Power system contingency analysis using complex valued neural networks.

Chary, D. Venu Madhava ; Amarnath, J.

Introduction

Contingency evaluation is one of the important tasks encountered by planning and operation engineers. Static security assessment of a power system enables to detect line flow violation or an out of limit voltage following a given list of contingencies. Contingency analysis plays an important role in real time power system security evaluation. This analysis comprises the simulation of a set of contingencies in which the system behavior is observed. Each post contingency scenario is evaluated in order to detect operational problems. These contingency cases which are potentially harmful to the system must be identified and detailed analysis is carried out, which is known as contingency selection. Finding these contingencies and determining the corrective action often involve exhaustive load flow calculations.

Large power systems require the analysis of all credible contingencies with in a very short time. There exist many fast approximate methods for contingency evaluation. The popular DC load flow method yield only real power loading and ignore the post contingency effects on bus voltage and reactive power. Peterson et al [1] introduced an iterable linear AC power flow solution for a fast approximate outage study. An Artificial neural networks attracted many researchers from energy system side to seek solutions to some of the complex problems [2]. This survey indicates that ANNs may be appropriate for assisting dispatchers in operating electric power systems. A.U Narendranath et al [3] proposed an ANN based voltage stability assessment of network in an energy control centre. J.A Refaee [4] presented RBFN model for contingency evaluation of bulk power system which uses two separate ANNs one for line flows and the other for bus voltage magnitude. The line flow neural network is trained to map the Y bus matrix, bus load and generator injections to the line flow while the bus voltage maps same inputs to the bus voltage, but the estimation of real and reactive power under different outages are not considered. D.Devaraj et al [5] proposed RBFN for fast contingency ranking in which the severity levels for each contingency are estimated focusing on single line outages. However most of the neural networks have been developed using real numbers. One of the challenges in designing a complex artificial neural network is to process complex valued inputs. In recent years complex valued neural networks have attracted considerable attention in the applications related to electrical power engineering, signal processing, image and vision processing. L.Chan et al [6] described a complex artificial neural network to estimate the bus voltages in a load flow problem in this paper the complex power (real and reactive powers in complex form) are used as inputs to the network the results obtained were encouraging. In this paper a new approach using complex valued neural network (CVNN) is introduced for contingency analysis in planning studies. The proposed CVNN is implemented to estimate post outage voltages, line flows in complex form.

Complex Valued Neural Network

A complex valued neural network uses a complex valued input, threshold and activation function. The need for such neural network is wide spread, when using existing method for real number we must apply the method individually to their real part and imaginary parts. On the other hand, complex valued neural networks allow us to directly process data. Moreover complex valued neural networks allow us to automatically capture good relational behavior of complex numbers. In electrical engineering the signal are i.e. admittance, powers etc are complex valued. For processing such signals design and implementation of new complex valued neural network architectures are required. Complex valued neural network have attracted considerable attention in different applications in the past decade. The conventional Back Propagation algorithm has been widely recognized as an effective method for training feed forward neural network. The complex-BP algorithm is a complex version of Real-BP In this proposed method, which was proposed by several researchers independently in the early 1990's.[7] complex valued version of the real BP i.e. complex back propagation is used. The inherent advantages of using CVNN are superior performance on operations and computations of complex numbers when compared with conventional real counter parts, lesser number of neurons there by faster convergence rates are obtained. Unlike real Back propagation, the complex algorithm does not suffer from local minima problem. A three layered complex valued neural network is shown in fig. 1

[FIGURE 1 OMITTED]

In this complex valued neural network

I                 number of input layer neurons
J                 number of hidden layer neurons
K                 number of output layer neurons
[x.sub.i]         output value of input neuron i
[h.sub.j]         output of hidden neuron j
[O.sub.k]         output of output neuron k
[w.sub.ji]        weight between input neuron i and
                  hidden layer neuron j
[v.sub.kj]        weight between hidden layer neuron i
                  and output layer neuron j
[theta]j and      are the thresholds at hidden and
[[gamma].sub.k]   output layers

The network is trained by using complex back propagation algorithm learning rule using steepest descent method given by Nitta [8]. The weights are initialized to small complex values. The error between actual output and desired output is calculated to update the weights using back propagation. For these new values of weights outputs are calculated again. Foe each data pair to be learned a forward pass and a backward pass is performed. This is repeated until the error becomes less than the minimum defined.

The internal potential of hidden neuron j

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

The output of hidden neuron j is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

Internal potential of output neuron k is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

At output layer

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

Error

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

The learning rule for complex--BP model is as follows. The weights are modified according to the following equations.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)

The error signal terms of output layer

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7) Output layer weights are adjusted:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

The error signal terms of hidden layer

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

Hidden layer weights are adjusted:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

Test Results

To demonstrate its suitability, complex valued neural network is employed to calculate real and reactive line flows and bus voltages in complex form. A 6-bus, 11 line system shown in fig.2 is tested using the proposed CVNN. This method can also be applied to a large scale power system. The system is divided in to subsystems according to geographical regions and each one can be handled separately by using sub problem approach. A bulk power system requires extensive studies to be carried out according to number of contingencies, their combinations, load and generation margins. A limited set of training instances should be selected to cover all these boundary conditions.

[FIGURE 2 OMITTED]

The training and test patterns are created using standard AC load flow algorithm. Loads are varied from 70% and 120 %of the base case values. A total of 48 load flow patterns were generated using different line outages. The network has 21 input neurons, 80 hidden neurons and 25 output neurons in complex form. The inputs represent the lower or upper triangle elements of complex Y bus matrix and loads in complex form at the load buses to represent a system with fluctuating loads. After 28000 iterations the mean squared error was 0.2. All the outputs are in per unit. The comparisons of variations in real and imaginary components of bus voltages are represented graphically in fig 3. Some sample graphs are also shown to compare the real and reactive line flows in fig 4.

The x-coordinates represent different contingencies as specified in table 1. Tables 2 -6 represent the outputs of the proposed method in comparison with Newton Raphson method.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Conclusion

In this paper we propose application of complex valued neural network with complex back propagation algorithm to contingency analysis.. To demonstrate the superiority a numerical example using 6-bus test system is considered for evaluation. To generate 48 training samples Newton rapphson method is used. A test pattern of 12 samples is used to evaluate the complex bus voltages and line flows. The results show that CVNN can be used for contingency evaluation purpose. It is concluded that CVNN is superior to the conventional real ANN as it requires less number of input neurons, the local minima problem does not arise during learning and it can manipulate complex data efficiently.

Acknowledgement

The authors would like to thank JNTUH, Hyderabad for providing the required facilities and also MVSREC, Hyderabad for the support.

References

[1] Peterson N.M., Tinney W.F and Bree D.W, 1972 " Iterative linear AC power flow solution for fast approximation outage studies," IEEE Trans. Power appar. & sys, PAS -91, pp. 2048-2056.

[2] R.C Bansal., 2006, "Overview and literature survey of artificial neural networks applications to power systems (1992-2004)," IE (I), Vol. 86 pp. 282-296.

[3] A U Narendranath., G K Purushothama., K.Partha sarathy and D.Thukaram, 2002 "An ANN based approach for voltage stability assessment" IE(I), Vol.83, pp. 33-43.

[4] J A Refaee., M.Mohandas and H. Maghrabi., 1999, "Radial basis function networks for contingency analysis of bulk power systems," IEEE Trans. on Power Systems, Vol. 14, pp. 772-778.

[5] D. Devaraj., J. Preetha Roselyn., R.Uma rani., 2007, "Artificial neural network model for voltage security based contingency ranking", Applied soft computing, Vol. 7, pp. 722-727.

[6] L.Chan., A.T.P.So and L.L.Lai, 2000, "Initial applications of complex artificial neural networks to load flow analysis," IEE Proc. Gener. Transm. Distrib., Vol.147, pp.361-366.

[7] M.S. Kim and C.C. Guest, 1990, "Modification of back propagation for complex valued signal processing in frequency domain," Proc. Internat. Joint Conf. Neural Networks, Vol. 3 pp.27-31.

[8] Akira Horose., 2004, Complex Valued Neural Networks, World Scientific publishing company.

(1) D. Venu Madhava Chary and (2) J. Amarnath

(1) Dept. of Electrical and Electronics Engineering M V S R Engineering College Hyderabad-501510, India. E-mail: dvm_chary@yahoo.com

(2) Dept. of Electrical and Electronics Engg., J.N.T.U.H College of Engineering, Hyderabad-85, India. E-mail: amarnathjinka@yahoo.com

Table 1: Contingencies

1    Base case
2    Line--1 out
3    Line--2 out
4    Line--3 out
5    Line--4 out
6    Line--5 out
7    Line--6 out
8    Line--7 out
9    Line--8 out
10   Line--9 out
11   Line--10 out
12   Line--11 out

Table 2: Bus Voltages

Outage Condition   BUS VOLTAGE (p.u)

                       Bus--4           Bus--5           Bus--6

Base case NR       0.9835-0.0775i   0.9770-0.0917i   0.9861-0.1040i
CVNN               0.9677-0.0805i   0.9792-0.0969i   0.9977-0.1117i

Line 1             0.9763-0.1134i   0.9676-0.1297i   0.9770-0.1546i
                   0.9748-0.1190i   0.9744-0.1369i   0.9905-0.1639i

Line 2             0.9914-0.1655i   0.9876-0.1494i   0.9928-0.1731i
                   0.9197-0.1738i   0.9642-0.1573i   0.9861-0.1834i

Line 3             0.9764-0.1073i   0.9658-0.1618i   0.9747-0.1656i
                   0.9811-0.1081i   0.9454-0.1611i   0.9853-0.1647i

Line 4             0.9759-0.0731i   0.9790-0.0923i   0.9870-0.1078i
                   0.9864-0.0721i   0.9807-0.0919i   0.9980-0.1065i

Line 5             0.8929-0.0888i   0.9539-0.0779i   0.9774-0.0803i
                   0.8876-0.0887i   0.9668-0.0783i   0.9977-0.0812i

Line 6             0.9778-0.0698i   0.9701-0.0953i   0.9845-0.1048i
                   0.9835-0.0705i   0.9581-0.0969i   0.9943-0.1065i

Line 7             0.9893-0.0710i   0.9833-0.1021i   0.9922-0.1389i
                   0.9855-0.0707i   0.9757-0.1023i   0.9749-0.1405i

Line 8             0.9900-0.0692i   0.9838-0.0906i   0.9918-0.0903i
                   0.9815-0.0704i   0.9496-0.0911i   0.9949-0.0906i

Line 9             0.9620-0.0765i   0.9377-0.0849i   0.8991-0.1246i
                   0.9820-0.0762i   0.9532-0.0852i   0.8811-0.1253i

Line 10            0.9829-0.0687i   0.9752-0.0929i   0.9850-0.1055i
                   0.9846-0.0689i   0.9775-0.0940i   0.9983-0.1065i

Line 11            0.9842-0.0693i   0.9764-0.0869i   0.9854-0.1106i
                   0.9853-0.0726i   0.9740-0.0872i   1.0010-0.1076i

Tables 3: Real and Reactive line flows

Line flows                     Base Case

                       NR                 CVNN

Line flow 1-2    0.3266 - 0.1655i    0.3240 - 0.1700i
Line flow 1-4    0.4530 + 0.2459i    0.4880 + 0.2390i
Line flow 1-5    0.3459 + 0.1382i    0.3780 + 0.1140i
Line flow 2-1   -0.3129 + 0.1521i    0.3130 + 0.1490i
Line flow 2-3    0.0545 - 0.1142i    0.0270 - 0.1220i
Line flow 2-4    0.3343 + 0.5315i    0.3740 + 0.5330i
Line flow 2-5    0.1839 + 0.1890i    0.1510 + 0.1600i
Line flow 2-6    0.2456 + 0.1562i    0.2600 + 0.1260i
Line flow 3-2   -0.0543 + 0.2281i    0.0270 + 0.5700i
Line flow 3-5    0.1875 + 0.2448i    0.1890 + 0.2380i
Line flow 3-6    0.4643 + 0.1739i    0.4380 - 0.6100i
Line flow 4-1   -0.4409 - 0.2164i   -0.4740 - 0.2240i
Line flow 4-2   -0.3716 - 0.5155i   -0.3540 - 0.5140i
Line flow 4-5    0.0190 - 0.0516i    0.0280 - 0.0610i
Line flow 5-1   -0.3507 - 0.1454i   -0.3660 - 0.1310i
Line flow 5-2   -0.1741 - 0.2122i   -0.1460 - 0.1860i
Line flow 5-3   -0.1685 - 0.2689i   -0.1780 - 0.2670i
Line flow 5-4   -0.0139 - 0.0303i    0.0280 - 0.0150i
Line flow 5-6    0.0153 - 0.0903i   -0.0180 - 0.1010i
Line flow 6-2   -0.1917 - 0.1935i   -0.2550 - 0.1620i
Line flow 6-3   -0.3912 - 0.5855i   -0.4280 - 0.5810i
Line flow 6-5   -0.0147 + 0.0478i   -0.0180 + 0.0430i

Line flows                     Line-1 out

                       NR                 CVNN

Line flow 1-2   -0.0017 - 0.0006i           0
Line flow 1-4    0.6418 + 0.1882i    0.6810 + 0.2030i
Line flow 1-5    0.4855 + 0.0954i    0.5130 + 0.0940i
Line flow 2-1    0.0003 + 0.0006i           0
Line flow 2-3   -0.0005 - 0.1093i   -0.0080 - 0.1150i
Line flow 2-4    0.1792 + 0.5596i    0.1940 + 0.6070i
Line flow 2-5    0.1109 + 0.1835i    0.0910 + 0.1780i
Line flow 2-6    0.2126 + 0.1397i    0.2230 + 0.1370i
Line flow 3-2    0.0005 + 0.0495i    0.0080 + 0.0500i
Line flow 3-5    0.1464 + 0.2460i    0.1510 + 0.2550i
Line flow 3-6    0.4523 + 0.6511i    0.4400 + 0.6090i
Line flow 4-1   -0.6208 - 0.1456i   -0.6580 - 0.1510i
Line flow 4-2   -0.1814 - 0.5468i   -0.1750 - 0.5890i
Line flow 4-5    0.0527 - 0.0581i    0.0330 - 0.0600i
Line flow 5-1   -0.4617 - 0.0880i   -0.4930 - 0.0800i
Line flow 5-2   -0.0991 - 0.2124i   -0.0860 - 0.2070i
Line flow 5-3   -0.1314 - 0.2757i   -0.1410 - 0.2840i
Line flow 5-4    0.0332 - 0.0267i    0.0330 - 0.0170i
Line flow 5-6   -0.0014 - 0.0989i   -0.0530 - 0.1120i
Line flow 6-2   -0.1769 - 0.1801i   -0.2180 - 0.1760i
Line flow 6-3   -0.4066 - 0.5678i   -0.4300 - 0.5800i
Line flow 6-5    0.0032 + 0.0570i    0.0520 + 0.0560i

Line flows                     Line-2 Out

                       NR                 CVNN

Line flow 1-2    0.6322 - 0.2846i    0.6820 - 0.3020i
Line flow 1-4    0.0061 + 0.0085i           0
Line flow 1-5    0.5548 + 0.1138i    0.5890 + 0.1100i
Line flow 2-1    0.5562 + 0.3034i    0.5610 + 0.3060i
Line flow 2-3    0.0134 - 0.1180i    0.0110 - 0.1190i
Line flow 2-4    0.6711 + 0.6858i    0.6840 + 0.6930i
Line flow 2-5    0.1376 + 0.1908i    0.1220 + 0.1840i
Line flow 2-6    0.2333 + 0.1417i    0.2430 + 0.1360i
Line flow 3-2    0.0135 + 0.0402i    0.0110 + 0.0540i
Line flow 3-5    0.1668 + 0.2591i    0.1730 + 0.2640i
Line flow 3-6    0.4461 + 0.6717i    0.4380 + 0.6190i
Line flow 4-1   -0.0059 - 0.0045i           0
Line flow 4-2   -0.6642 - 0.6197i   -0.6400 - 0.6260i
Line flow 4-5   -0.0546 - 0.0767i   -0.0600 - 0.0740i
Line flow 5-1   -0.5107 - 0.0877i   -0.5180 - 0.0830i
Line flow 5-2   -0.1308 - 0.2163i   -0.1170 - 0.2100i
Line flow 5-3    0.0205 - 0.2852i    0.0161 - 0.2910i
Line flow 5-4    0.0597 - 0.0024i    0.0610 + 0.0020i
Line flow 5-6    0.0373 - 0.1121i    0.0350 - 0.1180i
Line flow 6-2   -0.2089 - 0.1791i   -0.2380 - 0.1720i
Line flow 6-3   -0.4112 - 0.5881i   -0.4280 - 0.5890i
Line flow 6-5    0.0302 + 0.0636i    0.0340 + 0.0620i

Table 4: Real and Reactive line flows

Line flows                     Line-3 out

                       NR                 CVNN

Line flow 1-2    0.5001 - 0.2414i    0.5010 - 0.2390i
Line flow 1-4    0.6181 - 0.1994i    0.6190 - 0.1860i
Line flow 1-5   -0.0271 - 0.0047i           0
Line flow 2-1   -0.4760 + 0.2527i   -0.4740 + 0.2490i
Line flow 2-3    0.1084 - 0.1392i    0.1030 - 0.1360i
Line flow 2-4    0.2461 + 0.5301i    0.2570 + 0.5240i
Line flow 2-5    0.2883 + 0.2127i    0.2720 + 0.2110i
Line flow 2-6    0.3375 + 0.1250i    0.3420 + 0.1280i
Line flow 3-2   -0.1074 + 0.0757i   -0.1020 + 0.0730i
Line flow 3-5    0.2745 + 0.3064i    0.2740 + 0.3030i
Line flow 3-6    0.4316 + 0.7196i    0.4280 + 0.6660i
Line flow 4-1   -0.5996 - 0.1461i   -0.6000 - 0.1500i
Line flow 4-2   -0.2680 - 0.5188i   -0.2410 - 0.5130i
Line flow 4-5    0.1472 - 0.0350i    0.1410 - 0.0370i
Line flow 5-1    0.0228 + 0.0041i           0
Line flow 5-2   -0.2738 - 0.2189i   -0.2610 - 0.2170i
Line flow 5-3   -0.2545 - 0.3169i   -0.2550 - 0.3130i
Line flow 5-4   -0.1395 - 0.0306i   -0.1370 - 0.0310i
Line flow 5-6   -0.0440 - 0.1393i   -0.0480 - 0.1400i
Line flow 6-2   -0.2990 - 0.1512i   -0.3330 - 0.1550i
Line flow 6-3   -0.3781 - 0.6360i   -0.4170 - 0.6320i
Line flow 6-5    0.0520 + 0.0872i    0.0490 + 0.0870i

Line flows                     Line-4 out

                       NR                 CVNN

Line flow 1-2    0.2900 - 0.1471i    0.2840 - 0.1530i
Line flow 1-4    0.4421 + 0.2051i    0.4350 + 0.2010i
Line flow 1-5    0.3453 + 0.1107i    0.3600 + 0.1120i
Line flow 2-1   -0.2826 + 0.1222i   -0.2750 + 0.1260i
Line flow 2-3    0.0023 - 0.0022i           0
Line flow 2-4    0.3308 + 0.4622i    0.3350 + 0.4590i
Line flow 2-5    0.1686 + 0.1452i    0.1620 + 0.1520i
Line flow 2-6    0.2806 + 0.1203i    0.2780 + 0.1200i
Line flow 3-2   -0.0023 + 0.0111i           0
Line flow 3-5    0.1827 + 0.2459i    0.1810 + 0.2360i
Line flow 3-6    0.4179 + 0.6806i    0.4190 + 0.6120i
Line flow 4-1   -0.4306 - 0.1980i   -0.4240 - 0.2000i
Line flow 4-2   -0.3194 - 0.4515i   -0.3200 - 0.4490i
Line flow 4-5    0.0508 - 0.0549i    0.0440 - 0.0510i
Line flow 5-1   -0.3331 - 0.1306i   -0.3490 - 0.1330i
Line flow 5-2   -0.1626 - 0.1713i   -0.1560 - 0.1780i
Line flow 5-3   -0.1744 - 0.2767i   -0.1700 - 0.2660i
Line flow 5-4    0.0388 - 0.0254i    0.0440 - 0.0260i
Line flow 5-6    0.0208 - 0.0979i    0.0200 - 0.0970i
Line flow 6-2    0.3012 - 0.1546i    0.2720 - 0.1550i
Line flow 6-3    0.4517 - 0.5963i    0.4090 - 0.5840i
Line flow 6-5    0.0221 - 0.0374i    0.0190 - 0.0390i

Line flows                     Line-5 out

                       NR                 CVNN

Line flow 1-2    0.1345 - 0.0870i    0.1370 - 0.0880i
Line flow 1-4    0.6438 + 0.6666i    0.6390 + 0.6690i
Line flow 1-5    0.3212 + 0.1738i    0.3280 + 0.1710i
Line flow 2-1   -0.1333 + 0.0486i   -0.1350 + 0.0480i
Line flow 2-3    0.0821 - 0.1338i    0.0800 - 0.1320i
Line flow 2-4   -0.0052 + 0.0019i           0
Line flow 2-5    0.2441 + 0.1848i    0.2380 + 0.1830i
Line flow 2-6    0.3138 + 0.1270i    0.3170 + 0.1240i
Line flow 3-2   -0.0811 + 0.0670i   -0.0790 + 0.0680i
Line flow 3-5    0.2495 + 0.2683i    0.2480 + 0.2700i
Line flow 3-6    0.4346 + 0.6678i    0.4310 + 0.6420i
Line flow 4-1   -0.6021 - 0.5450i   -0.5980 - 0.5460i
Line flow 4-2   -0.0052 - 0.0017i           0
Line flow 4-5    0.1071 - 0.1550i    0.1020 - 0.1540i
Line flow 5-1   -0.3126 - 0.1937i   -0.3180 - 0.1920i
Line flow 5-2    0.2252 - 0.1983i    0.2290 - 0.1970i
Line flow 5-3    0.2326 - 0.2858i    0.2320 - 0.2880i
Line flow 5-4    0.1083 + 0.0945i    0.1080 + 0.0970i
Line flow 5-6   -0.0252 - 0.1205i   -0.0290 - 0.1200i
Line flow 6-2   -0.2848 - 0.1573i   -0.3100 - 0.1540i
Line flow 6-3   -0.3940 - 0.6146i   -0.4210 - 0.6100i
Line flow 6-5    0.0315 + 0.0689i    0.0300 + 0.0650i

Table 5: Real and Reactive line flows

Line flows                     Line-6 out

                       NR                 CVNN

Line flow 1-2    0.2418 - 0.1468i    0.2630 - 0.1440i
Line flow 1-4    0.4272 + 0.2200i    0.4310 + 0.2200i
Line flow 1-5    0.3898 + 0.1831i    0.3970 + 0.1840i
Line flow 2-1   -0.2373 + 0.1170i   -0.2550 + 0.1150i
Line flow 2-3    0.0729 - 0.1228i    0.0710 - 0.1300i
Line flow 2-4    0.3601 + 0.4713i    0.3760 + 0.4760i
Line flow 2-5    0.0037 - 0.0094i           0
Line flow 2-6    0.2981 + 0.1354i    0.3080 + 0.1340i
Line flow 3-2   -0.0720 + 0.0641i   -0.0700 + 0.0660i
Line flow 3-5    0.2412 + 0.3128i    0.2400 + 0.3030i
Line flow 3-6    0.4295 + 0.7274i    0.4310 + 0.6570i
Line flow 4-1   -0.4163 - 0.2178i   -0.4200 - 0.2180i
Line flow 4-2    0.3417 - 0.4577i    0.3590 - 0.4620i
Line flow 4-5    0.0749 - 0.0220i    0.0790 - 0.0200i
Line flow 5-1   -0.3887 - 0.1896i   -0.3820 - 0.1890i
Line flow 5-2   -0.0054 + 0.0094i           0
Line flow 5-3   -0.2267 - 0.3278i   -0.2220 - 0.3170i
Line flow 5-4   -0.0662 - 0.0538i   -0.0770 - 0.0530i
Line flow 5-6   -0.0135 - 0.1356i   -0.0190 - 0.1410i
Line flow 6-2   -0.2655 - 0.1659i   -0.3000 - 0.1640i
Line flow 6-3   -0.3874 - 0.6162i   -0.4200 - 0.6230i
Line flow 6-5    0.0209 + 0.0805i    0.0200 + 0.0870i

Line flows                     Line-7 out

                       NR                 CVNN

Line flow 1-2    0.2623 - 0.1389i    0.2600 - 0.1430i
Line flow 1-4    0.4320 + 0.2004i    0.4290 + 0.2070i
Line flow 1-5    0.3933 + 0.1266i    0.3990 + 0.1220i
Line flow 2-1   -0.2568 + 0.1086i   -0.2530 + 0.1140i
Line flow 2-3    0.1699 - 0.1428i    0.1660 - 0.1460i
Line flow 2-4    0.3682 + 0.4445i    0.3700 + 0.4510i
Line flow 2-5    0.2217 + 0.1567i    0.2170 + 0.1530i
Line flow 2-6   -0.0018 + 0.0043i           0
Line flow 3-2   -0.1689 + 0.0615i   -0.1650 + 0.0880i
Line flow 3-5    0.1557 + 0.2573i    0.1540 + 0.2690i
Line flow 3-6    0.6129 + 0.8194i    0.6110 + 0.7900i
Line flow 4-1   -0.4209 - 0.2054i   -0.4180 - 0.2060i
Line flow 4-2   -0.3522 - 0.4345i   -0.3540 - 0.4400i
Line flow 4-5    0.0750 - 0.0494i    0.0720 - 0.0540i
Line flow 5-1   -0.3797 - 0.1396i   -0.3860 - 0.1340i
Line flow 5-2   -0.2139 - 0.1762i   -0.2100 - 0.1730i
Line flow 5-3   -0.1437 - 0.2829i   -0.1420 - 0.2960i
Line flow 5-4   -0.0738 - 0.0267i   -0.0710 - 0.0210i
Line flow 5-6    0.1132 - 0.0694i    0.1090 - 0.0760i
Line flow 6-2    0.0161 - 0.0071i           0
Line flow 6-3   -0.5860 - 0.6999i   -0.5930 - 0.7220i
Line flow 6-5   -0.1041 - 0.0204i   -0.1070 - 0.0220i

Line flows                     Line-8 out

                       NR                 CVNN

Line flow 1-2    0.2533 - 0.1500i    0.2680 - 0.1460i
Line flow 1-4    0.4377 + 0.2290i    0.4320 + 0.2270i
Line flow 1-5    0.3847 + 0.2179i    0.3850 + 0.2160i
Line flow 2-1   -0.2466 + 0.1211i   -0.2600 + 0.1180i
Line flow 2-3   -0.0462 - 0.1070i   -0.0510 - 0.1070i
Line flow 2-4    0.3606 + 0.4860i    0.3720 + 0.4910i
Line flow 2-5    0.2074 + 0.2405i    0.2090 + 0.2470i
Line flow 2-6    0.2182 + 0.1576i    0.2290 + 0.1620i
Line flow 3-2    0.0461 + 0.0371i    0.0510 + 0.0410i
Line flow 3-5    0.0014 + 0.0032i           0
Line flow 3-6    0.5525 + 0.6725i    0.5490 + 0.6460i
Line flow 4-1   -0.4266 - 0.2261i   -0.4210 - 0.2230i
Line flow 4-2   -0.3609 - 0.4719i   -0.3550 - 0.4770i
Line flow 4-5    0.0717 - 0.0010i         0.076
Line flow 5-1   -0.3775 - 0.2208i   -0.3700 - 0.2190i
Line flow 5-2   -0.2058 - 0.2484i   -0.1990 - 0.2560i
Line flow 5-3   -0.0052 - 0.0029i           0
Line flow 5-4   -0.0673 - 0.0723i   -0.0740 - 0.0720i
Line flow 5-6   -0.0535 - 0.1564i   -0.0570 - 0.1530i
Line flow 6-2   -0.1866 - 0.1930i   -0.2230 - 0.1990i
Line flow 6-3   -0.4982 - 0.6130i   -0.5360 - 0.6030i
Line flow 6-5    0.0663 + 0.1011i    0.0590 + 0.1020i

Table 6: Real and Reactive line flows

Line flows                     Line-9 out

                       NR                 CVNN

Line flow 1-2    0.3160 - 0.1664i    0.3220 - 0.1690i
Line flow 1-4    0.4642 + 0.2218i    0.4600 + 0.2180i
Line flow 1-5    0.3564 + 0.2110i    0.3630 + 0.2080i
Line flow 2-1   -0.3049 + 0.1452i   -0.3100 + 0.1470i
Line flow 2-3   -0.2084 - 0.0710i   -0.2130 - 0.0700i
Line flow 2-4    0.3114 + 0.5077i    0.3180 + 0.5080i
Line flow 2-5    0.1527 + 0.2556i    0.1500 + 0.2530i
Line flow 2-6    0.5486 + 0.6324i    0.5550 + 0.6310i
Line flow 3-2    0.2104 + 0.0086i    0.2150 + 0.0130i
Line flow 3-5    0.3849 + 0.2613i    0.3850 + 0.2650i
Line flow 3-6    0.0055 + 0.0213i           0
Line flow 4-1   -0.4520 - 0.2117i   -0.4480 - 0.2100i
Line flow 4-2   -0.3078 - 0.4946i   -0.3020 - 0.4950i
Line flow 4-5    0.0508 + 0.0041i    0.0500 + 0.0050i
Line flow 5-1   -0.3468 - 0.2193i   -0.3490 - 0.2170i
Line flow 5-2   -0.1464 - 0.2682i   -0.1410 - 0.2660i
Line flow 5-3   -0.3624 - 0.2597i   -0.3600 - 0.2640i
Line flow 5-4   -0.0461 - 0.0808i   -0.0490 - 0.0790i
Line flow 5-6    0.2002 + 0.1266i    0.1990 + 0.1260i
Line flow 6-2   -0.4870 - 0.5460i   -0.5080 - 0.5440i
Line flow 6-3    0.0208 + 0.0006i           0
Line flow 6-5   -0.1862 - 0.1535i   -0.1920 - 0.1560i

Line flows                     Line-10 out

                       NR                 CVNN

Line flow 1-2    0.3304 - 0.1540i    0.2910 - 0.1560i
Line flow 1-4    0.4403 + 0.2169i    0.4210 + 0.2150i
Line flow 1-5    0.3287 + 0.1187i    0.3000 + 0.1220i
Line flow 2-1   -0.3183 + 0.1281i   -0.2810 + 0.1300i
Line flow 2-3    0.0358 - 0.1221i    0.0370 - 0.1240i
Line flow 2-4    0.3352 + 0.4875i    0.3050 + 0.4970i
Line flow 2-5    0.1549 + 0.1579i    0.1680 + 0.1620i
Line flow 2-6    0.2924 + 0.1264i    0.2710 + 0.1250i
Line flow 3-2   -0.0356 + 0.0500i   -0.0370 + 0.0590i
Line flow 3-5    0.2002 + 0.2576i    0.2000 + 0.2420i
Line flow 3-6    0.4389 + 0.6373i    0.4370 + 0.6150i
Line flow 4-1   -0.4285 - 0.2165i   -0.4110 - 0.2140i
Line flow 4-2   -0.2558 - 0.4774i   -0.2890 - 0.4860i
Line flow 4-5    0.0257 - 0.0012i           0
Line flow 5-1   -0.3408 - 0.1398i   -0.3590 - 0.1410i
Line flow 5-2   -0.1419 - 0.1838i   -0.1620 - 0.1870i
Line flow 5-3   -0.1784 - 0.2858i   -0.1880 - 0.2690i
Line flow 5-4   -0.0339 + 0.0002i           0
Line flow 5-6    0.0010 - 0.0898i    0.0090 - 0.1030i
Line flow 6-2   -0.3000 - 0.1635i   -0.2650 - 0.1600i
Line flow 6-3   -0.4514 - 0.5720i   -0.4260 - 0.5860i
Line flow 6-5   -0.0018 + 0.0402i   -0.0090 + 0.0460i

Line flows                     Line-11 out

                       NR                 CVNN

Line flow 1-2    0.3081 - 0.1675i    0.2920 - 0.1560i
Line flow 1-4    0.4487 + 0.2014i    0.4380 + 0.2070i
Line flow 1-5    0.3497 + 0.1307i    0.3510 + 0.1400i
Line flow 2-1   -0.2990 + 0.1425i   -0.2830 + 0.1310i
Line flow 2-3    0.0269 - 0.1318i    0.0330 - 0.1230i
Line flow 2-4    0.3284 + 0.4828i    0.3290 + 0.4730i
Line flow 2-5    0.1629 + 0.1696i    0.1510 + 0.1820i
Line flow 2-6    0.2819 + 0.0945i    0.2690 + 0.1100i
Line flow 3-2   -0.0267 + 0.0934i   -0.0330 + 0.0580i
Line flow 3-5    0.1867 + 0.2903i    0.1870 + 0.2660i
Line flow 3-6    0.4439 + 0.6118i    0.4460 + 0.5800i
Line flow 4-1   -0.4371 - 0.2004i   -0.4270 - 0.2040i
Line flow 4-2   -0.3226 - 0.4718i   -0.3130 - 0.4630i
Line flow 4-5    0.0196 - 0.0342i    0.0410 - 0.0330i
Line flow 5-1   -0.3579 - 0.1512i   -0.3400 - 0.1600i
Line flow 5-2   -0.1552 - 0.1943i   -0.1450 - 0.2060i
Line flow 5-3   -0.1758 - 0.3180i   -0.1740 - 0.2910i
Line flow 5-4   -0.0298 - 0.0364i   -0.0410 - 0.0440i
Line flow 5-6    0.0064 - 0.0057i           0
Line flow 6-2   -0.2928 - 0.1240i   -0.2640 - 0.1460i
Line flow 6-3   -0.4588 - 0.6022i   -0.4360 - 0.5540i
Line flow 6-5   -0.0093 - 0.0066i           0