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Voltage stability monitoring of power systems using reduced network and artificial neural network
Electrical Power and Energy Systems 87 (2017) 43–51
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Voltage stability monitoring of power systems using reduced network and artificial neural network Syed Mohammad Ashraf ⇑, Ankur Gupta, Dinesh Kumar Choudhary, Saikat Chakrabarti Department of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur, India
a r t i c l e
i n f o
Article history: Received 13 January 2016 Received in revised form 26 September 2016 Accepted 16 November 2016
Keywords: Artificial neural network Continuation power flow Network reduction Voltage stability
a b s t r a c t This paper presents network reduction based methodologies to monitor voltage stability of power systems using limited number of measurements. In a multi-area power system, artificial neural networks (ANNs) are used to estimate the loading margin of the overall system, based on measurements from the internal area only. Information regarding the important measurements from the external areas is considered in measurement transformation through the network reduction process, to enhance the estimation accuracy of the ANNs. A Z-score based bad or missing data processing algorithm is implemented to make the methodologies robust. To account for changing operating conditions, adaptive training of the ANNs is also suggested. The proposed methods are successfully implemented on IEEE 14-bus and 118bus test systems. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction The electrical power demand is increasing day-by-day, and the generation is limited. Deregulated and open electricity markets are norms of the day, and it is essential to maintain reliable and good quality of the electricity supply. All this has resulted in large interconnected power networks that are operated under heavily loaded conditions, and are often close to their stability limits. Power system voltage instability is now one of the challenging problems faced by the utilities. Modern day energy management systems (EMSs) have strong focus on online voltage stability monitoring [1–3]. The maximum power loadability limit of the transmission network is one of the widely used indices to represent the voltage security of a power system [4]. It is critical for the utilities to track how close the transmission network is, to its maximum loading limit, so that in case of emergency, proper control actions can be taken. A large amount of literature exists on the use of analytical methods for voltage stability monitoring [5–8]. The conventional P-V curves are extensively used by the utilities for determination of the maximum permissible loading [9]. Continuation power flow (CPF) method is frequently used for obtaining the P-V curves [4–6]. The loadability limit is determined by increasing the system load in a particular direction, representing the most probable stressing
⇑ Corresponding author. E-mail addresses: [email protected] (S.M. Ashraf), [email protected] (A. Gupta), [email protected] (D.K. Choudhary), [email protected] (S. Chakrabarti). http://dx.doi.org/10.1016/j.ijepes.2016.11.008 0142-0615/Ó 2016 Elsevier Ltd. All rights reserved.
scenario. In order to monitor the voltage security in real-time, it is necessary that the process of measurement and estimation of the state variables and the analysis be performed within a desired time frame. Conventionally, remote terminal units (RTUs) have been used to collect measurements from various locations in a power system. The refresh rate of RTU measurements is typically a few seconds. With the advent of phasor measurement units (PMUs), it is now possible to obtain the measurement data at a sub-second rate [10–12]. In the presence of conventional measurements (from RTUs), the voltage phasors at the buses are obtained from the traditional supervisory control and data acquisition (SCADA) based state estimator (SE), typically every few minutes. Real-time monitoring of the power system, therefore, is not possible with the help of conventional SCADA based measurements. Because of high refresh rate and better accuracy, PMUs are increasingly being deployed in modern power systems. This paper presents an algorithm for fast monitoring of the voltage stability of the system, utilizing PMU measurements. A reduced network containing the buses observed by the PMUs is used to develop the proposed methodologies. Many works have been reported in the literature, exploring the capability of artificial neural network (ANN) for voltage stability monitoring [13–17]. ANN is used in these works, typically to establish a relationship between a voltage stability indicator and the measurable power system parameters affecting the indicator. These methodologies require a large number of ANN inputs which significantly increases the ANN size and diminishes its accuracy. Refs. [18–20] discuss the assessment of voltage stability using
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artificial neural network with reduced set of inputs. The focus in these methodologies is on eliminating the redundant measurements and thus reducing the number of variables needed to assess the system voltage stability. It is assumed in these methodologies that a large number of measurement variables are available. The present work considers a more realistic scenario, in which only a few nodes in the system are assumed to have PMUs installed because of economic constraints. Only the data measured by these PMUs is utilized for stability analysis. The computational burden, as well as, the communication requirement for the stability analysis of large interconnected power systems can be reduced by using a reduced power system model. The reduced model also renders the advantage of: (i) monitoring the system by using only a limited number of measuring instruments; (ii) eliminating the need for detailed model in electrically remote areas; and (iii) monitoring the interconnected system, in which participating utilities are reluctant to share vital data. Usually, a utility’s own system is called the internal area. The rest of the system is called the external area for the internal subsystem under consideration. For running power system analysis functions in the internal area, the internal subsystem is typically modeled in detail. The external areas are usually represented by simple models, referred to as the external equivalent system. Numerous techniques for determining the reduced equivalent have been proposed in the literature [21–23]. In this paper, Ward equivalent technique is used and it is assumed that measurements of all the nodes in the internal area are available via PMUs. Two schemes for estimating the voltage instability in an interconnected power system are presented in this paper, using the reduced equivalent of the entire network. The fundamental motive of this research is fast and accurate assessment of the voltage stability of an entire network based on the data measured by PMUs at certain critical nodes. A feed-forward back-propagation network (FFBPN) is used to estimate the maximum loadability of the network. In the first scheme, complex bus voltages measured by the PMUs are used as the input to the FFBPN, and the available loadability margin is used as an indicator of the system voltage stability. In the second scheme, before estimating the system voltage stability margin, an FFBPN is first used to estimate the external network bus voltages by using the internal network bus voltages. The proposed networks have the ability to get adaptive training, when subjected to any new training pattern, following a change in the system operating condition. The proposed strategy is applied to IEEE 14-bus and 118-bus systems. The complex bus voltages measured by using PMUs may be subject to data packet loss or bad data. The bad data are recognized using Z-score algorithm [24]. The dropped data packets are compensated by using polynomial curve fitting. A method to update the ANN weights to incorporate a new training pattern is also presented. The key contributions of this paper are the following. Utilizing the voltage phasors measured by PMUs located at internal buses of the system to determine the system voltage stability. Although changes in the internal area parameters are reflected by PMU measurements, changes in loading condition for external area are unaccounted. For more accurate estimation of voltage stability, an ANN and ward reduction based method is presented to account for changes in the loading in the external area. To further strengthen the reliability in determination of system voltage stability, a bad data detection and correction method based on Z-score algorithm is presented. The power system may undergo changes after installation of ANN. To accommodate for these changes in the system, an adaptive training technique for ANN is also presented.
The paper is organized as follows. Section 2 illustrates the ward reduction technique, Section 3 defines ANN scheme for the proposed methods of voltage stability monitoring with load variations, Section 4 proposes a scheme to address bad or missing data, and the problem of updation of ANN weights is discussed in Section 5. The analysis of the results is given in Section 6, and Section 7 concludes the paper. A description of voltage stability indicator is presented in Appendix A. 2. Ward reduction One of the important aspects of the proposed methodology is the use of reduced network representation of the power system. This enables the monitoring of the voltage stability of the system by observing fewer number of nodes and in reduced amount of computational time. Ward reduction technique [21,25] is adopted in this paper for carrying out the network reduction function. The construction of Ward’s equivalent starts from the solved model of the entire interconnected power system. The injected current iðiÞ at each bus i is determined from the bus’s known complex power injection sðiÞ and voltage v ðiÞ.
iðiÞ ¼ s ðiÞ=v ðiÞ
ð1Þ
Gaussian elimination technique is used in this method to get the reduced network and current vectors. The nodal equations describing the power system are given by,
Ybus v ¼ i
ð2Þ
where Ybus is the n n bus admittance matrix, v is the n 1 vector of complex voltages at all nodes, and i is the n 1 vector of complex currents injected at all nodes. After elimination of the kth node, Ybus is modified as,
Y 0ij ¼ Y ij
Y ik Y kj Y kk
; 8i; j ¼ 1; . . . ; n; i; j–k
ð3Þ
where Y 0ij are the elements of the new ðn 1Þ ðn 1ÞY bus matrix. 0
The modified current vector, i , is given by, 0
i ðiÞ ¼ iðiÞ
Y ik iðkÞ; 8i ¼ 1; . . . ; n; i–k Y kk
ð4Þ
If the network is reduced to r nodes, a new bus admittance matrix of dimension r r, and a new current injection vector of dimension r 1 is obtained. The modified current vector is converted back into power injections for analysis purpose [21,25– 27]. This reduced network carries full information of the original power system at the base case. 3. Proposed ANN architecture The feed-forward ANNs employing back-propagation learning algorithm are used in this work as a mapping tool to estimate the available loadability margin of the system. The feed-forward networks are capable of approximating any measurable function to the desired accuracy level [28]. The back-propagation steps repeatedly adjust the weights of the connections in the network to minimize the mean squared error between the desired output and the output of the ANN. In this work, the ANNs are trained by using Levenberg-Marquardt algorithm [29]. The maximum loadability limit is considered in this paper as a measure of the voltage stability of the power system. If the total system load exceeds this limit, it will result in voltage collapse and possibly blackout. The objective here is to determine the point of maximum loadability so that corrective and/or preventive actions can be taken to avoid any voltage instability problem. In
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order to reduce computational burden and fast assessment of the loadability limit, measurements from the retained buses in the reduced network are used. Two separate methods using different ANN configurations are investigated in this paper. Both of these methods use the same training data set. The training data comprises of a set of measured voltage phasors at the buses that are directly or indirectly observable by PMUs installed in the system, and the corresponding loading margins. A large number of different operating conditions are generated by randomly altering the base case real and reactive power of the loads in the entire system in the range of ±30%, and correspondingly adjusting the generation. The output data set comprises of the maximum loadability limits for these operating conditions, which are obtained by running CPF up to the nose point. The objective here is to estimate the loadability margin based on a limited number of PMU measurements. The original system may not be completely observable with PMU measurements. A subset of the buses is first specified, keeping only the buses that are directly or indirectly observable by PMUs. Both the methods discussed below utilizes the voltage phasors measured at these buses, for estimating the loadability margin of the entire system. The main advantage of using PMU measurements is their high refresh rate and accuracy, compared to the conventional measurements. 3.1. Method 1 In Method 1, the loadability margin of the entire system is estimated, based on the voltage phasor measurements from the buses observed by the PMUs. Since only a small number of PMU measurements are used (the system is not required to be completely observable by PMUs), and the PMU measurements are obtained at sub-second rate (50 or 60 times per second, for 50 and 60 Hz systems, respectively), the loadability margin can be estimated at a sub-second rate. An ANN, hereafter called as ANN 1, is trained to estimate the maximum loadability margin based on the voltage phasors measured by the PMUs in the system. There is a single hidden layer in ANN 1. The number of neurons in the hidden layer is chosen according to the following thumb rule [30].
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3.2. Method 2 Although the methodology discussed in the previous section is easy to implement, its performance may deteriorate in case of significant change in the loading conditions in the buses not retained in the reduced network. To overcome this limitation, in the second method, i.e., in Method 2, the bus voltage phasors in the external system (part of the system outside the reduced network) are first estimated by using the available PMU measurements. An ANN, designated as ANN 2, shown in Fig. 2, is used for this purpose. It estimates the voltage phasors of the external buses by using the voltage phasors measured in the reduced network (i.e., the internal area) as the input. This makes available the information regarding the voltage phasors at all buses in the system. Some of these voltage phasors are directly observable by the PMUs; the remaining are estimated by ANN 2. Using these measured and estimated voltage phasors, known network parameters, and topology; the Ward reduction method discussed in Section 2 is used to obtain the power injections in the internal area. The main motivation for this step is to include the information regarding the power injections in the external area, in the power injections in the internal area or the reduced network. Due to the transformations of the power injections as a result of network reduction, the information regarding the power injections in the external area is now embedded in the power injections in the internal area [25,31]. An ANN, referred as ANN 3, is now trained with the real and reactive power injections in the internal area as the input, obtained after transformation resulting from the network reduction step discussed above. For each pattern of the input data, the output is the corresponding maximum loadability margin, which is obtained by running CPF for the entire network. Fig. 3 describes the configuration of ANN 3, which has a single hidden layer with the number of neurons determined by (5).
4. Bad data detection and correction
where nh is the number of neurons in the hidden layer, niv is the number of elements in the input vector, and nov is the number of elements in the output vector. The inputs to ANN 1 are the complex voltage phasors measured by the PMUs, and the output is the system loadability limit. Fig. 1 gives a schematic representation of ANN 1. Since the inputs to the ANN are the complex voltages measured by the PMUs, which are available at sub-second rate, the calculation of system loadability limit is also possible to complete at sub-second rate.
Once trained properly, the ANNs described above can be implemented in actual power systems to estimate the system loadability margin. Measurements from PMUs are to be used for these loading margin estimators. PMU measurements, however, may be subjected to the problems of bad data, loss of data packets, communication link failure, noise, etc. In case of such erroneous data, the estimated loading margin may not be accurate. Pre-processing of the PMU data i.e., processing of the measurement data coming from the PMUs before utilizing it as input data for ANNs, is therefore required to ensure the reliability and robustness of the ANNbased estimators. A commonly encountered type of bad data in field measurements is the outliers. An outlier is an observation that appears to deviate markedly from other observations in the sample. A number of techniques are reported in the literature to spot the outliers in
Fig. 1. Block diagram of ANN 1.
Fig. 2. Block diagram of ANN 2.
nh ¼
2 niv þ nov 3
ð5Þ
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32 3 2 3 n n X X a0 m x ðkÞ yðkÞ 76 7 6 7 6 76 7 6 k¼1 7 6 k¼1 k¼1 76 7 6 7 6 76 7 6 7 6 7 7 7 6 X 6 6 n n n n X X 76 7 6 X 7 6 2 mþ1 a 7 7 6 6 6 1 xðkÞ x ðkÞ x ðkÞ xðkÞyðkÞ 7 76 7 6 7 6 76 7 ¼ 6 k¼1 7 6 k¼1 k¼1 k¼1 76 7 6 7 6 76 . 7 6 7 6 7 7 7 6 6 6 . . . . . .. . .. . . 7 7 7 6 6 6 . . 76 7 6 7 6 76 7 6 7 6 76 7 6 X 7 6X n n n n X X 54 5 4 5 4 m mþ1 mþm m x ðkÞ x ðkÞ x ðkÞ x ðkÞyðkÞ a k¼1 k¼1 k¼1 k¼1 m ð11Þ 2
n
Fig. 3. Block diagram of ANN 3.
the measured data [24,32–34]. A simple and fast way to identify the outliers is the Z-score [24]. The Z-score of an observation, Z i , is defined as,
Zi ¼
Yi Y
r
;
ð6Þ
where Y i , Y, and r denotes the observation at the ith instant, the sample mean, and the sample standard deviation, respectively. The mean and standard deviations are calculated from the observed values over a window consisting of certain number of past samples. The necessary condition for being an outlier is,
jZ i j > Z th ;
ð7Þ
where Z th is the threshold value for the Z-score. The variation in Y i should be within a specific percentage of the first sample in the observed window. The condition described (7) ensures that the Zscore at the ith instant is Z th units of standard deviations away from its mean, i.e. the measurement at the ith instant is significantly different from the other observed values during the inspected period. Once the measurement is determined as a possible outlier, proper care should be taken so that it does not affect the case study. Loss of packets of PMU data can also be treated as outliers, and the method described above can be used to detect such events. A simple and efficient way to replace missing data is to use the polynomial curve fitting technique [35,36]. The idea behind polynomial curve fitting is to fit a polynomial to a set of n data points by utilizing least squares error approach. Consider a general form of polynomial of mth order,
f ðxÞ ¼ a0 þ a1 x1 þ þ am xm ¼ a0 þ
m X
Eq. (11) is solved to determine the coefficients of the curve that best fits the given data. The polynomial, thus derived, is used to estimate the missing data points or to replace the outliers. 5. Adaptive training of the ANNs In practice, the weights obtained after completion of the ANN training are saved and used during the operational phases as constant values. However, the parameters of a power system do not remain static. This implies that the training dataset may not be adequate to handle the unexpected changes in the system parameters. The weights are required to be dynamic and changing in an adaptive manner, based on the recent changes of the system. To address the issue, an adaptive weight update strategy is proposed in this paper. The initial training is carried out in conventional manner, and the ANN is deployed in the field. When the prediction starts, the weights are updated based on the error in the previous performance. Once the complete data is available for the system, the loadability limit is calculated using CPF and the error in estimation is computed. If the error is in excess of a pre-specified threshold value, the parameters, and the error are recorded. If the errors are found in excess for more than n predictions (n depends on the system), the n recorded errors are used to train the ANN again, and updated weights are obtained. This means that an ongoing training for the ANNs is implemented in the field to account for the changes in the operating conditions of the system. This is best explained with the following flowchart (Fig. 4): 6. Results
aj x j
ð8Þ
j¼1
The precise form of function f ðxÞ is determined based on the values of the coefficients, a0 ; . . . ; am . The curve which gives minimum error between data, y, and the fit, f ðxÞ, is the most appropriate. The residual, R, is obtained by,
R2 ¼
n X xðkÞ
n X 2 ½yðkÞ fa0 þ a1 xðkÞ þ a2 x2 ðkÞ þ þ am xm ðkÞg
ð9Þ
k¼1
where y(k) is the desired output at the kth instant, x(k) is the input at kth instant, and m is the order of the polynomial. The partial derivatives obtained from (9) are, n m X X @R2 ¼ 2 ½yðkÞ fa0 þ aj x j ðkÞg ¼ 0 @a0 j¼1 k¼1 n m X X @R2 ¼ 2 ½yðkÞ fa0 þ aj x j ðkÞgxðkÞ ¼ 0 @a1 j¼1 k¼1
ð10Þ
n m X X @R2 ¼ 2 ½yðkÞ fa0 þ aj x j ðkÞgxm ðkÞ ¼ 0 @am j¼1 k¼1
Eq. (10) can be re-written and a Vandermonde matrix is obtained as shown below [35,36],
The proposed methodologies are tested on standard IEEE 14bus and 118-bus systems [37]. The test systems are partitioned into internal and external areas. The partitioning of the system is based on a sensitivity analysis which identifies the most significant measurements that have to be retained in the reduced network [38]. The partitioning of IEEE 14-bus system is demonstrated in Fig. 5 [38]. Tables 1 and 2 present the internal and external buses considered in IEEE 14-bus and 118-bus systems, respectively. All the buses in the internal area are directly or indirectly observable by placing the PMUs at critical nodes. The nodes in the internal area which have PMUs installed are mentioned in Table 3. The time-series data for the studied cases were generated using PSAT software [39]. Simulation is performed using a variable time step integration option and later on data is interpolated at every 0.1 s. To simulate the phasor measurement noise, a random noise with standard deviation r = 0.01 p.u. is also added onto the obtained time series data. Utilizing the real and reactive loads and the real power generation of the base case, 240 random sets of loads and corresponding generations are created with in a limit of ±30% of the base case loading. The CPF analysis is carried on all the 240 sets to determine a set of maximum loadability points for the system. The test systems are then divided into internal and external areas.
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S.M. Ashraf et al. / Electrical Power and Energy Systems 87 (2017) 43–51 Table 1 Distribution of the IEEE 14-bus system. Internal system buses
External system buses
1, 2, 3, 4, 5
6, 7, 8, 9, 10, 11, 12, 13, 14
Table 2 Distribution of the IEEE 118-bus system. Internal system buses
External system buses
24, 45–77, 116, 118
1–23, 25–44, 78–115, 117
Table 3 Nodes in the internal subsystem having PMUs installed. Test system
Number of PMUs
Location of PMUs
IEEE 14-Bus System IEEE 118-Bus System
1 10
2 47, 49, 52, 56, 62, 64, 68, 70, 71, 76
6.1. Proposed ANN architecture
Fig. 4. Flow chart for adaptive update of ANN weights.
Fig. 5. Internal and external areas of IEEE 14-bus system.
The Neural Network Toolbox Version 8.2.1 provided with MATLAB 8.4 [40] was used to develop the neural network models. The data obtained from the CPF analysis was called in a MATLAB script for training, testing, and validation purposes. In the first scenario, the set of complex voltages measured by PMUs serve as input for ANN 1, and the set of maximum loadability points serve as the targets. 70% of the data set is used for training, 15% for validation, and 15% for testing. The number of neurons in all the ANNs is determined by first applying (5) and then through rigorous testing using trial and error. ANN 1 comprises of a single hidden layer. For IEEE 14-bus system, the number of neurons in input, hidden, and output layers are 10, 8, and 1, respectively. For IEEE 118-bus system, the number of neurons in input, hidden, and output layer is 72, 38, and 1, respectively. In the second scenario, the set of real and imaginary parts of the internal area complex voltage are fed as input for ANN 2 and the set of real and imaginary parts of external area complex voltage are its targets. Real and reactive power injections corresponding to the reduced system are fed as input to ANN 3, and the set of maximum loadability points are the targets. In both ANN 2 and ANN 3, 70% of the data is used for training, 15% for validation and 15% for testing. ANN 2 and ANN 3 also comprise of a single hidden layer. For IEEE 14-bus system, the number of neurons in input, hidden, and output layers are 10, 22, and 18, respectively. For IEEE 118-bus system, the number of neurons in input, hidden, and output layer is 72, 87, and 164, respectively. ANN 3 has a structure similar to that of ANN 1. For testing of these ANNs, the dataset of 240 samples is used as input. Figs. 6 and 7 show the performance of ANN 1 for IEEE 14-bus and 118-bus, respectively. Figs. 8 and 9 show the performance of ANN 3 for IEEE 14-bus and 118-bus, respectively. The target is the actual set of maximum loadability point and output is the set of maximum loadability point determined form ANNs. In the first scenario, for IEEE 14-bus system, mean squared error is determined as 3.370exp-04 and maximum error is 7.770exp-02. For IEEE 118-bus system, mean squared error is determined as 1.249exp-03 and maximum error is 1.757exp-01, as can be seen in Table 4. In the second scenario, for IEEE 14-bus system, mean squared error is determined as 3.265exp-04, and maximum error is 8.926exp-02. For IEEE 118-bus system, mean square error is
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Percentage error in maximum loadability
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Table 4 Results for Method 1.
4 3
Test system
Mean squared error
Maximum error
2
IEEE 14-bus IEEE 118-bus
3.370exp-04 1.249exp-03
7.770exp-02 1.757exp-01
Test system
Mean squared error
Maximum error
IEEE 14-bus IEEE 118-bus
3.265exp-04 8.015exp-04
8.926exp-02 3.059exp-01
1 0 0
50
100
150
200
250
Test case Fig. 6. Percentage error in maximum loadability for IEEE 14-bus system using ANN 1.
Table 5 Results for Method 2.
better. However, for IEEE 118-bus system, scenario 2 outperforms scenario 1. The reason for better performance of scenario 2 for bigger system is that, with the system getting bigger, and lesser number of monitored or retained nodes, it is better to incorporate the loading details of external system via network reduction process. The maximum loadability point for smaller systems with moderate number of retained or monitored nodes can found out by both the scenarios with similar accuracies. 6.2. Bad data detection and correction
Fig. 7. Percentage error in maximum loadability for IEEE 118-bus system using ANN 1.
Fig. 8. Percentage error in maximum loadability for IEEE 14-bus system using ANN 3.
The Z-score algorithm discussed earlier is used for detection of bad data. For testing of the algorithm, random error (<35%) is introduced into the complex voltage measurement coming from PMUs, for preselected buses, at a particular instant. The normal variation in voltage at the buses is considered to be ±5%. The output sequence, thus obtained is tested for outliers by Z-score algorithm. For IEEE 14-bus system test case, random errors of 32.7%, 31.56%, 30.36%, and 30.91% were introduced in the voltage magnitude and angle measurements coming from bus number 4 and 5, respectively; at the 13th instant of a 30 sample observation window. In case of IEEE 118-bus test system, random errors of 34.16%, 33.09%, 32.6%, 34.32%, 30.49%, and 34.54% were introduced in the voltage magnitude and angle measurements coming from bus number 45, 50, and 51, respectively; at the 23rd instant of the observation window. The plots of the Z-score test are presented in Figs. 10–13. It is observed that the Z-score of the 13th and 23rd instant of IEEE 14-bus and IEEE 118-bus, respectively, are very high in comparison to the Z-score of previous instances, indicating an outlier. Based on the observation, the threshold value of the Zscore for bad data detection was selected to be 2.5. Both angle and magnitude of the power systems can be used for outlier detection. Once bad data is detected, it is corrected using curve fitting technique. In this work, a second order curve fit is applied to data
Fig. 9. Percentage error in maximum loadability for IEEE 118-bus system using ANN 3.
determined as 8.015exp-04, and maximum error is 3.059exp-01, as can be seen in Table 5. It can be seen that both the mean squared error and the maximum error for IEEE 14-bus system are comparable in both the scenarios, with scenario 2 coming out to be slightly
Fig. 10. Plot of Z-score obtained from bus voltage angle against PMU sampling instant for IEEE 14-bus system.
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collected in the observation window. A tabulated output of the curve fitting technique is presented in Tables 6 and 7. 6.3. Adaptive update of ANN weights
Fig. 11. Plot of Z-score obtained from bus voltage magnitude against PMU sampling instant for IEEE 14-bus system.
To test the adaptive update of ANN weights, some loads are introduced in the external system. The ANNs trained earlier for testing of proposed architecture are then retrained using the new error. In IEEE 14-bus system, an active load of 20 MW is applied at bus 7, whereas, in IEEE 118-bus test system, an active load of 70 MW is applied at bus 9, and training data set are obtained as in the previous case. A comparison of ANN output error without weight update and with weight update is presented in Tables 8 and 9. It is observed that the ANN architecture with adaptive weight gives better results in comparison to the ANN architecture without an adaptive weight scheme. The update of ANN weights is not reflected well in performance of Method 2, as the update of weights are done for ANN 2 only, and the input of ANN 3 depends on output of ANN 2. Since ANN 2 will be facing a new scenario, its output will have some error. However, an improvement is seen in both the discussed methods. 7. Conclusion
Fig. 12. Plot of Z-score obtained from bus voltage angle against PMU sampling instant for IEEE 118-bus system.
Fig. 13. Plot of Z-score obtained from bus voltage magnitude against PMU sampling instant for IEEE 118-bus system.
Because of growing interconnections and geographical spread, today’s power systems are becoming increasingly larger and more complex. Online voltage stability assessment of such multi-area power systems in near real-time is becoming a challenging task. In this paper, a methodology is proposed to monitor a large power system with limited number of measurements. The loadability limit is considered as the stability margin. Assuming the power system to be consisting of multiple areas, the security of the entire system is determined through measurements from one of the areas, referred as the internal area. Adaptively trained ANNs are used to estimate the loading margin. Information regarding the measurements from the external areas is incorporated into the internal measurements through the process of network reduction. With this additional embedded information regarding the operating conditions of the external areas, the estimation accuracy of the ANNs is improved. Lower computational burden is still maintained due to the use of only limited number of measurements from the internal area, while assessing the voltage stability of the entire power system. A Z-score based methodology is also implemented to detect and process any bad data in the training data set for the ANNs. The proposed methodologies are successfully tested on IEEE 14-bus and 118-bus test systems.
Table 6 Bad data correction for IEEE 14-bus. Bus no.
Actual voltage magnitude
Observed voltage magnitude
Corrected voltage magnitude
Actual voltage angle
Observed voltage angle
Corrected voltage angle
4 5
1.0149 1.0489
1.3230 1.3731
1.0449 1.0429
9.8474 7.8112
13.0675 10.2761
9.5793 8.1841
Table 7 Bad data correction for IEEE 118-bus. Bus no.
Actual voltage magnitude
Observed voltage magnitude
Corrected voltage magnitude
Actual voltage angle
Observed voltage angle
Corrected voltage angle
45 50 51
1.0313 1.0123 0.9848
1.3852 1.3209 1.3250
1.0093 1.0124 0.9840
13.8800 16.7470 13.6442
18.6216 22.2880 18.0923
14.6555 17.1741 13.9143
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Table 8 Adaptive update of ANN weights for Method 1. Test system
IEEE 14-bus IEEE 118-bus
Without weight update
With weight update
Mean squared error
Maximum error
Mean squared error
Maximum error
5.427exp-02 3.646exp-02
3.598exp-01 4.916exp-01
1.613exp-03 8.392exp-03
5.500exp-02 2.540exp-01
Mean squared error
Maximum error
Mean squared error
Maximum error
2.237exp-01 8.488exp-02
1.507exp-01 1.852exp-01
2.989exp-03 1.286exp-02
7.354exp-02 2.368exp-01
Table 9 Adaptive update of ANN weights for Method 2. Test system
IEEE 14-bus IEEE 118-bus
Without weight update
With weight update
Appendix A. Voltage stability indicator The loadability limit is used as the voltage instability indicator in this paper. It indicates, how much the system can be stressed, from the current operating condition, before it becomes unstable. Continuation power flow (CPF) is performed to determine the loadability limit [4]. A brief discussion on the CPF is included in the following to make the paper self-contained. The power flow equations can be expressed as,
fðx; kÞ ¼ 0
ðA:1Þ n
where f represents the full set of power flow equations; x 2 R is the vector of bus voltage magnitudes and angles; and k 2 [0, kcritical] is the varying loading parameter (its maximum value being kcritical) [41]. It is assumed here that all the loads in the system change with the common factor. By varying the parameter k continuously, a path of solutions can be obtained by solving (A.1), and the nose curve depicting the trajectory of solutions can be plotted. The solution corresponding to kcritical is called the critical point. The nose curve is obtained by employing a predictor-corrector scheme. A.1. Parameterization and selection of continuation parameter In the method of parameterization, the loading parameter is included as an additional variable, and the original set of equations is augmented by an additional equation specifying the value of the loading parameter [41–43]. Let the augmented set of variables be defined as,
y :¼ ðx; kÞ 2 Rnþ1
ðA:2Þ
If the kth element of yk is chosen as the continuation parameter, the additional equation can be written as,
yk g ¼ 0
ðA:3Þ
where g is an appropriate value for the kth element of y. The augmented equation combining (A.1), (A.2) and (A.3) is,
fðyÞ yk g
¼ ½0
ðA:4Þ
To apply the CPF, the power flow equations are reformulated to introduce a load parameter, k. The load and generation profiles with variation are formulated as [9],
PLi ¼ PLi0 þ kK Li
ðA:5Þ
Q Li ¼ Q Li0 þ k tanðwi ÞK Li
ðA:6Þ
PGi ¼ P Gi0 þ kK Gi
ðA:7Þ
where PLi, QLi, PGi represent the active load, the reactive load, and the active generation at bus i, respectively; PLi0, QLi0, PGi0 represent the nominal values of the active load, the reactive load, and the active generation at bus i, respectively; wi is the power factor angle (assumed to be constant) of the load at bus i; KLi, KGi are constants specifying the rate of change in load and generation at bus i, respectively, as k changes. Initially, the parameter k is selected as the continuation parameter. In the vicinity of the nose point, parameterization with respect to k may not give any solution. In that case, one of the components of x may be selected as the continuation parameter. The parameter yk , corresponding to which the element in the tangent vector has the maximum absolute value, is selected as the continuation parameter [42].
jdyk j ¼ maxfjdy1 j; jdy2 j; . . . ; jdynþ1 jg
ðA:8Þ
n+1
where dy 2 R is the tangent vector along the direction of the predictor, and d yk is its kth component. A.2. Predictor The predictor is used to obtain an approximate solution, starting from the current operating point [42]. Differentiating (A.4), with the state variables corresponding to the initial solution, will result in the following equation:
"
@f @x
@f @k
#
ek
dx dk
¼
0 1
ðA:9Þ
T
where dy ¼ ½dxT dk is the tangent to be calculated; ek is an appropriately dimensioned row vector, with all elements equal to zero except the kth, which is equal to one, and corresponds to dyk . The sign of the state variable that has the greatest rate of change near the given solution determines the sign of ek . Proper selection of ek ensures non-singularity of the Jacobian in (A.9). The prediction for the starting point of the next solution, h iT b y¼ b xT b k , is given by,
^ x dx x ¼ þr ^ k dk k
ðA:10Þ
T where y ¼ xT k is a solution of the previous iteration or initial operating point for the current iteration, and r is the step size. A.3. Corrector The corrector is used to obtain a solution point from the starting b , by solving (A.4), using slightly modified Newton-Raphson point, y
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power flow method. The introduction of the additional equation specifying yk makes the Jacobian non-singular at the nose point, so that CPF analysis continues beyond the nose point.
[21] [22]
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Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Voltage stability monitoring of power systems using reduced network and artificial neural network Syed Mohammad Ashraf ⇑, Ankur Gupta, Dinesh Kumar Choudhary, Saikat Chakrabarti Department of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur, India
a r t i c l e
i n f o
Article history: Received 13 January 2016 Received in revised form 26 September 2016 Accepted 16 November 2016
Keywords: Artificial neural network Continuation power flow Network reduction Voltage stability
a b s t r a c t This paper presents network reduction based methodologies to monitor voltage stability of power systems using limited number of measurements. In a multi-area power system, artificial neural networks (ANNs) are used to estimate the loading margin of the overall system, based on measurements from the internal area only. Information regarding the important measurements from the external areas is considered in measurement transformation through the network reduction process, to enhance the estimation accuracy of the ANNs. A Z-score based bad or missing data processing algorithm is implemented to make the methodologies robust. To account for changing operating conditions, adaptive training of the ANNs is also suggested. The proposed methods are successfully implemented on IEEE 14-bus and 118bus test systems. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction The electrical power demand is increasing day-by-day, and the generation is limited. Deregulated and open electricity markets are norms of the day, and it is essential to maintain reliable and good quality of the electricity supply. All this has resulted in large interconnected power networks that are operated under heavily loaded conditions, and are often close to their stability limits. Power system voltage instability is now one of the challenging problems faced by the utilities. Modern day energy management systems (EMSs) have strong focus on online voltage stability monitoring [1–3]. The maximum power loadability limit of the transmission network is one of the widely used indices to represent the voltage security of a power system [4]. It is critical for the utilities to track how close the transmission network is, to its maximum loading limit, so that in case of emergency, proper control actions can be taken. A large amount of literature exists on the use of analytical methods for voltage stability monitoring [5–8]. The conventional P-V curves are extensively used by the utilities for determination of the maximum permissible loading [9]. Continuation power flow (CPF) method is frequently used for obtaining the P-V curves [4–6]. The loadability limit is determined by increasing the system load in a particular direction, representing the most probable stressing
⇑ Corresponding author. E-mail addresses: [email protected] (S.M. Ashraf), [email protected] (A. Gupta), [email protected] (D.K. Choudhary), [email protected] (S. Chakrabarti). http://dx.doi.org/10.1016/j.ijepes.2016.11.008 0142-0615/Ó 2016 Elsevier Ltd. All rights reserved.
scenario. In order to monitor the voltage security in real-time, it is necessary that the process of measurement and estimation of the state variables and the analysis be performed within a desired time frame. Conventionally, remote terminal units (RTUs) have been used to collect measurements from various locations in a power system. The refresh rate of RTU measurements is typically a few seconds. With the advent of phasor measurement units (PMUs), it is now possible to obtain the measurement data at a sub-second rate [10–12]. In the presence of conventional measurements (from RTUs), the voltage phasors at the buses are obtained from the traditional supervisory control and data acquisition (SCADA) based state estimator (SE), typically every few minutes. Real-time monitoring of the power system, therefore, is not possible with the help of conventional SCADA based measurements. Because of high refresh rate and better accuracy, PMUs are increasingly being deployed in modern power systems. This paper presents an algorithm for fast monitoring of the voltage stability of the system, utilizing PMU measurements. A reduced network containing the buses observed by the PMUs is used to develop the proposed methodologies. Many works have been reported in the literature, exploring the capability of artificial neural network (ANN) for voltage stability monitoring [13–17]. ANN is used in these works, typically to establish a relationship between a voltage stability indicator and the measurable power system parameters affecting the indicator. These methodologies require a large number of ANN inputs which significantly increases the ANN size and diminishes its accuracy. Refs. [18–20] discuss the assessment of voltage stability using
44
S.M. Ashraf et al. / Electrical Power and Energy Systems 87 (2017) 43–51
artificial neural network with reduced set of inputs. The focus in these methodologies is on eliminating the redundant measurements and thus reducing the number of variables needed to assess the system voltage stability. It is assumed in these methodologies that a large number of measurement variables are available. The present work considers a more realistic scenario, in which only a few nodes in the system are assumed to have PMUs installed because of economic constraints. Only the data measured by these PMUs is utilized for stability analysis. The computational burden, as well as, the communication requirement for the stability analysis of large interconnected power systems can be reduced by using a reduced power system model. The reduced model also renders the advantage of: (i) monitoring the system by using only a limited number of measuring instruments; (ii) eliminating the need for detailed model in electrically remote areas; and (iii) monitoring the interconnected system, in which participating utilities are reluctant to share vital data. Usually, a utility’s own system is called the internal area. The rest of the system is called the external area for the internal subsystem under consideration. For running power system analysis functions in the internal area, the internal subsystem is typically modeled in detail. The external areas are usually represented by simple models, referred to as the external equivalent system. Numerous techniques for determining the reduced equivalent have been proposed in the literature [21–23]. In this paper, Ward equivalent technique is used and it is assumed that measurements of all the nodes in the internal area are available via PMUs. Two schemes for estimating the voltage instability in an interconnected power system are presented in this paper, using the reduced equivalent of the entire network. The fundamental motive of this research is fast and accurate assessment of the voltage stability of an entire network based on the data measured by PMUs at certain critical nodes. A feed-forward back-propagation network (FFBPN) is used to estimate the maximum loadability of the network. In the first scheme, complex bus voltages measured by the PMUs are used as the input to the FFBPN, and the available loadability margin is used as an indicator of the system voltage stability. In the second scheme, before estimating the system voltage stability margin, an FFBPN is first used to estimate the external network bus voltages by using the internal network bus voltages. The proposed networks have the ability to get adaptive training, when subjected to any new training pattern, following a change in the system operating condition. The proposed strategy is applied to IEEE 14-bus and 118-bus systems. The complex bus voltages measured by using PMUs may be subject to data packet loss or bad data. The bad data are recognized using Z-score algorithm [24]. The dropped data packets are compensated by using polynomial curve fitting. A method to update the ANN weights to incorporate a new training pattern is also presented. The key contributions of this paper are the following. Utilizing the voltage phasors measured by PMUs located at internal buses of the system to determine the system voltage stability. Although changes in the internal area parameters are reflected by PMU measurements, changes in loading condition for external area are unaccounted. For more accurate estimation of voltage stability, an ANN and ward reduction based method is presented to account for changes in the loading in the external area. To further strengthen the reliability in determination of system voltage stability, a bad data detection and correction method based on Z-score algorithm is presented. The power system may undergo changes after installation of ANN. To accommodate for these changes in the system, an adaptive training technique for ANN is also presented.
The paper is organized as follows. Section 2 illustrates the ward reduction technique, Section 3 defines ANN scheme for the proposed methods of voltage stability monitoring with load variations, Section 4 proposes a scheme to address bad or missing data, and the problem of updation of ANN weights is discussed in Section 5. The analysis of the results is given in Section 6, and Section 7 concludes the paper. A description of voltage stability indicator is presented in Appendix A. 2. Ward reduction One of the important aspects of the proposed methodology is the use of reduced network representation of the power system. This enables the monitoring of the voltage stability of the system by observing fewer number of nodes and in reduced amount of computational time. Ward reduction technique [21,25] is adopted in this paper for carrying out the network reduction function. The construction of Ward’s equivalent starts from the solved model of the entire interconnected power system. The injected current iðiÞ at each bus i is determined from the bus’s known complex power injection sðiÞ and voltage v ðiÞ.
iðiÞ ¼ s ðiÞ=v ðiÞ
ð1Þ
Gaussian elimination technique is used in this method to get the reduced network and current vectors. The nodal equations describing the power system are given by,
Ybus v ¼ i
ð2Þ
where Ybus is the n n bus admittance matrix, v is the n 1 vector of complex voltages at all nodes, and i is the n 1 vector of complex currents injected at all nodes. After elimination of the kth node, Ybus is modified as,
Y 0ij ¼ Y ij
Y ik Y kj Y kk
; 8i; j ¼ 1; . . . ; n; i; j–k
ð3Þ
where Y 0ij are the elements of the new ðn 1Þ ðn 1ÞY bus matrix. 0
The modified current vector, i , is given by, 0
i ðiÞ ¼ iðiÞ
Y ik iðkÞ; 8i ¼ 1; . . . ; n; i–k Y kk
ð4Þ
If the network is reduced to r nodes, a new bus admittance matrix of dimension r r, and a new current injection vector of dimension r 1 is obtained. The modified current vector is converted back into power injections for analysis purpose [21,25– 27]. This reduced network carries full information of the original power system at the base case. 3. Proposed ANN architecture The feed-forward ANNs employing back-propagation learning algorithm are used in this work as a mapping tool to estimate the available loadability margin of the system. The feed-forward networks are capable of approximating any measurable function to the desired accuracy level [28]. The back-propagation steps repeatedly adjust the weights of the connections in the network to minimize the mean squared error between the desired output and the output of the ANN. In this work, the ANNs are trained by using Levenberg-Marquardt algorithm [29]. The maximum loadability limit is considered in this paper as a measure of the voltage stability of the power system. If the total system load exceeds this limit, it will result in voltage collapse and possibly blackout. The objective here is to determine the point of maximum loadability so that corrective and/or preventive actions can be taken to avoid any voltage instability problem. In
S.M. Ashraf et al. / Electrical Power and Energy Systems 87 (2017) 43–51
order to reduce computational burden and fast assessment of the loadability limit, measurements from the retained buses in the reduced network are used. Two separate methods using different ANN configurations are investigated in this paper. Both of these methods use the same training data set. The training data comprises of a set of measured voltage phasors at the buses that are directly or indirectly observable by PMUs installed in the system, and the corresponding loading margins. A large number of different operating conditions are generated by randomly altering the base case real and reactive power of the loads in the entire system in the range of ±30%, and correspondingly adjusting the generation. The output data set comprises of the maximum loadability limits for these operating conditions, which are obtained by running CPF up to the nose point. The objective here is to estimate the loadability margin based on a limited number of PMU measurements. The original system may not be completely observable with PMU measurements. A subset of the buses is first specified, keeping only the buses that are directly or indirectly observable by PMUs. Both the methods discussed below utilizes the voltage phasors measured at these buses, for estimating the loadability margin of the entire system. The main advantage of using PMU measurements is their high refresh rate and accuracy, compared to the conventional measurements. 3.1. Method 1 In Method 1, the loadability margin of the entire system is estimated, based on the voltage phasor measurements from the buses observed by the PMUs. Since only a small number of PMU measurements are used (the system is not required to be completely observable by PMUs), and the PMU measurements are obtained at sub-second rate (50 or 60 times per second, for 50 and 60 Hz systems, respectively), the loadability margin can be estimated at a sub-second rate. An ANN, hereafter called as ANN 1, is trained to estimate the maximum loadability margin based on the voltage phasors measured by the PMUs in the system. There is a single hidden layer in ANN 1. The number of neurons in the hidden layer is chosen according to the following thumb rule [30].
45
3.2. Method 2 Although the methodology discussed in the previous section is easy to implement, its performance may deteriorate in case of significant change in the loading conditions in the buses not retained in the reduced network. To overcome this limitation, in the second method, i.e., in Method 2, the bus voltage phasors in the external system (part of the system outside the reduced network) are first estimated by using the available PMU measurements. An ANN, designated as ANN 2, shown in Fig. 2, is used for this purpose. It estimates the voltage phasors of the external buses by using the voltage phasors measured in the reduced network (i.e., the internal area) as the input. This makes available the information regarding the voltage phasors at all buses in the system. Some of these voltage phasors are directly observable by the PMUs; the remaining are estimated by ANN 2. Using these measured and estimated voltage phasors, known network parameters, and topology; the Ward reduction method discussed in Section 2 is used to obtain the power injections in the internal area. The main motivation for this step is to include the information regarding the power injections in the external area, in the power injections in the internal area or the reduced network. Due to the transformations of the power injections as a result of network reduction, the information regarding the power injections in the external area is now embedded in the power injections in the internal area [25,31]. An ANN, referred as ANN 3, is now trained with the real and reactive power injections in the internal area as the input, obtained after transformation resulting from the network reduction step discussed above. For each pattern of the input data, the output is the corresponding maximum loadability margin, which is obtained by running CPF for the entire network. Fig. 3 describes the configuration of ANN 3, which has a single hidden layer with the number of neurons determined by (5).
4. Bad data detection and correction
where nh is the number of neurons in the hidden layer, niv is the number of elements in the input vector, and nov is the number of elements in the output vector. The inputs to ANN 1 are the complex voltage phasors measured by the PMUs, and the output is the system loadability limit. Fig. 1 gives a schematic representation of ANN 1. Since the inputs to the ANN are the complex voltages measured by the PMUs, which are available at sub-second rate, the calculation of system loadability limit is also possible to complete at sub-second rate.
Once trained properly, the ANNs described above can be implemented in actual power systems to estimate the system loadability margin. Measurements from PMUs are to be used for these loading margin estimators. PMU measurements, however, may be subjected to the problems of bad data, loss of data packets, communication link failure, noise, etc. In case of such erroneous data, the estimated loading margin may not be accurate. Pre-processing of the PMU data i.e., processing of the measurement data coming from the PMUs before utilizing it as input data for ANNs, is therefore required to ensure the reliability and robustness of the ANNbased estimators. A commonly encountered type of bad data in field measurements is the outliers. An outlier is an observation that appears to deviate markedly from other observations in the sample. A number of techniques are reported in the literature to spot the outliers in
Fig. 1. Block diagram of ANN 1.
Fig. 2. Block diagram of ANN 2.
nh ¼
2 niv þ nov 3
ð5Þ
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S.M. Ashraf et al. / Electrical Power and Energy Systems 87 (2017) 43–51
32 3 2 3 n n X X a0 m x ðkÞ yðkÞ 76 7 6 7 6 76 7 6 k¼1 7 6 k¼1 k¼1 76 7 6 7 6 76 7 6 7 6 7 7 7 6 X 6 6 n n n n X X 76 7 6 X 7 6 2 mþ1 a 7 7 6 6 6 1 xðkÞ x ðkÞ x ðkÞ xðkÞyðkÞ 7 76 7 6 7 6 76 7 ¼ 6 k¼1 7 6 k¼1 k¼1 k¼1 76 7 6 7 6 76 . 7 6 7 6 7 7 7 6 6 6 . . . . . .. . .. . . 7 7 7 6 6 6 . . 76 7 6 7 6 76 7 6 7 6 76 7 6 X 7 6X n n n n X X 54 5 4 5 4 m mþ1 mþm m x ðkÞ x ðkÞ x ðkÞ x ðkÞyðkÞ a k¼1 k¼1 k¼1 k¼1 m ð11Þ 2
n
Fig. 3. Block diagram of ANN 3.
the measured data [24,32–34]. A simple and fast way to identify the outliers is the Z-score [24]. The Z-score of an observation, Z i , is defined as,
Zi ¼
Yi Y
r
;
ð6Þ
where Y i , Y, and r denotes the observation at the ith instant, the sample mean, and the sample standard deviation, respectively. The mean and standard deviations are calculated from the observed values over a window consisting of certain number of past samples. The necessary condition for being an outlier is,
jZ i j > Z th ;
ð7Þ
where Z th is the threshold value for the Z-score. The variation in Y i should be within a specific percentage of the first sample in the observed window. The condition described (7) ensures that the Zscore at the ith instant is Z th units of standard deviations away from its mean, i.e. the measurement at the ith instant is significantly different from the other observed values during the inspected period. Once the measurement is determined as a possible outlier, proper care should be taken so that it does not affect the case study. Loss of packets of PMU data can also be treated as outliers, and the method described above can be used to detect such events. A simple and efficient way to replace missing data is to use the polynomial curve fitting technique [35,36]. The idea behind polynomial curve fitting is to fit a polynomial to a set of n data points by utilizing least squares error approach. Consider a general form of polynomial of mth order,
f ðxÞ ¼ a0 þ a1 x1 þ þ am xm ¼ a0 þ
m X
Eq. (11) is solved to determine the coefficients of the curve that best fits the given data. The polynomial, thus derived, is used to estimate the missing data points or to replace the outliers. 5. Adaptive training of the ANNs In practice, the weights obtained after completion of the ANN training are saved and used during the operational phases as constant values. However, the parameters of a power system do not remain static. This implies that the training dataset may not be adequate to handle the unexpected changes in the system parameters. The weights are required to be dynamic and changing in an adaptive manner, based on the recent changes of the system. To address the issue, an adaptive weight update strategy is proposed in this paper. The initial training is carried out in conventional manner, and the ANN is deployed in the field. When the prediction starts, the weights are updated based on the error in the previous performance. Once the complete data is available for the system, the loadability limit is calculated using CPF and the error in estimation is computed. If the error is in excess of a pre-specified threshold value, the parameters, and the error are recorded. If the errors are found in excess for more than n predictions (n depends on the system), the n recorded errors are used to train the ANN again, and updated weights are obtained. This means that an ongoing training for the ANNs is implemented in the field to account for the changes in the operating conditions of the system. This is best explained with the following flowchart (Fig. 4): 6. Results
aj x j
ð8Þ
j¼1
The precise form of function f ðxÞ is determined based on the values of the coefficients, a0 ; . . . ; am . The curve which gives minimum error between data, y, and the fit, f ðxÞ, is the most appropriate. The residual, R, is obtained by,
R2 ¼
n X xðkÞ
n X 2 ½yðkÞ fa0 þ a1 xðkÞ þ a2 x2 ðkÞ þ þ am xm ðkÞg
ð9Þ
k¼1
where y(k) is the desired output at the kth instant, x(k) is the input at kth instant, and m is the order of the polynomial. The partial derivatives obtained from (9) are, n m X X @R2 ¼ 2 ½yðkÞ fa0 þ aj x j ðkÞg ¼ 0 @a0 j¼1 k¼1 n m X X @R2 ¼ 2 ½yðkÞ fa0 þ aj x j ðkÞgxðkÞ ¼ 0 @a1 j¼1 k¼1
ð10Þ
n m X X @R2 ¼ 2 ½yðkÞ fa0 þ aj x j ðkÞgxm ðkÞ ¼ 0 @am j¼1 k¼1
Eq. (10) can be re-written and a Vandermonde matrix is obtained as shown below [35,36],
The proposed methodologies are tested on standard IEEE 14bus and 118-bus systems [37]. The test systems are partitioned into internal and external areas. The partitioning of the system is based on a sensitivity analysis which identifies the most significant measurements that have to be retained in the reduced network [38]. The partitioning of IEEE 14-bus system is demonstrated in Fig. 5 [38]. Tables 1 and 2 present the internal and external buses considered in IEEE 14-bus and 118-bus systems, respectively. All the buses in the internal area are directly or indirectly observable by placing the PMUs at critical nodes. The nodes in the internal area which have PMUs installed are mentioned in Table 3. The time-series data for the studied cases were generated using PSAT software [39]. Simulation is performed using a variable time step integration option and later on data is interpolated at every 0.1 s. To simulate the phasor measurement noise, a random noise with standard deviation r = 0.01 p.u. is also added onto the obtained time series data. Utilizing the real and reactive loads and the real power generation of the base case, 240 random sets of loads and corresponding generations are created with in a limit of ±30% of the base case loading. The CPF analysis is carried on all the 240 sets to determine a set of maximum loadability points for the system. The test systems are then divided into internal and external areas.
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S.M. Ashraf et al. / Electrical Power and Energy Systems 87 (2017) 43–51 Table 1 Distribution of the IEEE 14-bus system. Internal system buses
External system buses
1, 2, 3, 4, 5
6, 7, 8, 9, 10, 11, 12, 13, 14
Table 2 Distribution of the IEEE 118-bus system. Internal system buses
External system buses
24, 45–77, 116, 118
1–23, 25–44, 78–115, 117
Table 3 Nodes in the internal subsystem having PMUs installed. Test system
Number of PMUs
Location of PMUs
IEEE 14-Bus System IEEE 118-Bus System
1 10
2 47, 49, 52, 56, 62, 64, 68, 70, 71, 76
6.1. Proposed ANN architecture
Fig. 4. Flow chart for adaptive update of ANN weights.
Fig. 5. Internal and external areas of IEEE 14-bus system.
The Neural Network Toolbox Version 8.2.1 provided with MATLAB 8.4 [40] was used to develop the neural network models. The data obtained from the CPF analysis was called in a MATLAB script for training, testing, and validation purposes. In the first scenario, the set of complex voltages measured by PMUs serve as input for ANN 1, and the set of maximum loadability points serve as the targets. 70% of the data set is used for training, 15% for validation, and 15% for testing. The number of neurons in all the ANNs is determined by first applying (5) and then through rigorous testing using trial and error. ANN 1 comprises of a single hidden layer. For IEEE 14-bus system, the number of neurons in input, hidden, and output layers are 10, 8, and 1, respectively. For IEEE 118-bus system, the number of neurons in input, hidden, and output layer is 72, 38, and 1, respectively. In the second scenario, the set of real and imaginary parts of the internal area complex voltage are fed as input for ANN 2 and the set of real and imaginary parts of external area complex voltage are its targets. Real and reactive power injections corresponding to the reduced system are fed as input to ANN 3, and the set of maximum loadability points are the targets. In both ANN 2 and ANN 3, 70% of the data is used for training, 15% for validation and 15% for testing. ANN 2 and ANN 3 also comprise of a single hidden layer. For IEEE 14-bus system, the number of neurons in input, hidden, and output layers are 10, 22, and 18, respectively. For IEEE 118-bus system, the number of neurons in input, hidden, and output layer is 72, 87, and 164, respectively. ANN 3 has a structure similar to that of ANN 1. For testing of these ANNs, the dataset of 240 samples is used as input. Figs. 6 and 7 show the performance of ANN 1 for IEEE 14-bus and 118-bus, respectively. Figs. 8 and 9 show the performance of ANN 3 for IEEE 14-bus and 118-bus, respectively. The target is the actual set of maximum loadability point and output is the set of maximum loadability point determined form ANNs. In the first scenario, for IEEE 14-bus system, mean squared error is determined as 3.370exp-04 and maximum error is 7.770exp-02. For IEEE 118-bus system, mean squared error is determined as 1.249exp-03 and maximum error is 1.757exp-01, as can be seen in Table 4. In the second scenario, for IEEE 14-bus system, mean squared error is determined as 3.265exp-04, and maximum error is 8.926exp-02. For IEEE 118-bus system, mean square error is
S.M. Ashraf et al. / Electrical Power and Energy Systems 87 (2017) 43–51
Percentage error in maximum loadability
48
Table 4 Results for Method 1.
4 3
Test system
Mean squared error
Maximum error
2
IEEE 14-bus IEEE 118-bus
3.370exp-04 1.249exp-03
7.770exp-02 1.757exp-01
Test system
Mean squared error
Maximum error
IEEE 14-bus IEEE 118-bus
3.265exp-04 8.015exp-04
8.926exp-02 3.059exp-01
1 0 0
50
100
150
200
250
Test case Fig. 6. Percentage error in maximum loadability for IEEE 14-bus system using ANN 1.
Table 5 Results for Method 2.
better. However, for IEEE 118-bus system, scenario 2 outperforms scenario 1. The reason for better performance of scenario 2 for bigger system is that, with the system getting bigger, and lesser number of monitored or retained nodes, it is better to incorporate the loading details of external system via network reduction process. The maximum loadability point for smaller systems with moderate number of retained or monitored nodes can found out by both the scenarios with similar accuracies. 6.2. Bad data detection and correction
Fig. 7. Percentage error in maximum loadability for IEEE 118-bus system using ANN 1.
Fig. 8. Percentage error in maximum loadability for IEEE 14-bus system using ANN 3.
The Z-score algorithm discussed earlier is used for detection of bad data. For testing of the algorithm, random error (<35%) is introduced into the complex voltage measurement coming from PMUs, for preselected buses, at a particular instant. The normal variation in voltage at the buses is considered to be ±5%. The output sequence, thus obtained is tested for outliers by Z-score algorithm. For IEEE 14-bus system test case, random errors of 32.7%, 31.56%, 30.36%, and 30.91% were introduced in the voltage magnitude and angle measurements coming from bus number 4 and 5, respectively; at the 13th instant of a 30 sample observation window. In case of IEEE 118-bus test system, random errors of 34.16%, 33.09%, 32.6%, 34.32%, 30.49%, and 34.54% were introduced in the voltage magnitude and angle measurements coming from bus number 45, 50, and 51, respectively; at the 23rd instant of the observation window. The plots of the Z-score test are presented in Figs. 10–13. It is observed that the Z-score of the 13th and 23rd instant of IEEE 14-bus and IEEE 118-bus, respectively, are very high in comparison to the Z-score of previous instances, indicating an outlier. Based on the observation, the threshold value of the Zscore for bad data detection was selected to be 2.5. Both angle and magnitude of the power systems can be used for outlier detection. Once bad data is detected, it is corrected using curve fitting technique. In this work, a second order curve fit is applied to data
Fig. 9. Percentage error in maximum loadability for IEEE 118-bus system using ANN 3.
determined as 8.015exp-04, and maximum error is 3.059exp-01, as can be seen in Table 5. It can be seen that both the mean squared error and the maximum error for IEEE 14-bus system are comparable in both the scenarios, with scenario 2 coming out to be slightly
Fig. 10. Plot of Z-score obtained from bus voltage angle against PMU sampling instant for IEEE 14-bus system.
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S.M. Ashraf et al. / Electrical Power and Energy Systems 87 (2017) 43–51
collected in the observation window. A tabulated output of the curve fitting technique is presented in Tables 6 and 7. 6.3. Adaptive update of ANN weights
Fig. 11. Plot of Z-score obtained from bus voltage magnitude against PMU sampling instant for IEEE 14-bus system.
To test the adaptive update of ANN weights, some loads are introduced in the external system. The ANNs trained earlier for testing of proposed architecture are then retrained using the new error. In IEEE 14-bus system, an active load of 20 MW is applied at bus 7, whereas, in IEEE 118-bus test system, an active load of 70 MW is applied at bus 9, and training data set are obtained as in the previous case. A comparison of ANN output error without weight update and with weight update is presented in Tables 8 and 9. It is observed that the ANN architecture with adaptive weight gives better results in comparison to the ANN architecture without an adaptive weight scheme. The update of ANN weights is not reflected well in performance of Method 2, as the update of weights are done for ANN 2 only, and the input of ANN 3 depends on output of ANN 2. Since ANN 2 will be facing a new scenario, its output will have some error. However, an improvement is seen in both the discussed methods. 7. Conclusion
Fig. 12. Plot of Z-score obtained from bus voltage angle against PMU sampling instant for IEEE 118-bus system.
Fig. 13. Plot of Z-score obtained from bus voltage magnitude against PMU sampling instant for IEEE 118-bus system.
Because of growing interconnections and geographical spread, today’s power systems are becoming increasingly larger and more complex. Online voltage stability assessment of such multi-area power systems in near real-time is becoming a challenging task. In this paper, a methodology is proposed to monitor a large power system with limited number of measurements. The loadability limit is considered as the stability margin. Assuming the power system to be consisting of multiple areas, the security of the entire system is determined through measurements from one of the areas, referred as the internal area. Adaptively trained ANNs are used to estimate the loading margin. Information regarding the measurements from the external areas is incorporated into the internal measurements through the process of network reduction. With this additional embedded information regarding the operating conditions of the external areas, the estimation accuracy of the ANNs is improved. Lower computational burden is still maintained due to the use of only limited number of measurements from the internal area, while assessing the voltage stability of the entire power system. A Z-score based methodology is also implemented to detect and process any bad data in the training data set for the ANNs. The proposed methodologies are successfully tested on IEEE 14-bus and 118-bus test systems.
Table 6 Bad data correction for IEEE 14-bus. Bus no.
Actual voltage magnitude
Observed voltage magnitude
Corrected voltage magnitude
Actual voltage angle
Observed voltage angle
Corrected voltage angle
4 5
1.0149 1.0489
1.3230 1.3731
1.0449 1.0429
9.8474 7.8112
13.0675 10.2761
9.5793 8.1841
Table 7 Bad data correction for IEEE 118-bus. Bus no.
Actual voltage magnitude
Observed voltage magnitude
Corrected voltage magnitude
Actual voltage angle
Observed voltage angle
Corrected voltage angle
45 50 51
1.0313 1.0123 0.9848
1.3852 1.3209 1.3250
1.0093 1.0124 0.9840
13.8800 16.7470 13.6442
18.6216 22.2880 18.0923
14.6555 17.1741 13.9143
50
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Table 8 Adaptive update of ANN weights for Method 1. Test system
IEEE 14-bus IEEE 118-bus
Without weight update
With weight update
Mean squared error
Maximum error
Mean squared error
Maximum error
5.427exp-02 3.646exp-02
3.598exp-01 4.916exp-01
1.613exp-03 8.392exp-03
5.500exp-02 2.540exp-01
Mean squared error
Maximum error
Mean squared error
Maximum error
2.237exp-01 8.488exp-02
1.507exp-01 1.852exp-01
2.989exp-03 1.286exp-02
7.354exp-02 2.368exp-01
Table 9 Adaptive update of ANN weights for Method 2. Test system
IEEE 14-bus IEEE 118-bus
Without weight update
With weight update
Appendix A. Voltage stability indicator The loadability limit is used as the voltage instability indicator in this paper. It indicates, how much the system can be stressed, from the current operating condition, before it becomes unstable. Continuation power flow (CPF) is performed to determine the loadability limit [4]. A brief discussion on the CPF is included in the following to make the paper self-contained. The power flow equations can be expressed as,
fðx; kÞ ¼ 0
ðA:1Þ n
where f represents the full set of power flow equations; x 2 R is the vector of bus voltage magnitudes and angles; and k 2 [0, kcritical] is the varying loading parameter (its maximum value being kcritical) [41]. It is assumed here that all the loads in the system change with the common factor. By varying the parameter k continuously, a path of solutions can be obtained by solving (A.1), and the nose curve depicting the trajectory of solutions can be plotted. The solution corresponding to kcritical is called the critical point. The nose curve is obtained by employing a predictor-corrector scheme. A.1. Parameterization and selection of continuation parameter In the method of parameterization, the loading parameter is included as an additional variable, and the original set of equations is augmented by an additional equation specifying the value of the loading parameter [41–43]. Let the augmented set of variables be defined as,
y :¼ ðx; kÞ 2 Rnþ1
ðA:2Þ
If the kth element of yk is chosen as the continuation parameter, the additional equation can be written as,
yk g ¼ 0
ðA:3Þ
where g is an appropriate value for the kth element of y. The augmented equation combining (A.1), (A.2) and (A.3) is,
fðyÞ yk g
¼ ½0
ðA:4Þ
To apply the CPF, the power flow equations are reformulated to introduce a load parameter, k. The load and generation profiles with variation are formulated as [9],
PLi ¼ PLi0 þ kK Li
ðA:5Þ
Q Li ¼ Q Li0 þ k tanðwi ÞK Li
ðA:6Þ
PGi ¼ P Gi0 þ kK Gi
ðA:7Þ
where PLi, QLi, PGi represent the active load, the reactive load, and the active generation at bus i, respectively; PLi0, QLi0, PGi0 represent the nominal values of the active load, the reactive load, and the active generation at bus i, respectively; wi is the power factor angle (assumed to be constant) of the load at bus i; KLi, KGi are constants specifying the rate of change in load and generation at bus i, respectively, as k changes. Initially, the parameter k is selected as the continuation parameter. In the vicinity of the nose point, parameterization with respect to k may not give any solution. In that case, one of the components of x may be selected as the continuation parameter. The parameter yk , corresponding to which the element in the tangent vector has the maximum absolute value, is selected as the continuation parameter [42].
jdyk j ¼ maxfjdy1 j; jdy2 j; . . . ; jdynþ1 jg
ðA:8Þ
n+1
where dy 2 R is the tangent vector along the direction of the predictor, and d yk is its kth component. A.2. Predictor The predictor is used to obtain an approximate solution, starting from the current operating point [42]. Differentiating (A.4), with the state variables corresponding to the initial solution, will result in the following equation:
"
@f @x
@f @k
#
ek
dx dk
¼
0 1
ðA:9Þ
T
where dy ¼ ½dxT dk is the tangent to be calculated; ek is an appropriately dimensioned row vector, with all elements equal to zero except the kth, which is equal to one, and corresponds to dyk . The sign of the state variable that has the greatest rate of change near the given solution determines the sign of ek . Proper selection of ek ensures non-singularity of the Jacobian in (A.9). The prediction for the starting point of the next solution, h iT b y¼ b xT b k , is given by,
^ x dx x ¼ þr ^ k dk k
ðA:10Þ
T where y ¼ xT k is a solution of the previous iteration or initial operating point for the current iteration, and r is the step size. A.3. Corrector The corrector is used to obtain a solution point from the starting b , by solving (A.4), using slightly modified Newton-Raphson point, y
S.M. Ashraf et al. / Electrical Power and Energy Systems 87 (2017) 43–51
power flow method. The introduction of the additional equation specifying yk makes the Jacobian non-singular at the nose point, so that CPF analysis continues beyond the nose point.
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