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Power Flow Calculations


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POWER FLOW CALCULATION

Name:____郝攀_______________ Class:____电气 153 班________ ID: _____152247____________

> electric power system

郝攀

152247

POWER FLOW CALCULATION
Abstract: Power flow calculation has been valued as the basis of power system analysis. Flow calculation problems of complex power system are can be included as to solve a set of multivariate nonlinear equations. And we can use the iteration to solve multivariate nonlinear equations. Three approaches will be present to solve the power flow calculation. Through Newton method, fast-decoupled method, DC power flow method, we can get some conclusions which will benefit us on power analysis. Different methods have different characteristics and advantages and shortcomings. Keywords: Power flow calculation; Newton iteration; fast-decoupled; DC method. Introduction: The power flow problem is a very well-known problem in the field of power systems engineering, where voltage magnitudes and angles for one set of buses are desired, given that voltage magnitudes and power levels for another set of buses are known and that a model of the network configuration (unit commitment and circuit topology) is available. The power flow solution contains the voltages and angles at all buses, and from this information, we may compute the active and reactive generation and load levels at all buses and the active and reactive flows across all circuits. Power balance equations and types of buses: When analyzing power systems we know neither the complex bus voltages nor the complex current injections. However, we know the complex power being consumed by the load and the power being injected by the generators plus their voltage magnitudes. Therefore, we can’t directly use the Y-bus equations, but we may use the power balance equations.
=


=1 =

? ? = +

=
=1 =

( + ) ( ? )
=1

= Where,

= + There are three main types of power flow buses: PQ,PV, Slack bus. Load (PQ) at which P/Q are fixed; iteration solves for voltage magnitude and angle. Generator (PV) at which P and V are fixed; iteration solves for voltage angle and Q injection.Slack at which the voltage magnitude and angle are fixed; iteration solves for P/Q injections. Basic principles of three algorithms: Three main methods are used in the project to solve thepower flow calculations of this

electric power system

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152247

24-Yus-network: a full AC Newton approach, a fast-decoupled AC Newton approach and a DC power flow approach. (a) Newton approach If we use theNewton approach, we need to get the Jacobian matrix, and the correction equation: ? = ?
=

? ?

? = ?
=1 =

( + ) ( ? )
=1

? = ? Where,

(b) fast-decoupled Newton approach If we use thefast-decoupled AC Newton approach, we need to get the B’ and B’’ matrix, and the correction equation: ? = ?′? ? = ?′′ When we calculate B’ matrix, we ignore the effect of transformers and the admittance. But we take them into consideration when we calculate the B’’ matrix. (c) DC power flow approach If we use theDC power flow approach, we need to get the B matrix, and the correction equation: ? P = ? = Power of branch: When we get all the voltages and angles of each bus, we should calculate the power flow of each branch. P = U 2 ? U U ( cos + sin ) Q = ?U 2 ( + ( 0 ) + U U ( cos ? sin )

? ? = ? = ? = =

electric power system

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152247

How to cope with the transformer: We use the π equivalent circuit diagram to show the influence of transformers to voltage. If there is a transformer between node I and node j, the ratio is k, and the admittance of 1 side is . Y = ? 1 Y = 1 ? Y = 2 Power system structure and the bus and branch data:

electric power system

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Bus data:

electric power system

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name 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124

V 138 138 138 138 138 138 138 138 138 138 230 230 230 230 230 230 230 230 230 230 230 230 230 230

V.pu phase angle PG 1.035 0 1.035 0 1 0 1 0 1 0 1 0 1.025 0 1 0 1 0 1 0 1 0 1 0 1.02 0 0.98 0 1.014 0 1.017 0 1 0 1.05 0 1 0 1 0 1.05 0 1.05 0 1.05 0 1 0

QG 172 172 0 0 0 0 240 0 0 0 0 0 0 0 215 155 0 400 0 0 400 300 660 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

PL(75%) QL(75%) BL 81 16.5 72.75 15 135 27.75 55.5 11.25 53.25 10.5 102 21 93.75 18.75 128.25 26.25 131.25 27 146.25 30 0 0 0 0 198.75 40.5 145.5 29.25 237.75 48 75 15 0 0 249.75 51 135.75 27.75 96 19.5 0 0 0 0 0 0 0 0

-1

Branch data:

electric power system

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from number from bus to number to bus r pu x pu b pu tr pu 14 101 15 102 0.003 0.014 0.461 0 14 101 1 103 0.055 0.211 0.057 0 14 101 3 105 0.022 0.085 0.023 0 15 102 2 104 0.033 0.127 0.034 0 15 102 4 106 0.05 0.192 0.052 0 1 103 6 109 0.031 0.119 0.032 0 1 103 13 124 0.002 0.084 0 1.015 2 104 6 109 0.027 0.104 0.028 0 3 105 7 110 0.023 0.088 0.024 0 4 106 7 110 0.014 0.061 2.459 0 16 107 5 108 0.016 0.061 0.017 0 5 108 6 109 0.043 0.165 0.045 0 5 108 7 110 0.043 0.165 0.045 0 6 109 8 111 0.002 0.084 0 1.03 6 109 9 112 0.002 0.084 0 1.03 7 110 8 111 0.002 0.084 0 1.015 7 110 9 112 0.002 0.084 0 1.015 8 111 24 113 0.006 0.048 0.1 0 8 111 17 114 0.005 0.042 0.088 0 9 112 24 113 0.006 0.048 0.1 0 9 112 23 123 0.012 0.097 0.203 0 24 113 23 123 0.011 0.087 0.182 0 17 114 19 116 0.005 0.059 0.082 0 18 115 19 116 0.002 0.017 0.036 0 18 115 21 121 0.006 0.049 0.103 0 18 115 21 121 0.006 0.049 0.103 0 18 115 13 124 0.007 0.052 0.109 0 19 116 10 117 0.003 0.026 0.055 0 19 116 11 119 0.003 0.023 0.049 0 10 117 20 118 0.002 0.014 0.03 0 10 117 22 122 0.014 0.105 0.221 0 20 118 21 121 0.003 0.026 0.055 0 20 118 21 121 0.003 0.026 0.055 0 11 119 12 120 0.005 0.04 0.083 0 11 119 12 120 0.005 0.04 0.083 0 12 120 23 123 0.003 0.022 0.046 0 12 120 23 123 0.003 0.022 0.046 0 21 121 22 122 0.009 0.068 0.142 0

Main results:

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As we can see from the table, it’s the voltages and angles of each bus, and also the active and reactive power of the generators and the load.
name 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 V(75%) 142.83 142.83 133.7678 136.0178 140.2702 139.6057 141.45 136.7024 133.9526 140.5927 230.4001 233.9745 234.6 225.4 233.22 233.91 238.6004 241.5 235.6187 239.1135 241.5 241.5 241.5 226.6016 V.pu(75%) phase angle(75%) PG 1.035 6.035611746 172 1.035 5.892936567 172 0.96933182 8.340466866 0 0.9856363 2.444460421 0 1.01645055 2.24960373 0 1.01163531 -0.641711335 0 1.025 7.355011853 240 0.99059706 2.543733801 0 0.97067092 2.758334708 0 1.01878802 0.760127628 0 1.0017394 4.176676813 0 1.01728047 4.396987 0 1.02 0 -500.326 0.98 10.52256188 0 1.014 26.11512934 215 1.017 24.13694858 155 1.03739311 29.53145882 0 1.05 31.28533797 400 1.02442914 20.70582305 0 1.03962377 19.23160828 0 1.05 31.94892388 400 1.05 37.51327736 300 1.05 18.98419433 660 0.98522446 19.61058554 0 QG PL(75%) QL(75%) Pin(75%) Qin(75%) 22.70809 81 16.5 91 6.208089514 16.89291 72.75 15 99.25 1.892905922 0 135 27.75 -135 -27.75 0 55.5 11.25 -55.5 -11.25 0 53.25 10.5 -53.25 -10.5 0 102 21 -102 -21 43.16971 93.75 18.75 146.25 24.41970598 0 128.25 26.25 -128.25 -26.25 0 131.25 27 -131.25 -27 0 146.25 30 -146.25 -30 0 0 0 0 0 0 0 0 0 0 196.9577 198.75 40.5 -699.0762577 156.457665 -25.5708 145.5 29.25 -145.5 -54.8207876 -41.2751 237.75 48 -22.75 -89.2751082 49.44755 75 15 80 34.44755425 0 0 0 0 0 117.9226 249.75 51 150.25 66.92255331 0 135.75 27.75 -135.75 -27.75 0 96 19.5 -96 -19.5 112.9798 0 0 400 112.9797791 -29.5612 0 0 300 -29.5611503 164.5283 0 0 660 164.5283446 0 0 0 0 0

This is a comparison between the 3 methods. As we can see from the results of MATLAB code, the Newton approach has the same accuracy with the PQ approach.
number name V.pu(nr-pq)75% phase angle(nr-pq)75% phase angle (DC)75% 1 103 0.969331819 8.340466866 9.707419702 2 104 0.985636303 2.444460421 3.643887235 3 105 1.016450553 2.24960373 3.52999014 4 106 1.011635307 -0.641711335 0.361386509 5 108 0.990597065 2.543733801 3.664485167 6 109 0.97067092 2.758334708 3.739420402 7 110 1.018788023 0.760127628 1.772294484 8 111 1.001739402 4.176676813 4.946532316 9 112 1.017280474 4.396987 5.013597013 10 117 1.037393107 29.53145882 31.61760042 11 119 1.024429137 20.70582305 22.23519434 12 120 1.03962377 19.23160828 20.77858852 13 124 0.985224459 19.61058554 20.79522229 14 101 1.035 6.035611746 7.99274224 15 102 1.035 5.892936567 7.838378181 16 107 1.025 7.355011853 9.127649484 17 114 0.98 10.52256188 11.55655211 18 115 1.014 26.11512934 27.77952313 19 116 1.017 24.13694858 25.73193343 20 118 1.05 31.28533797 33.62318152 21 121 1.05 31.94892388 34.33852925 22 122 1.05 37.51327736 40.48827127 23 123 1.05 18.98419433 20.59140676 24 113 1.02 0 0

electric power system

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from number to number Pij (nr-pq) Pij (DC) 1 6 72.67180609 81.9679094 1 13 -218.701144 -230.24883 1 14 10.63597311 13.2809235 2 6 -1.268270459 -1.5020035 2 15 -54.23172954 -53.997996 3 7 28.06874084 32.6317748 3 14 -81.31874084 -85.881775 4 7 -41.74195433 -38.348848 4 15 -60.25804567 -63.651152 5 6 0.876449427 -0.742237 5 7 13.76862608 18.742237 5 16 -142.8950755 -146.25 6 1 -70.84203982 -81.967909 6 2 1.327362698 1.5020035 6 5 -0.815731125 0.742237 6 8 -28.62337171 -25.066834 6 9 -33.85105165 -26.459497 7 3 -27.87267138 -32.631775 7 4 41.98074176 38.3488477 7 5 -13.50645546 -18.742237 7 8 -70.7589021 -65.916087 7 9 -76.96176963 -67.308749 8 6 28.66630686 25.0668342 8 7 70.86826527 65.9160867 8 17 -246.9747353 -270.84391 8 24 148.657924 0 9 6 33.93301426 26.4594968 9 7 77.07833837 67.3087493 9 23 -273.011287 -276.06778 9 24 163.2557731 0 10 19 387.7854191 389.902329 10 20 -245.9352531 -245.02796 10 22 -141.850166 -144.87437 11 12 125.5295785 125.157256 11 19 -261.2795785 -260.90726 12 11 -124.953633 -125.15726 12 23 28.95363298 29.1572556 13 1 219.7361573 230.248833 13 18 -219.3357935 -230.24883 14 1 -9.949357981 -13.280924 14 3 82.72735669 85.8817748

electric power system

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152247

(with the above)
14 15 15 15 16 17 17 18 18 18 19 19 19 19 20 20 21 21 21 22 22 23 23 23 24 24 24 15 2 4 14 5 8 19 13 19 21 10 11 17 18 10 21 18 20 22 10 21 9 12 24 8 9 23 18.22200129 55.40681546 62.05546509 -18.21228055 146.25 250.4691143 -395.9691143 222.8166303 204.8797674 -450.4463977 -383.5103654 263.3931185 404.1518654 -204.0346185 247.1147804 -96.86478038 456.5149958 96.99431429 -153.5093101 144.503203 155.496797 281.7959114 -28.79782648 407.0019151 -147.180824 -161.6877453 -390.2076884 18.3991488 53.9979965 63.6511523 -18.399149 146.25 270.843911 -416.34391 230.248833 207.348838 -460.34767 -389.90233 260.907256 416.343911 -207.34884 245.027956 -94.777956 460.347671 94.7779564 -155.12563 144.874373 155.125627 276.067778 -29.157256 0 0 0 0

electric power system

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This is the power of each branch:
ID# 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 from number to number Pij(75%) 1 6 72.67180609 1 13 -218.701144 1 14 10.63597311 2 6 -1.268270459 2 15 -54.23172954 3 7 28.06874084 3 14 -81.31874084 4 7 -41.74195433 4 15 -60.25804567 5 6 0.876449427 5 7 13.76862608 5 16 -142.8950755 6 1 -70.84203982 6 2 1.327362698 6 5 -0.815731125 6 8 -28.62337171 6 9 -33.85105165 7 3 -27.87267138 7 4 41.98074176 7 5 -13.50645546 7 8 -70.7589021 7 9 -76.96176963 8 6 28.66630686 8 7 70.86826527 8 17 -246.9747353 8 24 148.657924 9 6 33.93301426 9 7 77.07833837 9 23 -273.011287 9 24 163.2557731 10 19 387.7854191 10 20 -245.9352531 10 22 -141.850166 11 12 125.5295785 11 19 -261.2795785 12 11 -124.953633 12 23 28.95363298 13 1 219.7361573 13 18 -219.3357935 14 1 -9.949357981 Qij(75%) -17.7759141 25.25942475 -35.2335106 13.16609583 -24.4160958 -10.8782589 0.378258919 -127.604162 4.26356214 9.530691035 -22.4245133 -13.3561778 21.78896566 -15.6176346 -13.6255425 -1.17853685 -18.3672518 9.143116785 -124.79655 18.88729075 42.4341277 24.33201441 1.871063928 -36.9744239 91.15871771 -56.0553577 19.64322641 -19.1132461 24.45246528 -24.9824455 51.63608169 -56.252147 4.616065311 -100.468211 72.71821112 87.39451224 -106.894512 17.91717463 -17.9171747 32.13675281

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(with the above) 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 ploss_sum

14 14 15 15 15 16 17 17 18 18 18 19 19 19 19 20 20 21 21 21 22 22 23 23 23 24 24 24

3 15 2 4 14 5 8 19 13 19 21 10 11 17 18 10 21 18 20 22 10 21 9 12 24 8 9 23

82.72735669 18.22200129 55.40681546 62.05546509 -18.21228055 146.25 250.4691143 -395.9691143 222.8166303 204.8797674 -450.4463977 -383.5103654 263.3931185 404.1518654 -204.0346185 247.1147804 -96.86478038 456.5149958 96.99431429 -153.5093101 144.503203 155.496797 281.7959114 -28.79782648 407.0019151 -147.180824 -161.6877453 -390.2076884 76.23045261

2.644064274 -28.5727276 25.46579952 -2.80751214 -20.7653815 24.41970598 -70.4470142 15.6262266 32.88101509 -40.2332457 -81.9228776 -20.3894188 -61.6195773 72.75201223 43.70453817 61.24081117 5.681742142 109.5369267 -16.6866149 20.12946732 -8.79275055 -20.7683998 24.86238269 97.99383234 41.67212956 57.65274876 27.15037048 71.65454575

Conclusions: From the results of the project, we can draw a lot of conclusions: First, the Newton approach has the same accuracy of voltage magnitudes, voltage angles, active flows, and reactive flows with the fast-decoupled AC Newton approach. And the results from the DC power flow approach is a little different from Newton approach and fast-decoupled AC Newton approach. Second, as we can see from the table1, the PQ-method has more iterations than the NR-method, but it is faster than the NR-method. The DC-method is the fastest from the three methods. If we want to have a fast calculation and we can allow some error, we must turn tothe DC-method first especially when the network has many buses. It’s because we don’t take the to-ground capacitors and the transformers into consideration. Third, the parameter of BL has a great influence on the voltage phasors.

electric power system

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Table 1 Accuracy(1e-7) NR-method PQ-method DC-method Iterations 5 12 1 Time 0.051502s 0.011400s 0.007457s All the results is given in the attachment. It includes all input and outputdata of buses and branches and a comparison table(in DC-method) with voltage phasors and all circuit flows (active & reactive power) for all three ways of solving the network. To make a reference, we also give the results under a 100% load. (When you run the program you can change the pq to pq1 and p to p1 to get results undera 100% load.) Attachment 1:NR-PQ(75%) Attachment 2:DC(75%)(with a comparison) Attachment 3:NR-PQ(100%) Attachment 4:DC(100%)(with a comparison)


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