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ANN based online voltage estimation
Applied Soft Computing 12 (2012) 313–319
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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
ANN based online voltage estimation P. Aravindhababu ∗ , G. Balamurugan Electrical Engineering, Annamalai University, Annamalainagar 608 002, Tamil Nadu, India
a r t i c l e
i n f o
Article history: Received 15 April 2009 Received in revised form 22 January 2011 Accepted 14 August 2011 Available online 25 August 2011 Keywords: Artificial neural network Voltage stability Voltage collapse proximity indicator VAR support
a b s t r a c t A new ANN based algorithm that requires only a minimum number of inputs is proposed to estimate the voltage magnitude of each critical bus in a power system under normal and/or contingent states. The basic philosophy is to develop an online tool, with a view to predict the occurrence of voltage collapse due to either an outage and/or projected changes in load. Besides the scheme contemplates to aid in the process of computing the approximate VAR support/sheddable load that is required to prevent voltage instability, which is imminent on account of the stressed operating nature of the present day power networks. The performance is evaluated through standard IEEE 14, 30 and 57 bus test systems and the simulation results are presented to demonstrate its validity and suitability for practical applications. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Voltage stability analysis is concerned with the ability of assessing the power system to maintain acceptable voltages at all system buses under normal conditions and after being subjected to disturbances. Electric power systems have become larger, more complex and found to be operating dangerously close to their stability limits. The lack of new transmission facilities, inadequate reactive power support, cutbacks in system maintenance, workforce downsizing, unpredicted power flow patterns, just to name a few, are some of the important factors that affect the stability of the system. A voltage collapse can appear quite abruptly in systems or sub-systems due to the continuously changing operating conditions and various unforeseen factors associated with large power systems and hence offline stability studies can no longer be sufficient to ensure a secure operation of the power system. It is well known that unlike angle instability, voltage instability often starts in a local network and gradually extends to the whole system. This feature slows down the process of system losing voltage stability compared to that of losing angle stability, and allows time for a mechanism to predict static voltage stability [1–9]. The bus voltages are used in voltage stability studies [10,11] for computing the voltage margin in terms of bus voltage variations and in steady state security assessment [12,13] for checking the bus voltage limit violations of a power system. The prediction of bus voltages is essential for different loading and/or contingent conditions in these studies. Online estimation of bus voltage
∗ Corresponding author. Tel.: +91 9842565093 (mobile). E-mail address: [email protected] (P. Aravindhababu). 1568-4946/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2011.08.041
magnitudes (VM) at selected load buses for each contingency and/or projected load changes facilitates to foresee the possible occurrence of voltage instability and initiates precautionary measures. Therefore, there is a definite need to develop a fast bus voltage estimation technique that is ideally suitable for on-line applications. A number of methods are in vogue for bus voltage estimation in the literature [14–17]. Real time use of conventional methods in an energy management centre is difficult due to their significant requirement of large computational times. Recently artificial neural networks (ANN) appear to find its role in the area of power systems because of its usefulness in online applications. Though several ANN based methods are available for the prediction of voltage stability and proximity to voltage collapse conditions [6–9], hardly any work is reported for the online estimation of bus voltages [18–21]. Most of these methods require large number of inputs, which causes difficulty in training the network especially for larger power systems. However, efforts are being taken to reduce the number of input variables for ANN based approach. Once trained, the execution time of ANNs subjected to any input is very less, which makes it more suitable for online estimation of VMs compared to conventional methods. It is in the prelude that an attempt is made to create an ANN based model that requires as minimum a number of inputs as possible for estimating VM. The main objective is to develop an ANN based approach that requires only a minimum number of inputs in order to estimate the VM of each vulnerable bus in a power system under normal and/or under contingent condition over a wide range of load variations. The scheme is to be evaluated through simulation and the results of standard IEEE 14, 30 and 57 bus systems are compared with that of a conventional approach (CA). The scope includes to predict
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presented with test data will be able to predict the correct output in the operational phase.
H1 X1
3. Proposed approach H2
Y1
X2 H3
Y2
X3 H4 Input layer
Hidden layer
Output layer
Fig. 1. Architecture of feed forward neural network.
the VAR support/sheddable load that may be necessary to ensure voltage stability of the system. 2. Artificial neural network In recent years, ANNs seems to offer an alternative scheme for solving certain difficult power system problems, where the conventional techniques have not achieved the desired speed, accuracy and efficiency. One of the main features, which can be attributed to ANN, is its ability to learn non-linear problem offline with selective training that can pave the way to offer a sufficiently accurate online response. A multi-layer feed forward network trained by backpropagation is the most popular and versatile form of neural network for pattern mapping or function approximation [22]. It consists of an input layer, an output layer and at least one hidden layer, with each layer consisting of a set of neurons as shown in Fig. 1. The input vector representing the pattern to be recognised is presented to the input layer and distributed to the subsequent hidden layers and finally to the output layer via the weighted connections. Each neuron in the network operates by taking the sum of weighted inputs and passing the result through a nonlinear activation function, which is usually a sigmoid or hyperbolic tangent function. This is mathematically represented as,
⎡ ⎤ n wij outj + bi ⎦ outi = f (neti ) = f ⎣
(1)
j=1
where outi is the output of the ith neuron in the layer under consideration. outj is the output of the jth neuron in the preceding layer. wij are the connection weights between ith neuron and jth inputs and bi is a constant called bias. The knowledge required to map the input patterns and output is embodied in the form of weights. Initially the weights appropriate to a given problem domain are unknown. The network does not have the ability to deal with the problem to be solved until a set of appropriate weights is found. The process of finding a useful set of weights is called training. Training begins with tutoring a set consisting of specimen inputs with associated outputs. Training the network involves adjusting the connection weights to correctly map the training set vectors at least to within some defined error limit. If the training set is good and the algorithm is effective, the network will be able to correctly estimate the output even for the inputs not belonging to the training set. The application of neural network to a recognition problem involves two distinct phases: training phase and operational phase. The network weights are adopted to reflect the problem domain during the training phase. The weights are frozen and the network when
The aim of the proposed method is to determine the VM at the most vulnerable bus-k for a projected load pattern under normal network conditions as well as under credible contingent conditions using ANN with minimum number of input variables. However, if there are a large number of vulnerable buses, a separate ANN model for each bus has to be developed. The VM at any bus depends on the real and reactive power loads connected at that bus and also on the loads at the other buses. If the real and reactive load powers at all the buses are treated as inputs, which are available as measurements in a typical power system, it may be time-consuming to train a network as the number of connection weights and neurons would be extremely large. It is therefore essential to reduce the number of inputs to a neural network and select an optimum number of mutually independent inputs, which clearly establish the required input–output relationship. The effect on the VM by the real and reactive load powers of all the buses at the chosen vulnerable bus can be investigated from the sensitivity factors that relate the VM of the chosen bus-k with real and reactive load powers of all buses. Though it is observed from these factors that the effect of local bus powers on its VM is considerably large, still it is comparatively lower for the neighbouring buses and insignificant for the remote buses, which is further elaborated in Appendix A. However, the remote buses are indirectly connected to bus-k and their influence on the VM of bus-k cannot be neglected. So in the proposed approach (PA), the following four input variables, irrespective of the system size, are chosen in order to account the effects of all bus powers for the ANN model of the vulnerable bus k as • PL k , the real power demand at the interested vulnerable bus k, • QL k , the reactive power demand at the interested vulnerable bus k, • PNET , the system net real power demand excluding PL k , • QNET , the system net reactive power demand excluding Q L k , The standard procedure in most utilities is to examine the effects of all possible outages, one at a time. It is rather time consuming, if every single outage is analysed for a large power system. Therefore, a selected number of worst-case contingencies, called credible contingencies, are considered in developing the proposed model. If there are n-credible contingencies, the network then contains one output for normal network conditions and n-output for n-credible contingencies, resulting in n + 1 output variables as
V0 k , V1 k , V2 k , V3 k ...., Vn k
(2)
where V0 k represents VM at bus k under normal network conditions, Vj k represents VM at bus k under j-line/generator outage conditions. The block diagram, shown in Fig. 2, shows the ANN model for determination of VM at the most vulnerable load bus-k under normal as well as under selected contingent conditions. The training data is the only available information to build the ANN model and care is taken to include the complete possible operating conditions of the system. The testing data is used to evaluate the performance of the model. The input-target patterns are generated using the following procedure for each of the vulnerable bus k. A range of load patterns are first generated by randomly perturbing the load from the reasonable lower expected loading level to the point of voltage instability. Initially one variable preferably
P. Aravindhababu, G. Balamurugan / Applied Soft Computing 12 (2012) 313–319
PL k
P NET
Table 1 List of credible outages.
k
Vo
Q Lk ANN model for bus-k
Q NET
V1
k
V2
k
Vn
k
IEEE 14 IEEE 30 IEEE 57
PL k is chosen and set at a lower loading level while the remaining three are perturbed in small steps one by one in order to cover the entire data range. Then PL k is incremented and the remaining three are again adjusted. This process is repeated till PL k reaches the maximum loading level. The second variable QL k is chosen and the remaining three variables are adjusted accordingly. The values of PNET and QNET are distributed to all load buses other than bus k in proportion to their base case values in order to obtain widely varying combinations of load patterns with different power factors. The VM at bus k is computed after carrying out the precontingency load flow for each load pattern and then the single line/generator outage specified in the credible contingency list is simulated one by one. The corresponding VMs at bus k are computed for forming the training/testing data set as follows: PL k , QL k , PNET , QNET ⇔ V0 k , V1 k , V2 k , V3 k ...., Vn k
(3)
The generated input-target data are split into two partitions: the first is the training data, which is used to train the network and the second the testing data, serves to assess how well the network is generalised. There is a possibility of obtaining good performance on the training data followed by much poor performance on the test data. This can be avoided by ensuring that the training data is uniformly distributed. During training of the neural network, higher valued input variables may tend to suppress the influence of smaller ones. Besides, if the raw data is directly applied to the network, there is a risk of the simulated neurons reaching the saturated states. If the neuron becomes saturated, then the changes in the output value are likely to produce a very small change or no change in the output value. This affects the network training to a great extent. The raw data is therefore normalised, before it is applied to the neural network. One way to normalise the data x is by using the expression: xn =
(x − xmin ) × (UR − LR ) + LR xmax − xmin
Outage of generator bus no.
Outage of line between buses
2 6 5
4–13 14–15 26–27
4. Simulation results
Fig. 2. Block diagram of the proposed ANN model.
315
(4)
where xn is the normalised value, xmin and xmax are the minimum and maximum values of the variable x, respectively, LR and UR lower and upper range for normalisation, respectively. The developed ANN model with four input neurons and n + 1 output neurons is trained by back-propagation algorithm using the training data and then assessed as to how well the network is generalised using the testing data. Tangent hyperbolic function and linear activation functions are chosen as the activation function for the hidden layer neurons and the output neurons, respectively. The connection weights are adjusted to correctly map the training set vectors at least to within a defined error limit during training. A trial and error procedure is adopted in selecting the number of hidden layers and hidden neurons in such a way as to obtain the ANN model that correctly estimates the VMs for the testing data.
The PA is tested on IEEE 14, 30 and 57 bus systems. Initially the fast decoupled load flow (FDLF) [23] is carried out for the basecase load demands. It is identified that Buses-14, 30 and 31 are the most vulnerable buses in terms of voltage instability in the IEEE 14, 30 and 57 bus systems, respectively [24]. These buses are thus considered for developing the ANN model in this paper. The list of credible contingencies, as given in Table 1, that includes an outage of a line and an outage of a generator for each system are chosen to design the neural network. There are thus four inputs and three outputs for the proposed ANN model to estimate VMs for the chosen bus. The input-target database for both training and testing data are generated by perturbing the real and reactive power demands at the chosen bus as well as the net power demand of the system as described in Section 3. A sample of the 4096 training data is given in Table A.2 of Appendix for IEEE 14 bus system. The data are normalised and thereafter the ANN is trained many times with different combinations of hidden layers and hidden neurons. Repeated trial and error studies with training and testing data reveal that two hidden layers, each with five hidden neurons, are quite satisfactory for the proposed ANN model. The network is then tested using projected input data that corresponds to different loading patterns and the results are compared with that of the of the CA in Tables 2–4 for 14, 30 and 57 bus systems, respectively. The analysis of these tables clearly reveals that the PA quickly provides VMs for the chosen buses with reasonable accuracy under normal and contingent conditions in the credible outage list for the projected load pattern. In a similar fashion, a number of models can be developed for each of the vulnerable bus. It is to be noted that the fall of VM below a certain critical value indicates that the system is likely to enter the region of voltage instability. The threshold voltage of 0.9 is chosen from stability point of view in this study. It is further observed from Table 2 that the system is safe both in the normal and in the contingent states for both the test cases 1 and 2. The system though safe under normal state for test case-3 becomes unstable on the occurrence of outage-1 or -2. However, the operating state for the remaining test cases appear to be in the critical region and demands immediate corrective actions such as reactive power compensation and load curtailment to bring the system into the safe operating zone in the 14 bus test system. The results follow the same pattern in the case of 30 and 57 bus systems as evident from Tables 4 and 5. The effect on voltage stability of the system for a projected change in load pattern from the current operating point may be studied by changing the input values as {PL k + PL k , QL k + QL k , PNET + PNET , QNET + QNET } and presenting them to the neural network. Table 5 presents the VMs for change in load patterns, treating test case-2 in Table 2 as the current operating point. It is very clear that the VMs decrease with increase in load demand and the operator may initiate decisions based on the obtained VM values. An additional load of up to (0.15 + j0.075) can be applied safely to the system, when the system is expected to be operated under normal network condition. It follows that the proposed ANN model is ideally suitable for estimation of VMs of vulnerable buses without considering the entire power system or without using any rigorous
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Table 2 Comparison of results obtained by PA with CA for 14 bus system. Test case
Input (all quantities in per unit)
Method
PL 14
QL 14
PNET
QNET
1
0.200
0.070
2.450
0.830
2
0.300
0.110
2.600
0.740
3
0.400
0.150
2.600
0.800
4
0.250
0.120
3.050
1.080
5
0.500
0.200
2.400
1.000
Output V0 14
V1 14
V2 14
CA PA CA PA CA PA CA PA CA PA
0.992 0.992 0.968 0.967 0.924 0.923 0.878 0.878 0.846 0.846
0.967 0.968 0.937 0.937 0.875 0.876 0.825 0.826 0.761 0.762
0.971 0.971 0.940 0.939 0.887 0.887 0.838 0.839 0.800 0.801
Method
Output
Table 3 Comparison of results obtained by PA with CA for 30 bus system. Test case
Input (all quantities in per unit) PL 30
QL 30
PNET
QNET
1
0.100
0.030
1.700
1.070
2
0.120
0.040
1.880
1.210
3
0.140
0.060
2.360
1.340
4
0.210
0.070
1.790
1.030
5
0.250
0.070
2.650
1.230
V0 30
V1 30
V2 30
CA PA CA PA CA PA CA PA CA PA
0.970 0.970 0.952 0.953 0.924 0.924 0.900 0.901 0.869 0.867
0.970 0.970 0.951 0.952 0.923 0.924 0.899 0.897 0.869 0.868
0.957 0.958 0.937 0.937 0.907 0.906 0.885 0.886 0.847 0.846
Method
Output V0 31
V1 31
V2 31
0.972 0.971 0.961 0.961 0.934 0.934 0.901 0.899 0.867 0.868
0.968 0.967 0.955 0.954 0.924 0.923 0.882 0.881 0.839 0.837
0.951 0.952 0.934 0.935 0.934 0.933 0.828 0.826 nc 0.001
Table 4 Comparison of results obtained by PA with CA for 57 bus system. Test case
Input (all quantities in per unit) PL 31
QL 31
1
0.040
0.012
5.960
1.088
2
0.045
0.015
6.450
1.185
3
0.050
0.020
7.450
1.580
4
0.043
0.011
9.957
2.389
5
0.055
0.012
11.445
2.888
PNET
QNET CA PA CA PA CA PA CA PA CA PA
nc: not converged.
Table 5 Effect on VM for projected load changes for 14 bus system. PL 14 = 0.30 QL 14 = 0.11 PNET = 2.60 QNET = 0.74
Current operating point (test case-2 of Table 2)
Sl. no.
Projected change in load demand PL
1 2 3 4 5 6 7 8 9
14
– 0.05 0.10 0.15 0.20 0.25 – – –
QL
14
– 0.025 0.050 0.075 0.100 0.125 – – –
Expected values of VM PNET
QNET
V0 14
V1 14
V2 14
– – – – – – 0.5 1.0 1.5
– – – – – – 0.25 0.50 0.75
0.968 0.951 0.930 0.905 0.877 0.845 0.884 0.736 0.000
0.937 0.913 0.882 0.847 0.805 0.749 0.830 0.000 0.000
0.940 0.918 0.893 0.866 0.835 0.798 0.844 0.656 0.000
P. Aravindhababu, G. Balamurugan / Applied Soft Computing 12 (2012) 313–319
317
Table 6 Effect on VM for VAR compensation for 14 bus system. PL 14 = 0.25 QL 14 = 0.12 PNET = 3.05 QNET = 1.08
Current operating point (test case-4 of Table 2)
Location of VAR support
Change in Input data
Expected values of VM
QL 14
QNET
V0 14
V1 14
V2 14
No VAR support
–
–
0.878
0.825
0.838
VAR support only at bus-14
0.015 0.030 0.045 0.060
– – – –
0.886 0.894 0.902 0.909
0.836 0.846 0.855 0.865
0.847 0.855 0.863 0.871
Net VAR support in the system other than bus-14
– – – – – – – –
0.120 0.150 0.180 0.210 0.240 0.270 0.300 0.330
0.903 0.908 0.914 0.920 0.925 0.931 0.936 0.941
0.855 0.862 0.869 0.876 0.883 0.890 0.896 0.902
0.865 0.871 0.877 0.883 0.889 0.895 0.901 0.907
VAR support at bus-14 and also at other buses in the system
0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06
0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27
0.915 0.920 0.926 0.932 0.937 0.942 0.948 0.953 0.958
0.872 0.879 0.885 0.892 0.899 0.905 0.911 0.917 0.923
0.878 0.884 0.890 0.896 0.902 0.908 0.913 0.919 0.924
iterative calculations, which facilitates the operator to foresee the system behaviour from a stability point of view for projected load patterns. Preventive measures similar to switching of capacitor banks must be considered in the event of system entering the unstable region. The effect of VAR support on VM of vulnerable bus in order to enhance voltage stability under critical situation may be studied by providing VAR support in steps either at local bus or at buses other than the local bus by changing the input values as {PL k , QL k − QL k , PNET , QNET − QNET }. Three series of tests are generated to study the effect of VAR support on VM, one by varying the VAR support only at local bus, the second by changing the net VAR compensation in the system and the last by providing fixed compensation at local bus and altering the net VAR support in the system. Table 6 presents the VMs for change in VAR support, treating test case-4 in Table 2 as the current operating point. It is very clear from these results that 0.045 per unit of VAR at bus-14 can bring the system to a safe operating condition but fails to improve the situation under contingent conditions. However, a net VAR support of 0.330 per unit without any local compensation or a much lower value of 0.18 per unit with a local VAR support of 0.06 per unit ensures the system to be safe under normal as well as under contingent conditions. The normalised execution time required by CA for each test case and for each outage of the system under study is given in Table 7. These are seemingly very large and do not favour CA to be suitable for online computations. But the ANN based PA obviously does not
require even a fraction of a second and therefore finds it suitable for online applications. It is summarised that the model ideally serves to provide approximate VAR compensation required to retain the system in the stable region in addition to enabling estimation of VMs of vulnerable buses.
5. Conclusion A novel ANN based approach that requires four inputs, irrespective of the system size, has been formulated for estimation of VM for each critical bus under normal and contingent conditions. The strategy has been coined to investigate the effect on VM for projected changes in load patterns under normal and/or credible contingencies, besides being able to compute the approximate amount of VAR compensation/sheddable load in order to ensure the stability of the system. The PA on account of its smaller execution time will be suitable for online practical implementation on systems of any size.
Acknowledgements The authors gratefully acknowledge the authorities of Annamalai University for the facilities offered to carry out this work.
Appendix A.
Table 7 Execution time. Test system
Time (ms)
IEEE 14 IEEE 30 IEEE 57
66.7 177.8 510.8
The aim of this section is to provide justifications and a methodology for obtaining the reduced number of inputs for the proposed ANN model through a sensitivity relationship between the chosen bus VM to the real and reactive power loads of all the buses. The sensitivity factors ∂V/∂P and ∂V/∂Q can be obtained from the
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P. Aravindhababu, G. Balamurugan / Applied Soft Computing 12 (2012) 313–319 Table A.2 Sample training data for IEEE 14 bus system.
Table A.1 Sensitivity factors. Bus no.
∂V ∂P
∂V ∂Q
PL 14
QL 14
PNET
QNET
V0 14
V1 14
V2 14
2 3 4 5 6 7 8 9 10 11 12 13 14
0.017 0.045 0.057 0.071 0.049 0.071 0.037 0.082 0.068 0.071 0.064 0.069 0.091
0.098 0.224 0.083 0.195 0.186 0.162 0.142 0.246 0.177 0.042 0.073 0.235 0.366
0.050 0.050 – 0.100 0.100 – 0.150 0.150 – 0.250 0.250 – 0.350 0.350 – 0.400 0.400
0.025 0.025 – 0.175 0.175 – 0.200 0.200 – 0.025 0.025 – 0.025 0.025 – 0.200 0.200
0.550 0.550 – 0.500 0.500 – 2.850 2.850 – 0.350 0.950 – 3.250 3.250 – 3.800 3.800
0.275 0.575 – 1.325 1.625 – 0.100 0.400 – 2.375 0.275 – 0.575 0.875 – 0.400 0.700
1.068 1.061 – 0.944 0.910 – 1.009 0.983 – 0.844 1.056 – 0.974 0.920 – 0.823 0.726
1.065 1.052 – 0.916 0.876 – 0.988 0.954 – 0.795 1.047 – 0.939 0.873 – 0.719 0.000
1.071 1.062 – 0.937 0.898 – 1.001 0.965 – 0.803 1.057 – 0.942 0.883 – 0.794 0.671
gradient matrix of the NR load flow. The NR load flow expression [23] can be written as
[J]
ı V/V
=
P Q
(A.1)
where J is the Jacobian matrix, ı is the voltage angle correction vector, V is the VM correction vector, P is the real power mismatch vector, and Q is the reactive power mismatch vector. Assuming V = 1.0 per unit for all the buses, the Eq. (A.1) can be written as
ı V
= [ ]
P Q
(A.2)
where = J−1 , consists of the sensitivity factors ∂V/∂P and ∂V/∂Q. The expression for V at the chosen bus-k can be written as Vk =
ki Pi +
i ∈ ˚P
ki Qj
(A.3)
j ∈ ˚Q
where ˚P and ˚Q are the set of buses corresponding to P and Q, respectively. The sensitivity terms corresponding to P and Q of the chosen bus-k is in general larger than that of the terms of the neighbouring buses. These values are very small for remote buses. These sensitivity terms for the bus-14 are obtained from NR load flow Jacobian without omitting the generator buses for IEEE-14 bus test system and given in Table A.1. Fig. A.1 shows the connectivity of bus-14 to the neighbouring buses, which are shown inside the dotted ellipse. It is very clear from this table that the sensitivity factors are larger for bus-14, comparatively lower for the adjacent buses of 13 and 9 and further decrease for the remote buses that are not directly connected to the bus-k. However, the remote buses are indirectly connected to bus-k and the influence of their loads on the VM of
14
12
13
4
9 6
7 10 Fig. A.1. Connectivity of the chosen bus-14 to neighbouring buses.
bus-k cannot be neglected. Therefore, the local real and reactive power loads at bus-k are considered as two inputs for ANN model, as the corresponding sensitivity factors are the largest. Though the sensitivity factors of the other buses are comparatively smaller, their accumulated effects can be taken into account by considering the net real and reactive load powers of the system as another two inputs for the ANN model. References [1] A.K. Sinha, D. Hazarika, A comparative study of voltage stability indices in a power system, Electrical Power and Energy Systems 22 (1) (2000) 589–596. [2] A.C. Zambroni de Souza, Discussion on some voltage collapse indices, Electric Power Systems Research 53 (1) (2000) 53–58. [3] D.B. Bedoya, C.A. Castro, L.C.P. Da Silva, A method for computing minimum voltage stability margins of power systems, IEE Proceedings-Generation Transmission and Distribution 2 (5) (2008) 676–689. [4] A. Berizzi, C. Bovo, D. Crio, M. Delfanti, M. Merlo, M. Pozzi, Online fuzzy voltage collapse risk quantification, Electric Power Systems Research 79 (5) (2009) 740–749. [5] Y. Wang, L. Wenyuan, L. Jiping, A new node voltage stability index based on local voltage phasors, Electric Power Systems Research 79 (1) (2009) 265–271. [6] S. Chakrabarti, Voltage stability monitoring by artificial neural network using regression-based feature selection method, Expert Systems with Applications 35 (4) (2008) 1802–1808. [7] Mohammad-Taghi, Vakil-Baghmisheh, R. Hadi, Dynamic voltage stability assessment of power transmission systems using neural networks, Energy Conversion and Management 49 (1) (2008) 1–7. [8] P.K. Modi, S.P. Singh, J.D. Sharma, Fuzzy neural network based voltage stability evaluation of power systems with SVC, Applied Soft Computing 8 (1) (2008) 657–665. [9] S. Kamalasadan, Ttukaram, A.K. Srivastava, A new intelligent algorithm for online voltage stability assessment and monitoring, Electrical Power and Energy Systems 31 (2–3) (2009) 100–110. [10] O.O. Obadina, G.J. Berg, Identifying electrically weak and strong segments of a power system from a voltage stability view point, IEE Proceedings-C Generation Transmission and Distribution 137 (3) (1990) 205–212. [11] K. Ramar, M.S. Raviprakasha, Determination of weak bus ordering vector in multimachine power systems from voltage stability view point, Electric Machines and Power Systems (1995) 597–608. [12] B. Stott, O. Alsac, A.C. Monticelli, Security analysis and optimisation, Proceedings of the IEEE 75 (12) (1987) 1623–1644. [13] A.J. Wood, B.F. Wollenberg, Power Generation, Operation and Control, John Wiley & Sons, New York, 1984. [14] C.Y. Lee, N. Chen, Distribution factors of reactive power flow in transmission line and transformer outage studies, Presented at IEEE/PES 1991 Summer Meeting, Paper No. 91 SM 448-7 PWRS, 1991. [15] S.N. Singh, S.C. Srivastava, Improved voltage and reactive power distribution factor for outage studies, Presented at IEEE/PES 1996 Summer Meeting, Paper No. 96 SM 471-3 PWRS, 1996. [16] V. Brandwajn, M.G. Lauby, Complete bounding method for a.c. contingency selection, IEEE Transactions on Power Systems 4 (2) (1989) 724–729. [17] J. Zaborszky, K.W. Hwang, K. Prasad, Fast contingency evaluation using concentric relaxation, IEEE Transactions on Power Apparatus and Systems PAS-99 (1) (1980) 28–36.
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[21] L. Srivastava, S.N. Singh, J. Sharma, Comparison of feature selection techniques for ANN-based voltage estimation, Electric Power Systems Research 53 (3) (2000) 187–195. [22] S. Haykin, Neural Networks: A Comprehensive Foundation, Pearson Education, 2002. [23] B. Stott, O. Alsac, Fast decoupled load flow, IEEE Transactions on Power Apparatus and Systems PAS-93 (1974) 859–869. [24] P. Kessel, H. Glavitsch, Estimating the voltage stability of a power system, IEEE Transactions on Power Delivery 1 (3) (1986) 346–354.
Contents lists available at SciVerse ScienceDirect
Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
ANN based online voltage estimation P. Aravindhababu ∗ , G. Balamurugan Electrical Engineering, Annamalai University, Annamalainagar 608 002, Tamil Nadu, India
a r t i c l e
i n f o
Article history: Received 15 April 2009 Received in revised form 22 January 2011 Accepted 14 August 2011 Available online 25 August 2011 Keywords: Artificial neural network Voltage stability Voltage collapse proximity indicator VAR support
a b s t r a c t A new ANN based algorithm that requires only a minimum number of inputs is proposed to estimate the voltage magnitude of each critical bus in a power system under normal and/or contingent states. The basic philosophy is to develop an online tool, with a view to predict the occurrence of voltage collapse due to either an outage and/or projected changes in load. Besides the scheme contemplates to aid in the process of computing the approximate VAR support/sheddable load that is required to prevent voltage instability, which is imminent on account of the stressed operating nature of the present day power networks. The performance is evaluated through standard IEEE 14, 30 and 57 bus test systems and the simulation results are presented to demonstrate its validity and suitability for practical applications. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Voltage stability analysis is concerned with the ability of assessing the power system to maintain acceptable voltages at all system buses under normal conditions and after being subjected to disturbances. Electric power systems have become larger, more complex and found to be operating dangerously close to their stability limits. The lack of new transmission facilities, inadequate reactive power support, cutbacks in system maintenance, workforce downsizing, unpredicted power flow patterns, just to name a few, are some of the important factors that affect the stability of the system. A voltage collapse can appear quite abruptly in systems or sub-systems due to the continuously changing operating conditions and various unforeseen factors associated with large power systems and hence offline stability studies can no longer be sufficient to ensure a secure operation of the power system. It is well known that unlike angle instability, voltage instability often starts in a local network and gradually extends to the whole system. This feature slows down the process of system losing voltage stability compared to that of losing angle stability, and allows time for a mechanism to predict static voltage stability [1–9]. The bus voltages are used in voltage stability studies [10,11] for computing the voltage margin in terms of bus voltage variations and in steady state security assessment [12,13] for checking the bus voltage limit violations of a power system. The prediction of bus voltages is essential for different loading and/or contingent conditions in these studies. Online estimation of bus voltage
∗ Corresponding author. Tel.: +91 9842565093 (mobile). E-mail address: [email protected] (P. Aravindhababu). 1568-4946/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2011.08.041
magnitudes (VM) at selected load buses for each contingency and/or projected load changes facilitates to foresee the possible occurrence of voltage instability and initiates precautionary measures. Therefore, there is a definite need to develop a fast bus voltage estimation technique that is ideally suitable for on-line applications. A number of methods are in vogue for bus voltage estimation in the literature [14–17]. Real time use of conventional methods in an energy management centre is difficult due to their significant requirement of large computational times. Recently artificial neural networks (ANN) appear to find its role in the area of power systems because of its usefulness in online applications. Though several ANN based methods are available for the prediction of voltage stability and proximity to voltage collapse conditions [6–9], hardly any work is reported for the online estimation of bus voltages [18–21]. Most of these methods require large number of inputs, which causes difficulty in training the network especially for larger power systems. However, efforts are being taken to reduce the number of input variables for ANN based approach. Once trained, the execution time of ANNs subjected to any input is very less, which makes it more suitable for online estimation of VMs compared to conventional methods. It is in the prelude that an attempt is made to create an ANN based model that requires as minimum a number of inputs as possible for estimating VM. The main objective is to develop an ANN based approach that requires only a minimum number of inputs in order to estimate the VM of each vulnerable bus in a power system under normal and/or under contingent condition over a wide range of load variations. The scheme is to be evaluated through simulation and the results of standard IEEE 14, 30 and 57 bus systems are compared with that of a conventional approach (CA). The scope includes to predict
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presented with test data will be able to predict the correct output in the operational phase.
H1 X1
3. Proposed approach H2
Y1
X2 H3
Y2
X3 H4 Input layer
Hidden layer
Output layer
Fig. 1. Architecture of feed forward neural network.
the VAR support/sheddable load that may be necessary to ensure voltage stability of the system. 2. Artificial neural network In recent years, ANNs seems to offer an alternative scheme for solving certain difficult power system problems, where the conventional techniques have not achieved the desired speed, accuracy and efficiency. One of the main features, which can be attributed to ANN, is its ability to learn non-linear problem offline with selective training that can pave the way to offer a sufficiently accurate online response. A multi-layer feed forward network trained by backpropagation is the most popular and versatile form of neural network for pattern mapping or function approximation [22]. It consists of an input layer, an output layer and at least one hidden layer, with each layer consisting of a set of neurons as shown in Fig. 1. The input vector representing the pattern to be recognised is presented to the input layer and distributed to the subsequent hidden layers and finally to the output layer via the weighted connections. Each neuron in the network operates by taking the sum of weighted inputs and passing the result through a nonlinear activation function, which is usually a sigmoid or hyperbolic tangent function. This is mathematically represented as,
⎡ ⎤ n wij outj + bi ⎦ outi = f (neti ) = f ⎣
(1)
j=1
where outi is the output of the ith neuron in the layer under consideration. outj is the output of the jth neuron in the preceding layer. wij are the connection weights between ith neuron and jth inputs and bi is a constant called bias. The knowledge required to map the input patterns and output is embodied in the form of weights. Initially the weights appropriate to a given problem domain are unknown. The network does not have the ability to deal with the problem to be solved until a set of appropriate weights is found. The process of finding a useful set of weights is called training. Training begins with tutoring a set consisting of specimen inputs with associated outputs. Training the network involves adjusting the connection weights to correctly map the training set vectors at least to within some defined error limit. If the training set is good and the algorithm is effective, the network will be able to correctly estimate the output even for the inputs not belonging to the training set. The application of neural network to a recognition problem involves two distinct phases: training phase and operational phase. The network weights are adopted to reflect the problem domain during the training phase. The weights are frozen and the network when
The aim of the proposed method is to determine the VM at the most vulnerable bus-k for a projected load pattern under normal network conditions as well as under credible contingent conditions using ANN with minimum number of input variables. However, if there are a large number of vulnerable buses, a separate ANN model for each bus has to be developed. The VM at any bus depends on the real and reactive power loads connected at that bus and also on the loads at the other buses. If the real and reactive load powers at all the buses are treated as inputs, which are available as measurements in a typical power system, it may be time-consuming to train a network as the number of connection weights and neurons would be extremely large. It is therefore essential to reduce the number of inputs to a neural network and select an optimum number of mutually independent inputs, which clearly establish the required input–output relationship. The effect on the VM by the real and reactive load powers of all the buses at the chosen vulnerable bus can be investigated from the sensitivity factors that relate the VM of the chosen bus-k with real and reactive load powers of all buses. Though it is observed from these factors that the effect of local bus powers on its VM is considerably large, still it is comparatively lower for the neighbouring buses and insignificant for the remote buses, which is further elaborated in Appendix A. However, the remote buses are indirectly connected to bus-k and their influence on the VM of bus-k cannot be neglected. So in the proposed approach (PA), the following four input variables, irrespective of the system size, are chosen in order to account the effects of all bus powers for the ANN model of the vulnerable bus k as • PL k , the real power demand at the interested vulnerable bus k, • QL k , the reactive power demand at the interested vulnerable bus k, • PNET , the system net real power demand excluding PL k , • QNET , the system net reactive power demand excluding Q L k , The standard procedure in most utilities is to examine the effects of all possible outages, one at a time. It is rather time consuming, if every single outage is analysed for a large power system. Therefore, a selected number of worst-case contingencies, called credible contingencies, are considered in developing the proposed model. If there are n-credible contingencies, the network then contains one output for normal network conditions and n-output for n-credible contingencies, resulting in n + 1 output variables as
V0 k , V1 k , V2 k , V3 k ...., Vn k
(2)
where V0 k represents VM at bus k under normal network conditions, Vj k represents VM at bus k under j-line/generator outage conditions. The block diagram, shown in Fig. 2, shows the ANN model for determination of VM at the most vulnerable load bus-k under normal as well as under selected contingent conditions. The training data is the only available information to build the ANN model and care is taken to include the complete possible operating conditions of the system. The testing data is used to evaluate the performance of the model. The input-target patterns are generated using the following procedure for each of the vulnerable bus k. A range of load patterns are first generated by randomly perturbing the load from the reasonable lower expected loading level to the point of voltage instability. Initially one variable preferably
P. Aravindhababu, G. Balamurugan / Applied Soft Computing 12 (2012) 313–319
PL k
P NET
Table 1 List of credible outages.
k
Vo
Q Lk ANN model for bus-k
Q NET
V1
k
V2
k
Vn
k
IEEE 14 IEEE 30 IEEE 57
PL k is chosen and set at a lower loading level while the remaining three are perturbed in small steps one by one in order to cover the entire data range. Then PL k is incremented and the remaining three are again adjusted. This process is repeated till PL k reaches the maximum loading level. The second variable QL k is chosen and the remaining three variables are adjusted accordingly. The values of PNET and QNET are distributed to all load buses other than bus k in proportion to their base case values in order to obtain widely varying combinations of load patterns with different power factors. The VM at bus k is computed after carrying out the precontingency load flow for each load pattern and then the single line/generator outage specified in the credible contingency list is simulated one by one. The corresponding VMs at bus k are computed for forming the training/testing data set as follows: PL k , QL k , PNET , QNET ⇔ V0 k , V1 k , V2 k , V3 k ...., Vn k
(3)
The generated input-target data are split into two partitions: the first is the training data, which is used to train the network and the second the testing data, serves to assess how well the network is generalised. There is a possibility of obtaining good performance on the training data followed by much poor performance on the test data. This can be avoided by ensuring that the training data is uniformly distributed. During training of the neural network, higher valued input variables may tend to suppress the influence of smaller ones. Besides, if the raw data is directly applied to the network, there is a risk of the simulated neurons reaching the saturated states. If the neuron becomes saturated, then the changes in the output value are likely to produce a very small change or no change in the output value. This affects the network training to a great extent. The raw data is therefore normalised, before it is applied to the neural network. One way to normalise the data x is by using the expression: xn =
(x − xmin ) × (UR − LR ) + LR xmax − xmin
Outage of generator bus no.
Outage of line between buses
2 6 5
4–13 14–15 26–27
4. Simulation results
Fig. 2. Block diagram of the proposed ANN model.
315
(4)
where xn is the normalised value, xmin and xmax are the minimum and maximum values of the variable x, respectively, LR and UR lower and upper range for normalisation, respectively. The developed ANN model with four input neurons and n + 1 output neurons is trained by back-propagation algorithm using the training data and then assessed as to how well the network is generalised using the testing data. Tangent hyperbolic function and linear activation functions are chosen as the activation function for the hidden layer neurons and the output neurons, respectively. The connection weights are adjusted to correctly map the training set vectors at least to within a defined error limit during training. A trial and error procedure is adopted in selecting the number of hidden layers and hidden neurons in such a way as to obtain the ANN model that correctly estimates the VMs for the testing data.
The PA is tested on IEEE 14, 30 and 57 bus systems. Initially the fast decoupled load flow (FDLF) [23] is carried out for the basecase load demands. It is identified that Buses-14, 30 and 31 are the most vulnerable buses in terms of voltage instability in the IEEE 14, 30 and 57 bus systems, respectively [24]. These buses are thus considered for developing the ANN model in this paper. The list of credible contingencies, as given in Table 1, that includes an outage of a line and an outage of a generator for each system are chosen to design the neural network. There are thus four inputs and three outputs for the proposed ANN model to estimate VMs for the chosen bus. The input-target database for both training and testing data are generated by perturbing the real and reactive power demands at the chosen bus as well as the net power demand of the system as described in Section 3. A sample of the 4096 training data is given in Table A.2 of Appendix for IEEE 14 bus system. The data are normalised and thereafter the ANN is trained many times with different combinations of hidden layers and hidden neurons. Repeated trial and error studies with training and testing data reveal that two hidden layers, each with five hidden neurons, are quite satisfactory for the proposed ANN model. The network is then tested using projected input data that corresponds to different loading patterns and the results are compared with that of the of the CA in Tables 2–4 for 14, 30 and 57 bus systems, respectively. The analysis of these tables clearly reveals that the PA quickly provides VMs for the chosen buses with reasonable accuracy under normal and contingent conditions in the credible outage list for the projected load pattern. In a similar fashion, a number of models can be developed for each of the vulnerable bus. It is to be noted that the fall of VM below a certain critical value indicates that the system is likely to enter the region of voltage instability. The threshold voltage of 0.9 is chosen from stability point of view in this study. It is further observed from Table 2 that the system is safe both in the normal and in the contingent states for both the test cases 1 and 2. The system though safe under normal state for test case-3 becomes unstable on the occurrence of outage-1 or -2. However, the operating state for the remaining test cases appear to be in the critical region and demands immediate corrective actions such as reactive power compensation and load curtailment to bring the system into the safe operating zone in the 14 bus test system. The results follow the same pattern in the case of 30 and 57 bus systems as evident from Tables 4 and 5. The effect on voltage stability of the system for a projected change in load pattern from the current operating point may be studied by changing the input values as {PL k + PL k , QL k + QL k , PNET + PNET , QNET + QNET } and presenting them to the neural network. Table 5 presents the VMs for change in load patterns, treating test case-2 in Table 2 as the current operating point. It is very clear that the VMs decrease with increase in load demand and the operator may initiate decisions based on the obtained VM values. An additional load of up to (0.15 + j0.075) can be applied safely to the system, when the system is expected to be operated under normal network condition. It follows that the proposed ANN model is ideally suitable for estimation of VMs of vulnerable buses without considering the entire power system or without using any rigorous
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Table 2 Comparison of results obtained by PA with CA for 14 bus system. Test case
Input (all quantities in per unit)
Method
PL 14
QL 14
PNET
QNET
1
0.200
0.070
2.450
0.830
2
0.300
0.110
2.600
0.740
3
0.400
0.150
2.600
0.800
4
0.250
0.120
3.050
1.080
5
0.500
0.200
2.400
1.000
Output V0 14
V1 14
V2 14
CA PA CA PA CA PA CA PA CA PA
0.992 0.992 0.968 0.967 0.924 0.923 0.878 0.878 0.846 0.846
0.967 0.968 0.937 0.937 0.875 0.876 0.825 0.826 0.761 0.762
0.971 0.971 0.940 0.939 0.887 0.887 0.838 0.839 0.800 0.801
Method
Output
Table 3 Comparison of results obtained by PA with CA for 30 bus system. Test case
Input (all quantities in per unit) PL 30
QL 30
PNET
QNET
1
0.100
0.030
1.700
1.070
2
0.120
0.040
1.880
1.210
3
0.140
0.060
2.360
1.340
4
0.210
0.070
1.790
1.030
5
0.250
0.070
2.650
1.230
V0 30
V1 30
V2 30
CA PA CA PA CA PA CA PA CA PA
0.970 0.970 0.952 0.953 0.924 0.924 0.900 0.901 0.869 0.867
0.970 0.970 0.951 0.952 0.923 0.924 0.899 0.897 0.869 0.868
0.957 0.958 0.937 0.937 0.907 0.906 0.885 0.886 0.847 0.846
Method
Output V0 31
V1 31
V2 31
0.972 0.971 0.961 0.961 0.934 0.934 0.901 0.899 0.867 0.868
0.968 0.967 0.955 0.954 0.924 0.923 0.882 0.881 0.839 0.837
0.951 0.952 0.934 0.935 0.934 0.933 0.828 0.826 nc 0.001
Table 4 Comparison of results obtained by PA with CA for 57 bus system. Test case
Input (all quantities in per unit) PL 31
QL 31
1
0.040
0.012
5.960
1.088
2
0.045
0.015
6.450
1.185
3
0.050
0.020
7.450
1.580
4
0.043
0.011
9.957
2.389
5
0.055
0.012
11.445
2.888
PNET
QNET CA PA CA PA CA PA CA PA CA PA
nc: not converged.
Table 5 Effect on VM for projected load changes for 14 bus system. PL 14 = 0.30 QL 14 = 0.11 PNET = 2.60 QNET = 0.74
Current operating point (test case-2 of Table 2)
Sl. no.
Projected change in load demand PL
1 2 3 4 5 6 7 8 9
14
– 0.05 0.10 0.15 0.20 0.25 – – –
QL
14
– 0.025 0.050 0.075 0.100 0.125 – – –
Expected values of VM PNET
QNET
V0 14
V1 14
V2 14
– – – – – – 0.5 1.0 1.5
– – – – – – 0.25 0.50 0.75
0.968 0.951 0.930 0.905 0.877 0.845 0.884 0.736 0.000
0.937 0.913 0.882 0.847 0.805 0.749 0.830 0.000 0.000
0.940 0.918 0.893 0.866 0.835 0.798 0.844 0.656 0.000
P. Aravindhababu, G. Balamurugan / Applied Soft Computing 12 (2012) 313–319
317
Table 6 Effect on VM for VAR compensation for 14 bus system. PL 14 = 0.25 QL 14 = 0.12 PNET = 3.05 QNET = 1.08
Current operating point (test case-4 of Table 2)
Location of VAR support
Change in Input data
Expected values of VM
QL 14
QNET
V0 14
V1 14
V2 14
No VAR support
–
–
0.878
0.825
0.838
VAR support only at bus-14
0.015 0.030 0.045 0.060
– – – –
0.886 0.894 0.902 0.909
0.836 0.846 0.855 0.865
0.847 0.855 0.863 0.871
Net VAR support in the system other than bus-14
– – – – – – – –
0.120 0.150 0.180 0.210 0.240 0.270 0.300 0.330
0.903 0.908 0.914 0.920 0.925 0.931 0.936 0.941
0.855 0.862 0.869 0.876 0.883 0.890 0.896 0.902
0.865 0.871 0.877 0.883 0.889 0.895 0.901 0.907
VAR support at bus-14 and also at other buses in the system
0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06
0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27
0.915 0.920 0.926 0.932 0.937 0.942 0.948 0.953 0.958
0.872 0.879 0.885 0.892 0.899 0.905 0.911 0.917 0.923
0.878 0.884 0.890 0.896 0.902 0.908 0.913 0.919 0.924
iterative calculations, which facilitates the operator to foresee the system behaviour from a stability point of view for projected load patterns. Preventive measures similar to switching of capacitor banks must be considered in the event of system entering the unstable region. The effect of VAR support on VM of vulnerable bus in order to enhance voltage stability under critical situation may be studied by providing VAR support in steps either at local bus or at buses other than the local bus by changing the input values as {PL k , QL k − QL k , PNET , QNET − QNET }. Three series of tests are generated to study the effect of VAR support on VM, one by varying the VAR support only at local bus, the second by changing the net VAR compensation in the system and the last by providing fixed compensation at local bus and altering the net VAR support in the system. Table 6 presents the VMs for change in VAR support, treating test case-4 in Table 2 as the current operating point. It is very clear from these results that 0.045 per unit of VAR at bus-14 can bring the system to a safe operating condition but fails to improve the situation under contingent conditions. However, a net VAR support of 0.330 per unit without any local compensation or a much lower value of 0.18 per unit with a local VAR support of 0.06 per unit ensures the system to be safe under normal as well as under contingent conditions. The normalised execution time required by CA for each test case and for each outage of the system under study is given in Table 7. These are seemingly very large and do not favour CA to be suitable for online computations. But the ANN based PA obviously does not
require even a fraction of a second and therefore finds it suitable for online applications. It is summarised that the model ideally serves to provide approximate VAR compensation required to retain the system in the stable region in addition to enabling estimation of VMs of vulnerable buses.
5. Conclusion A novel ANN based approach that requires four inputs, irrespective of the system size, has been formulated for estimation of VM for each critical bus under normal and contingent conditions. The strategy has been coined to investigate the effect on VM for projected changes in load patterns under normal and/or credible contingencies, besides being able to compute the approximate amount of VAR compensation/sheddable load in order to ensure the stability of the system. The PA on account of its smaller execution time will be suitable for online practical implementation on systems of any size.
Acknowledgements The authors gratefully acknowledge the authorities of Annamalai University for the facilities offered to carry out this work.
Appendix A.
Table 7 Execution time. Test system
Time (ms)
IEEE 14 IEEE 30 IEEE 57
66.7 177.8 510.8
The aim of this section is to provide justifications and a methodology for obtaining the reduced number of inputs for the proposed ANN model through a sensitivity relationship between the chosen bus VM to the real and reactive power loads of all the buses. The sensitivity factors ∂V/∂P and ∂V/∂Q can be obtained from the
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P. Aravindhababu, G. Balamurugan / Applied Soft Computing 12 (2012) 313–319 Table A.2 Sample training data for IEEE 14 bus system.
Table A.1 Sensitivity factors. Bus no.
∂V ∂P
∂V ∂Q
PL 14
QL 14
PNET
QNET
V0 14
V1 14
V2 14
2 3 4 5 6 7 8 9 10 11 12 13 14
0.017 0.045 0.057 0.071 0.049 0.071 0.037 0.082 0.068 0.071 0.064 0.069 0.091
0.098 0.224 0.083 0.195 0.186 0.162 0.142 0.246 0.177 0.042 0.073 0.235 0.366
0.050 0.050 – 0.100 0.100 – 0.150 0.150 – 0.250 0.250 – 0.350 0.350 – 0.400 0.400
0.025 0.025 – 0.175 0.175 – 0.200 0.200 – 0.025 0.025 – 0.025 0.025 – 0.200 0.200
0.550 0.550 – 0.500 0.500 – 2.850 2.850 – 0.350 0.950 – 3.250 3.250 – 3.800 3.800
0.275 0.575 – 1.325 1.625 – 0.100 0.400 – 2.375 0.275 – 0.575 0.875 – 0.400 0.700
1.068 1.061 – 0.944 0.910 – 1.009 0.983 – 0.844 1.056 – 0.974 0.920 – 0.823 0.726
1.065 1.052 – 0.916 0.876 – 0.988 0.954 – 0.795 1.047 – 0.939 0.873 – 0.719 0.000
1.071 1.062 – 0.937 0.898 – 1.001 0.965 – 0.803 1.057 – 0.942 0.883 – 0.794 0.671
gradient matrix of the NR load flow. The NR load flow expression [23] can be written as
[J]
ı V/V
=
P Q
(A.1)
where J is the Jacobian matrix, ı is the voltage angle correction vector, V is the VM correction vector, P is the real power mismatch vector, and Q is the reactive power mismatch vector. Assuming V = 1.0 per unit for all the buses, the Eq. (A.1) can be written as
ı V
= [ ]
P Q
(A.2)
where = J−1 , consists of the sensitivity factors ∂V/∂P and ∂V/∂Q. The expression for V at the chosen bus-k can be written as Vk =
ki Pi +
i ∈ ˚P
ki Qj
(A.3)
j ∈ ˚Q
where ˚P and ˚Q are the set of buses corresponding to P and Q, respectively. The sensitivity terms corresponding to P and Q of the chosen bus-k is in general larger than that of the terms of the neighbouring buses. These values are very small for remote buses. These sensitivity terms for the bus-14 are obtained from NR load flow Jacobian without omitting the generator buses for IEEE-14 bus test system and given in Table A.1. Fig. A.1 shows the connectivity of bus-14 to the neighbouring buses, which are shown inside the dotted ellipse. It is very clear from this table that the sensitivity factors are larger for bus-14, comparatively lower for the adjacent buses of 13 and 9 and further decrease for the remote buses that are not directly connected to the bus-k. However, the remote buses are indirectly connected to bus-k and the influence of their loads on the VM of
14
12
13
4
9 6
7 10 Fig. A.1. Connectivity of the chosen bus-14 to neighbouring buses.
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