A Novel Frequency Tracking Method Based on Complex Adaptive Linear Neural Network State Vector in Power Systems 269 rule. It was developed by ProfessorBernard Widrow and his graduate student Ted Hoff at Stanford University in 1960. It is based on the McCulloch–Pitts neuron. It consists of a weight, a bias and a summation function.The difference between Adaline and the standard (McCulloch-Pitts) perceptron is that in the learning phase the weights are adjusted according to the weighted sum of the inputs (the net). In the standard perceptron, the net is passed to the activation (transfer) function and the function's output is used for adjusting the weights. The main functional difference with the perceptron training rule is the way the output of the system is used in the learning rule. The perceptron learning rule uses the output of the threshold function (either -1 or +1) for learning. The delta-rule uses the net output without further mapping into output values -1 or +1. The ADALINE network shown below has one layer of S neurons connected to R inputs through a matrix of weights W. This network is sometimes called a MADALINE for Many ADALINEs. Note that the figure on the right defines an S-length output vector a. The Widrow-Hoff rule can only train single-layer linear networks. This is not much of a disadvantage, however, as single-layer linear networks are just as capable as multilayer linear networks. For every multilayer linear network, there is an equivalent single-layer linear network. 5.1 Single ADALINE Consider a single ADALINE with two inputs. The following figure shows the diagram for this network. The weight matrix W in this case has only one row. The network output is: ( ) ( ) ( ) a p urelin n p urelin W p bW p b = =+=+ (39) Equation a can be written as follows: 1,1 1 1,2 2 aw p w p b = ++ (40) Like the perceptron, the ADALINE has a decision boundary that is determined by the input vectors for which the net input n is zero. For n = 0 the equation Wp + b = 0 specifies such a decision boundary, as shown below: Artificial Neural Networks - Industrial and Control Engineering Applications 270 Input vectors in the upper right gray area lead to an output greater than 0. Input vectors in the lower left white area lead to an output less than 0. Thus, the ADALINE can be used to classify objects into two categories. However, ADALINE can classify objects in this way only when the objects are linearly separable. Thus, ADALINE has the same limitation as the perceptron. 5.2 Networks with linear activation functions: the delta rule For a single layer network with an output unit with a linear activation function the output is simply given by: 1 n ii i ywx θ = = + ∑ (41) Such a simple network is able to represent a linear relationship between the value of the output unit and the value of the input units. By thresholding the output value, a classifier can be constructed (such as Widrow's Adaline), but here we focus on the linear relationship and use the network for a function approximation task. In high dimensional input spaces the network represents a (hyper) plane and it will be clear that also multiple output units may be defined. Suppose we want to train the network such that a hyper plane is fitted as well as possible to a set of training samples consisting of input values p d and desired (or target) output values p d . For every given input sample, the output of the network differs from the target value p d by ( ) pp dy− where p y is the actual output for this pattern. The delta-rule now uses a cost- or error-function based on these differences to adjust the weights. The error function, as indicated by the name least mean square, is the summed squared error. That is, the total error E is denoted to be: () 2 1 2 ppp pp EE dy== − ∑∑ (42) Where the index p ranges over the set of input patterns and p E represents the error on pattern p . The LMS procedure finds the values of all the weights that minimize the error function by a method called gradient descent. The idea is to make a change in the weight proportional to the negative of the derivative of the error as measured on the current pattern with respect to each weight: A Novel Frequency Tracking Method Based on Complex Adaptive Linear Neural Network State Vector in Power Systems 271 p pj j E w w γ ∂ Δ=− ∂ (43) where γ is a constant of proportionality. The derivative is p pp p jj y EE ww y ∂ ∂∂ = ∂ ∂ ∂ (44) p i j y x w ∂ = ∂ (45) Because of the linearity, p j E w ∂ ∂ is as follows: ( ) p pp j E dE w ∂ =− − ∂ (46) Where ppp dE δ =− is the difference between the target output and the actual output for pattern p .The delta rule modifies weight appropriately for target and actual outputs of either polarity and for both continuous and binary input and output units. These characteristics have opened up a wealth of new applications. 6. Simulation results Simulation examples include the following three categories. Numerical simulations are represented in Section 5.1. for two cases, simulation in PSCAD/EMTDC software is presented in Section 5.2. Lastly, Section 5.3. presents practical measurement of a real fault incidence in Fars province, Iran. 6.1 Simulated signals Herein, a disturbance is simulated at time 0.3 sec. Three-phase non-sinusoidal unbalanced signals, including decaying DC offset and third harmonic, are produced as: 0 0 0 220sin( t) 2 220sin( t- ) 0 t 0.3 3 2 220sin( t+ ) 3 A B C V V V ω π ω π ω ⎧ ⎪ = ⎪ ⎪ =≤≤ ⎨ ⎪ ⎪ = ⎪ ⎩ (47) After disturbance at 0.3 sec, signals are: (-10t) A (-10t) V =400sin( t)+40sin(3 t)+400 2 800sin( t- )+60sin(3 t)+800 0.3 t 0.6 3 2 800sin( t+ )+20sin(3 t) 3 xx Bx x Cx x e Ve V ωω π ωω π ωω ⎧ ⎪ ⎪ ⎪ =≤≤ ⎨ ⎪ ⎪ = ⎪ ⎩ (48) Artificial Neural Networks - Industrial and Control Engineering Applications 272 where 0 ω is the base angular frequency and x ω is the actual angular frequency after disturbance. 6.1.1 Case 1 In this case, a 1-Hz frequency deviation occurs and tracked frequency using CADALINE, ADALINE, Kalman, and DFT approaches is revealed in Fig. 4; three-phase signals are shown in Fig. 5. Estimation error percentage according to the samples fed to each algorithm after frequency drift is shown in Fig. 6. Second set of samples including 100 samples, equivalent to two and half cycles, which is fed to all algorithms is magnified in Fig. 6. It can be seen that CADALINE converges to the real value after first 116 samples, less than three power cycles, with error of -0.4 %; and reaches a perfect estimation after having more few samples. Other methods’ estimations are too fare from real value in this snapshoot. DFT, ADALINE and Kalman respectively need 120, 200 and 360 samples to reach less than one percent error in estimating the frequency drift. It should be considered that for 2.4-kHz sampling frequency and power system frequency of 60 Hz, each power cycle includes 40 samples. The complex normalized rotating state vector 1 () s An kT with respect to time and in d-q frame is shown in Fig. 7. It has been seen that for 1-Hz frequency deviation ( 1 1f = Hz), CADALINE has the best convergence response in terms of speed and over/under shoot. ADALINE method convergence speed is half that in the CADALINE and shows a really high overshoot. Besides, Kalman approach shows the biggest error. in the first 7 power system cycles, it converges to 61.7 Hz instead of 61 Hz and its computational burden is considerably higher than other methods. In this case, presence of a long-lasting decaying DC offset affects the DFT performance. Consequently, its convergence speed and overshoot are not as improved as CADALINE. Fig. 4. Tracked frequency (Hz) A Novel Frequency Tracking Method Based on Complex Adaptive Linear Neural Network State Vector in Power Systems 273 Fig. 5. Three-phase signals Fig. 6. Estimation error percentage according to samples fed to each algorithm after frequency drift As can be seen in Fig. 7, 1 () s An kT starts rotation simultaneously when the frequency changes at time 0.3 sec. Fig. 7. Complex normalized rotating state vector ( 1 An ) Artificial Neural Networks - Industrial and Control Engineering Applications 274 6.1.2 Case 2 In this case, a three-phase balanced voltage is simulated numerically. The only change applied is a step-by-step 1-Hz change in fundamental frequency to study the steady-state response of the proposed method when the power system operates under/over frequency conditions. The three-phase signals are: 220sin( t) 2 220sin( t- ) 3 2 220sin( t+ ) 3 Ax Bx Cx V V V ω π ω π ω ⎧ ⎪ = ⎪ ⎪ = ⎨ ⎪ ⎪ = ⎪ ⎩ (49) where 2 xx f ω π = , and values of x f are shown in Table I. The range of frequency that has been studied here is 50–70 Hz. Results are revealed in Table I and average convergence time is shown in Fig. 8 for CADALINE, ADALINE, Kalman filter and DFT approaches. The results from this section can give an insight into the number of samples that each algorithm needs to converge to a reasonable estimation. According to the fact that each power cycle is equivalent to 40 samples, average number of samples that is needed for each algorithm to have estimation with less than one percent error is represented in Table I. Fig. 8. Average convergence time (cycles) to track static frequency changes 6.2 Simulation in PSCAD/EMTDC software In this case, a three-machine system controlled by governors is simulated in PSCAD/EMTDC software, shown in Fig. 9. Information of the simulated system is given in Appendix I. A three-phase fault occurs at 1 sec. Real frequency changes, estimation by use of ADALINE, CADALINE and Kalman approaches are shown in Fig. 10. Instead of DFT method, the frequency measurement module (FMM) performance which exists in PSCAD library is compared with the presented methods. Phase-A voltage signal is shown in Fig. 11. A Novel Frequency Tracking Method Based on Complex Adaptive Linear Neural Network State Vector in Power Systems 275 Approaches (Hz) x f CADALINE KALMAN ADALINE DFT 70 95 360 202 111 69 97 421 188 114 68 93 358 186 118 67 90 384 187 114 66 95 385 178 114 65 97 305 138 139 64 92 361 211 114 63 93 328 193 116 62 98 430 206 115 61 96 360 231 116 60 92 385 220 112 59 83 234 155 97 58 81 281 181 116 57 88 313 197 117 56 98 216 178 123 55 97 377 192 117 54 96 336 206 122 53 90 331 195 114 52 96 290 190 108 51 96 374 184 120 50 105 405 113 112 Table I Samples needed to estimate with 1 percent error for 50-70 frequency range The complex normalized rotating state vector ( 1 () s An kT ) is shown in Fig. 12. The best transient response and accuracy belongs to ADALINE and CADALINE, but CADALINE has faster response with a considerable lower overshoot, as can be seen in Fig. 10. Kalman Artificial Neural Networks - Industrial and Control Engineering Applications 276 approach has a suitable response in this case, but its error and overshoot in estimating frequency are bigger than that in CADALINE. The PSCAD FMM shows drastic fluctuations in comparison with other methods proposed and reviewed here. Fig. 9. A three-machine connected system simulated in PSCAD/EMTDC software A Novel Frequency Tracking Method Based on Complex Adaptive Linear Neural Network State Vector in Power Systems 277 Fig. 10. Tracked frequency (Hz) Fig. 11. Phase-A voltage (kV) Fig. 12. Complex normalized rotating state vector ( 1 A n ) Artificial Neural Networks - Industrial and Control Engineering Applications 278 6.3 Practical study In this case, a practical example is represented. Voltage signal measurements are applied from the Marvdasht power station in Fars province, Iran. The recorder’s sampling frequency ( s f ) is 6.39 kHz and fundamental frequency of power system is 50 Hz. A fault between pahse-C and groung occurred on 4 March 2006. The fault location was 46.557 km from Arsanjan substation. Main information on the Marvdasht 230/66 kV station and other substation supplied by this station is given in Tables II and III, presented in Appendix II. Fig. 13 shows the performance of CADALINE, ADALINE, Kaman and DFT approaches. Besides, phase-C voltage and residual voltage are revealed in Fig. 14 (A) and Fig. 14 (B) respectively. Complex normalized rotating state vector ( 1 A n ) is shown in Fig. 15. Fig. 13. Tracked frequency (Hz), case V.C. Fig. 14. (A): phase-C voltage and (B): residual voltage, case V.C. 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Gen- 19 Gen-20 Gen-21 Gen-22 Gen-23 Gen-24 Gen-25 Total Load Actual Load 1 0.31 492 0.31 492 0.74181 0.73775 0 .97 821 0.5 791 4 0 .99 2 49 0.28846 0.28845 1.2756 1.24 79 1. 294 1 ANN Output 2 4 5 17.18 0 .96 386 1.5686 17.18 0 .96 386 1.5687 1.2167 36.6 89 3.4786 1.2058 36.835 3.4 49 1.6865 3.2764 4. 792 7 0 .93 22 30.05 2.6715 1.7 194 3.3288 4.8834 3.24 89 0. 896 34 1 .92 23 3.2487 0. 896 32 1 .92 22 2.2431 4. 297 1 6.3 599 2.1685 4.1 894 ... 101 97 .088 3.857 2.371 1 .91 1 2.107 0 .90 18 3.5 69 3.857 2.371 1 .91 1 2.107 0 .90 182 3.5 69 0. 092 18 0.150 69 -0.13734 -1.125 0.007336 0.15065 -0.13734 -1.125 -2 .92 3 30.24 0.13265 102 11 .93 7 0.007438 -0.34445 -0.34441 0. 092 196 106 34.2 19 42.4 39 -0.57113 1.485 1. 895 -0.5 492 3 -2 .92 3 42.438 -0.57107 1.485 1. 895 -0.5 493 108 89. 13 12. 297 3.114 2.377 2.603 0.57067 1 .91 5 12. 296 3.114 2.377 2.602 0.57067 1 .91 5 1 09 14.324... 2.1685 4.1 894 6.15 69 2. 292 8 4.3687 6. 494 9 Load bus no Modified Nodal Equations Method 6 1 2 4 5 6 4.5436 0.31 492 17.18 0 .96 3 89 1.5687 4.5436 4.5435 0.31 492 17.18 0 .96 3 89 1.5687 4.5436 4.0286 0.74182 1.2167 36.688 3.4787 4.0287 3 .99 78 0.73775 1.2058 36.835 3.4 491 3 .99 78 5.4841 0 .97 8 19 1.6864 3.2764 4. 792 6 5.484 3.1082 0.5 791 3 0 .93 221 30.051 2.6715 3.1082 5.5815 0 .99 247 1.7 194 3.32 89 4.8834 5.5814 5.2152... 4.8834 5.5814 5.2152 0.28846 3.2488 0. 896 33 1 .92 22 5.21 49 5.2147 0.28846 3.2488 0. 896 33 1 .92 22 5.21 49 7.2433 1.2757 2.2432 4. 297 1 6.3601 7.2436 7.03 19 1.248 2.1686 4.1 895 6.1571 7.0321 7.38 39 1. 294 1 2. 292 8 4.3687 6. 495 1 7.3842 9. 05375 54.3226 126.755 45.2687 63.3763 9. 05 392 54.32271 126.755 45.2 694 63.377 9. 05 39 54.323 126.75 63.377 126.75 45.2 69 63.377 45.2 69 9.05 39 54.323 Table 7 Analysis of reactive... 0.71278 0.33 29 0.08 195 1 0.056484 0.032257 0.033 798 0.7128 0.332 89 33 42.177 0. 590 22 0.66717 1 .95 6 0. 590 24 0.26853 0.66711 1 .95 6 35 26.261 -0.22153 -0.11404 -0. 090 94 -0.22151 -0.11404 -0. 090 99 -0.10018 -0. 194 97 -0.78503 39 29. 445 0.15404 0.18401 0.15401 0.18401 0.2685 0.24 595 0.275 -0.1002 -0. 194 87 -0.78481 0.05 295 4 0.045775 -0.16485 0 .90 055 0.24 594 0.05 296 0.275 0.045736 -0.16 498 Table 11 Analysis... 12.308 15 .98 2 20.564 3 .93 1 Bialek's Method 2 4 5 71.274 0 0 71.274 0 0 0 85.144 0 0 82. 090 0 0 16. 593 21.805 0 56. 392 0 0 17 .90 3 23.527 19. 446 0 0 19. 446 0 0 0 27. 292 35.863 0 23.362 30. 699 0 29. 912 39. 306 6 0 0 0 0 13.444 0 14.505 51.670 51.670 22.111 18 .92 7 24.234 15.120 181.443 338. 691 151.202 196 .564 15.121 181.440 338.688 151.200 196 .561 15.12 181.44 338. 69 151.2 196 .56 181.44 196 .56 15.12 338. 69 151.2 . 65 97 305 138 1 39 64 92 361 211 114 63 93 328 193 116 62 98 430 206 115 61 96 360 231 116 60 92 385 220 112 59 83 234 155 97 58 81 281 181 116 57 88 313 197 117 56 98 216 178 123 55 97 . simplicity and to reduce the training time of the neural networks. Artificial Neural Networks - Industrial and Control Engineering Applications 288 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2122 23. (Sept. 199 4), pp. 5 29 536. J. Szafran and W. Rebizant, Power system frequency estimation, IEE Gen. Trans. Dist. Proc., vol. 145, no. 5, (Sep. 199 8), pp. 578–582. Artificial Neural Networks - Industrial