2.11.1
Example for Fast-decoupled load-flow technique
As an example, the 5 bus system described earlier is considered again. Starting from the flat start,
the initial calculations are shown in Table 2.29. As the mismatch is more than the tolerance, the
algorithm proceeds. The ȲBUS matrix of this system is shown in Table 2.30. In this table, the
notation ȲBUS (∶, m ∶ n) represents the elements (of the ȲBUS matrix) corresponding to all rows
(denoted by the notation ‘:’) and columns spanning from mth column to nth column (denoted by the
′
′′
notation ‘m:n’). From this ȲBUS matrix, the matrices [B ] and [B ] are constructed as shown in
′
Table 2.30. Note that the size of the matrix [B ] is (4 × 4) (corresponding to the non slack buses,
′′
i.e. 2, 3, 4 and 5) and the size of the [B ] matrix is (2 × 2) (corresponding to the PV buses 4
′
′′
and 5). Also note that the matrices [B ] and [B ] have been formed by taking the negative of the
imaginary parts of the corresponding elements of the ȲBUS matrix. With these constant matrices,
the vectors ∆θ and ∆V are calculated and subsequently, the vectors θ and V have been updated.
With these updated values of θ and V, the final mismatch (error) is again calculated as shown in
Table 2.30. As the error is still more than the tolerance, the algorithm proceeds further and finally
converges in 19 iterations for a tolerance value of 10−12 p.u. The final solution is shown in Table
2.31, which happens to be the same as the results obtained by the other methods described earlier.
Table 2.29: Initial calculation with FDLF in the 5 bus system
Pcal = [−0.0444 −0.1776 −0.0333 −0.0444] × 10−14 ;
T
T
Qcal = [−0.1430 −0.1430] ;
T
∆M = [0.5000 1.0000 −1.1500 −0.8500 −0.4570 −0.2570] ;
error = 1.15 ;
Now, let us impose the reactive power limit on the generation at bus 3. Starting from the flat
start, the calculation up to 1st iteration are same and hence are not shown here. The calculation
pertaining to 2nd iteration are shown in Table 2.32. From this set, observe that the calculated value
of Q3 has exceeded the limit of 50 MVAR and hence it should be now treated as PQ-bus. Thus,
′′
the dimension of the matrix [B ] increases to (3 × 3) (corresponding to the buses 4, 5 and 3).
′
′′
′
This new, augmented matrix is shown in Table 2.33. With these [B ] and [B ] matrices ([B ]
matrix remains the same), the calculations are further carried out and the algorithm converges in
20 iterations. The final solution is shown in Table 2.31. Comparison of this result with the earlier
results (obtained with other methods) shows that because of the approximations made in FDLF, the
results obtained with FDLF are not identically the same with those obtained by the other methods
but are very close.
The results corresponding to 14-bus and 30-bus systems are shown in Tables 2.34 and 2.35
respectively. From these tables note that the number of iterations taken by FDLF is much higher than
those required by NRLF. This is because of approximations adopted by FDLF (to achieve decoupling)
due to which, the convergence is slower. However, because of the constant Jacobrian matrices, the
68
Table 2.30: Calculations at 1st iteration with FDLF in the 5 bus system
⎤
⎡ 3.2417 − 13.0138i −1.4006 + 5.6022i
0
⎥
⎢
⎥
⎢
⎢−1.4006 + 5.6022i 3.2417 − 13.0138i −1.8412 + 7.4835i⎥
⎥
⎢
⎥
⎢
⎢
0
−1.8412 + 7.4835i 4.2294 − 18.9271i ⎥⎥;
ȲBUS (∶, 1 ∶ 3) = ⎢
⎥
⎢
⎥
⎢
0
0
−1.2584
+
7.1309i
⎥
⎢
⎥
⎢
⎥
⎢−1.8412 + 7.4835i
0
−1.1298
+
4.4768i
⎦
⎣
⎡
0
−1.8412 + 7.4835i⎤⎥
⎢
⎥
⎢
⎥
⎢
0
0
⎥
⎢
⎥
⎢
ȲBUS (∶, 4 ∶ 5) = ⎢⎢−1.2584 + 7.1309i −1.1298 + 4.4768i⎥⎥;
⎥
⎢
⎢ 2.1921 − 10.7227i −0.9337 + 3.7348i⎥
⎥
⎢
⎥
⎢
⎢−0.9337 + 3.7348i 3.9047 − 15.5521i ⎥
⎦
⎣
⎡ 13.0138 −7.4835
0
0 ⎤⎥
⎢
⎢
⎥
⎢−7.4835 18.9271 −7.1309 −4.4768⎥
10.7227 −3.7348
′
′′
⎢
⎥
⎥ ; [B ] = [
[B ] = ⎢
];
⎢ 0
−7.1309 10.7227 −3.7348⎥⎥
−3.7348 15.5521
⎢
⎢
⎥
⎢ 0
−4.4768 −3.7348 15.5521 ⎥⎦
⎣
T
∆θ = [0.0306 −0.0136 −0.1492 −0.0944] ;
T
∆V = [−0.0528 −0.0292] ;
T
θ = [0 0.0306 −0.0136 −0.1492 −0.0944] ;
T
V = [1.0 1.0 1.0 0.9472 0.9708] ;
T
Pcal = [0.5045 1.0488 −1.1725 −0.8975] ;
T
Qcal = [−0.2931 −0.1267] ;
T
∆M = [−0.0045 −0.0488 0.0225 0.0475 −0.3069 −0.2733] ;
error = 0.3069;
execution of each iteration is much faster (as the Jacobrian matrix need not be recomputed and
reversed at each iteration) and hence, the total time taken by FDLF is quiet comparable to that
needed by NRLF. From the two tables it is further noted that, in the absence of any generation
reactive power limit, the results obtained by FDLF are almost identical to those obtained by other
methods. However, in the presence of generation reactive power limits, FDLF results are quiet close
to the results obtained by other methods, though not identical.
We are now at the end of our discussion of AC load flow techniques. In the next lecture, we will
study the method of load flow analysis of an AC system in which a HVDC link is also embedded
(namely, the AC-DC load flow technique).
69
Table 2.31: Final Results of the 5 bus system with FDLF
Bus no.
1
2
3
4
5
∣V ∣
Without generator Q limit
θ
Pinj
∣V ∣
Qinj
With generator Q limit
θ
Pinj
Qinj
(p.u)
(deg)
(p.u)
(p.u)
(p.u)
(deg)
(p.u)
(p.u)
1.0
0
0.56743 0.26505
1.0
0
0.56985 0.34069
1.0
1.65757
0.5
-0.18519
1.0
1.69742
0.5
-0.04522
1.0
-0.91206
1.0
0.68875 0.98219 -0.63507
1.0
0.49668
0.90594 -8.35088
-1.15
-0.6
0.88888 -8.35938
-1.15
-0.6
0.94397 -5.02735
-0.85
-0.4
0.93428 -4.9861
-0.85
-0.4
Total iteration = 19
Total iteration = 20
Table 2.32: Calculations at 2nd iteration with FDLF in the 5 bus system with limit on Q3
T
∆θ = [−0.0010 −0.0012 0.0026 0.0034] ;
T
∆V = [−0.0399 −0.0277] ;
T
θ = [0 0.0296 −0.0148 −0.1465 −0.0910] ;
T
V = [1.0 1.0 1.0 0.9074 0.9431] ;
T
Pcal = [0.4999 1.0353 −1.1526 −0.9012] ;
T
Qcal = [−0.5830 −0.4020 0.6775] ;
T
∆M = [0.0001 −0.0353 0.0026 0.0512 −0.0170 0.0020 −0.1775] ;
error = 0.1775;
70
Table 2.33: Calculations at 3rd iteration with FDLF in the 5 bus system with limit on Q3
⎡ 13.0138
⎢
⎢
⎢−7.4835
′
⎢
[B ] = ⎢
⎢ 0
⎢
⎢
⎢ 0
⎣
−7.4835
0
0 ⎤⎥
⎥
18.9271 −7.1309 −4.4768⎥⎥
⎥;
−7.1309 10.7227 −3.7348⎥⎥
⎥
−4.4768 −3.7348 15.5521 ⎥⎦
T
∆θ = [−0.0006 −0.0010 0.0008 0.0034] ;
⎡ 10.7227 −3.7348 −7.1309⎤
⎥
⎢
⎥
⎢
′′
⎢
[B ] = ⎢−3.7348 15.5521 −4.4768⎥⎥;
⎥
⎢
⎢−7.1309 −4.4768 18.9271 ⎥
⎦
⎣
T
∆V = [−0.0167 −0.0090 −0.0178] ;
T
θ = [0 0.0290 −0.0157 −0.1457 −0.0875] ;
T
V = [1.0 1.0 0.9822 0.8906 0.9341] ;
T
T
Qcal = [−0.5939 −0.4125] ;
Pcal = [0.5268 0.9223 −1.1174 −0.8408] ;
T
∆M = [−0.0268 0.0777 −0.0326 −0.0092 −0.0061 0.0125] ;
error = 0.0777;
Table 2.34: Final Results of the 14 bus system with FDLF
Without generator Q limit
Bus no.
∣V ∣
1
2
3
4
5
6
7
8
9
10
11
12
13
14
(p.u)
1.06
1.045
1.04932
1.03299
1.04015
1.07
1.02076
1.0224
1.0201
1.0211
1.04144
1.0526
1.04494
1.01249
θ
Pinj
(deg)
(p.u)
0
2.37259
-5.17113
0.183
-14.54246
-1.19
-10.39269 -0.4779
-8.76418 -0.07599
-12.52265
0.112
-13.44781
0
-13.47154
0
-13.60908 -0.29499
-13.69541
-0.09
-13.22158 -0.03501
-13.42868 -0.06099
-13.50388 -0.135
-14.60128 -0.14901
Total iteration = 123
Qinj
∣V ∣
(p.u)
-0.3308
-0.166
-0.08762
-0.039
-0.01599
0.37278
0
-0.129
-0.16599
-0.05799
-0.018
-0.01599
-0.05799
-0.05001
(p.u)
1.06
1.045
1.04697
1.02902
1.03615
1.05497
1.01266
1.01391
1.0118
1.01154
1.02915
1.03787
1.03063
1.00136
71
With generator Q limit
θ
Pinj
(deg)
(p.u)
0
2.37188
-5.17845
0.183
-14.55556
-1.19
-10.35987 -0.4779
-8.71026 -0.07599
-12.4587
0.112
-13.49478
0
-13.5185
0
-13.66102 -0.29499
-13.73679
-0.09
-13.21814 -0.03501
-13.39144 -0.06099
-13.48166 -0.135
-14.64504 -0.14901
Total iteration = 119
Qinj
(p.u)
-0.31249
-0.1066
-0.08762
-0.039
-0.01599
0.29999
0
-0.129
-0.16599
-0.05799
-0.018
-0.01599
-0.05799
-0.05001
Table 2.35: Final Results of the 30 bus system with FDLF
Without generator Q limit
Bus no.
∣V ∣
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
(p.u)
1.05
1.0338
1.03128
1.02578
1.005 8
1.02178
1.00111
1.023
1.04608
1.03606
1.0913
1.04859
1.0883
1.03346
1.02825
1.0359
1.0306
1.01873
1.01626
1.02041
1.02305
1.02343
1.0165
1.00939
1.00048
0.9825
1.00379
1.02049
0.98353
0.97181
θ
Pinj
(deg)
(p.u)
0
2.38673
-4.97945 0.3586
-7.96653
-0.024
-9.58235
-0.076
-13.60103 -0.6964
-11.50296
0
-13.9994
-0.628
-12.56853
-0.45
-13.04088
0
-14.88589 -0.058
-11.16876 0.1793
-13.74947 -0.112
-12.56078 0.1691
-14.71704 -0.062
-14.86737 -0.082
-14.50539 -0.035
-14.98291
-0.09
-15.58107 -0.032
-15.81066 -0.095
-15.63819 -0.022
-15.35955 -0.175
-15.35222
0
-15.41998 -0.032
-15.81043 -0.087
-15.84004
0
-16.27422 -0.035
-15.59587
0
-12.1474
0
-16.87497 -0.024
-17.79427 -0.106
Total iteration = 115
Qinj
∣V ∣
(p.u)
-0.29842
-0.05698
-0.012
-0.016
0.05042
0
-0.109
0.12343
0
-0.02
0.24018
-0.075
0.31043
-0.016
-0.025
-0.018
-0.058
-0.009
-0.034
-0.007
-0.112
0
-0.016
-0.067
0
-0.023
0
0
-0.009
-0.019
(p.u)
1.05
1.0338
1.03026
1.02455
1.0058
1.0206
1.00041
1.023
1.03842
1.02995
1.07426
1.04384
1.0824
1.02853
1.0232
1.03054
1.02469
1.01323
1.01052
1.01457
1.01706
1.0175
1.01136
1.00418
0.99677
0.97872
1.00105
1.01937
0.98073
0.96897
72
With generator Q limit
θ
Pinj
(deg)
(p.u)
0
2.38678
-4.98182 0.3586
-7.95403
-0.024
-9.56795
-0.076
-13.6111 -0.6964
-11.49172
0
-13.9986
-0.628
-12.5786
-0.45
-13.02612
0
-14.88705 -0.058
-11.11026 0.1793
-13.76458 -0.112
-12.56395 0.1691
-14.73984 -0.062
-14.88791 -0.082
-14.51633 -0.035
-14.98908
-0.09
-15.60231 -0.032
-15.83038 -0.095
-15.65374 -0.022
-15.36723 -0.175
-15.36018
0
-15.44311 -0.032
-15.83356 -0.087
-15.87684
0
-16.31432 -0.035
-15.63737
0
-12.14228
0
-16.92364 -0.024
-17.84827 -0.106
Total iteration = 112
Qinj
(p.u)
-0.29289
-0.0429
-0.012
-0.016
0.05654
0
-0.109
0.15797
0
-0.02
0.18809
-0.075
0.29991
-0.016
-0.025
-0.018
-0.058
-0.009
-0.034
-0.007
-0.112
0
-0.016
-0.067
0
-0.023
0
0
-0.009
-0.019