Improved Fast Decoupled Power Flow

The power flow analysis is a very important and tundamental tool in power system analysis. Its results play the major role during the operational stages of any system for its control and economic schedule, as well as during expansion and design stages The purpose of any load flow analysis is to compute precise steady-state voltages and voltage angles of all buses in the network, the real and reactive power flows into every line and transformer, under the assumption of known generation and load.During the second half of the twentieth century, and after the large technological evelopments in the fields of digital computers and high-level programming languages, many methods for solving the load flow problem have been developed, such as indirect Gauss-Siedel (bus admittance matrix). direct Gauss-Siedel (bus impedance matrix).Newton-Raphson (NR) and its decoupled versions Nowadays, many Improvements have been added to all these methods involving assumptions and approximations of the trans mission lines and bus data, based on real systems conditions The Fast Decoupled Power Flow Method (FDPFM) is one of these improved methods, which was based on a simplification of the Newton-Raphson method and reported by Stott and Alsac in 1974[4]. This method and due to its calculations simplifications, fast convergence and reliable results became the most widely used method in load flow analysis.However, FDPFM for some cases, where high RA ratios or heavy loading (Low Voltage) at some buses are present, does not converge well. For these cases, many efforts and developments have been made to overcome these convergence obstacles. some of them targeted the convergence of systems with hgh RIX ratios, others those with low voltage buses However, one of the most recent developments is a Robust Fast Decoupled Power Flow developed by Wang and u; it Is ased on heuristic justification and general voltage normalization methods [171 and solves both high RIX ratios and low bus voltages problem s simultaneously.Though many efforts and elaborations have been achieved in order to improve the and simulations are becoming more developed and are now able to handle and analyze large size system. Today, and after reaching processor's speeds higher than 3 GHz, any improvement in the speed of convergence of the power flow method, provided it leads to reliable results, is of great value. This speed improvement is very important when involved in operational stages of power distribution, where any illisecond saving can hugely increase the probability of the right decision, of the control and dispatch computerized system.This paper works on providing computing savings (in flops) and thus higher speed of convergence of the FDPFM based on the initial approximation in which real power changes are considered to be most sensitive to variations in voltage angle and much less to those of voltage magnitude, as well as on the high sensitivity of reactive power changes to variations in voltage m agnitude and much less to those of voltage angle. In this paper, the attention was focused on the update of the voltage angle (6) and oltage magnitude (V) in each iteration, based on the improvement of flops achieved, and obviously on the results obtained.The results of these improvements and the comparative analysis with the Newton-Raphson and classical FDPFM will be presented using the three IEEE bus systems of 14, 30 and 57-bus, although the IFDPFM can be applied to any size bus system. II. Fast Decoupled Power Flow Method As the FDPFM is derived from the Newton-Raphson we will start from the matrix representation of NR, apply some simplifications and approximations, to reach the equations of the FDPFM.The matrix representation of the N-R method [17] is: O APOOH Where I IVJI IYiJl +6]) And -2 cos Bit +2 cos -6i +6]) Nii – = I VI II YiJ I cos (B iJ- 6i + 6]) Nil (7) -2 IYiil stn +2 IVJI IYiJl cos -6i +6]) Now, for typical power system branches: XIR and ; 200 (10) between AQ and A6, hence N and J entries of the initial matrix of (1) can be ignored leading to the following decoupled equations: (12) Now, the diagonal elements of H according to Stott and Alsac [4] can be written as: IVi12Bii (13) Where Bii = I Yill sin Bii is the imaginary part of the diagonal elements of the bus admittance matrix Ybus.Further simplifications can be applied to equation (12), by considering Bii Qi and I Vil 2 z I Vil yielding to the following simplified Hit: Hii=- (14) Also, as under normal operating conditions 6] – 6i is quite small, thus Bii – 6i + 6] Bit, and IVJI 1, the off-diagonal elements of the matrix H can be written as: HIJ I Vil (15) Similarly, the diagonal elements of the L matrix can be written as: Lil † (16) And its off- diagonal elements as: LiJ=-lVll (17) Applying these assumptions to equations (11) and (12) we get: =-B'A6 I vil (18) (19) where B' and B† are the imaginary part of the bus admittance matrix Ybus , such thatB' contain s all buses admittances except those related to the slack bus, and B† is B' deprived from all voltage-controlled buses related admittances. Finally, all these approximations and simplifications lead to the following successive voltage magnitude and voltage angle updating equations: (20) IVI (21) These equations formed the basis of the iteration scheme upon which the Matlab software written and then updated. Ill.Updated Algorithm The algorithm written according to the equations derived in the previous section is as follows: Step 1: Creation of the bus admittance Ybus according to the lines data given y the IEEE standard bus test systems. Step 2: Detection of all kinds and numbers of buses according to the bus data given by the IEEE standard bus test systems, setting all bus voltages to an initial value of 1 pu, all voltage angles to O, and the iteration counter iter to O.Step 3: Creation of the matrices B' and B† according to equations (18) and (19). Step 4: If max (AP, A Q) accuracy then Go to Step 6 else 1. Calculation of the H and L elements of equations (14), (1 5), (16), (17). 2. Calculation of the real and reactive power at each bus, and checking if Mvar of generator buses re within the limits, otherwise update the voltage magnitude at these buses by ? ±2 3. Calculation of the power residuals, AP and AQ. 4.Calculation of the bus voltage and voltage angle updates AV and A6 according to equations (19) and (20). 5. Update of the voltage magnitude V and the voltage angle 6 at each bus. 6. Increment of the iteration counter iter = iter + 1 then Go to Step 4 Print out ‘Solution did not converge' and go to Step 6 Step 6: Print out of the power flow solution, computation and display of the line flow and losses. The update of this algorithm was based on the weak coupling between AP and AV, nd between AQ and A6, explained in the previous section.Specifically, in the fourth subroutine of Step 4 of the initial algorithm, and instead of updating the voltage magnitude and the voltage angle once and simultaneously in each iteration, the improved algorithm updated either the voltage angle or the voltage magnitude at each bus, Jumped to subroutine 1 to recalculate the real and reactive power and then updated the second variable based on what was updated first.Moreover, and for more speed improvements and convergence reliability, the update of one of the two variables was repeated several times, holding the other ariable at its last calculated value, which reduced the number of floating point operations of the algorithm and thus lead to the faster convergence of the IFDPFM. IV. Numerical Analysis The performance of the IFDPFM was tested on IEEE 14, 30 and 57-bus systems with a convergence accuracy of 10-3 on a MVA base of 100 or equivalently 10-1 MVA for both power residuals AP and AQ.This numerical analysis involved a speed comparison between the NR method, the FDPFM and the IFDPFM based on the number of flops (floating point oper ations) of each algorithm implementing each method, rather than on any other basis, because he flops count is independent from the CPU speed or the specific programming language used. In addition, as mentioned in the previous part, the algorithm of this paper updated the voltage angle several times before updating the voltage magnitude or vice versa which resulted in a different flops count for each combination used for the same IEEE bus system.These combinations will be noted according to the number of loops of update of each variable. For instance, updating twice the voltage angle (6) and then once the voltage magnitude (V) in the same iteration will be written as (2;1). Note that any flops number without the previous notation will be the one of the best case of the updated algorithm. Moreover, for any combination to be listed in this paper it should have satisfied the condition of no more than 3 % deviation of its results from that of the NR method.The bar graph in Figure 1 shows a comparison based on the number of flops between the NR, FDPFM and the best case of IFDPFM for the three IEEE standard bus systems used in this paper. Number of flops per method per system 934. 573 305. 126 314. 925 157. 310 System 57 4,421. 752 2,841. 646 14 30. 823 56. 829 24. 574 1 ,ooo ,500 2,000 2,500 3,000 Flops IFDPFM FDPFM 4,000 4,500 (Thousands) Fig. 1: Flops Comparison between the 3 methods. It is clearly seen that the IFDPFM requires much less flops to converge as compared to FDPFM or NR.This flops saving is proportional to the system size and as shown, increases with the increase of the number of buses. Obviously, this improvement in the number of flops will make the IFDPFM converge much faster than the two other methods whatever CPU used. Numerically, and for the biggest system involved in this paper (IEEE 57-Bus System), the IFDPFM revealed a flops saving of about 67 % when ompared with the FDPFM and about 78 % when compared with the NR.Normally, and as mentioned bef ore, this saving goes down to the order of 50 % for the two smaller bus systems. In addition, and in order to reach the best case presented above, different strategies of updating the voltage angle (6) and the voltage magnitude (V) were tested and compared first with the FDPFM then with the NR. Figure 2 below the percentage of flops of IFDPFM versus that of the FDPFM, for 10 different updating strategies and for the three IEEE systems.Percentage Flops IFDPFM vs FDPFM 75 50 25 Delta;Voltage Loops IFDPFM14 IFDPFM30 IFDPFM57 Fig. 2: % of flops of IFDPFM vs. FDPFM for different voltage angle and voltage magnitude updating strategies. At the first look, it is seen that for the three systems, three parallel curves are sketched with most values less then 75 % of the FDPFM. This parallel property of this graph shows the consistency of the algorithm in its number of flops variation for each strategy for each system studied.Also, it is seen that for low number of voltage magnitude and voltage angle loops the IFDPFM can't be more efficient than FDPFM, but for a slightly higher number the IFDPFM shows great improvement in flops saving nd reaches the highest improvement at the point (4;3), where in each iteration, the voltage angle was updated four times while the voltage was kept at its initial value and then 6 was kept at its last value and V updated three times.Numerically, and for the best case of IFDPFM (4;3), the new algorithm showed a flops saving of 57 % for the 14-bus system, 50% for the 30-bus system, and 68% for the 57-bus system. Figure 3 below shows the percentage of flops of IFDPFM versus that of the NR, for 10 different updating strategies and for the three IEEE systems. IFDPFM vs NR 175 150 25 Fig. 3: % of flops of IFDPFM vs. NR for different voltage angle and voltage magnitude updating strategies.Basically, the same comments of the comparison of IFDPFM with FDPFM apply in this comparison. However, here the flops saving is much more significant and is propo rtional to the system size. Numerically, we have a 21 % flops saving for the 14-bus system, 49 % for the 30-bus system and 78% for the 57-bus system. Finally, it is remarked that when compared with NR, IFDPFM savings showed a high variation in their percentage, mainly because they are highly proportional to the

Free Essays on Survival In Auschwitz Review

Survival in Auschwitz In 1944, Primo Levi was abducted by the fascist government in Italy and shipped off to Auschwitz, Nazi death camp. Many people do not know that the concentration camps were not only for those of Jewish descent. Unfortunately, political prisoners, criminals, gypsies, and homosexuals were also tossed away. Until reading this book, I had a vague idea of what went on inside of the Nazi concentration camps. Most of my knowledge of the Holocaust stems from textbooks or from firsthand accounts that do not center around camp life. This book gives a vivid account of the atrocities committed by the Nazis. One of the more striking aspects of this work is the attitudes toward each other the â€Å"inmates† have. Every man is really for himself, and the newer detainees have to learn the camp rules the hard way. Levi is even left wanting his spoon, bowl, and gloves upon entrance to the Ka-Be, because no one told him that personal effects were forbidden. He speaks of the practice, which everyone learns, of bundling everything one owns in their jacket and keeping it under their head as a pillow so that no one would steal it. I would have thought there would have been more a sense of camaraderie among the inmates. Levi states, however, that when one’s humanity is stripped from them, they live for themselves and themselves only. He finally catches a glimpse of humanity toward the end of the novel, when a young Franco-Pole, Towarowski, offers those who helped carry the stove to their hut a little extra of his bread and suggests that everyone else do the same. Levi is touch ed by this move, and mused â€Å"only a day before a similar event would have been inconceivable. The law of the Lager said: ‘eat your own bread, and if you can, that of your neighbour’, and left no room for gratitude. It really meant that the Lager was dead.† (Levi, 160). This book affected me a great deal. I have never read a firsthand ... Free Essays on Survival In Auschwitz Review Free Essays on Survival In Auschwitz Review Survival in Auschwitz In 1944, Primo Levi was abducted by the fascist government in Italy and shipped off to Auschwitz, Nazi death camp. Many people do not know that the concentration camps were not only for those of Jewish descent. Unfortunately, political prisoners, criminals, gypsies, and homosexuals were also tossed away. Until reading this book, I had a vague idea of what went on inside of the Nazi concentration camps. Most of my knowledge of the Holocaust stems from textbooks or from firsthand accounts that do not center around camp life. This book gives a vivid account of the atrocities committed by the Nazis. One of the more striking aspects of this work is the attitudes toward each other the â€Å"inmates† have. Every man is really for himself, and the newer detainees have to learn the camp rules the hard way. Levi is even left wanting his spoon, bowl, and gloves upon entrance to the Ka-Be, because no one told him that personal effects were forbidden. He speaks of the practice, which everyone learns, of bundling everything one owns in their jacket and keeping it under their head as a pillow so that no one would steal it. I would have thought there would have been more a sense of camaraderie among the inmates. Levi states, however, that when one’s humanity is stripped from them, they live for themselves and themselves only. He finally catches a glimpse of humanity toward the end of the novel, when a young Franco-Pole, Towarowski, offers those who helped carry the stove to their hut a little extra of his bread and suggests that everyone else do the same. Levi is touch ed by this move, and mused â€Å"only a day before a similar event would have been inconceivable. The law of the Lager said: ‘eat your own bread, and if you can, that of your neighbour’, and left no room for gratitude. It really meant that the Lager was dead.† (Levi, 160). This book affected me a great deal. I have never read a firsthand ...

How Many Years a President Can Serve

How Many Years a President Can Serve U.S. presidents are limited to serving two elected four-year terms in the White House and as many as two years of another presidents term. That means the longest any president could serve is 10 years, though no one has been in the White House that long since Congress passed the constitutional amendment on term limits. How many years a president can serve in the White House is spelled out in the 22nd amendment  to the  U.S. Constitution, which states no person shall be elected to the office of the President more than twice. However, if an individual becomes president through the  order of succession they are allowed to serve an additional two years. Why Limited to 2 Terms The amendment defining limits on how many terms can a president serve was approved by Congress on March 21, 1947, during the administration of President Harry S. Truman. It was ratified by the states on Feb. 27, 1951. Before the 22nd Amendment, the Constitution did not limit the number of presidential terms to two, though many early presidents including George Washington imposed such a limit on themselves. Many argue that the 22nd Amendment merely put on paper the unwritten tradition held by presidents of retiring after two terms. Before the ratification of the 22nd Amendment, Democrat Franklin Delano Roosevelt was elected to four terms in the White House in 1932, 1936, 1940, and 1944. Roosevelt died less than a year into his fourth term, but he is the only president to have served more than two terms. Congressional Republicans proposed the 22nd Amendment in response to Roosevelts four election victories. Historians have written that the party felt such a move was the best way to invalidate and discredit the popular progressives legacy. Defined In 22nd Amendment The relevant section of the 22nd Amendment defining presidential terms  reads: No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of President more than once. American presidents are elected for four-year terms. While the 22nd Amendment limits presidents to two full terms in office, it also allows them to serve two years at most of another presidents term. That means the most any president can serve in the White House is 10 years. History The framers of the Constitution originally considered a lifetime appointment by Congress for the president. When this proposal failed, they discussed whether the president should be elected by either the Congress, the people, or something in between, such as the Electoral College (which was eventually chosen) and whether term limits should be imposed. The idea of an appointment by Congress, with the option for re-appointment, failed on the fear that a president could make an underhanded deal with Congress to get re-appointed. Cant Serve a 3rd Term Conspiracy theorists have through the years perpetuated the notion that power-hungry presidents try to find ways to win a third election. A few people even point to the wording of the 22nd Amendment, noting that it says no person shall be eligible to be elected to the office more than twice. Would this disqualify a former president from being elected vice president, then serving as president if the president died or resigned? Its unlikely any former president would run for the lower office of vice president to test the theory. Over the years, several lawmakers have proposed repealing the 22nd Amendment. Congressional opponents of the 22nd Amendment argue that it restricts voters from exercising their will. As Democratic U.S. Rep. John McCormack of Massachusetts proclaimed during a debate over the proposal: The framers of the Constitution considered the question and did not think they should tie the hands of future generations. I dont think we should. Although Thomas Jefferson favored only two terms, he specifically recognized the fact that situations could arise where a longer tenure would be necessary. One of the most high-profile opponents of the two-term limit for presidents was Republican President Ronald Reagan, who was elected to and served two terms in office. In a 1986 interview with The Washington Post, Reagan lamented the lack of focus on important issues and lame-duck presidents. I have come to the conclusion that the 22nd Amendment was a mistake, Reagan said. Shouldnt the people have the right to vote for someone as many times as they want to vote for him? They send senators up there for 30 or 40 years, congressmen the same. Sources Buckley, F.H. and Metzer, Gillian. â€Å"The 22nd Amendment of the U.S. Constitution.†Ã‚  National Constitution CenterCannon, Lou. â€Å"Short-Sighted Amendment.†Ã‚  The Washington Post, WP Company, 16 June 1986