OPTIMAL PLACEMENT AND UTILIZATION OF PHASOR MEASUREMENTS FOR STATE ESTIMATION

Transcription

1 OPTIMAL PLACEMENT AND UTILIZATION OF PHASOR MEASUREMENTS FOR STATE ESTIMATION Xu Bei, Yeo Jun Yoon and Ali Abur Teas A&M University College Station, Teas, U.S.A. Abstract This paper presents a procedure by which new PMU locations can be systematically determined in order to render an observable system. The procedure is then etended to account or cases o loss o a single phasor measurement unit (PMU). Buses with zero and nonzero injections, and branches with power low measurements are also accounted or in this generalized procedure. Several cases involving dierent power system and measurement conigurations are presented where introducing ew etra strategically placed PMUs minimizes vulnerability o the measurement system against the loss o single PMUs. The paper also develops a linear estimator based on strictly PMU measurements and investigates the computational perormance as well as the bad data processing problem. Detection and identiication o PMU ailures are demonstrated via simulations. Keywords: State estimation, phasor measurement units, network observability, optimal meter placement, bad data processing, integer programming, graph theory. INTRODUCTION State estimators provide optimal estimates o bus voltage phasors based on the available measurements and knowledge about the network topology. Until recently, available measurement sets did not contain phase angle measurements due to the technical diiculties associated with the synchronization o measurements at remote locations. Global positioning satellite (GPS) technology alleviated these diiculties and lead to the development o phasor measurement units (PMU). As the PMUs become more and more aordable, their utilization will increase not only or substation applications but also at the control centers or the EMS applications. One o the applications, which will be signiicantly aected by the introduction o PMUs, is the state estimator. This paper is concerned with two aspects o this issue. One is related to the proper placement and the other to the eective utilization o these new devices or state estimation. The idea o using direct phasor measurements or system monitoring applications including the speciic case o state estimation is not new. Earlier work done by Phadke and his co-workers [-] introduces the use o PMUs or such applications. This work is later etended to the investigation o optimal location o PMUs where each PMU is assumed to provide voltage and current measurements at its associated bus and all incident branches []. It is thereore possible to ully monitor the system by using relatively less number o PMUs than the number o system buses. This problem is solved by using a graph theoretic observability analysis and an optimization method based on Simulated Annealing in []. Possible loss or ailure o PMUs is not considered in that study. In this paper, a numerical ormulation o the optimal PMU placement problem will be presented. Preliminary results, which do not account or the loss o PMUs are presented in [] earlier. This ormulation leads to a solution based on integer programming and also acilitates analysis o network observability. Furthermore, it is general enough to account or loss o single PMUs, eistence o zero and non-zero power injections and power low measurements. Using the measurement design ound by this method, the paper investigates the perormance o linear estimators that eclusively use PMUs and describes detection and identiication o ailed PMUs by using the residual based bad data processing methods. STATE ESTIMATION USING PHASOR MEASUREMENTS. State Estimation Measurements that are telemetered rom the substations are processed at the control centers by the state estimator. State estimator provides the optimal estimate o the system state based on the received measurements and the knowledge o the network model. Measurements may include the ollowing: - Power injections (real/reactive), - Power lows (real/reactive), - Bus voltage magnitude, - Line current magnitude, - Current injection magnitude. PMUs provide two other types o measurements, namely bus voltage phasors and branch current phasors. Depending on the type o PMUs used the number o channels used or measuring voltage and current phasors will vary. In this study, it is assumed that each PMU has enough channels to record the bus voltage phasor at its associated bus and current phasors along all branches that are incident to this bus.

2 State estimation can be ormulated as an optimization problem and solved using an appropriate numerical method. A common choice or the objective unction is the weighted sum o the measurement residual squares, which leads to the well known weighted least squares (WLS) state estimation solution. Formulation and solution methods or WLS state estimation when using the above-itemized conventional measurements are well documented in the literature and can be ound or instance in []. Incorporating phasor measurements to an eisting state estimator will require augmenting various arrays such as the measurement vector and the measurement Jacobian while not aecting the state vector. In that sense, the required modiications are not substantial and the overall problem ormulation does not change signiicantly. On the other hand, i enough PMUs eist to make the entire system observable based eclusively on PMU measurements, then the state estimation problem can be ormulated in a slightly simpler manner. In this case, the relation between the measured phasors and the system states will become linear yielding the ollowing linear measurement model: z = H + e () where: z : is the measurement vector containing the real and imaginary parts o the measured voltage and current phasors. H : is the measurement Jacobian, which is constant and a unction o the network model parameters only. : is the state vector containing the real and imaginary parts o bus voltage phasors. e: is the measurement error vector. This measurement model will lead to a linear state estimator, which will be given by the ollowing equation: T ˆ = G H R z () where: ˆ : is the estimated system state. T G = H R H : is the constant gain matri. T R = E{ e e ) : is the diagonal error covariance matri.. Bad Data Processing Note that several measurements will be associated with a single PMU in the measurement vector, z. In case o an error or ailure o a PMU, this will contaminate all measurements provided by that PMU. Bad data detection and identiication schemes typically process individual measurements and thereore they need to be used repeatedly on the aected measurements o the PMU in order to identiy the aulty PMU. This is accomplished by irst applying the largest normalized residual test to the measurements and then eliminating bad measurements one at a time in a cyclic manner. Due to the interactive nature o the measurements associated with a single PMU, they are vulnerable to error masking and possible misidentiication o bad data. Fortunately, this occurs when errors are conorming which is typically the case i there are topology errors. PLACEMENT OF PHASOR MEASUREMENTS The objective o PMU placement is to make the entire system observable using a minimum number o PMUs. This objective will be slightly modiied later in section. to account or loss o single PMUs. One o the assumptions about the considered PMUs is that each PMU has enough channels to measure bus voltage and all incident branch current phasors at a given bus. Discrete nature o the problem naturally results in deinition o a vector X o binary (/) decision variables, i as given below: i = i a PMU is installed at bus i otherwise Inner product o this binary vector and a vector containing corresponding installation costs o these PMUs, will be deined as the objective unction o the optimization problem. Constraints will be added in order to ensure ull network observability while minimizing the total installation cost o the PMUs. For an n-bus system, this optimization problem takes the ollowing orm: minimize subject to n w i i i ( X ) ˆ where: w i : is the installation cost o the PMU at bus i. ( X ) : is a vector unction representing the constraints. Its entries are non-zero i the corresponding bus voltage is solvable using the given measurement set and zero otherwise. ˆ : is a vector whose entries are all equal to. Proper deinition o the constraint vector unction, (X) is the key to the presented ormulation. Depending on the type o measurement system, this deinition will change, while the objective unction will remain the same. In order to simpliy the description o the constraint unction, it will be given using a small tutorial eample containing buses. Three cases will be discussed, where the eisting system has: - No measurements at all and no zero injections, - Zero and nonzero power injections, - Power injections and power lows. Network diagram or the -bus eample system is shown in Figure. Initially, or case, both o the conventional measurements, namely the power injection measurement at bus (dot shown net to bus ) and the power low measurement on line - (solid () ()

3 square shown on line -) will be ignored. They will be gradually taken into account in cases and. Figure : Network diagram or the -bus system. Case : A system, which has no conventional measurements and/or zero injections Since it is assumed that a given PMU will provide phasors or the currents along branches that connect this bus to all its neighbors, once a bus is assigned a PMU, phasor voltages at all o its neighbors will be assumed to be known or solvable. An easy way to determine all such solvable buses is by using the binary connectivity matri A as deined below: i k = m A k, m = i k and m are connected () i otherwise This yields the ollowing matri or the -bus system o Figure : A = Taking the product o this matri and the binary decision vector X will provide the desired vector unction. Elements o this vector unction will be at least equal to one, i at least one neighbor o the corresponding bus is assigned a PMU. Hence, the constraint equations or the above eample or this case will be ormed as: ( X ) = A X = = + + () The operator + serves as the logical OR and the use o in the right hand side o the inequality ensures that at least one o the variables appearing in the sum will be non-zero. For eample, consider the constraints associated with bus and as given below: The irst constraint implies that at least one PMU must be placed at either one o buses or (or both) in order to make bus observable. Similarly, the second constraint indicates that at least one PMU should be installed at any one o the buses,,,, or in order to make bus observable.. Case : A system, which contains zero and nonzero injection measurements In this case, the injection measurement at bus will be taken into account when determining the PMU locations. Note that i the phasor voltages at any three out o our buses,, and are known, then the ourth one can be solved using the Kirchho s Current Law applied at bus where the net injected current is known. This observation allows a topology transormation where the bus, which has the injection measurement can be merged with any one o its neighbors. Injection measurements whether they are actual measurements or zero injections, are treated the same way. A word o caution needs to be added here in that, i the optimal solution chooses the newly ormed ictitious bus (merger o two actual buses) as a candidate bus, it may place one PMU on one o these two buses or two PMUs on both. In this paper, a topology analysis is applied to check the observability o the system once this happens. This also assures that the minimum number o PMUs will be placed. Figure shows the updated system diagram ater the merger o buses and into a new bus. The newly created branch - relects the original connection between buses and. Binary connection matri A or this case is given below: : A = () Hence, the constraints vector unction will be given

4 = = ( X ) (8) = + Figure : System diagram ater the merger o buses and.. Case : A system, which contains injections as well as low measurements. This case considers the most general situation where both injection and low measurements may be present, but not enough to make the entire system observable. So, the objective is to place PMUs in order to merge the observable islands ormed by the eisting conventional measurements and render a ully observable system. Flow measurement on branch - in the -bus eample system will be used to illustrate the approach. Eistence o this low measurement will lead to the modiication o the constraints or buses and accordingly. Modiication ollows the observation that having a low measurement along a given branch allows the calculation o one o the terminal bus voltage phasors when the other one is known. Hence, the constraint equations associated with the terminal buses o the measured branch can be merged into a single constraint. In the case o the eample system, the constraints or buses and are merged into a joint constraint as ollows: _ new = + Note that this new constraint ensures that i either one o the voltage phasors at buses or is observable, then the other one will also be observable. Applying this modiication to the constraints the ollowing set o inal constraints will be obtained: _ new = ( X ) = = + +. Placement strategy against loss o a single PMU So ar it is assumed that those PMUs which are placed by the proposed method, will unction perectly. While PMUs are highly reliable, they are prone to ailure just like any other measuring device. In order to guard against such unepected ailures o PMUs, the above placement strategy is etended to account or single PMU loss. In this paper, this objective is achieved by choosing two independent PMU sets, a primary set and a backup set, each o which can make the system observable on its own. I any PMU is lost, the other set o PMUs will guarantee the observability o the system. The primary set o PMUs is chosen by building the constraint unctions according to the procedures described in subsections above and solving the integerprogramming problem. The backup set is chosen by removing all the terms in the constraint unctions, i where bus i is in the primary set, in order to avoid picking up the same bus which appears in primary set. Then the integer-programming problem is solved to obtain the backup set. SIMULATION RESULTS Simulations are carried out on the IEEE -bus, IEEE -bus and IEEE 8-bus systems.. Case. In case, two groups o simulations are carried out on the three test systems, which initially have no low measurements. Loss o PMUs is not considered in this case. In the irst group o simulations, zero injections are simply ignored while in the second group, they are used as eisting measurements. Comparative simulation results are shown in Table. Having zero injections will reduce the number o required PMUs as can be seen rom these results. SYSTEMS NO. OF ZERO NUMBER OF PMUS IGNORING ZERO USING ZERO IEEE-BUS IEEE -BUS IEEE 8-BUS 9 Table : Results with and without considering zero injections

5 . Case. In this case, simulations o case are repeated by accounting or loss o single PMUs. Primary and backup locations are ound and combined results are shown in Table. SYSTEMS NO. OF ZERO NUMBER OF PMUS IGNORING ZERO USING ZERO IEEE -BUS 9 IEEE -BUS IEEE 8-BUS Table : Results considering loss o single PMUs. Case. In this case simulations are carried out using IEEE 8-bus system. Three sets o low measurements (P and Q) containing low measurements each, are added in the system one at a time. The locations o these low measurements in the system are given in Table. Simulation results are shown in Table. The required number o PMUs is reduced rom 9 to. Hence, as epected, having conventional measurements reduces the number o required PMUs to make the entire system observable. MEAS. FLOW MEASUREMENTS IN THE SET SET NO Table : Locations o low measurements used or case. SYSTEM IEEE 8-BUS FLOW MEASUREMENT SET Table : Simulation results or case. NUMBER OF PMUS NONE 9 AND, AND. Linear State Estimation using PMUs Perormance o the linear state estimator, which uses only PMU measurements, is evaluated by simulations. Due to space limitations PMU locations are shown only or the -bus system in Figure. Computation times, the number o PMUs and state variables are given or dierent size systems in Table. It is noted that, zero injections are not used in these simulations. Figure : PMU locations or the bus system (ignoring zero injections) Following these simulations, these test systems are used to simulate single PMU ailure cases. Bad PMU detection and identiication requires redundancy. This is accomplished by adding etra PMUs at buses and or the -bus system and PMU at bus 9 or the 8-bus system. System No o States No o PMUs CPU time (sec) Table : CPU times or the linear state estimator. Bad PMU is simulated by introducing errors to all measurements provided by the same PMU. Denoting voltage phasor components as V=E+jF and current phasor components as I=C+jD, the ollowing measurements are simulated as bad data: For the -bus system: V(), I(,), I(,) For the 8-bus system: V(9), I(9,), I(9,). Bad data processing is carried out using the cyclic application o the largest normalized residual test as discussed in section. above. Test results or both systems are shown in Tables and. BD ident. cycle Eliminated meas. Sorted Normalized Residuals st nd rd th E () /. E () / 98. E () / 9. C (,) /. C (,) /. C (,) /. C (,) /. C (,) / 8. C (,) /. C (,9) /.8 C (9,) /.8 C (,) /. V () I (,) I (,) No More Bad data

6 Table : Bad data identiication cycles or -bus system with a bad PMU at bus. Bad components o voltage and current phasors are correctly identiied and eliminated thanks to locally redundant measurement conigurations or both cases. As discussed in section., it is possible to guarantee bad PMU detection and identiication by making sure that none o the PMUs are critical. This can be achieved by the measurement design proposed in section.. Cost associated with such a design versus the beneits o bad PMU detection capability can be evaluated. CONCLUSIONS This paper is concerned with two issues related to the phasor measurement units. The irst issue is proper placement o these devices or a given budget. This issue is addressed via a - integer programming method, where injections and power low measurements are also considered. As a urther consideration, loss o single PMUs is also taken into account to minimize the vulnerability o state estimation to PMU ailures. Once placed, the utilization o PMUs or state estimation is investigated. Potential beneits o incorporating these devices into the eisting measurement set, both rom the point o view o bad data elimination as well as computational perormance, are identiied and illustrated by simulated eamples. REFERENCES BD identiication cycle Eliminated meas. Sorted Normalized Residuals in Descending Order st nd rd th E (9) / 88.8 C(9,) /. E () /. C(9,) /.8 C (9,) /. C(,9) /. C (9.) /.8 C (,9) / 9. C (,9) /. E (8) /. E () /. E (8) /.98 V (9) I (9,) I (9,) No More Bad Data Table : Bad data identiication cycles or 8-bus system with a bad PMU at bus 9. [] A. G. Phadke, Synchronized phasor measurements in power systems, IEEE Computer Applications in Power, Vol., Issue, pp. -, April 99. [] A. G. Phadke, J. S. Thorp, and K. J. Karimi, State Estimation with Phasor Measurements, IEEE Transactions on Power Systems, Vol., No., pp. -, February 98. [] T. L. Baldwin, L. Mili, M. B. Boisen, and R. Adapa, Power System Observability With Minimal Phasor Measurement Placement, IEEE Transactions on Power Systems, Vol. 8, No., pp. -, May 99. [] Bei Xu and A. Abur, Observability Analysis and Measurement Placement or Systems with PMUs, Proceedings o the IEEE PES Power Systems Conerence and Eposition, October -,, New York, NY. [] A. Abur and A.G. Eposito, Power System State Estimation, Marcel Dekker,.