2 /2) allows us to calculate r 2 /2 e? s dx = 1 ? e? r G(r) = 2 /2 . s=0 Therefore, we may sample R by solving the distribution function equation1 G(R) = 1 ? e? R 2 /2 = 1 ?U2 , whose solution is R = ? 2 ln(U2 ). Altogether, the Box Muller method takes independent standard uniform random variables U1 and U2 and produces independent standard normals X and Y using the formulas ? = 2? U1 , R = ?2 ln(U2 ) , X = R cos(? ) , Y = R sin(? ) . (1) It may seem odd that X and Y in (13) are independent given that they use the same R and ?. Not only does our algebra shows that this is true, but we can test the independence computationally, and it will be con? rmed. Part of this method was generating a point â€Å"at random† on the unit circle. We sug gested doing this by choosing ? niformly in the interval [0, 2? ] then taking the point on the circle to be (cos(? ), sin(? )). This has the possible drawback that the computer must evaluate the sine and cosine functions. Another way to do this2 is to choose a point uniformly in the 2 ? 2 square ? 1 ? x ? 1, 1 ? y ? 1 then rejecting it if it falls outside the unit circle. The ? rst accepted point will be uniformly distributed in the unit disk x2 + y 2 ? 1, so its angle will be random and uniformly distributed. The ? nal step is to get a point on the unit circle x2 + y 2 = 1 by dividing by the length.The methods have equal accuracy (both are exact in exact arithmetic). What distinguishes them is computer performance (a topic discussed more in a later lecture, hopefully). The rejection method, with an acceptance probability ? ? 4 78%, seems e? cient, but rejection can break the instruction pipeline and slow a computation by a factor of ten. Also, the square root needed to compute 1 Re call that 1 ? U2 is a standard uniform if U2 is. for example, in the dubious book Numerical Recipies. 2 Suggested, 2 the length may not be faster to evaluate than sine and cosine.Moreover, the rejection method uses two uniforms while the ? method uses just one. The method can be reversed to solve another sampling problem, generating a random point on the â€Å"unit spnere† in Rn . If we generate n independent standard normals, then the vector X = (X1 , . . . , Xn ) has all angles equally n likely (because the probability density is f (x) = v1 ? exp(? (x2 + ·  ·  ·+x2 )/2), n 1 2 which is radially symmetric. Therefore X/ X is uniformly distributed on the unit sphere, as desired. 1. 1 Other methods for univariate normals The Box Muller method is elegant and reasonably fast and is ? ne for casual omputations, but it may not be the best method for hard core users. Many software packages have native standard normal random number generators, which (if they are any good) use e xpertly optimized methods. There is very fast and accurate software on the web for directly inverting the normal distribution function N (x). This is particularly important for quasi Monte Carlo, which substitutes equidistributed sequences for random sequences (see a later lecture). 2 Multivariate normals An n component multivariate normal, X , is characterized by its mean  µ = E [X ] and its covariance matrix C = E [(X ?  µ)(X ?  µ)t ].We discuss the problem of generating such an X with mean zero, since we achieve mean  µ by adding  µ to a mean zero multivariate normal. The key to generating such an X is the fact that if Y is an m component mean zero multivariate normal with covariance D and X = AY , then X is a mean zero multivariate normal with covariance t C = E X X t = E AY (AY ) = AE Y Y t At = ADAt . We know how to sample the n component multivariate normal with D = I , just take the components of Y to be independent univariate standard normals. The formula X = AY w ill produce the desired covariance matrix if we ? nd A with AAt = C .A simple way to do this in practice is to use the Choleski decomposition from numerical linear algebra. This is a simple algorithm that produces a lower triangular matrix, L, so that LLt = C . It works for any positive de? nite C . In physical applications it is common that one has not C but its inverse, H . This would happen, for example, if X had the Gibbs-Boltzmann distribution with kT = 1 (it’s easy to change this) and energy 1 X t HX , and probability 2 1 density Z exp(? 1 X t HX ). In large scale physical problems it may be impracti2 cal to calculate and store the covariance matrix C = H ? though the Choleski factorization H = LLt is available. Note that3 H ? 1 = L? t L? 1 , so the choice 3 It is traditional to write L? t for the transpose of L? 1 , which also is the inverse of Lt . 3 A = L? t works. Computing X = L? t Y is the same as solving for X in the equation Y = Lt X , which is the process of ba ck substitution in numerical linear algebra. In some applications one knows the eigenvectors of C (which also are the eigenvectors of H ), and the corresponding eigenvalues. These (either the eigenvectors or the eigenvectors and eigenvalues) sometimes are called principal com2 ponents.Let qj be the eigenvectors, normalized to be orthonormal, and ? j the corresponding eigenvalues of C , so that 2 Cqj = ? j qj , t qj qk = ? jk . t Denote the qj component of X by Zj = qj X . This is a linear function of X and t therefore Gaussian with mean zero. It’s variance (note: Zj = Zj = X t qj ) is 2 t t t 2 E [Zj ] = E [Zj  · Zj ] = qj E [XX t ]qj = qj Cqj = ? j . A similar calculation shows that Zj and Zk are uncorrelated and hence (as components of a multivariate normal) independent. Therefore, we can generate Yj as independent standard normals and sample the Zj using Zj = ? j Yj . (2) After that, we can get an X using Zj qj . X= (3) j =1 We restate this in matrix terms. Let Q be the orthogonal matrix whose columns are the orthonormal eigenvectors of C , and let ? 2 be the diagonal ma2 trix with ? j in the (j, j ) diagonal position. The eigenvalue/eigenvector relations are CQ = Q? 2 , Qt Q = I = QQt . (4) The multivariate normal vector Z = Qt X then has covariance matrix E [ZZ t ] = E [Qt XX t Q] = Qt CQ = ? 2 . This says that the Zj , the components of Z , are 2 independent univariate normals with variances ? j . Therefore, we may sample Z by choosing its components by (14) and then reconstruct X by X = QZ , which s the same as (15). Alternatively, we can calculate, using (17) that t C = Q? 2 Qt = Q Qt = (Q? ) (Q? ) . Therefore A = Q? satis? es AAt = C and X = AY = Q? Y = QZ has covariance C if the components of Y are independent standard univariate normals or 2 the components of Z are independent univariate normals with variance ? j . 3 Brownian motion examples We illustrate these ideas for various kids of Brownian motion. Let X (t) be a Brownian motion path. Choose a ? nal time t and a time step ? t = T /n. The 4 observation times will be tj = j ? t and the observations (or observation values) will be Xj = X (tj ).These observations may be assembled into a vector X = (X1 , . . . , Xn )t . We seek to generate sample observation vectors (or observation paths). How we do this depends on the boundary conditions. The simplest case is standard Brownian motion. Specifying X (0) = 0 is a Dirichlet boundary condition at t = 0. Saying nothing about X (T ) is a free (or Neumann) condition at t = T . The joint probability density for the observation vector, f (x) = f (x1 , . . . , xn ), is found by multiplying the conditional densities. Given Xk = X (tk ), the next observation Xk+1 = X (tk + ? ) is Gaussian with mean Xk and variance ? t, so its conditional density is v 2 1 e? (xk+1 ? Xk ) /2? t . 2? ?t Multiply these together and use X0 = 0 and you ? nd (with the convention x0 = 0) f (x1 , . . . , xn ) = 3. 1 1 2? ?t n/2 exp ?1 2 ? Deltat n? 1 (xk+ 1 ? xk )2 . (5) k=0 The random walk method The simplest and possibly best way to generate a sample observation path, X , comes from the derivation of (1). First generate X1 = X (? t) as a mean zero v univariate normal with mean zero and variance ? t, i. e. X1 = ? tY1 . Given X1 , X2 is a univariate normal with mean X1 and variance ? , so we may v take X2 = X1 + ? tY2 , and so on. This is the random walk method. If you just want to make standard Brownian motion paths, stop here. We push on for pedigogical purposes and to develop strategies that apply to other types of Brownian motion. We describe the random walk method in terms of the matrices above, starting by identifying the matrices C and H . Examining (1) leads to ? 2 ? 1 0  ·Ã‚ ·Ã‚ · ? ? ? 1 2 ? 1 0  ·Ã‚ ·Ã‚ · ? ? .. .. .. . . . 1 ? 0 ? 1 ? H= ?. .. ?t ? . . 2 ? 1 ?. ? .. ? . ? 1 2 0  ·Ã‚ ·Ã‚ · 0 ? 1 ? 0 .? .? .? ? ? ? ? 0? ? ? ?1 ? 1 This is a tridiagonal matrix with pattern ? 1, 2, ? except at the bottom right corner. O ne can calculate the covariances Cjk from the random walk representation v Xk = ? t (Y1 +  ·  ·  · + Yk ) . 5 Since the Yj are independent, we have Ckk = var(Xk ) = ? t  · k  · var(Yj ) = tk , and, supposing j < k , Cjk = E [Xj Xk ] = ? tE [((Y1 +  ·  ·  · + Yj ) + (Yj +1 +  ·  ·  · + Yk ))  · (Y1 +  ·  ·  · + Yj )] = 2 ?tE (Y1 +  ·  ·  · + Yj ) = tj . These combine into the familiar formula Cjk = cov(X (tj ), X (tk )) = min(tj , tk ) . This is the same as saying that the ? 1 ?1 ? ?. ?. C = ? t ? . ? ? ? 1 matrix C is 1  ·Ã‚ ·Ã‚ · 2 2  ·Ã‚ ·Ã‚ · 2 . . . 3  ·Ã‚ ·Ã‚ · . . . 2 3  ·Ã‚ ·Ã‚ · ? 1 2? ? ? 3? .? .? .? .. . (6) The random walk method for generating X may be expresses as ? ? ? Y ? X1 1 1 0  ·Ã‚ ·Ã‚ · 01 ? ? ? ?1 1 0  ·Ã‚ ·Ã‚ · 0 ? ? . ? ?.? ?.? v? ? . ? ?.? 1 0 . . ? . .? ? . ? = ? t ? 1 1 ? ? ? ? ?. . .. ? ? ? ?. . . .. ? ? ? ? 11 1  ·Ã‚ ·Ã‚ · 1 Yn Xn Thus, X = AY with ? ? 1 0  ·Ã‚ ·Ã‚ · 01 ?1 1 0  ·Ã‚ ·Ã‚ · 0 ? ? ? v? .? .? . ?1 1 1 0 .? A = ? t ? ?. . ? .. .. ?. . ? . 11 1  ·Ã‚ ·Ã‚ · 1 (7) The reader should do the matrix multiplication to check that indeed C = AAt for (6) and (7). Notice that H is a sparse matrix indicating short range interactions while C is full indicating long range correlations.This is true of in great number of physical applications, though it is rare to have an explicit formula for C . 6 We also can calculate the Choleski factorization of H . The reader can convince herself or himself that the Choleski factor, L, is bidiagonal, with nonzeros only on or immediately below the diagonal. However, the formulas are simpler if we reverse the order of the coordinates. Therefore we de? ne the coordinate reversed observation vector t X = (Xn , xn? 1 , . . . , Xn ) and whose covariance matrix is ? tn ? tn? 1 ? C=? . ?. . t1 tn? 1 tn? 1  ·  ·  · t1 t1 .. .  ·Ã‚ ·Ã‚ · ? ? ? , ? t1 and energy matrix ? 1 ? 1  ·Ã‚ ·Ã‚ · 0 ? 0 .? .? .? ? ? ?. ? 0? ? ? ?1 ? 2 ? ? ? 1 2 ? 1 0  ·Ã‚ ·Ã‚ · ? ? .. .. .. . . . 1 ? 0 ? 1 ? H= .. ?t ? . . ?. . 2 ? 1 ? ? .. ? . ? 1 2 0  ·Ã‚ ·Ã‚ · 0 ? 1 We seek the Choleski factorization H = LLt ? l1 0 ? m2 l2 1? L= v ? m3 ?t ? 0 ? . .. . . . with bidiagonal ?  ·Ã‚ ·Ã‚ · ? 0 ? ?. .. ? . ? .. . Multiplying out H = LLt leads to equations that successively determine the lk and mk : 2 l1 l 1 m2 2 2 l1 + l 2 l 2 m3 = 1 =? l1 = 1 , = ? 1 =? m2 = ? 1 , = 2 =? l2 = 1 , = 1 =? m3 = ? 1 , etc. , The result is H = LLt with L simply ? 1 0  ·Ã‚ ·Ã‚ · ? ? 1 10 1? .. L= v ? . ?t ? ? 1 ? . .. .. . . . . 7 ? ? ? ?. ? ? The sampling algorithm using this Y = Lt X : ? ? ? 1 Yn ? Yn? 1 ? ? ? ? ?0 ? ? 1? ? ? ? ? . ?= v ? ?.? ?t ? ?.? ?. ? ? ?. . Y1 0 information is to ? nd X from Y by solving ?1 0 1 .. . ?1 .. .  ·Ã‚ ·Ã‚ ·  ·Ã‚ ·Ã‚ · .. . 0 0 Xn . ? ? Xn? 1 . . . 0 . . ?1 X1 1 ? ? ? ? ? ? ? ? ? Solving from the bottom up (back substitution), we have Y1 = Y2 = v 1 v X1 =? X1 = ? tY1 , ?t v 1 v (X2 ? X1 ) =? X2 = X1 + ? tY2 , etc. ? t This whole process turns out to give the same random walk sampling method. Had we not gone to the time reversed (X , etc. variables, we could have calculated the bidiagonal Choleski factor L numerically. This works for any problem with a tridiagonal energy matrix H and has a name in the control theory/estimation literature that escapes me. In particular, it will allow to ? nd sample Brownian motion paths with other boundary conditions. 3. 2 The Brownian bridge construction The Brownian bridge construction is useful in the mathematical theory of Brownian motion. It also is the basis for the success of quasi Monte Carlo methods in ? nance. Suppose n is a power of 2: n = 2L . We will construct the observation path X through a sequence of L re? ements. First, notice that Xn is a univariate normal with mean zero and variance T , so we may take (with Yk,l being independent standard normals) v Xn = T Y1,1 . Given the value of Xn , the midoint observation, Xn/2 , is a univariate normal4 w ith mean 1 Xn and variance T /4, so we may take 2 Xn 2 v 1 T = Xn + Y2,1 . 2 2 At the ? rst level, we chose the endpoint value for X . We could draw a ? rst level path by connenting Xn to zero with a straight line. At the second level, or ? rst re? nement, we created a midpoint value. The second level path could be piecewise linear, connecting 0 to X n to Xn . 4 We assign this and related claims below as exercises for the student. 8 The second re? nement level creates values for the â€Å"quarter points†. Given n X n , X n is a normal with mean 1 X n and variance 1 T . Similarly, X 34 is a 2 42 2 4 2 1 1T normal with mean 2 (X n + Xn ) and variance 4 2 . Therefore, we may take 2 Xn = 4 1 1 Xn + 22 2 T Y3,1 2 and n X 34 = 1 1 (X n + Xn ) + 2 2 2 T Y3,2 . 2 1 The level three path would be piecewise linear with breakpoints at 1 , 2 , and 3 . 4 4 Note that in each case we add a mean zero normal of the appropriate variance to the linear interpolation value.In the general step, we go from the level k ? 1 path to the level k paths by creating values for the midpoints of the level k ? 1 intervals. The level k observations are X j . The values with even j are known from the previous 2k? 1 level, so we need values for odd j . That is, we want to interpolate between the j = 2m value and the j = 2m + 2 value and add a mean zero normal of the appropriate variance: X (2m+1)n = 2k? 1 1 2 mn X 2k? 1 + X (2m+2)n 2 2k? 1 + 1 2(k? 2)/2 T Ym,k . 2 The reader should check that the vector of standard normals Y = (Y1,1 , Y2,1 , Y3,1 , Y3,2 , . . . t indeed has n = 2L components. The value of this method for quasi Monte Carlo comes from the fact that the most important values that determine the large scale structure of X are the ? rst components of Y . As we will see, the components of the Y vectors of quasi Monte Carlo have uneven quality, with the ? rst components being the best. 3. 3 Principle components The principle component eigenvalues and eigenvectors for many types of Brownian motion are known in closed form. In many of these cases, the Fast Fourier Transform (FFT) algorithm leads to a reasonably fast sampling method.These FFT based methods are slower than random walk or Brownian bridge sampling for standard random walk, but they sometimes are the most e? cient for fractional Brownian motion. They may be better than Brownian bridge sampling with quasi Monte Carlo (I’m not sure about this). The eigenvectors of H are known5 to have components (qj,k is the k th component of eigenvector qj . ) qj,k = const  · sin(? j tk ) . 5 See e. g. Numerical Analysis by Eugene Isaacson and Herbert Keller. 9 (8) The n eigenvectors and eigenvalues then are determined by the allowed values of ? j , which, in turn, are determined throught the boundary conditions.We 2 2 can ? nd ? j in terms of ? j using the eigenvalue equation Hqj = ? j qj evaluated at any of the interior components 1 < k < n: 1 2 [? sin(? j (tk ? ?t)) + 2 sin(? j tk ) ? sin(? j (tk + ? t)) ] = ? j sin(? j tk ) . ?t Doing the math shown that the eigenvalue equation is satis? ed and that 2 ?j = 2 1 ? cos(? j ? t) . ?t (9) The eigenvalue equation also is satis? ed at k = 1 because the form (8) automatically satis? es the boundary condition qj,0 = 0. This is why we used the sine and not the cosine. Only special values ? j give qj,k that satisfy the eigenvalue equation at the right boundary point k = n. 10

Healthy environment Essay

A healthy environment to me would be a community with low pollution, stable economical levels, clean water and communicates who communicate well with one another. The residents in this community would be happy, and would encourage others to come live in their community (Maurer & Smith, 2013). There would be quality health care services available and good schools for the children to promote knowledge. This would help provide education to promote successful lifestyles (Maurer & Smith, 2013). Nursing cares could be provided depending on what the needs were. Nurses could hold seminars to provide the community ways to better their health. They can provide teaching to assist with decreasing pollution in their communities. If there is a high amount of respiratory illness in that community, then minimizing the pollutants would be beneficial. They could start smoke-free areas, teach about decreasing litter to prevent pests, then the pollutant of exterminating gases would be decreased. Nursing could gather statistics about the community’s health to provide information to help promote good health. Maurer, F.A. & Smith, C.M. 2013. Community/Public Health Nursing Practice: Health for Families and Populations, 5th edition. Retrieved from: http//pagebursts.elsevier.com

SAT Score Tables - Compare Admissions Data for Colleges

SAT Score Tables - Compare Admissions Data for Colleges Below youll find links to dozens of articles that can help you put your SAT scores in context for a wide range of colleges and universities. Always keep in mind that the SAT is just one part of your application, and less-than-ideal scores dont need to torpedo your chances of admission if you have strengths in other areas. Top College and University SAT Tables: See how the countrys most prestigious colleges and universities compare on the SAT front (or you can check out the ACT comparison charts). The Ivy LeagueTop Universities (non-Ivy)Top 10 Liberal Arts CollegesTop 10 Public Universities22 More Top Public UniversitiesTop Public Liberal Arts CollegesTop Engineering Schools (PhD Granting)Top Engineering Schools (Bachelors and Masters)Top Womens CollegesTop Catholic Colleges and Universities State University SAT Data: Admissions criteria vary widely from campus to campus within state university systems. These charts can help you find schools that match your SAT scores. Alabama: Four-Year Alabama Colleges and UniversitiesAlaska: Four-Year Alaska Colleges and UniversitiesArizona: Four-Year Arizona Colleges and UniversitiesArkansas: Four-Year Arkansas Colleges and UniversitiesCalifornia: Cal State SystemCalifornia: UC SystemCalifornia: Top California Colleges and UniversitiesColorado: Four-Year Colorado CollegesConnecticut: Four-Year Colleges and UniversitiesDelaware: Four-Year Delaware Colleges and UniversitiesDistrict of Columbia: Four-Year Washington D.C. Colleges and UniversitiesFlorida: State University SystemFlorida: Top Florida Colleges and UniversitiesGeorgia: Top Georgia Colleges and UniversitiesHawaii: Four-Year Hawaii Colleges and UniversitiesIdaho: Four-Year Idaho Colleges and UniversitiesIllinois: Top Illinois Colleges and UniversitiesIndiana: 15 Top Indiana Colleges and UniversitiesIowas: Four-Year Iowa Colleges and UniversitiesKansas: Four-Year Kansas Colleges and UniversitiesKentucky: Four-Year Kentucky Colleges and UniversitiesLouisia na: Four-Year Louisiana Colleges and Universities Maine: Four-Year Maine Collegs and UniversitiesMaryland: Top Maryland Collegs and UniversitiesMassachusetts: Top Massachusetts Collegs and UniversitiesMichigan: 13 Top Michigan Colleges and UniversitiesMinnesota: 13 Top Minnesota Colleges and UniversitiesMississippi: Four-Year Mississippi Colleges and UniversitiesMissouri: 12 Top Missouri Colleges and UniversitiesMontana: Four-Year Montana Colleges and UniversitiesNebraska: Four-Year Nebraska Colleges and UniversitiesNevada: Four-Year Nevada Colleges and UniversitiesNew Hampshire: New Hampshire Colleges and UniversitiesNew Jersey: Four-Year New Jersey Colleges and UniversitiesNew Mexico: Four-Year New Mexico Colleges and UniversitiesNew York: CUNY Senior CollegesNew York: SUNY SystemNew York: Top New York Colleges and UniversitiesNorth Carolina: 16 Public UniversitiesNorth Carolina: Top North Carolina Colleges and UniversitiesNorth Dakota: Four-Year North Dakota Colleges and UniversitiesOhio: 10 Top Ohio Colleges and UniversitiesOhio : 13 University System of Ohio campuses Oklahoma: Four-Year Oklahoma Colleges and UniversitiesOregon: Selective Oregon Colleges and UniversitiesPennsylvania: Top Pennsylania Colleges and UniversitiesRhode Island: Four-Year Rhode Island Colleges and UniversitiesSouth Carolina: Four-Year South Carolina Colleges and UniversitiesSouth Dakota: Four-Year South Dakota Colleges and UniversitiesTennessee: Top Tennessee Collegs and UniversitiesTexas: 13 Top Texas Colleges and UniversitiesUtah: Four-Year Utah Colleges and UniversitiesVermont: Four-Year Vermont Colleges and UniversitiesVirginia: 15 Public UniversitiesVirginia: 17 Top Virginia Colleges and UniversitiesWashington: 11 Top Washington Colleges and UniversitiesWest Virginia: Four-Year West Virginia Colleges and UniversitiesWisconsin: Four-Year Wisconsin Colleges and Universities SAT Scores for Division I Athletic Conferences: For students interested in the excitement of Division I sports, these charts make some of the admissions distinctions between universities clear. America East ConferenceAtlantic 10 ConferenceAtlantic Coast ConferenceAtlantic Sun ConferenceBig East ConferenceBig Sky ConferenceBig South ConferenceBig Ten ConferenceBig 12 ConferenceConference USA (C-USA)Horizon LeagueMetro Atlantic Athletic ConferenceMid-American ConferenceMissouri Valley ConferenceMountain West ConferenceNortheast ConferenceOhio Valley ConferencePac 12Â  ConferenceSoutheastern ConferenceSouthern ConferenceSun Belt ConferenceWestern Athletic Conference More SAT Information: Here are some more articles to help you make sense of the SAT. Whats a Good SAT Score?What Schools Dont Require Scores?When is the SAT?Should I use SAT Score Choice?What Schools Require SAT Subject Tests?Are SAT Prep Courses Worth the Cost?Does the SAT Writing Section Matter?