EE523_Load Flow.pdf - LOAD FLOW ANALYSIS Load Flow or Power Flow Analysis is a useful power system tool It is a steady state simulation of a power

EE523_Load Flow.pdf - LOAD FLOW ANALYSIS Load Flow or Power...

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Unformatted text preview: LOAD FLOW ANALYSIS Load Flow or Power Flow Analysis is a useful power system tool. It is a steady state simulation of a power system operating under normal conditions that provides several useful information including, but not limited to, voltage profile magnitude and phase angle , power flows, and power losses. The information obtained from the analysis may then be used for system planning and design of a new system or expansion of an existing one, transformer tap settings for transformers with on-load tap changer, capacitor sizing for power factor correction and compensation for long transmission lines, etc. Bus Admittance Matrix The Bus Admittance Matrix or YBUS is intensively used in Load Flow solutions. Its elements are the coefficients of the Nodal Equation which assumes the matrix form The diagonal elements, Yjj, are called the self-admittances and are computed by summing up the admittances of each branch or line connected to Bus j. The off-diagonal elements, Yjk, are called mutual admittances and are computed as the negative of the admittance in between Buses j and k. There must be no bus in between the considered buses. In case of parallel branches between Buses j and k, they are computed as the negative of the sum of the admittances of all the parallel branches between Buses j and k or simply the negative of the equivalent admittance of the parallel branches. The current injection, I, is the current going in to generated or going out from load a specific bus aside from the currents through the lines. The bus voltage, V, is the potential at a specific bus with reference to the ground zero potential . General Power Equations ∗ ∗ ∗ ∗ ∗ ∗ Such that expressing P and Q separately: | | | | ℎ ℎ ℎ ℎ where: ∗ 2 Types of Buses There are three types of buses in the power system for power flow studies. Each type is treated differently depending on which values are specified and unknown. The load bus has the real and reactive power of the specified load while the bus voltage magnitude and phase angle are computed. Thus, it is also known as PQ bus. For a generator bus, the voltage magnitude and the generated real power are specified, hence, it is also called the PV bus. The voltage angle and reactive power of the generated are calculated. It is noteworthy to mention that a PV bus can have real and reactive power load. Slack or swing bus is a special kind of generator bus. The voltage magnitude and phase angle of the bus are specified. It provides the real and reactive power balance in the power system such that the sum of the real and reactive power generated in the power system equals the sum of all real and reactive power loads in all the busses plus losses in the transmission and distribution lines and power transformers real and reactive . Type of Bus Load PQ Generator/Voltagecontrolled PV Slack/Swing Voltage || Generation Load * * * * Note: * Optional 1. The blank cells denote that the specified parameters are calculated and the cells with the “” marks are given. 2. For PQ busses, the initial voltage is assumed to be equal to 1.0∠0° pu when not specified. 3. In most cases for a PV bus, the minimum and maximum limits of the generated reactive power, Qg are specified. For Gauss-Seidel Load flow solution, if the calculated generated reactive power is beyond the limit, then use the violated limit Qsched Qmin if Qg Qmin or Qsched Qmax if Qg Qmax in recalculating the new bus voltage using the equations: 1 ∗, ∗ On the other hand, if the calculated Qg is within the limit, the bus voltage shall be recalculated using the foregoing equation with Qsched as the computed Qg and the bus voltage shall be corrected as follows: , , 3 Gauss-Seidel Load Flow , ∗, , , , , , ∗, ∗ 1 , ∗, 1 ∗ , Newton-Raphson Load Flow Polar Δ Δ Δ Δ|| || ⎡ ⎢ ⎢ ⎣ ⎤ Δ ||⎥ Δ|| ⎥ || || ||⎦ || Power Injected and Power Mismatch , , ∗ , Δ , , , Δ Δ ∗ , , Elements of the Jacobian Matrix ∗ ∗ ∗ ∗ | | cos sin cos | | | | sin | | cos sin 4 For J1: | | | | sin sin For J2: 2 | | For J3: | | | | cos | | | | cos cos cos cos For J4: 2 | | | | sin | | ∗ sin ∗, 2 ∗ | | | | | | | | | | sin ∗ ∗, ∗ 5 Example 1. A simple power system is shown below. The line data are impedances in per unit. The limits for the reactive powers of the generators are set at –0.5 pu Q 0.5 pu for bus 3, and –0.3 pu Q 0.3 pu for bus 4. All quantities provided are in per unit. Using a tolerance, ε 0.002, determine the voltages at each bus using a Gauss-Seidel YBUS Load Flow and b Newton-Raphson Polar Load Flow. Solution: Calculating first the elements of the Ybus matrix: 1 1 24 56 . . 0.0125 0.0375 1 1 1 1 36 82 . . 0.0125 0.025 0.06 0.12 0.03 0.06 0.01 0.03 1 1 1 1 38 84 . . 0.0125 0.0375 0.06 0.12 0.03 0.06 0.01 0.02 1 1 30 70 . . 0.01 0.03 0.01 0.02 1 16 32 0.0125 0.025 1 8 24 . . 0.0125 0.0375 1 0 . . ∞ 1 1 10 20 . . 0.06 0.12 0.03 0.06 1 10 30 . . 0.01 0.03 1 20 40 . . 0.01 0.02 0.0125 0.025 6 In matrix form: 24 56 16 32 8 24 0 16 32 36 82 10 20 10 30 8 24 10 20 38 84 20 40 0 10 30 20 40 30 70 For better visualization, the foregoing table shows the bus data: Bus Voltage |V| θ 1.03 0° 1.02 1.02 - Bus No. 1 2 3 4 Generated PG QG 0 0 0.9 0.6 - Load PL 0 2.8 0 0 QL 0 2.1 0 0 Thus, the initial values of the voltages are as follow: 1.03∠0° . . , ∗, , , 1∠0° . . , 0 2.8 2.1 0.9 0 0.9 0.6 0 0.6 2.8 1.02∠0° . . , 1.02∠0° . . 2.1 Gauss-Seidel 1 ∗, 1 1 , ∗ , , , ∗ ∗ First Iteration 1 36 82 2.8 2.1 1∠0° ∗ 1.02∠0° Δ| | |0.99013 1.02∠0° 0.5 . . , 1| 1.03∠0° 10 30 32 0.99013∠ 1.02∠0° 10 20 1.09968° . . 0.00987 1.03∠0° 8 1.02∠0° 20 24 40 0.17439 . . 0.17439 . . 0.17439 . . 16 0.5 . . ∗ 0.99013∠ 1.09968° 10 0.61256 0.17439 . . 20 1.02∠0° 38 84 7 1 38 0.9 0.17439 84 1.02∠0° ∗ 1.02∠0° Δ| | |1.02126 1.03∠0° 20 1.02| 40 8 24 0.99013∠ 1.09968° 10 20 1.02126∠0.15624° . . 0.00126 The voltage of Bus 3, being a PV bus, must be corrected as Q3 is within limits: 1.02126∠0.15624° 1.02 1.02∠0.15624° . . 1.02126 1.02∠0° 1.03∠0° 0 0.99013∠ 1.09968° 10 30 1.02126∠0.15624° 20 40 1.02∠0° 30 70 , 0.3 . . Since 0.77458 0.78260 . . 0.78260 . . ≮ 0.3 . . 0.78260 . . ≮ 0.3 . . is higher than the upper limit for Qg, then , 0.3 . . Thus, calculating V4: 1 0.6 0.3 30 70 1.02∠0° ∗ 1.03∠0° 0 1.02∠0.15624° Δ| | ∗ |1.01340 1.02| 20 0.99013∠ 40 1.09968° 10 30 1.01340∠0.02157° . . 0.00660 Second Iteration 1 0.5 . . , Δ| | 1 ∗ Δ| | , , 0.98706∠ 1.01523° . . 0.00307 ∗ 0.96876 0.42339 . . 1.01970∠0.11881° . . 0.42339 . . 0.42339 . . 0.42339 . . , ∗ , 0.5 . . , 0.00030 8 Again, the voltage of Bus 3, being a PV bus, must be corrected since computed Q3 is within limits: , , 0.3 . . 1.02∠0.11881° . . ∗ 1.01226∠0.04308° . . 0.39271 . . ≮ 0.3 . . is higher than the upper limit for Qg, then 0.3 . . 1 0.39271 . . 0.39271 . . Since , 0.60782 , Δ| | , ∗ 0.00114 Third Iteration 1 ∗, Δ| | 0.5 . . Δ| | 0.98660∠ 1.02182° . . 0.92653 0.51141 . . 1.01977∠0.10717° . . 0.00046 ∗ 0.5 . . , ∗ , , Δ| | 0.51141 . . 1 0.3 . . 0.51141 . . ≮ 0.5 . . Again, set ∗ , , 0.00023 ∗ 0.62005 0.31782 . . 0.31782 . . 0.31782 . . ≮ 0.3 . . 0.3 . . 1 , ∗ , 0.00031 1.01195∠0.03470° . . 9 Therefore, since Δ|V2|, Δ|V3| and Δ|V4| are all less than the convergence criterion ε Seidel load flow solution has converged. The foregoing table of iteration shows the calculated voltages and angles using ε Seidel load flow solution converges after 10 iterations: i V1 V2 V3 V4 0.99013 1.02 1.01340 1 1.03∠0° ∠–1.09968° ∠0.15624° ∠0.02157° 0.98706 1.02 1.01226 2 1.03∠0° ∠–1.01523° ∠0.11881° ∠0.04308° 0.98660 1.01977 1.01195 3 1.03∠0° ∠–1.02182° ∠0.10717° ∠0.03470° 0.98641 1.01958 1.01176 4 1.03∠0° ∠–1.02790° ∠ 0.10132° ∠ 0.02891° 0.98629 1.01947 1.01164 5 1.03∠0° ∠–1.03141° ∠ 0.09751° ∠ 0.02532° 1.01939 1.01157 0.98621 6 1.03∠0° ∠–1.03365° ∠ 0.09513° ∠ 0.02306° 1.01935 1.01152 0.98617 7 1.03∠0° ∠–1.03505° ∠ 0.09363° ∠ 0.02164° 1.01932 1.01150 0.98614 8 1.03∠0° ∠–1.03593° ∠ 0.09269° ∠ 0.02075° 0.98612 1.01930 1.01148 9 1.03∠0° ∠–1.03648° ∠ 0.09210° ∠ 0.02020° 1.01929 1.01147 0.98611 10 1.03∠0° ∠–1.03683° ∠ 0.09173° ∠ 0.01985° 0.002, the Gauss0.00001, the Gauss Newton-Raphson 1.03∠0° 16 32 1∠0° 36 82 1.02∠0° 10 20 ∗ 1.02∠0° 10 30 0.88 1.96 . . 1.02∠0° 1.03∠0° 8 24 1∠0° 10 20 1.02∠0° 38 84 ∗ 1.02∠0° 20 40 0.1224 0.1632 . . 0.5 . . 0.1632 . . 0.5 . . 1.02∠0° 1.03∠0° 0 1∠0° 10 30 1.02∠0° 20 40 1.02∠0° 30 0.204 0.612 . . 0.3 . . 0.612 . . ≮ 0.3 . . , 0.3 . . Δ 1∠0° 2.8 0.88 Δ 0.9 0.1224 Δ 0.6 0.204 Δ Δ 2.1 0.3 0.7776 . . 0.396 . . 1.96 0.612 1.92 . . 0.14 . . 0.312 . . 70 ∗ 10 First Iteration | | | | | | | | | | | | | | | | | | | | | | | | ⎡ ⎢ ⎢ ⎢ ⎣ 1.92 ⎡ 0.7776 ⎤ ⎢ ⎥ ⎢ 0.396 ⎥ ⎢ 0.14 ⎥ ⎣ 0.312⎦ 0.88 1.96 ∗ 1 36 | | 0.88 0.1224 0.1632 | | 0.204 0.612 | | 0.204 | | | | 1.02 30 ∗ 84 39.4128 70 31.008 1.02 30 70 10 20 10.2 20.4 1∠0° ∗ 1.02∠0° 10 30 10.2 30.6 | | 10.2 ∗ | | ∗ 20 20.4 1.02∠0° ∗ ∗ 20.4 87.2304 41.616 10.2 20.808 20 10 30.6 1.02∠0° 30.6 41.616 72.216 10.2 31.008 20.4 87.2304 41.616 10.2 20.808 10.2 20 20.4 40 20.808 20.808 10.2 72.216 31.416 73.44 30.6 40 20.808 35.12 10.2 10.2 80.04 30.6 41.616 41.616 30.6 10.2 | | 35.12 10.2 10.2 80.04 30.6 30.6 41.616 72.216 10.2 31.008 87.2304 20.4 10.2 30 80.04 30.6 41.616 1∠0° 10.2 1.02∠0° 10.2 10 20.808 | | 83.96 ⎡ 20.4 ⎢ ⎢ 30.6 ⎢ 36.88 ⎣ 10.2 30.6 1∠0° 10.2 | | 83.96 20.4 30.6 36.88 10.2 35.12 1.02∠0° 1.02∠0° | | 82 1 36 1.02 38 0.612 83.96 ∗ 1.02∠0° | | ∗ 36.88 1∠0° 1.02∠0° | | ∗ 1.96 ∗ 82 41.616 | | 10.2 20.808⎤ ⎥ 31.416 ⎥ 30.6 ⎥ 73.44 ⎦ 10.2 ⎡ Δ ⎤ 20.808⎤ ⎢ Δ ⎥ ⎥ 31.416 ⎥ ⎢ Δ ⎥ 30.6 ⎥ ⎢Δ| |⁄| |⎥ 73.44 ⎦ ⎣ Δ| |⁄| | ⎦ 11 The foregoing Jacobian matrix has to be inverted to determine required variables. Since the Jacobian matrix is a 5x5 matrix and beyond the capability of the Casio FX-991ES or FX-991EX calculator, you have to get the inverse matrix using the manual way. However, for your convenience, you can use MS Excel’s matrix inversion functionality to calculate the inverse of the 5x5 Jacobian matrix. Δ ⎡ ⎤ Δ ⎢ ⎥ ⎢ Δ ⎥ ⎢Δ| |⁄| |⎥ ⎣ Δ| |⁄| | ⎦ Δ ⎡ ⎤ Δ ⎢ ⎥ ⎢ Δ ⎥ ⎢Δ| |⁄| |⎥ ⎣ Δ| |⁄| | ⎦ 0.01812 ⎡0.01250 ⎢ ⎢0.01460 ⎢0.00587 ⎣0.00255 0.01786 ⎡ 0.00151 ⎤ ⎢ ⎥ ⎢ 0.00022 ⎥ ⎢ 0.01309⎥ ⎣ 0.00755⎦ 1 1 Δ Δ|| || 0.01240 0.01454 0.02332 0.01885 0.01884 0.02869 0.00024 0.00292 0.00027 0.00597 180° 0.01309 ∠ 0° 1.02∠ 0° 1.02 1 180° 0.00151 0.00771 ∠ 0° , Δ 0.30966 . . ≮ 0.3 . . 0.3 . . 2.8 2.76439 0.03561 . . Δ 0.9 0.89631 0.00369 . . Δ 0.6 0.59143 0.00857. . Δ 1.92 0.00246 ⎤ ⎡ 0.00028 0.7776 ⎤ ⎥⎢ ⎥ 0.00601⎥ ⎢ 0.396 ⎥ 0.00468 ⎥ ⎢ 0.14 ⎥ 0.01329 ⎦ ⎣ 0.312⎦ 0.98691∠ 1.02344° . . ∗ ∗ 1.01229∠0.01279° . . 2.76439 2.08538 . . 0.89631 0.51417 . . 0.59143 0.30966 . . 0.51417 . . ≮ 0.5 . . 0.5 . . 0.3 . . 0.00546 0.00001 0.00265 0.01211 0.00469 180° 0.00022 , Δ Δ 1.02∠0.08656° . . 0.01786 0.5 . . 2.1 2.08538 0.01462 . . Δ 0.5 0.51417 0.01417 Δ 0.3 0.30966 0.00966 . . ∗ 12 Second Iteration ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 81.9529 20.3242 30.1471 37.8282 9.6746 9.4468 19.9342 86.8794 41.2750 10.4546 38.6389 20.7040 29.7858 41.3282 71.4221 10.5308 20.5976 30.1507 32.2994 9.6746 9.4468 77.7821 20.3242 30.1471 10.4546 40.4315 20.7040 19.9342 87.9078 41.2750 10.5308 20.5976⎤ ⎥ 31.3336 ⎥ 29.7858⎥ 41.3282⎥ 72.0414 ⎦ Getting the inverse of the foregoing Jacobian matrix and multiplying this with ∆Ps and ∆Qs to solve for the unknown parameters, we get: Δ ⎡ ⎤ Δ ⎢ ⎥ ⎢ Δ ⎥ | ⁄ | | Δ| ⎢ ⎥ ⎢Δ| |⁄| |⎥ ⎣ Δ| |⁄| | ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0.00024 0.00008 ⎤ ⎥ 0.00011 ⎥ 0.00083⎥ 0.00071⎥ 0.00083⎦ 0.98691 1 1.02 1 1.01229 1 0.5 . . , , Δ 0.00024 0.98609∠ 1.03739° . . 180° 0.00008 1.01927∠0.09110° . . 180° 0.00083 ∠ 0.01279° 0.00011 1.01145∠0.01924 . . 0.00071 ∠ 0.08656° ∗ ∗ 2.79995 2.09999 . . 0.89999 0.50001 . . ∗ 0.59998 0.30001 . . 0.30001 . . ≮ 0.3 . . 0.3 . . 2.8 2.79995 0.00005 . . Δ 0.9 0.89999 0.00001 . . Δ 0.6 0.59998 0.00002. . Δ 180° 1.02344° 0.50001 . . ≮ 0.5 . . 0.5 . . 0.3 . . 0.00083 ∠ 2.1 2.09999 0.00001 . . Δ 0.5 0.50001 0.00001 Δ 0.3 0.30001 0.00001 . . Since all the ∣∆P∣s and ∣∆Q∣s are now less than the convergence criterion ε load flow solution has converged. 0.002, the Newton Raphson 13 On the other hand, using a convergence criterion of ε 0.00001, the Newton Raphson load flow solution will converge in just 3 iterations compared with 10 iterations using Gauss-Seidel load flow solution. In general, Newton-Raphson load flow solution converges faster than the Gauss-Seidel load flow solution. i V1 1 1.03∠0° 2 1.03∠0° 3 1.03∠0° V2 V3 V4 0.98691 ∠–1.02344° 0.98609 ∠–1.03739° 0.98609 ∠–1.03741° 1.02000 ∠0.08656° 1.01927 ∠0.09110° 1.01927 ∠0.09111° 1.01229 ∠0.01279° 1.01145 ∠0.01924° 1.01145 ∠0.01926° The foregoing are some notes with regard to Newton Raphson load flow solution: Advantages fast convergence as long as initial guess is close to the solution large region of convergence Disadvantages each iteration takes much longer than a Gauss-Seidel iteration calculation of Jacobian matrix and subsequently getting the inverse in programming the Newton Raphson solution, it is more complicated to code, particularly when implementing sparse matrix algorithms Newton-Raphson algorithm is very common in power flow analysis some algorithms use a combination of the Gauss-Seidel load flow solution for the first few iterations 2 – 3 and then switches to Newton-Raphson load flow solution PLEASE FEEL FREE TO EMAIL US SHOULD YOU HAVE ANY QUESTIONS OR CLARIFICATIONS. SUGGESTIONS ON HOW TO IMPROVE THIS LECTURE MATERIAL ON LOADFLOW ARE ALSO WELCOME. Prepared by: FORTUNATO C. LEYNES JOHN CARLO G. PERION JOHN KENNETH T. SINSON ...
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