Power System Parameter Estimation for Enhanced Grid Stability Assessment in Systems with Renewable Energy Sources

Transcription

1 Power System Parameter Estimation for Enhanced Grid Stability Assessment in Systems with Renewable Energy Sources Andreas J. Schmitt Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical and Computer Engineering Jaime De La ReeLopez, Chair Virgilio Centeno, Co-Chair Vassilis Kekatos Werner Kohler Amos Abbott Kevin D. Jones April 24, 2018 Blacksburg, Virginia Keywords: Power System Topology Estimation, Low Observability State Estimation, System Identification, Inertia Estimation, Power System Stability Copyright 2018, Andreas J. Schmitt

2 Power System Parameter Estimation for Enhanced Grid Stability Assessment in Systems with Renewable Energy Sources Andreas J. Schmitt (ABSTRACT) The modern day power grid is a highly complex system; as such, maintaining stable operations of the grid relies on many factors. Additionally, the increased usage of renewable energy sources significantly complicates matters. Attempts to assess the current stability of the grid make use of several key parameters, however obtaining these parameters to make an assessment has its own challenges. Due to the limited number of measurements and the unavailability of information, it is often difficult to accurately know the current value of these parameters needed for stability assessment. This work attempts to estimate three of these parameters: the Inertia, Topology, and Voltage Phasors. Without these parameters, it is no longer possible to determine the current stability of the grid. Through the use of machine learning, empirical studies, and mathematical optimization it is possible to estimate these three parameters when previously this was not the case. These three methodologies perform estimations through measurement-based approaches. This allows for the obtaining of these parameters without required system knowledge, while improving results when systems information is known.

3 Power System Parameter Estimation for Enhanced Grid Stability Assessment in Systems with Renewable Energy Sources Andreas J. Schmitt (GENERAL AUDIENCE ABSTRACT) Stable grid operations means that electricity is supplied to all customers at any given time regardless of changes in the system. As the power grid grows and develops, the number of ways in which a grid can lose stability also grows. As a result, the metrics that are used to determine if a grid is stable at any given time have grown increasingly complex and rely on significantly more amounts of information. This information required in order to obtain the metrics which determine grid stability often has key limitations in when and how it can be obtained. The work presented details several methods for obtaining this information in situations were it was previously not possible to do so. The methods are all measurement based, which means that no prior knowledge about the grid is required in order to compute the values.

4 Acknowledgments There are many who have given me help and support during my studies, that deserve more appreciation than I am able to adequately convey. First, naturally my family needs to be mentioned. The constant support throughout my life has been a true blessing. Mom, Dad, Christian, Stefan, you really have been the best anyone could ask for and for that I cannot be more grateful. To all of my professors and colleagues, particularly Dr. De La Ree and Dr. Centeno, I truly am thankful for all of your guidance. You allowed me to search the world for what was really important to me, while still guiding me towards my goals. It is a rare gift that I have received and I will appreciate that forever. Finally, to my close and dear friends, Jacques and Elliott, who have been with me this entire stage of my life. Your friendship throughout these years has perhaps been the most rewarding aspect of my studies. It is hard to imagine what it would have been like had I not met you two and no matter where we end up on our now separate paths, I will always remember and cherish the time we spent together. iv

5 Contents 1 Introduction Inertia Estimation Topology Estimation Low Observability State Estimation Steady-State Inertia Estimation Modal Information Extraction Neural Networks Simulations and Results Empirical Topology Estimation Threshold Calculation Standard Substation Topologies Approach Synthetic-Mesh Approach Topology Estimation Level 0: Initialization v

6 3.2.2 Level 1: Read Breaker Status Telemetry(If Available) Level 2: Infer/Validate Breaker Status Level 3: Voltage Coherency Level 4: Complete Combination Simulation and Initial Results OpenECA Platform and Implementation Low Observability State Estimation Matrix Completion Low Observability Voltage Estimation Power System Model Matrix Set-Up Power Flow Constraints Simulation and Results Radial Network Results Mesh Network Results Model-Free Voltage Estimation Comparison to Traditional Methods Conclusions Inertia Estimation Conclusions Topology Estimation Conclusions vi

7 5.3 State Estimation Conclusions Bibliography 68 vii

8 List of Figures 2.1 Example of a Neural Network IEEE 118 Bus System with Regions Identified Accuracy of Neural Networks as Training Time Increases Neural Network Errors Distributions of Estimation Errors Example of Node Groupings IEEE 118 Bus System Standard Substation Configurations Histogram of Phase Angle Deviations in the Standard Substation System CDF of Phase Angle Deviations in the Standard Substation System Histogram of Phase Deviations for Each Substation Type CDF of Phase Deviations for Each Substation Type Threshold Result for Each Individual Substation IEEE Standard 14 Bus System Histogram of Phase Deviations for the 14 Bus System viii

9 3.11 CDF of Phase Deviations for the 14 Bus System Synthetic Mesh Substation Example Histogram of Angle Deviations for the Synthetic Mesh Approach Example of Level 0 Topology Estimation Example of Level 1 Topology Estimation Example of Level 2 Topology Estimation Example of Level 4 Topology Estimation Accuracy of Topology Estimation Procedure at Each Substation IEEE Standard 33 Bus System Voltage Estimation of Radial Network MAPE as Observability Increases Voltage Angle Estimation of Radial Network Impact of Measurement Errors Impact of Number of Phasor Measurements in Radial Network Impact of Phasor Measurement Device Placement IEEE Standard 39 Bus System Voltage Estimation of Mesh Network Voltage Angle Estimation of Mesh Network Impact of Number of Phasor Measurements in Mesh Network Impact of Measurement Errors in Mesh Network Voltage Estimation with No Missing Data ix

10 4.14 Angle Estimation with Missing Data Voltage Estimation with Missing Data MAPE as Observability Increases High PV Injection Voltage Estimation Impact of δ Value MAPE of WLS vs Matrix Completion as Observability Increases x

11 List of Tables 2.1 Mean Absolute Percentage Error of Inertia Estimations Phasor Deviation Threshold for Each Standard Substation Type Estimation Accuracy of Two Different Algorithm Levels with Varying Noise 39 xi

12 Chapter 1 Introduction The creation and maintaining of a robust and stable power grid has been of desire since the inception of the power system. However, as time passed and the power system has become much more complex, this task has similarly become increasingly complex. There are many factors which may affect the stability of a grid at any given time. The most common ways in which the system can become unstable are transient stability issues, voltage stability, frequency stability, and inter-area oscillations [1]. Because of all of these different ways in which the grid can become unstable, being able to determine if a given system is stable at any given time is of utmost importance. There are several ways in which grid stability has been attempted to be quantified. Among the most common metrics for determining current grid stability are the H and H 2 metrics [2]. While these metrics vary in their approach, they make use of similar input quantities to make decisions. Particularly, the voltage and frequency deviations between buses are used to provide a metric for current system stability [3]. It has also been shown that grid inertia has an enormous impact on grid stability through its impact on the frequency response of the system [4]. Studies have also shown that when the sources of inertia are not evenly spread throughout the system, then the topology of the system plays a large part in the stability of the system [5]. 1

13 2 With these parameters of inertia, topology, and voltage being necessary in order to quantify power system stability, the issue arises of how to obtain these quantities. Each quantity comes with its own separate challenge, yet all are needed in order to properly perform grid stability assessment. The major contributions of this work include: The development of a machine learning based procedure to estimate power system inertia constant values during steady-state conditions The ability to accurately estimate substation topology solely from synchrophasor data, without having to rely on breaker status information Improved estimation accuracy of voltage phasors in systems with reduced observability through the use of matrix completion optimization 1.1 Inertia Estimation Inertia has a significant impact on the stability of the power system. The system inertia aids in maintaining the current conditions of the system, preventing events and small disturbances from rapidly changing the state of the system. The major source that provides inertia to the system is the rotating mass of machinery that is used in traditional energy sources such as coal or gas power plants. However, renewable energy sources such as wind or solar energy do not contribute to the system inertia. This is becoming a large issue as ever increasing percentages of power generation are supplied by renewable energies. As these renewable energies get added to the system, the overall inertia will drop which makes the entire grid more susceptible to disturbances. Therefore, being able to estimate the value of the system inertia is becoming a necessity. Having the value of the inertia present in the system provides knowledge on the susceptibility of the current system and may enable the use of counter measures to prevent system instability.

14 3 Current methods used to determine inertia values, for example [6] and [7], rely on large system dynamic events, such as significant faults, to track the dynamic values in machines. While these techniques are accurate, their usage is very limited as large events do not occur frequently. Additionally, there are very large efforts being taken towards preventing large system events from occurring. Thus, using techniques which rely on these events is not ideal. With the increased usage of phasor measurement units or synchrophasors in the system, there is much more information available about the current state of the system. With an increased amount of data available, the ability to determine the system inertia values during steady state operating conditions becomes a possibility. However, extracting the correct information that can be used for inertia estimation from synchrophasors can be a difficult process. One possibility for the estimation of inertia is through the use of modal information. Modes of oscillation in electrical quantities arise in a system due to the slow mechanical oscillations of machinery against one another. These are the same machines that are responsible for adding inertia to the system; hence, using modal information as an input to the estimation procedure should be viable. There are techniques such as [8] which make use of modal information to determine the parameters, such as inertia, of the generators in a system. However, the aim of this work is to be able to incorporate all components in the system and obtain regional and complete system inertia values as well, not only generator parameter values. The issue then becomes how to best use this modal information to estimate the desired parameters. When considering the complexity of the system, neural networks are a great tool for handling this problem. The field of machine learning has been growing rapidly and has made its way into a variety of fields, including power systems [9] or [10]. Most research, such as [10], focuses on using machine learning to predict future states of the system in techniques like load forecasting. The research proposed in [9], attempts to predict the load response of future events by using a neural network. The research into predicting the future

15 4 of the power system is extensive, yet the amount of machine learning used for real-time analysis is limited. While there are many viable machine learning techniques, a neural network approach was selected, as opposed to other techniques such as state vector machines, for several reasons.with the power system being as complex as it is, the method needs to be able to handle these complexities. Neural networks are well equipped to handle very complex and particularly non-linear systems. This makes them an ideal tool for use in power systems. The major drawback being that neural networks do not allow for the user to view how decisions or calculations were made. Through this approach, estimating the inertia during steady-state conditions becomes possible, as seen in Chapter Topology Estimation The state of a power grid is highly dependent on its topology. Analytics such as power flow or state estimation, as well as many others, require accurate knowledge about the connectivity of the system in order to obtain correct results. Normally the discussion of topology estimation revolves around the desire to determine which lines are in service and which are not. However, an equally important aspect of topology estimation is the connectivity of nodes within a substation. Current topology estimation techniques make use of breaker status telemetry in order to determine substation connectivity [11 13]. Topology processors, such as the one suggested in [14,15], determine the topology for those parts of the system which have changed as compared to the previously processed topology. However, there are limitations on the availability and accuracy of this data. Many synchrophasor data systems do not have access to time synchronized breaker status telemetry, and those that do, have to deal with errors such as latching or other measurement errors. Thus, there is a huge financial burden on the deployment of synchrophasor analytics that require topology due to the required installation

16 5 of circuit breaker telemetry into the synchrophasor data stream. Some systems make use of Supervisory Control and Data Acquisition (SCADA) system breaker telemetry in order to overcome this obstacle. However, there are several issues associated with this such as latency issues, lack of time synchronization, and the prevention of any redundancy with SCADA that would have been possible before. Additionally, the SCADA system is not time synchronized and the data obtained very infrequently. Through the increased availability of synchrophasor measurements, breaker statuses across nodes within a substation can be inferred or validated [16]. This can then be used in synchrophasor analytics, such as the Three Phase Linear State Estimator [17] or other state estimation applications, to improve their robustness. There have been several previous approaches taken in an attempt to estimate substation topology. The work proposed by [18] uses synchrophasor data to validate breaker statuses. The breaker statuses are then used with a search procedure to determine the topology. Other techniques, such as [19], attempt to use a L1-norm state estimator in conjunction with a search algorithm to determine the topology. Other approaches attempt to detect errors in the assumed topology rather than to estimate the topology [20, 21]. The approach developed is a novel empirical method for determining the bus-branch topology of a synchrophasor based substation relying upon a phasor deviation threshold value. This threshold is determined based upon the concept of equipotential nodes within a substation. The deviation threshold can then be used to reconstruct the bus-branch topology of the substation. This procedure is detailed in Chapter Low Observability State Estimation Observability in the power grid has been a constant issue since its inception. Knowledge of the present state of the grid allows for optimal grid operations. Traditional state estimation techniques require full network observability in order to produce accurate results. The most

17 6 common technique currently in use for state estimation is the Weighted Least Squares (WLS) algorithm [22]. While WLS does produce accurate results, it requires a high observability. Additionally, the WLS algorithm is limited to voltage, current, and power measurements. However, obtaining the required observability for traditional state estimation, particularly in the distribution grid, is not feasible due to the immense scale of the network and limited availability of phasor measurement units (PMUs) [23]. Therefore, it has become imperative to be capable of estimating/completing the state of the grid for cases of low-observability, which roughly speaking means that the number of measurements is less than that of the quantities being estimated. There has been much work done in an attempt to complete state estimation in situations where observability is an issue. [24] determines the optimal minimum measurement locations and then uses the difference between the measured and calculated voltages and complex powers to obtain the voltage profile of the whole network. [25] makes use of the voltage sensitivity at each bus to the load at all buses to determine the voltage given the measured load at each location. These approaches, however, still require the installation of significant numbers of PMUs when considering the scale of the system. In order to improve the observability, methods for the optimal placement of limited number of meters have recently been investigated in, e.g., [26 28]; and methods that use pseudomeasurements and historical load profiles are proposed in [29 31]. [32] makes use of smart meter data to enhance observability of the system by solving the power flow over several consecutive time instances. Machine-learning-based approaches are recently becoming popular for the task of state estimation in distribution networks. For example, [30, 33, 34] make use of neural networks to estimate the voltage in the network. These approaches have the benefit of not requiring a system model to obtain results, but retain the same consequences of all machine learning approaches, requiring significant amounts of data to obtain an accurate estimation. Matrix completion is a well-established tool in the signal processing community for estimat-

18 7 ing missing values in low rank matrices [35]. It has been used within the power systems previously, in an attempt to do load forecasting in networks with incomplete data. For instance, [36] uses a correlation structure completion to fill in missing data for load forecasting; and [37] uses a kernel-based matrix completion approach that allows for the estimation of missing data even in locations where no measurements are available. The methodology created in this work takes a matrix completion approach for estimating voltage phasors in networks with low observability. The original problem formulation for matrix completions is augmented with power-flow constraints which provide an additional link between parameter values. This structured approach has the following benefits compared to the existing techniques: It allows for voltage estimation from whichever measurements are available; as the number of measurement devices increase, regardless of quantity being measured, this data can be used as a supplement to estimate the voltage phasors in the entire system. The inclusion of power-flow constraints allow for smaller amounts of data required for accurate estimation compare to typical black box machine learning methods. The approach can be used to determine both voltage magnitude and phase angle in systems with few voltage phasor measurements. The point of emphasis is that the versatility of the matrix completion approach allows for the use of any measurement that may be available in the field in order to perform state estimation. This is in contrast to many available techniques, such as [31, 32, 38], that make use of only voltage and real/reactive power measurements for the purpose of estimation. This allows for state estimation to be performed in systems that previously were unable to do so due to a lack of these measurements. The process for voltage estimation is shown in Chapter 4.

19 Chapter 2 Steady-State Inertia Estimation Traditionally, the value of inertia is obtained through the equation H = P 2 df dt f 0 (2.1) where P is the change in power that caused an event, df is the resultant rate of change dt of frequency, and f 0 is the system frequency at the time of the event. The value of H is the inertia referenced to the power rating of the bus such that it is possible to compare machines or buses of differing power levels, as seen in the equation H = 1 Jω 2 (2.2) 2 S base where J is the moment of inertia, ω is the rated angular velocity, and S base is the rated power of the machine. Due to the physical nature of the quantity of inertia, it is much simpler to obtain its value during dynamic conditions due to the change in power and frequency being much easier to identify. However, this severely limits the opportunities for determining system inertia during normal power system operations since dynamic events are both infre- 8

20 9 quent and undesirable. Thus, estimating inertia during steady-state conditions has become necessary in order to be able to assess current system stability. When discussing inertia, there are several levels of inertia depending on the scope taken into consideration. The smallest scale is the inertia of each individual bus in the system. This may provide some information, but is not as useful as the regional or total system inertia. The regional and system inertias are found through the combination of all the individual bus inertias through the equation H sys = N i=1 S ih i S sys (2.3) where S sys is the summation of all rated powers within the scope. Through this it is possible to determine the inertia at various levels of scope. Thus, the method developed in this work attempts to determine the inertia at three different levels: the individual bus inertias, the regional inertias, and the system inertia [39]. 2.1 Modal Information Extraction The first step in the inertia estimation is to extract the modal information from the system. There are several methods for obtaining the modal information in a system. Many techniques, such as Prony analysis, require a disturbance in the system in order to obtain accurate results. Many of these techniques are thoroughly developed and are in widespread use. However, since the aim is to estimate values during steady state conditions these are not viable solutions for this specific problem. There are a few methods for determining the modal information which do not require fault conditions for accurate calculations. These methods make use of ambient data from devices, such as synchrophasors, in an attempt to calculate modal information during small signal periods. However, these techniques are less well established than those that require fault

21 10 conditions. The technique proposed by [40] allows for the computation of modal information during steady state using information readily available from synchrophasors. The technique proposed by [40] uses empirical mode decomposition in order to calculate the modes of oscillation. This method takes frequency data from the buses over a set time window and uses that to calculate the modal information. It is shown that if the window of data is large enough, the result of this method accurately represents the modal information in a system. The general algorithm for the technique is shown below. 1. Obtain the frequency value from each bus over a time window, removing the mean value from each data point, and compute the covariance matrix 2. Calculate the eigenvalues of this matrix and place into ascending order 3. Calculate the eigenvectors and normalize them 4. Determine the correct number of eigenvalues to ensure a significant amount of energy is captured by these vectors 5. Re-compute input data to guarantee accuracy of results During steady-state operating conditions, the frequency in the system will have minor fluctuations due to small load changes. Additionally, due to the nature of the power system the frequency at all buses will not be equal. The frequency between buses will oscillate against one another. These oscillations and frequency deviations are recorded over a period of time in a window. Once the frequency of each bus over the time period is captured, each bus s data is placed into a matrix. An example of the created matrix can be seen below

22 11 f 1,1 f 1,j f 1,N X = f i,1 f i,j f i,n f m,1 f m,j f m,n (2.4) where f i,j is j th measurement at bus i. From this matrix, once the mean value has been removed from each element, the covariance matrix C can be calculated through the equation: C = 1 N XT X (2.5) Where N is the number of data points in each window and X is the matrix created from the collected frequency data at each bus, as seen in equation 2.4. From the covariance matrix C, the eigenvalues and eigenvectors can be calculated. It is shown, that if the time window of the sampling is large enough these values will approach that of the real system. This window size required will vary for each application, but for power systems a window in the range of several seconds and larger is enough to obtain accurate results. The method described assumes that the system is linear, which is suitable for steady-state cases that have very small fluctuations. 2.2 Neural Networks Using the modal information as an input to the system, a neural network was used to determine the inertia values. Neural networks make use of a combination of multiple simple decision techniques (notated as a neuron) to output an accurate solution for complex systems. As can be seen in Figure 2.1, each input parameter is input into every neuron, which then outputs a result that is combined with all of the other paths taken to create an optimal

23 12 solution. Each path is assigned a weight depending on the paths impact on the accuracy of the output. Figure 2.1: Example of a Neural Network All of these decision paths are self-learned or trained using a portion of the data. When constructing the neural network, the complete data set is split into three categories: training data, validation data, and test data. As the name implies the training set is used to create or teach the neural network. The training set consists of giving the neural network both the input parameters and the outputs, allowing for it to construct the optimal path from the inputs to the outputs. The validation set is used to tune the parameters of the neural network, such as number of hidden layers and neurons. The network calculates the outputs given the set of input parameters and compares the outputs calculated to the actual outputs. The neural network is typically selected based on the best performance of the validation set. The test set is the final step of using the neural network on the desired data set and observing the accuracy. Additionally, the parameters of the neural network were tuned to obtain the most accurate results. The tuning was done through the use of the validation set by the neural network as well as empirical iterations to determine the optimal network. The largest influence on the functionality of a neural network is the number of hidden layers and neurons used by the network.

24 13 The number of hidden layers used by the neural network can be seen as the number of simple decision metrics each path goes through. Each hidden layer will contain neurons which make decisions, then sends the decisions to the next layer. The number of neurons determines how many decision techniques are used within each hidden layer that are used in combination to produce an output. Increasing the number of neurons will increase the variety in decision approaches, but also increases the complexity of the network. Increasing the complexity of the network does not always improve the results of the network. The most accurate results were obtained through the use of a single hidden layer with 10 neurons in it. Bagging was used to decrease the variance of the solution and to prevent overfitting. Since the result of neural networks is highly dependent on the starting conditions, results can vary highly. Thus, the variance of the output can be reduced through taking the average of several runs of the neural network. The final results that are shown in the following section are the average result of 10 runs of the neural network. Finally, another method of early stopping was used while training the neural network to prevent overfitting of the data. Neural networks have a tendency to overfit the calculation procedure based on the specific input data. This is because the training of the neural network is an iterative process, meaning it will continuously run until the error on the specific input data is as low as possible. This will cause overfitting to the data and creates errors when applying the network to more general data. To overcome this tendency, the iterative process is stopped after a desired accuracy or number of iterations has been reached. The accuracy of the estimation on training and validation data is recorded overtime. The result with the most accurate validation results is typically selected. An example of this can be seen in Figure 2.3.

25 Simulations and Results The methodology was tested on the IEEE 118-bus system using PSS/E. The system was split into three regions as seen in Figure 2.2. The regions were found based on the method described in [41], which makes use of graph theory to split the system. Figure 2.2: IEEE 118 Bus System with Regions Identified To begin each simulation, the system inertia value was selected at random within a range of realistic values. From there, the regional inertia values were split randomly from the system inertia. The inertia values of the generators and rotating loads within the system were then randomized, maintaining a range of 2s to 8s. This was to ensure that the system would maintain stability during simulation and that the inertia values are realistic. The system would then be simulated for a long duration under realistic steady-state conditions in PSSE. The minor fluctuations in the frequency present in realistic systems during steady-state were created through the use of minor load fluctuations during the simulation. At various random bus locations throughout the system, the existing load was altered to a new value by increasing or decreasing the value by a small percentage of the base load following a normal

26 15 distribution. This was done periodically such that the behavior of the system resembled a realistic system during steady-state. The frequency from each bus over a time window of two seconds was taken in order to calculate the modal information with the method described in section 2.1. To resemble actual PMU data, the measurements were taken 30 times per second. This time window was found to be large enough that the eigenvalues and vectors calculated through this method were an accurate approximation to the modal information of the system. For each bus in the system, the first four modes of oscillation were recorded. It was found that typically most of the energy present in the signal was contained in the first four modes of oscillation, thus this number of modes was selected. The inclusion of additional modes did not improve the accuracy of the output. Once the modal information was found, the inertia values of that specific simulation were stored in pair with the modal information of the system obtained. This process was repeated for hundreds of cases to achieve a sizeable data base to allow for accurate results from the neural network analysis. The data was split randomly into the three groups of train data, validation data, and test data. As the names imply, the neural network was creating and trained using the train data, attempting to have the results as close to the validation data while being cautious of overfitting. The results that are output can then be compared to the test data to see the accuracy of the network. The neural network was run with each mode of oscillation individually, and then with all modes included. The most accurate results were found when using all possible modal information as an input. The results of this procedure can be seen below. Figure 2.3 shows the accuracy of the neural network versus training time.

27 16 Figure 2.3: Accuracy of Neural Networks as Training Time Increases Each epoch is an iteration through the training procedure on the data. It can be seen that the accuracy of the training and validation sets reaches optimal accuracy at around 5 epochs, however the accuracy of the neural network on the test data begins to decrease after the 5th epoch as well. This is due to the problem of overtraining of neural networks. Each iteration tunes the neural network to work more accurately for the data set given, which reduces the generality of the network causing it to work less well on other data sets. As a result, the training of the neural network should be stopped between 5 and 15 epochs for optimal results, in this case 15 epochs was used due to having the highest accuracy in the validation set. Figure 2.4 shows a histogram of the errors between calculated values and the actual values. Most of the errors are near to zero, while a few are slightly larger. This is true for all of the data sets: training, validation, and test data. However, as the inertia values do not vary drastically in realistic systems, these small differences can signify significant errors.

28 17 Figure 2.4: Neural Network Errors Figure 2.5 shows the distribution of the errors between each estimate and the actual value. Figure 2.5: Distributions of Estimation Errors

29 18 Table 2.1: Mean Absolute Percentage Error of Inertia Estimations Individual Bus Inertias Regional Inertias System In MAPE 21.5% 2.8% 1.4% Standard Deviation of Percentage Errors 26.89% 3.62% 1.65% It can be seen from this, that the neural network accurately identifies that regional and system inertias, but assigns the average value of inertia to the individual buses such that these regional and system inertias are obtained. Table 2.1 shows the mean absolute percentage error (MAPE) and standard deviation for each case. MAPE is calculated through the following formula: MAP E = 100 n n i=1 ActualV alue i EstimatedV alue i ActualV alue i (2.6) Where n is the number of data points calculated. In these cases, bus inertia is defined by the combined inertia of any rotating masses attached to the bus, such as generators or machine loads. If a bus did not have any rotating masses connected to it, then it was not considered in this work. It can be seen from these results that the estimation of the individual bus inertias is not accurate. The mean absolute percentage error of the individual bus inertia estimation results is 21.5%. This is a very large error and as a result the estimates for individual bus inertia are not usable. It is also of note that the standard deviation of the errors for the individual bus estimations was very large. This indicates that this procedure was not able to find a strong enough relation between the calculated modal information and the inertia values at the buses. There are a few reasons for as to why the neural network is not able to estimate individual bus inertias accurately. The correlation between modal information and inertia may not be strong enough for accu-

30 19 rate estimation. That is to say, the inertia at the buses may not have had a large enough impact on the modal information in the system for the neural network to detect the changes in the modal information when different inertia values were used. It is possible that including other variables in addition to the modal information would be beneficial. Another alternative could be to attempt to determine under what circumstances the bus inertia estimations would be accurate. This could possibly be done through the use of participation factors or sensitivity analysis. However, when looking at the entire system inertia values or the regional inertia values the results are much more accurate. The mean absolute percentage errors for the regional and system inertia estimates are 2.8% and 1.4% respectively. Additionally, the standard deviation of the errors was low for both of these cases. While calculation of individual bus inertias is desirable, the results from the regional and system inertias are important and can be used for analysis on the system which, ultimately, is the objective. The results show that this method is usable for the estimation of regional and system values. However, it is not able to accurately estimate the values of the inertia at individual buses within the system.

31 Chapter 3 Empirical Topology Estimation When discussing topology, most people will first consider how all the substations are connected to each other through transmission lines. However, another factor which also has a significant impact is how each substation is connected internally at any given moment. Most topology estimation techniques make use of circuit breaker statuses in order to obtain the topology. For example, in Figure 3.1, it can be seen that since two breakers are open, while the other two are closed, two buses within the substation have formed. Figure 3.1: Example of Node Groupings However, the issue is that this information is not normally available in synchrophasor data 20

32 21 streams in the vast majority of systems. This means that any synchrophasor analytic, such as state estimation, that relies upon the knowledge of topology in order to accurately function has to obtain this information from other sources. This severely limits the functionality of synchrophasor analytics in these systems. Unfortunately, the reason that the circuit breaker status data is so often not available in synchrophasor data streams is because of the huge investment burden of the cost to install all of the equipment required. Therefore, to alleviate this burden a methodology was devised such that system topology can be estimated solely with the use of synchrophasors with relatively high accuracy. This allows for the use synchrophasor-only analytics, which do not need to rely on any other types of data sources [42]. 3.1 Threshold Calculation The first step in the procedure is to determine a voltage phasor deviation threshold, such that it is safe to say that any nodes which have a difference in their voltage phasors larger than this threshold are not connected. This deviation threshold is obtained empirically making use of the concept of voltage coherency [43]. Voltage coherency is the idea that nodes which are connected should have ideally the same, but at the very least extremely close, voltage phasors. This can be seen best while observing phase angles, thus going forward all discussion will be focused on phase angles rather than magnitudes. With this in mind, it should be possible to observe in a system the normal differences in phase angle that disconnected nodes have. If this value can be obtained, then it should be possible to say whether two nodes are connected or not with relative certainty. The way in which the threshold is obtained is through the running of simulations in order to obtain the difference of angles across disconnected nodes for all possible system configurations and then run a statistical analysis on the acquired data to best select a value for the threshold. This does, however, mean that the phasor deviation threshold value obtained is specific to

33 22 the system upon which the analysis is done. The issue of how to determine the voltage phase angle deviation threshold from this information must next be solved. While ideally any two nodes that are connected by a breaker will have the same voltage phasor, this is not the case in a real system. There are several factors that may cause the voltage phasors to differ. The most likely cause and factor of highest impact is measurement errors. Any measurement in a real system will have an associated error with it due to various issues, such as noise, poor calibration, quantization, etc., that will make comparisons of values more difficult. Another cause of disconnected nodes maintaining similar voltage phasors is the connectivity of the system. It is possible for very short paths between two nodes to exist outside of the substation which will cause the voltage phasor values to stay relatively close even if they are disconnected. Thus, because of these issues, when determining the phasor deviation threshold, it needs to both be small enough to accurately determine the connectivity state of nodes in the vast majority of cases, while being large enough to avoid misclassifying connections due to these issues. As such, it was chosen to use a value such that 95% of all phase deviations recorded were above the threshold. It will be shown later, that even with this criteria, the voltage phase deviation threshold is relatively large, which confirms the assumptions made that nodes that are disconnected will have differing voltage phasor values. The system that is used for this work is the IEEE standard 118 bus system, as seen in Figure 3.2. However, since this topology estimation procedure attempts to determine the connections within a substation, this system needs to be modified to be of use. As such, there are two approaches used for the creation of a system that contains substations. The first method transforms each bus within the IEEE 118 bus system into a substation with one of four different standard layouts, while the other method is a more general black box approach, which has been named the synthetic mesh method.

34 23 Figure 3.2: IEEE 118 Bus System Standard Substation Topologies Approach Within power systems, there are several standard substation topologies or layouts in use. These topologies are: Double-Breaker, Double-Bus Breaker and a Half Ring Bus Single Breaker, Single Bus and can be seen in Figure 3.3.

35 24 Figure 3.3: Standard Substation Configurations With this information, every bus within the 118 bus system was modified into one of these substation topologies, such that each component(transmission line, machine, load, etc.) connected to the substation is connected to an individual node within the substation. With this system created, all possible connection cases the system substation topology connections can be tested. The simulations were run and the phasor variation threshold was found as described previously. Figure 3.4 and 3.5 show the histogram and cumulative distribution function of the recorded angle deviations, respectively.

36 25 Figure 3.4: Histogram of Phase Angle Deviations in the Standard Substation System Figure 3.5: CDF of Phase Angle Deviations in the Standard Substation System From these plots, it can be seen that when selecting a threshold value for which 95% of cases are larger, the threshold is This is a desirable value for a few reasons. First, it confirms the assumption that when two nodes are disconnected, the voltage phasors will

37 26 be different from one another. Secondly, the value is large enough to be able to account for noise with relatively high accuracy. From the synchrophasor measurement standard, it is required that all synchrophasors have a measurement total vector error under 1% [44]. In the worst case scenario, which should be stated is highly unlikely, that the entirety of this error is applied to the phase angle, the measurement error for phase angles can only be as large as.517 [44]. Therefore, for there to be a misclassification due to measurement errors, both measurements would need to have the highest worst case scenario error on their measurements in opposite directions, which is extremely unlikely. Additionally, since the individual substations are modeled both the threshold for each type of standard substation layout, as well as the potential threshold for each individual substation can be found. Figure 3.6 and 3.7 show the histogram and cumulative distribution function for each type of standard substation configuration respectively. Figure 3.6: Histogram of Phase Deviations for Each Substation Type

38 27 Table 3.1: Phasor Deviation Threshold for Each Standard Substation Type Substation Type Double-Breaker, Double-Bus Breaker and a Half Ring Bus Single-Breaker, Single-Bus Deviation Threshold Figure 3.7: CDF of Phase Deviations for Each Substation Type The threshold of each type of standard substation configuration can then be found and is shown in Table 3.1. From this it can be seen that while the type of substation layout does have an impact on the threshold value, it is not significant. There may be issues with the value of the breaker and a half threshold, but it is still not unreasonable. If each substation is considered individually, the resultant threshold for each individual substation can then be seen in Figure 3.8. Through this, the variations and importance

39 28 of substation location become more apparent. However, it can be seen that the phasor deviation threshold for the vast majority of substations can be set significantly large. Figure 3.8: Threshold Result for Each Individual Substation Finally, to verify that the results are plausible on other systems, similar studies were run for the IEEE 14 bus system as seen in Figure 3.9.

40 29 Figure 3.9: IEEE Standard 14 Bus System Figures 3.10 and 3.11 show the histogram and cumulative distribution function respectively. Figure 3.10: Histogram of Phase Deviations for the 14 Bus System

41 30 Figure 3.11: CDF of Phase Deviations for the 14 Bus System From these plots, it can be seen that the voltage phasor deviation threshold is 1.03, which is also a desirable value. This also shows that the size of the system is not a determining factor in the value of the phasor deviation threshold Synthetic-Mesh Approach The synthetic mesh approach does not assume knowledge of any connections between the nodes within the substation. It works under the assumption that any node within the substation can be connected to any other node, as seen in Figure Through this, every possible node connection case is tested and the phase angle between disconnected nodes are recorded.

42 31 Figure 3.12: Synthetic Mesh Substation Example Once again, all simulations were run and the results can be seen in Figure 3.13, which shows a histogram of the phase angle deviations. Figure 3.13: Histogram of Angle Deviations for the Synthetic Mesh Approach In the same process as before, the deviation threshold was selected such that 95% of all deviation cases were above the threshold. For the synthetic-mesh approach, this results in a deviation threshold of While still large, this value is lower than the standard

43 32 substation topologies method. This is due to the increased connectivity of the system being tested. However, the value is still viable and able to produce accurate results. 3.2 Topology Estimation Once the voltage phasor deviation threshold has been found, the topology of the substation can be estimated. The procedure has been split into several levels that build upon the output of the previous levels by making additional assumptions. This was done in this manner such that the user can select which assumptions they are comfortable with and have confidence in the result. While there will be much discussion about inferring breaker status correctly, it should be noted that this is not the end goal. There are cases where a specific breaker s status is irrelevant in correctly determining the topology of the substation. The objective of this procedure is to construct the bus-branch model of the system from the nodal-breaker model, thus the final output of the estimation is the topology of the substation, not necessarily the breaker statuses Level 0: Initialization The first level makes no assumptions about the connection between nodes. It solely takes into account the voltage phasor measurement where available and determines whether each node is energize or not. Each individual energized node forms its own bus, unenergized nodes are disconnected, and unmeasured nodes are considered unknown or unobserved.