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Event Detection and Location in Electric Power Systems using Constrained Optimization Michael J Smith Kevin Wedeward Student Member IEEE Albuquerque NM Department of Electrical Engineering New Mexico Institute of Mining and Technology Socorro NM 87801 Abstract Online monitoring and diagnostics are important functions in the operation and maintenance of electric power systems In this preliminary paper we present two new methods for electric power grid state and parameter estimation when a limited amount of grid information is available The methods are based on developing a constrained optimization problem whose solution provides a set of desired grid information The fi rst method attempts to estimate an approximate state of the grid from a set of measurements at a relatively small number of sites Constrained to the power balance manifold this method minimizes an objective function based on generic grid behavior as well as any available information about the particular state of the grid to estimate a selected set of grid states and parameters The second approach uses time data from a small number of sites to try to detect and localize an event such as a faulted line in the grid This method minimizes an objective function defi ned on the time varying power balance manifold that is designed to be sensitive to abrupt local state and parameter changes This detection and localization tool is intended as an early warning system or a supplemental check within a larger multi modal detection system The capabilities of the approaches are demonstrated through simulation on a IEEE power fl ow test grid I INTRODUCTION The day to day dependence of homes industry commerce etc on electricity requires a steady supply of electric power The interconnected grid uses Supervisory Control and Data Acquisition SCADA systems control systems and protective device systems to cope with changing load demands and grid failures Since the deregulation of electric power systems the government has been trying to help power system control areas understand their own grid behavior and status to ensure reliable power delivery A cost effective and easy way for monitoring grid status is to use a limited set of instrumentation already in place The use of limited SCADA measurement data can be used as a back up monitoring tool to quickly detect and locate device failures such as transmission line outages Detecting and locating such failures in a timely fashion would minimize damage to connecting equipment and reduce the possibility of cascading failures while improving situational awareness in electric power system operations In order to observe the status of the grid using limited measurement sites a new technique has been developed The new technique is cast as a nonlinear constrained optimiza tion problem in which the system states and parameters are forced to the power balance manifold and imposed constraints Beyond using SCADA instrumentation for the power system state estimator little research has been deployed in using an optimization approach with limited measurements to estimate grid behavior and device failures particularly outside of the local area of measurement sites The research articles that come close to this approach are found in references 1 2 3 The work presented in 1 is a new fault location algorithm based on bus voltages An error function is evaluated by dif ferencing the measured bus voltage magnitudes and predicted bus voltage magnitudes The predicted bus voltage magnitudes are calculated through the use of the impedance matrix and maximum available fault currents In work presented by 2 existing phasor measurement units PMUs on the grid are used to locate transmission line faults Following a system event the bus voltage phase angle differences at the observable buses in the system with respect to their pre event values can be determined After changes in voltage phase angles are determined an optimization problem is solved for detecting event occurrences Another fault detection algorithm is to train neural networks to learn on simulated or real data from an electric power system model or real system respectfully 3 After the power system at hand is trained measurement inputs to the system are compared to trained neural networks and classifi ed either as faults or some other anomaly on the grid The work presented in this paper describes a new algorithm for detecting and locating anomalous events such as device failures or line outages in large electric power systems given multiple time measurements from a limited number of remote sites Line outages are detected and located using an opti mization based approach where a time differencing objective function is being minimized to identify any grid changes A separate problem of this work is estimating state and parameter values in an electric power system given single time measurement information from a limited number of remote sites To estimate state and parameter values an optimization approach is used to minimize an objective function based upon grid operating principles II ESTIMATINGPOWERSYSTEMSTATE This section discusses solving a nonlinear constrained op timization NLCO problem to provide a set of desired grid information This method attempts to estimate an approximate state voltages and powers of the grid from a set of mea surements at a relatively small number of sites Constrained to the power balance manifold this method minimizes an 978 1 4244 4241 6 09 25 00 2009 IEEE objective function based on generic grid behavior as well as any available information about the particular state of the grid to estimate a selected set of grid states and parameters In contrast to NLCO an optimal power fl ow OPF approach has suffi ciently more known data that includes all the load real and reactive powers as well as measured data throughout the grid A Optimal Power Flow The mathematical problem formulation for the OPF problem is as follows Minimize f x subject to g x 0 h x 0 1 xmin x xmax where x the adjustable variables are the bus voltage magni tudes and phase angles as well as the fi xed parameters of the system The objective function f x is a scalar that represents the minimization of generator costs The equality constraints g x 0 represent the power fl ow equations The inequality constraints h x 0 bound functions of on variables such as line power fl ows In addition limits xmin x xmax may be placed directly on state variables or control variables This is a typical OPF problem formulation which can be solved using several methods as discussed in 4 B Objective Function In the NLCO problem considered the objective function was designed to capture the characteristics of normal grid operation F NBus k 1 WV k Vk 1 2 NLine k 1 W k 2 k WP NGen k 1 PGk NLoad k 1 PLk 2 WQ NGen k 1 QGk NLoad k 1 QLk 2 2 where WV k W k WP WQare weights given to each term to emphasize different levels of minimization Larger weights minimize specifi c terms more heavily than do smaller weights The Vkis the voltage magnitude at bus k NBusis number of buses present kis the difference of voltage phase angles across line k where NLineis number of lines The PGkand QGkare the real and reactive generator powers at bus k where NGenis number of generators The PLkand QLkare the real and reactive load powers at bus k where NLoadis number of loads In this problem the objective function was weighted as follows voltage magnitude and voltage phase angles with WV k W k 01 real power losses with WP 1 and reactive power losses with WQ 001 Note weights could be scaled up to give the same results The fi rst term Vk 1 is the voltage at each bus being designed near 1p u the nominal bus voltage magnitude the second term kis the difference of l jat each transmission line k between buses l and j used to keep ksmall for stability purposes and the third and fourth terms were put in the objective function to minimize the total transmission line losses for real and reactive power For further information on nominal operation of large electric power systems see 5 In this objective function each term is squared to yield a positive objective function In general the design of the objective function is an engineering look at how an electric power system should operate in normal conditions C Equality Constraints power fl ow constraints The power balance equations are the nonlinear equality con straints These equations defi ne the power balance manifold to which all feasible solutions are constrained GkV 2 k N j 1 j k VkVj Gjkcos kj Bjksin kj PGk PLk 0 3 BkV 2 k N j 1 j k VkVj Gjksin kj Bjkcos kj QGk QLk 0 4 Gkand Bkare the total line s shunt conductance and susceptance for all lines connected to bus k respectively Gjk and Bjkare the line s series conductance and susceptance from bus j to k kjis the difference of phase angles j kfrom bus j to k D Inequality Constraints The inequality constraints used in this problem are the real load powers ranked according to size For example if the real load power at bus k PLk is larger than that of bus j PLj i e PLk PLj then this inequality constraint can be written as PLj PLk 1 1 PLj PLk 0 In general for ranking of all loads matrix A can be constructed with 1 in the column of the larger real power and 1 in the column of the smaller real power All other elements in the row are 0 This leads to the general form in the equation below A PL1 PLNLoad 0 5 PL1through PLNLoadare all real load powers in the system E Side Constraints The side constraints impose limits on variables at each bus in the system PGk min PGk PGk maxk 1 NGen QGk min QGk QGk maxk 1 NGen Vk min Vk Vk maxk 1 NBus k min k k maxk 1 NBus 6 7 Where min and max denote the minimum and maximum values in the above terms The solver used for this nonlinear programming problem is the MATLAB function fmincon which is based on the SQP successive quadratic programming method 6 The software was developed using MATLAB 7 For analysis using the NLCO set up it is assumed that all transmission line parameters Gjk s Bjk s and Bk s are known throughout the grid and Gk s are practically zero and are ignored The capability to perturb these parameters is possible in the NLCO code In the upcoming test case system variables PGkand Vkare known at buses with generators throughout the grid The generator bus variables were chosen as knowns because they are the heart of a power system and therefore heavily instrumented It is important that our initial starting point be within a fea sible solution region hence the importance of grid knowledge An initial condition for all unknown Vk s and k s is 1p u and 0 rad respectively The load PLk s were calculated by taking the sum of the known generator PGk s and dividing by the number of loads in the grid Load QLk s and generator QGk s were initialized using an assumed power factor PF Because we know measurements of real power at all generators we can calculate the generator reactive power assuming a PF at 95 F Results In these results the IEEE 118 bus test sytem see 8 for details is used for simulation and analysis where 15 of known measurements are taken from the grid There are a total of 472 variables in this system This system has 118 buses 34 generators 91 loads and 186 lines There are 34 known generator real powers and 34 known voltage magnitudes at generator buses Unknown varibles include 118 voltage phase angle 84 voltage magnitudes 34 reactive generator powers and 91 real and reactive load powers An arbitrary load fl ow solution of this system was available with that data and is used only for comparison NLCO solutions can be varied by tuning the objective function weights on each term There are a total of four variables types plotted for this system voltage magnitudes voltage phase angles and load real powers and reactive powers Recall real powers and magnitudes of voltages at generator buses are assumed known measured where for every red that matches a black o is a known measurement A red marker that does not match a black o marker represents a variable solved for at its bus location All other fi gures can be similarly interpreted In Figure 1 NLCO s output for voltage magnitudes is represented by a red line with a The black dashed line with a o is a given load fl ow solution for both systems All reactive load powers are unknowns in the system The scaling in the Figures 3 and 4 may go up to 120 on the horizontal axis but keep in mind there are only 91 load buses for this system meaning there is not a or o for every bus number Using NLCO in this work is useful for solving for unknown variables in a highly underdetermined system for purposes of estimating grid conditions To improve the solution more information about the grid is needed in the form of measurements objective function and constraints 020406080100120 0 92 0 94 0 96 0 98 1 1 02 1 04 1 06 Voltage Magnitude Bus Number Voltage Magnitude V NLCO Load Flow Fig 1 Voltage magnitude for IEEE 118 bus system 020406080100120 0 2 0 1 0 0 1 0 2 0 3 0 4 0 5 Bus Number Voltage Phase Anlge Voltage Phase Angle NLCO Load Flow Fig 2 Voltage phase angle for IEEE 118 bus system 020406080100120 3 2 5 2 1 5 1 0 5 0 0 5 Bus Number Real Power P Output Load Power P NLCO Load Flow Fig 3 Real load power for IEEE 118 bus system 020406080100120 1 2 1 0 8 0 6 0 4 0 2 0 0 2 0 4 0 6 0 8 Bus Number Reactive Power Q Output Load Power Q NLCO Load Flow Fig 4 Reactive load power for IEEE 118 bus system III NLCO ANOMALYDETECTIONAPPROACH The work presented in this section is main focus of paper and describes a new algorithm for detecting and locating anomalous events such as line outages in large electric power systems given multiple time measurements from a limited number of remote sites Line outages are detected and lo cated using an optimization based approach where a time differencing objective function is minimized to be sensitive to abrupt state and parameter changes in the grid The sys tems nonlinear constraints inequality constraints and side constraints used in the previous section are identical The differences are the objective function measurements taken and remote site location Over time fmincon minimizes the objective function based upon a time series of measurements The output of the NLCO code is a vector fi lled state values for times at which measurements were taken For large electric power systems it is diffi cult to look at various plots of the time series data for detection and location of the line outage To overcome this problem it was crucial to understand grid behavior during a line outage A MATLAB based simulation and analysis code for electric power EP systems also known as the EP code 9 was used to simulate line outages and study how the grid reacted It was observed that the most common trend during a line outage was voltage magnitude and voltage phase angle would diverge across lines Obtaining this grid knowledge was important for the development of post processing tool sets necessary to analyze the output of the NLCO code The tools used are based off the bus voltage magnitudes and bus voltage phase angles Lacking data from a real power system the multiple time measurements used in the NLCO code are synthetic mea surements taken from the EP simulation and analysis code The method used for choosing measurements is to fi nd and rank the steady state sensitivities of the measured quantities with respect to the line parameters Figure 5 is a big picture look at the NLCO for detection and location of line outages on the grid The process starts by taking measurements from Generated Multiple Time Measurements Bus Voltage Magnitude Generator Real Power Load Real Power Optimization Process Optimization Output Bus Voltage Magnitudes Bus Voltage Phase Angles Normal Operating Procedures Constraint Based Grid Knowledge Time sec Time sec Time sec Fig 5 NLCO Process for 118 bus test system limited sites on the grid in this diagram the generated multiple time measurements are taken from the transformer with the green outline The bus voltage magnitude generator real power and load real power are then sent to the optimization block for processing Generic grid knowledge is also sent to the optimization process in the form of constraints The output of NLCO is time series data not only for bus voltage magnitude and voltage phase angle as shown in the fi gure but also for all other unknown variables such as load powers being solved A Time Differencing Objective Function A mathematical description of the time differencing objec tive function is shown in equation 8 All summation terms are simple time differences between the current set of unknowns at iteration it and their values at previous iteration it 1 F Nbus k 1 WV k Vk it Vk it 1 2 Nbus k 1 W k k it k it 1 2 NGen k 1 WPG k PG k it PG k it 1 2 NLoad k 1 WPL k PL k it PL k it 1 2 NGen k 1 WQG k QG k it QG k it 1 2 NLoad k 1 WQL k QL k it QL k it 1 2 8 Looking at terms in equation 8 all other variables at the it 1 iteration need to be fi lled with a feasible starting point to get the time differencing objective function started This fi rst solution is fi lled in with a best initial condition that satisfi es power balance equations 3 and 4 as well as constraints just like was done in with the objective function in section II The starting point for optimization is the solution at the previous iteration with new measurements taken as knowns The initial starting point from start to fi nish of this algorithm changes at every iteration due to the new set of changing measurements injected into equations 3 and 4 Looking at equation 8 it must be understood that the goal is to minimize any change meaning from start to fi nish the initial condition will not change but what has been found and will be shown in the following results is there is enough of a change to be able to detect and closely approximate where the line outage occurred B Results The results presented all come from tests run on the IEEE 118 bus system The meas