Non-Collocated Power Measurements in a Power System State Estimator

Transcription

1 Non-Collocated Power Measurements in a Power ystem tate Estimator B Mann, G Heydt, G trickler Abstract This paper addresses a problem in state estimators for power systems The issue of non-collocated s is studied The of P+jQ in a line or at a bus is usually accomplished by measuring voltage and current at an appropriate point A transducer converts the voltage and current to active and reactive power which are transmitted to the state estimator f the voltage and current s are not at the same point in the circuit, that is the s are non-collocated, error is introduced The paper reports a way to correct for non-collocation of s ndex Terms tate estimation, s, error, complex power NTRODUCTON TATE estimation is a widely used tool in power system energy management systems The essence of state estimation is that s are taken of active and reactive power, and system voltage magnitudes and phase angles (ie, the states ) are estimated The process usually uses minimum least squares methods Power system s sometimes possess a degree of error, due to anything from instruments installed with reversed polarities to information loss through analog to digital conversion tate estimation methods can flag and smooth out bad s, but state estimation has better accuracy overall when there is less error n a deregulated power market it is increasingly important to find cost effective ways to improve system visibility and account for errors f the error is due to an incorrectly installed instrument, then a software solution would likely save more money than instrument reinstallation One such power error is non-collocated complex power s Complex power is usually calculated from voltage and current s, which must be taken at the same place to have any meaning n the non-collocated case the current transformer (CT) and the potential transformer (PT) of a power instrument is separated by an amount of impedance This work was supported by the alt River Project (RP), Phoenix, AZ Authors Mann and Heydt are with the Department of Electrical Engineering at Arizona tate University, Tempe, AZ ( s: brianmann@asuedu, heydt@asuedu) Authors trickler is with RP, Phoenix, AZ ( gwstrick@srpnetcom) t will be shown that a more accurate power can be calculated utilizing a non-collocated value plus a correction procedure This correction calculation requires an accurate knowledge of the impedance between the instruments, the instruments location relative to the impedance and access to a PT reading Given these values, a non-collocated power can be corrected through calculation rather than a costly and interruptive reinstallation of hardware This paper gives detail on the nature of non-collocated s and presents a method to calculate more accurate power values An bus test bed has been created to demonstrate the effect non-collocated power correction has on improving state estimation results over a large system The general topic of power system state estimation is well documented in the literature References [-] are a small sampling of the available literature Concerning error in state estimation, the following are offered as samples of relevant resources: Reference [] describes network topology errors and how to identify and correct these Reference [5] relates to parameter errors Reference [6] describes error in circuit breaker status Reference [7] deals with bad data s NON COLLOCATED POWER MEAUREMENT Complex power is a function of the complex voltage and current, where Power measuring wattmeters are a very common part of power systems instrumentation These power devices work by sampling the voltage, v(t), and the current, i(t), which are turned into a digital signal by an analog / digital converter and the power is calculated by a transducer This value is provided to a remote system operator and sometimes the individual voltage or current as well When measuring or calculating complex power, the voltage and current must be taken from the same place in the system n the case of a non-collocated power, this is not the case The impedance between the CT and PT can be represented as the circuit shown in Fig n the case of a non-collocated it is assumed that the instruments are not far apart and well within the range of typical short line modeling limits Because the CT and PT are often not separated by even a kilometer, resistance in the impedance model has been assumed to be negligible An example of a non-collocated instrument placement is shown in Fig

2 Figure A model for the reactance between CT and PT in a non-collocated be changed to work with and being known values, which will be explained later First, the voltage magnitude can be made the reference voltage, so 0 ince and and is known, can be calculated immediately To calculate and, and are needed The following matrix equation can be formed to solve for the remaining unknowns, + j + () Equations () are a simple consequence of the bus impedance analysis of a linear AC circuit, namely bus Z bus bus, where Z bus is the bus impedance matrix referred to ground [8] Figure An example of a non-collocated power instrument placing When discussing power values in this paper, the notation ab will be used The subscript a will be the same subscript as the voltage used to calculate ab and the subscript b will be the subscript of the current used to calculate ab For instance, meaningful power values from the system in Fig would be and, which are calculated as follows, Examples of non-collocated power s from the system in Fig would be (shown with instrument placing in Fig ) or Each of these is calculated as follows, t is possible to calculate all of the power, voltage, and current values of the circuit in Fig (including and ) given only the systems reactance,,, and, a voltage magnitude, or, and a non-collocated complex power, or, which are values from the circuit equation shown in Fig This can be done using one of two methods: Case A and Case B, which will be detailed in this section Case A is for instances when the voltage of the non-collocated power is known and Case B is used when a voltage is know that isn t part of the non-collocated power value One method for correcting non-collocated s will be called the Case A method For Case A, the is known, as well as,,, and and can be calculated This method can also Equation () can be multiplied out to give two equations that can be solved for the two unknowns, and olving for and yields the following equations j j( ) j The matrix relationship simplifies into the following j ( Z Z + ZZ + Z Z ) j( ) j j( ) This Case A method can be changed to work when and are known The diagram in Fig shows how the polarities of these values are changed using Case A with these numbers The preceding method can be used, but is used in place of, is used in place as, is replaced by, is replaced by, is used for and is used for For instance, in the first step of Case A, it is that becomes the reference voltage instead of and next solve for -, and so on A detailed guide for parameter replacement is shown in Table Table A parameter replacement guide to use Case A with and known or Case B with and Parameter Replace With

3 Figure Circuit diagram for using Case A to calculate power given and The next method is called the Case B method This is for when and are known, along with the reactance,, and t will later be shown how Case B can solve for the instance where and are known ince one of the current cannot be immediately calculated with the given voltage and non-collocated power, Case B is a bit more difficult To solve for and, a new circuit model is needed, shown in Fig Figure A new notation for the non-collocated impedance model for use in Case B The given voltage is again the reference voltage, so using the new notation given in Fig 0 a Using basic power relationships, where, PReal{}, and Qmag{}, the following relationships can be made using the new notation P ab Q ac P hf + kg Q kf hg P hb + kc Q bk hc Currently a, P, and Q are known values To help solve for the rest, more equations can be made using the relationship between and, and and Kirchhoff s current law, d + je a ( b + h + b + jk d + je ( f + jg)( j ) d + je jc + f + j jc)( j ) Here,, and are known values, along with a Using these equations and the power relationships, there are eight equations with eight unknown parameters The eight equations can be simplified even more to the following d a + c e b h d + g k e f b f + e d c g P hb + kc Q bk + hc jg olving for the unknown parameters is a simple but time consuming process and left as an exercise for the reader Once all of the parameters in Fig are solved for, the real and unreal parts of and can be calculated with the following equations P P + a 0 b P P + P k ( ( + ) Q P P k b + P + Q + k + kbq j j The Case B method equations shown directly above can also be used when and are known The diagram in Fig shows how the polarities of these values are changed using Case B with these numbers The Case B method can be used, but is used in place of, is used in place as, is replaced by, is replaced by, is used for and is used for For instance, in the first step of Case A, it is that becomes the reference voltage instead of and )

4 next solve for -, and so on A detailed guide for parameter replacement is shown in Table, which works for Case A and Case B Case A and Case B show that complex power can be calculated from a non-collocated A local voltage and a detailed model of the local impedance are required along with the non-collocated power LLUTRATE EAMPLE To illustrate the effect of non-collocated power s, an bus test bed has been created The test bed has been loosely based on the 500 k transmission line power system in the United tates southwest The reactance and network configurations have been invented to obtain a convenient test bed The black diamonds in Fig 5 represent wattmeters Line parameters were calculated from the actual line configurations and approximate line length These parameters are shown in Table in per unit on a 00 MA, 500 k base Each bus was assigned a per unit voltage value Knowing the bus voltage and line parameters allowed calculation of the individual line currents and complex power flow The bus system and state estimation calculations were performed using MATLAB mathematical software The goal in this test bed is to introduce random error into the s and also a non-collocated to one of the buses to study the effect on state estimation The no-error case is used as a basis of comparison Table Line reactances for test bed, on a 00 MA, 500 k base Line Parameter Reactance (pu) a 007 b 000 c 005 d 0005 e 0006 f 008 g 0006 h 0006 j 0009 k l 0056 Figure 5 An bus test bed inspired by the 500 k lines of the U southwest

5 5 The test bed uses 8 power instruments to estimate the voltage angle and magnitude at each bus Least squares state estimation is the most common form of state estimation used by power utilities and will be used in this example Least squares state estimation multiplies a vector of s [z] to the pseudo-inverse of a matrix [H], which gives a state estimate vector [x] The matrix [H] is a matrix of coefficients that is N m (the number of s) by N s (the number of states) Least squares state estimation is improved by a large number of s and in power engineering the case is always overdetermined, where N m >N s The final unweighted, overdetermined case is shown here [ x] [ H ] T [ H ] [ H ] T [ z] For calculating the bus angles with real power s, a simplified version of the following power flow equation is used P sin( δ δ ) ince the voltages are all very close to per unit during the steady state and the angles very close to zero, the general power flow equation can be simplified The sine of small angles is near the angle itself The following power equation is used P ( δ δ ) n the context of least squares state estimation, the matrix [z] is made of real power s and the matrix [H] is made from the inverse of line reactance The state matrix [x] is composed of the bus reference angles to be estimated Now the state estimation equation for the imaginary power is created Again voltages are assumed to be very close to per unit value and the reference angles close to zero The equation can be simplified as follows + Δ + Δ cos( Δδ ) Q [ Δ Δ ] Whether calculating voltage magnitude or angle, the [H] matrix can be made from the inverse of each lines reactance Here the matrix [z] is the reactive power s and the state vector [x] is made of bus voltage magnitudes The test bed will have introduced a non-collocated at Bus for the non-collocation case studies The impedance used in all instances is shown in Table, relative to the general non-collocated impedance circuit shown in Fig The and impedance values could represent large series capacitors used to decrease the change in voltage angle over these lines that somehow got installed between the wattmeter CT and PT The non-collocated power instrument will calculate power from the and positions, relative to the diagram in Fig The power at instrument M should read 5 per unit, but in this non-collocated instance it reads 05 per unit Table Test bed non-collocated reactance values, on a 00 MA and 500 k base alue in per unit / The test will be conducted in five cases, called Cases 0,,,,, and 5 For each case, the complex power s are used to calculate the state variables δ and The state variables will have the no-error, no non-collocation Case 0 results subtracted from them, giving δ-δ no-error and - no-error for each Case The δ-δ no-error and - no-error values will be calculated 000 times and the average value will be analyzed by observing the -norm residual, mean, and variance The Case 0 will have no power s error and no instruments, so the δ-δ no-error and - no-error values are expected to be close to zero Case will have power s with a random amount of error, with a maximum error of 0%, and no non-collocated power instruments Case will have power s with a random amount of error, with a maximum error of 0%, and have no power s Case will have power s with a random amount of error, with a maximum error of 0%, and one non-collocated power instrument at Bus Case will have power s with a random amount of error, with a maximum error of 0%, and one non-collocated power instrument at Bus The percent error represents typical random error in power systems s t is hoped that by comparing Cases 0,,, and that the amount of error due to random noise (the percent error in cases,,, and ) and the non-collocated instrument (Cases and only) can be discerned The results of this test are shown in Tables, 5, 6, and 7 Table Test bed average δ after 000 runs compared to actual δ, for all cases tate δ-δ no-error Case Noise r of δ Mean ariance 0 none E- 0% E-09 0% E-08 0%, one E-05 0%, one E-05

6 6 Table 5 Test bed average after 000 runs compared to actual, for all cases tate - no-error Case Noise r of Mean ariance 0 none E-05 0% E-05 0% E-05 0%, one E-05 0%, one E-05 Table 6 Test bed average P after 000 runs compared to actual P, for all cases P-P no-error Case Noise r of P Mean ariance 0 none E-0 0% E-0 0% %, one %, one Table 7 Test bed average Q after 000 runs compared to actual Q, for all cases Q-Q no-error Case Noise r of Q Mean ariance 0 none % % %, one %, one OME OBERATON DRAWN FROM THE EAMPLE The complex power resulting from each test Case is shown in Tables 6 and 7 n general, the random error in Cases and increases the norm residual for P-P no-error and the mean difference for the real power moves away from zero, all relative to the no-error Case 0 For both the real and imaginary power comparisons, the variance increases due to error ubsequently, the error alone increases overall error and variance Cases and possess error and a single n Cases and the average P-P noerror and Q-Q no-error values result in a larger -norm residual, a mean difference further away from zero for P-P no-error, and a larger variance relative to Cases 0, and Because of this, it can be said that a single non-collocated instrument increased the error in the test bed and the variance The effect on the estimation of states is more subtle than the direct power s The state variable average differences from the no-error case are shown in Tables and 5 Table shows that Cases and, which contain only error, differ little from the no-error Case 0, the only exception being a marked increase in variance in the δ-δ no-error calculations in Cases and relative to Case 0 Measurement error alone then only introduces variability into the bus phase angle calculations When a single non-collocated is added during Cases and the result is a small - norm residual increase and variance increase for the δ-δ no-error and - no-error s relative to Cases 0, and The variance change is smaller for the - no-error than it is for the δ-δ no-error This shows that in power system state estimation, a single non-collocated instrument can widen the variability of calculated bus voltage angles and magnitudes and increase error Applying the non-collocated calculations discussed in ection would eliminate the non-collocated error n this case, since the non-collocated power is calculated from and, so Case B from ection would be the appropriate fix After the power adjustment, the Case and results would resemble the Cases and results By comparing Cases and to Cases and, the benefits become apparent: there is a lower amount of variance and a lower -norm residual in Cases and Therefore, accounting for non-collocated would increase state estimation confidence and will create more accurate s system-wide CONCLUON Non-collocated s in a power system can present a source of error However, with an accurate knowledge of reactance and transformer configuration at the power transducer instrument, it is possible to correct the faulty power and improve state estimation while avoiding costly instrument reinstallation / reconfiguration n cases where there are non-collocated power s, this calculation has the potential to improve state estimation results and system-wide confidence

7 7 ACKNOWLEDGEMENT The authors acknowledge our colleagues D Tylavsky and L Jauregui-Rivera from Arizona tate University, and N Logić and turgill from alt River Project Mr L Jarriel of Power oftware and Consulting also provided valuable input The auspices of the Power Engineering Research Center (Perc) are also acknowledged REFERENCE [] Ali Abur, Antonio Gomez Exposito, Power ystem tate Estimation: Theory and mplementation (Power Engineering, ), Marcel Decker, New York, 00 [] A Monticelli, tate Estimation in Electric Power ystems: A Generalized Approach (Power Electronics and Power ystems), pringer-erlag, 999 [] F chweppe, J Wildes, Power system static state estimation, Part : exact model, EEE Trans on Power Apparatus and ystems, v 89, January 970, pp 0-5 [] C N Lu, J H Teng, B Chang, Power system network topology error detection, EE Proceedings on Generation, Transmission and Distribution, ol, No 6, Nov 99, pp 6 69 [5] W-H E Liu, L wee-lian, Parameter error identification and estimation in power system state estimation, EEE Transactions on Power ystems, ol 0, No, Feb 995, pp [6] A Abur, H Kim; M K Celik, dentifying the unknown circuit breaker statuses in power networks, EEE Transactions on Power ystems, on ol 0, No, Nov 995, pp [7] G N Korres, G C Contaxis, A reduced model for bad data processing in state estimation, EEE Transactions on Power ystems, ol 6, ssue, May 99, pp [8] G Heydt, Computer Analysis of Power ystems, tars in a Circle Publications, cottsdale AZ, 998 BOGRAPHE Brian Charles Mann was born in Helena, MT His BE degree is from the University of Washington, eattle He is a Research Assistant at Arizona tate University, Tempe where is also pursuing the M E E degree Gerald Thomas Heydt is from Las egas, N He holds the PhD in Electrical Engineering from Purdue University His industrial experience is with the Commonwealth Edison Company, Chicago, and E G & G, Mercury, N He is a member of the National Academy of Engineering Dr Heydt is presently the director of a power engineering center program at Arizona tate University in Tempe, AZ where he is a Regents Professor Gary William trickler is from Nebraska City, NE He holds an MEE from New Mexico tate University and a BEE from llinois nstitute of Technology He is presently the Manager of ystem Operations Applications at alt River Project