On-line Calibration of Voltage Transformers Using Synchrophasor Measurements

Transcription

1 On-line Calibration of Voltage Transformers Using Synchrophasor Measurements Anamitra Pal, Member, IEEE, Paroma Chatterjee, Student Member, IEEE, James S. Thorp, Life Fellow, IEEE, and Virgilio A. Centeno, Senior Member, IEEE Abstract Un-calibrated instrument transformers present at the inputs of phasor measurement units (PMUs) can significantly degrade their outputs. This also causes problems in downstream applications that use PMU data. This paper presents a method for calibrating voltage transformers online using synchrophasor measurements. The proposed approach aims to find the optimal locations where good quality measurements must be added so as to bring the calibration error of all the measurements below a pre-defined threshold. The IEEE-8 bus system, the IEEE-300 bus system, and a 383 bus Polish system have been used as the test systems for this analysis. The advantage of the proposed approach is its effectiveness and robustness. Index Terms Binary integer programming (BIP), Calibration, Optimal placement, Phasor measurement units (PMUs), Voltage transformers. S I. INTRODUCTION INCE the introduction of phasor measurement units (PMUs) a variety of applications have been proposed that use synchrophasor measurements for operational decisionmaking. Some of these applications include state estimation (hybrid/linear), base-lining studies (computing alert/alarm limits), protection (security assessment, adaptive relaying), control (damping oscillations), etc. []-[9]. Most of these applications use voltage measurements (voltage magnitude and/or angle) obtained from PMUs for their successful functioning. Moreover, many of these applications suggest applying results obtained from simulated data to the field. However, results obtained from simulated data can cause problems when applied directly to the field as illustrated below with a base-lining example. Current and voltage instrument transformers located at the inputs of a PMU are high accuracy class electrical devices. They provide reduced levels of current and voltage This work was partially supported by Lawrence Berkeley National Lab (LBL) subcontract of prime contract DE-AC03-05CH3 between LBL, Department of Energy (DOE), and Dominion Virginia Power (DVP), Defense Threat Reduction Agency (DTRA) Grant HDTRA---006, DTRA Comprehensive National Incident Management System (CNIMS) Contract HDTRA--D , and National Science Foundation (NSF) Network Science and Engineering (NetSE) Grant CNS Anamitra Pal ( anam86@vbi.vt.edu) is with the Network Dynamics & Simulation Science Laboratory of Virginia Bioinformatics Institute at Virginia Tech, Blacksburg-406, USA. Paroma Chatterjee ( pchatt4@vt.edu) is a PhD student, James S. Thorp ( jsthorp@vt.edu) is a Research Professor, and Virgilio A. Centeno ( virgilio@vt.edu) is an Associate Professor at the Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA-406, USA. measurements to the PMU. But the nominal transformer ratios specified in the name plates of these devices may differ from the actual conversion ratios due to age, environmental conditions, and prevailing burdens [0]. Since the differences occur in both magnitude and angle, they are described by a complex quantity called ratio error. According to [0], the ratio errors can be as high as ±6% for voltage magnitudes, ±4 for voltage angles, ±0% for current magnitudes, and ±6.67 for current angles. Now, as these ratio errors are usually not included in the simulations, in the worst-case, an alert/alarm limit on voltage angle differences computed using simulated data may be off by 8 when applied to the field. One way to reduce this error is by calibrating the instrument transformers. Thus, in the context of the problem solved here, calibration is the process of estimating unknown ratio errors present in the instrument transformers used by PMUs. Interest in instrument transformer calibration has intensified in the last two decades []-[4]. A self-calibration method based on zero-point test, artificial offset test, and ratio meter test was proposed in []. Impedance synthesis methods applicable to active, hybrid and phantom burdens for instrument transformer calibration were discussed in []. Efficient and accurate methods for onsite calibration were developed in [3], [4]. But since most onsite calibration tests were time-consuming and expensive, they were difficult to perform on a system-wide basis. Calibration in relation to PMUs was also proposed in many papers [5-3]. However, most of the research was directed towards generator model validation or calibration of the PMU device itself [5-8]. One of the earliest attempts at calibrating instrument transformers using PMUs was made in [9]. But the model in [9] did not include the PMU errors that were present in the individual measurement sets. A methodology for calibrating instrument transformers under heavy load and light load conditions was proposed in [0]. However, because of its dependence on load conditions, its practical use was limited. An improvement to [0] was made in [] in which the dependence on load conditions was removed. But [0] and [] required at least one highly accurate pre-calibrated voltage measurement for calibrating the other devices. Combined with a highly accurate pre-calibrated current measurement, the logic developed in [] could also estimate line parameters [], [3]. The objective of this paper is to replace the need for one (voltage) or two (voltage and current) highly accurate measurements by several good quality measurements (such as those already available on tie-line buses) and optimally place them within the network so that the calibration error in the presence of both

2 ratio errors and PMU errors is below a given threshold. The rest of the paper is structured as follows. Section II describes the scope of the problem and the challenges associated with it. A brief summary of the previous research done to solve this problem is given in Section III. The proposed solution methodology is described in Section IV. The results obtained by applying the proposed technique on the IEEE-8 bus system, the IEEE-300 bus system, and a 383 bus Polish system are presented in Section V. Some clarifications regarding the proposed methodology are made in Section VI. The conclusions are provided in Section VII. II. PROBLEM SCOPE The focus of this paper is on the online calibration of voltage instrument transformers. Major control and measurement systems, such as the supervisory control and data acquisition (SCADA) system and the PMU based hybrid/ linear state estimation system depend on them for their proper functioning. The voltage instrument transformers are typically assumed to have very high accuracy levels [0]. However, as the error values vary with the manufacturer and increase over time and usage, the resulting biased measurements become an inherent component of input data errors for both SCADA and PMU data based applications. Since the accuracy of modern intelligent electronic devices (IEDs) is constantly improving, it becomes important to also calibrate the instrument transformers so as to take full advantage of these devices. The computation of ratio errors in the presence of PMU errors is not straightforward. The reason for this is that the two errors are multiplicative and when combined with the unknown true measurement results in three unknowns being multiplied together. Mathematically, for the voltage of the i th bus, this is described by () V measij = α ij β i V trueij () In (), j = {,,, N} where N is the total number of measurements made, V measij is the j th measured voltage, α ij is the unknown PMU error for the j th measurement, β i is the ratio error of the voltage of the i th bus, and V trueij is the j th true voltage. The symbol on top of a variable indicates that it is a complex number. If N is sufficiently large, then the PMU errors can be approximated by a Gaussian distribution having zero mean and a small standard deviation. However, since the ratio error (which follows a uniform distribution) and the true voltage values are not known, the number of unknowns still exceeds the number of equations. Under such circumstances, the previous researches either did not consider the PMU errors [9], or made the assumption that there is one measurement whose errors are known and computed the rest on the basis of that measurement [0]-[3]. In this paper, it is proved that by placing sufficient number of good quality measurements, the errors in the computation of the ratio errors of all the voltages can be kept below a pre-defined threshold. Since the proposed method is based on the algorithm developed in [3], a brief overview of that algorithm is provided in the next section. III. SUMMARY OF PREVIOUS RESEARCH Considering the challenges outlined in the previous section for solving the calibration problem, the approach developed in [3] was significant because with the least number of assumptions, it was able to simultaneously calibrate voltage and current transformers as well as estimate line parameters. A brief overview of that approach is provided here. Although, the algorithm developed in [3] works for both three-phase and positive sequence, for simplicity of notations, the positive sequence version is presented below. Consider the two-bus pi-network model pq shown in Fig.. If PMUs are placed on both ends of the line, then direct measurements of two complex voltages (V p and V q ) and two complex currents (I p and I q ) are obtained. Using Kirchhoff s laws in Fig., the following equations can be realized, I p = b pq V p + (V p V q )y pq I q = b pq V q (V p V q )y pq () On multiplying both equations of () by z pq we get (3). z pq I p = b pq z pq V p + (V p V q ) z pq I q = b pq z pq V q (V p V q ) (3) On multiplying the first equation of (3) by W pq where W pq = ( + b pq z pq ) and rearranging we get (4). W pq V p W pq V q W pq z pq I p = 0 (4) W pq V q z pq I q = V p The voltages and currents in (4) are the true voltages and currents of the network. Now, if voltage of the p side is known, then in absence of PMU errors, the measured voltages and currents can be written as shown in (5). V pm (j) = V p (j) ; V qm (j) = REV q V q (j) (5) I pm (j) = REI p I p (j) ; I qm (j) = REI q I q (j) In (5), REV q, REI p and REI q are the unknown voltage and current ratio errors, and j = {,,, N} where N is the total number of measurements made. Now, if KV q =, KI p = q REI REV and KI q = are the ratio correction factors, then using p q REI (5), for the j th measurement (4) can be rewritten as W pq V pm (j) W pq KV q V qm (j) W pq z pq KI p I pm (j) = 0 (6) W pq KV q V qm (j) z pq KI q I qm (j) = V pm (j) Now, if N such measurements are made, where N >, then an over-determined set of equations can be written for the pisection pq as shown in (7). Fig.. Two-bus pi-network model of the power system

3 3 V pm () V qm () I pm () 0 0 V qm () 0 I qm () V pm () V qm () I pm () 0 A 0 V qm () 0 I qm () V pm (N) V qm (N) I pm (N) 0 W pq W pq KV q z pq W pq KI p [ z pq KI q ] [ 0 V qm (N) 0 I qm (N)] In (7), b = [0 V pm () 0 V pm () 0 V pm (N)] T and the numbers within brackets represent different measurement sets. Eq. (7) is of the form Ax = b in which A and b are known and x is unknown. Hence, (7) can be solved in the least-square sense to compute for x. An example showing how (7) can be used for computing for the unknown ratio errors is provided below. x = b (7) Consider the case where all line parameters are known and all ratio errors except that of the pre-calibrated voltage measurement are unknown. That is, there is only one known voltage measurement (V pm = V p ) present in the system. Then, for the pi-section pq, z pq and b pq are known and hence W pq is known, while KV q, KI p, and KI q are unknown. Then, if x = [x () x () x (3) x (4)] T is the solution to (7), we get (8). x () x (3) x (4) KV q = ; KI p = ; KI q = (8) W pq z pq W pq z pq Using KV q obtained from (8), V q can be calculated using (5). Then, for the next pi-section, say qr, a similar procedure can be followed to compute V r. This process can then be repeated for subsequent pi-sections. It is important to note here that (5)-(8) do not contain PMU errors in the measured voltages and currents. In the presence of PMU errors in the measurements, the unknown ratio correction factors of pi-section pq cannot be computed accurately. This implies that V q also cannot be estimated precisely. Now, since V q is used in the computation of subsequent pi-sections, as one keeps moving further away from the pre-calibrated measurement, the errors in the estimates of the ratio errors keep growing. Moreover, since the true measurement is difficult to know in practice, (7) cannot be used directly in the field. A methodology that is capable of containing the growth in error of the estimates of the ratio errors is presented in the next section. IV. PROPOSED METHODOLOGY A. Practical aspects to be considered The problem associated with un-calibrated measurements surfaced during the research performed for a DOEdemonstration project of a three phase, linear PMU-only state estimator developed for Dominion Virginia Power (DVP) [4], [4]. Because of this, the characteristics of the solution match problems/constraints specific to DVP as described below. ) Focus is on calibrating voltage transformers: As DVP is interested in improving the accuracy of their linear state estimator, the primary focus of this research is on calibrating voltage instrument transformers. ) PMUs are placed on all buses that need to be calibrated: Since phasor measurements are used for calibrating the instrument transformers, it is necessary that a PMU is placed at the locations where calibration is needed. For the DVP system calibration was intended for the highvoltage network and PMUs were placed on all high-voltage buses. 3) A connected tree is present: If {i, j} B where B is the set of buses whose voltages need to be calibrated, then for all i and j there has to be at-least one path connecting them (called constraint-approved path) such that all buses that lie between i and j belong to B. 4) Some good quality measurement instruments are already present in the system: Typical examples are revenue quality meters that are placed on the tie-line buses that join two utilities. 5) Use of dual-use line relay PMUs instead of traditional PMUs: DVP is placing dual-use line relay PMUs in their highvoltage network. While a traditional PMU installed at a particular bus is assumed to measure currents of all lines connected to that bus and voltage of that bus, a dual use line relay acting as a PMU when placed on a line measures the voltage of the bus that it is protecting and the current in that line. Thus, when choosing locations for good measurements, end-points of the lines were selected rather than the nodes (For illustration purposes, see Fig. ). B. Formulation of Proposed Methodology The objective of the proposed methodology is to compute the voltage ratio errors when PMU errors are present and the practical constraints are satisfied. To do so, a modified version of (7) was derived as shown below. Since the known voltage measurement at p is replaced by an unknown but relatively good measurement, (5) becomes V pm (j) = REV p V p (j) ; V qm (j) = REV q V q (j) (9) I pm (j) = REI p I p (j) ; I qm (j) = REI q I q (j) However, since the p side voltage is a good quality measurement in comparison to the other normal measurements, REV p is smaller than the other ratio errors. Now, if KV p =, KV q =, KI p p = and KI q = q REV REV REI p REI q are the ratio correction factors, then using (9), for the j th measurement (3) can be re-written as shown in (0), z pq KI p I pm (j) = W pq KV p V pm (j) KV q V qm (j) (0a) z pq KI q I qm (j) = W pq KV q V qm (j) KV p V pm (j) (0b) where, W pq = ( + b pq z pq ). By dividing both equations in (0) by KV p and rearranging, we get (). KV q V KV qm (j) + KI p z p KV pq I pm (j) = W pq V pm (j) p KV q W KV pq V qm (j) KI () q z p KV pq I qm (j) = V pm (j) p

4 4 Fig.. Two pi-sections with dual-use line relays acting as PMUs Now, if N such measurements are made, where N >, then an over-determined set of equations can be written for the pisection pq as shown in (). A V qm () z pq I pm () 0 W pq V qm () 0 z pq I qm () V qm () z pq I pm () 0 W pq V qm () 0 z pq I qm () V qm (N) z pq I pm (N) 0 x KV q KV p KI p KV p KI q KV p] [ = b W pq V pm () V pm () W pq V pm () V pm () W pq V pm (N) [ W pq V qm (N) 0 z pq I qm (N)] [ V pm (N) ] () Eq. () solves for the other ratio errors in pi-section pq with respect to the ratio error in voltage of bus p. This implies that for subsequent pi-sections, the estimates of the ratio errors will also be a function of KV p. In order to accommodate for dual-use line relay PMUs and PMU errors, the following changes are made to (). ) When dual-use line relays are present: Let the two neighboring pi-sections be pq and qr as shown in Fig.. The small rectangles depict the dual-use line relays while the dotted shapes at q indicate the possibility of an injection. For the pi-section pq shown in Fig., () will become (3). A V qpm () z pq I pqm () 0 x KV qp W pq V qpm () 0 z pq I qpm () KV pq V qpm () z pq I pqm () 0 KI pq W pq V qpm () 0 z pq I qpm () = b KV (3) pq V qpm (N) z pq I pqm (N) 0 KI qp [ W pq V qpm (N) 0 z pq I qpm (N)] [ KV pq] In (3), b = [W pq V pqm () V pqm () W pq V pqm () V pqm () W pq V pqm (N) V pqm (N)] T. Now, if V q is the actual voltage of bus q then V q = KV qp V qpm (4) V q = KV qr V qrm Using (4), a variable γ q can be defined where, γ q = V qrm = KV qp (5) V qpm KV qr If N measurements are made, then the complex quantity γ q can be calculated experimentally from the measurements of V qpm and V qrm as shown in (6). N γ q = N V qrm (j) V qpm (j) j= (6) Then, since KV qp as a function of KV pq can be estimated from (3), using γ q obtained from (6), KV qr can be computed using (5). The KV qr thus obtained (also as a function of KV pq ) can be used in the next pi-section (qr of Fig. ). This process can then be repeated for subsequent pi-sections. ) When PMU errors are present: In presence of PMU errors in measurements, there will be further degradation in quality of the estimates. This degradation is quantified by the growth in standard deviation of the difference in the actual ratio error and its estimate (defined sigma from henceforth). One contributing factor that was identified for this was the illconditioning of the A matrix of (3). The ill-conditioning was reduced by splitting the impedance term in (3) between A and x. The resulting matrices A 3 and x 3 are shown in (7). The symbol over the measurements in (7) indicate that PMU errors are embedded in them. V qpm() V qpm() zpq I () pqm 0 zpq I () pqm 0 W pq V qpm() 0 zpq I () qpm (N) 0 V qpm(n) A 3 W pq V qpm() 0 zpq I () qpm zpq I pqm W [ pq V qpm(n) 0 zpq I (N)] qpm = b 3 (7) In (7), b 3 = [W pq V pqm() V pqm() W pq V pqm() V pqm() W pq V pqm(n) T V pqm(n)]. [ x 3 KV qp KV pq KI pq z pq KV pq KI qp z pq KV pq] As an example of the improvement in conditioning, typical condition numbers of the A matrices of (3) for different pisections were found to be above one thousand. By splitting the impedance term, as was done in (7), the condition numbers of the A 3 matrices for the same pi-sections were found to be below fifty. However, even after improving the conditioning of A 3, due to their being only one reliable measurement, the results did not improve significantly. This was especially found to be true for estimating the ratio errors that were more than 3-4 pi-sections away from that reliable measurement (see Fig. 6). Therefore, it was realized that more number of reliable measurements must be added to the system. The logic developed to do that is described in the next sub-section. C. Choosing optimal locations for adding good quality measurements Before identifying the optimal locations where good measurements can be added, the following Lemma is defined. Lemma : The advantage of adding multiple good measurements to a system is that as long as the measurements are independent, their combination will give a better estimate. Proof of Lemma : Let n measurements be made of an unknown quantity x. For the given problem, x is a scalar that

5 5 denotes the sigmas. Then for the k th measurement, we have z k = x + e k (8) where e~n(0, σ ). The residual is given by (9). [r] n = x [] n [z] n (9) In (9), x is the optimal estimate of x and [] n is a n vector of ones. In order to find x, weighted least squares approach is used which minimizes [r] T [Q] [r] where [Q] is the error covariance matrix. On solving, the desired value of x comes out to be x = []T n [Q] [z] n [] T n [Q] [] n (0) Then, the covariance of x is given by cov(x ) = E [ []T n [Q] [z] n [z] T n [Q] [] n ([] T n [Q] [] n ) ] () Simplifying the RHS of () gives cov(x ) = () ([] T n [Q] [] n ) Now, if measurements are independent, [Q] = diag(σ k ) and so [] T n [Q] [] n n = k= which gives (σ k ) cov(x ) independent = (σ ) + (σ ) + (3) (σ n ) Therefore, as long as the measurements are independent, the net variance will be lower than the variance of the individual measurements; thereby resulting in an improvement of the over-all estimate. For example, let measurements be made independently of end-point v from good measurements located at end-points u and w. Now, if the sigma of the difference between the actual ratio error and its estimate at v for u and w be σ vu and σ vw, respectively, then the combined sigma at end-point v denoted by σ v is σ v = (σ vu ) + (σ vw ) (4) From (4), it is easy to see that σ v < σ vu and σ v < σ vw. However, if the n measurements are not independent then the resulting covariance of x is a function of the correlation between the measurements. In the context of the given problem, dependence occurs only when there is more than one constraint-approved path (see Section IVA) to reach one endpoint from another end-point. However, since the degree of correlation between parallel paths is difficult to compute in practice, a definite upper-bound on the value of the net variance is obtained based on the premise that being able to observe a particular entity by a different path can only increase the over-all accuracy (and not decrease it). Therefore, the net-variance is set equal to the minimum variance obtained along any one of the parallel paths as shown in (5). cov(x ) dependent = min(σ k ) for k = {,, n} (5) For example, let measurements be made of end-point v from the good measurement located at end-point u via paths and. Now, if the sigma of the difference between the actual ratio error and its estimate at v for u via paths and are σ vu and σ vu, respectively, where σ vu > σ vu then the combined sigma at end-point v due to the good measurement at u denoted by σ vu is σ vu = σ vu (6) On the basis of Lemma, the calibration problem reduces to choosing optimal locations for adding good measurements to the network. This can be done through a binary integer programming (BIP) formulation as shown below. Let the connected undirected graph of buses whose voltages need to be calibrated be described by G(V, E) where V is the set of vertices, and E is the set of edges. Let [S] be a E E matrix whose columns correspond to the locations where the good measurements can be placed and whose rows denote the sigmas obtained for all the other locations with respect to a good measurement at the column to which the row belongs to. Let [X init ] be a E binary integer matrix such that the indices of the non-zero entries of [X init ] denote locations of initial good measurements. Also, let the variable T denote the value below which all the sigmas must lie. Then, the objective of the optimization problem is the minimization of the L norm of a E binary integer matrix [X] such that the indices of the non-zero entries of [X] denote locations of good measurements. The constraints further imposed on this objective are given by (7)-(8). [A] E E [X] E [B] E (7a) where A(i, j) = (S(i, j)) [B] E = (7b) T [] E [X init ] T E [X] E = nnz([x init ] E ) (8) Eq. (7) ensures that the combined sigmas are below the desired threshold while (8) guarantees that the locations of the initial good measurements are retained and accounted for in the final solution. The flowchart based on this BIP formulation for finding the optimal locations is shown in Fig. 3 and described below in more details. Step : Find good measurements initially present in the system. If there are no good measurements initially present, then GO TO Step 5. Step : Taking one good measurement at a time, find sigmas for end-points that are P pi-sections away from the end-point where the good measurement is placed. Step 3: For the initial good measurements, find combined sigmas for all end-points using (). Call initial set of good measurement end-points as the Starting Set. Step 4: Define T 0 as the pre-defined threshold for the sigmas. If the maximum of the combined sigmas for all endpoints computed based on the Starting Set is less than T 0, then no more good measurements need to be added to the system; GO TO Step. Step 5: For all the end-points that do not have good measurements initially, taking one end-point at a time, find sigmas for end-points that are P pi-sections away from that From the simulations, a suitable value of P was found to be 8. T 0 is a scalar constant that is independent of the test system.

6 6 end-point by assuming that a good measurement will be placed only on that end-point. Step 6: In presence of dependent measurements, find net sigmas using (5). Step 7: Set T = T 0. Step 8: Perform the optimization using (7)-(8). This gives the minimum number of good measurements that must be present in the system to keep the sigmas for all the endpoints below T 0. Step 9: Reduce T by a small amount (say % of itself). Start number of good measurements required is equal to the number obtained in Step 8, then GO TO Step 9. Step : The location set corresponding to the smallest value of T (called T min ) for which the number of good measurements is equal to the number of good measurements obtained in Step 8 is the best location set. Step : Stop. Steps -4 of the flowchart cater to the initial good measurements present in the system. The optimal number of good measurements is computed in Steps 5-8. Steps 9- increase the robustness of the solution. The results obtained by applying this methodology to different test systems are described next. No Are good measurements initially present? Yes For individual good measurements, find sigmas for other endpoints using (7) Find combined sigmas for all end-points using () Are combined sigmas for all end-points below? Yes V. SIMULATION AND RESULTS The IEEE-8 bus system, the IEEE-300 bus system, and a 383-bus Polish system were selected for the analysis done here. The characteristics of the test systems which make them relevant for this study are as follows. All three systems have a distinct high voltage (HV) network which forms the backbone of the respective systems. For illustration purposes, the HV network of the IEEE-8 and 300 bus systems are shown in Figs. 4 and 5. The IEEE-300 bus system and the Polish system have closed loops in them which make dependent measurements possible. The arrangement of the fifty HV buses of the Polish system also make it similar in size and structure to the HV network of the DVP system [4]. No Taking one end-point at a time, find sigmas for other endpoints by assuming a good measurement at only that end-point In presence of dependent measurements, find net sigmas using (5) Fig. 4. High-voltage network of IEEE-8 bus system Set T = and using (7)-(8) find minimum number of good measurements required to keep sigmas for all end-points below T Lower T by a small amount and re-do the optimization using (7)-(8) to find new number of good measurements required Fig. 5. High-voltage network of IEEE-300 bus system No Is new number of good measurements greater than minimum number? Location set corresponding to lowest T value ( ) for which new number of good measurements is equal to the minimum number is the best location set Stop Fig. 3. Flowchart for finding optimal locations of good quality measurements Step 0: Re-do the optimization using (7)-(8). If the Yes For each system, the simulation was run 000 times to generate the sigmas. During each run, for every measurement a different ratio error was picked at random from the range given in [0]. Moreover, for every run, different operating conditions were created, that is N was equal to. The perfect measurements were assumed to have no ratio errors or PMU errors in them. The smaller ratio errors of the good quality measurements were chosen from a uniform distribution having zero mean and standard deviation of 0.5% for magnitudes and 0. for angles [0]. All PMU errors were chosen from a Gaussian distribution having zero mean and standard deviation of 0.% for magnitudes and 0.04 for angles. It was assumed in the simulations that the sigmas were to be kept below %.

7 7 Accordingly, T 0 was set at 0.0. The test system data can be found in the MATPOWER [5] toolbox of MATLAB [6], while the integer optimization was performed using GUROBI [7]. The results obtained are as follows. Fig. 6 depicts the growth in sigmas if one reliable measurement (either good quality or perfect quality) is placed at different locations in the IEEE-8 and 300-bus systems. In order to create Fig. 6, all possible locations where a reliable measurement could be placed were initially identified. For instance, the IEEE-8 bus system had 0 such locations (either end of the ten HV lines shown in Fig. 4). Then taking one location at a time, the sigmas were computed for all the other locations by assuming that the reliable measurement was placed only at that location. So, for the IEEE-8 bus system, the whole set-up (000 runs with randomly chosen ratio errors for sets of measurements) was repeated 0 times with the reliable measurement at one of those 0 locations at each time. From Fig. 6, it is realized that good quality and perfect quality measurements have similar performances (almost all dots lay inside circles). Another observation that is made from Fig. 6 is that one reliable measurement is not able to keep most of the sigmas below 0.0 beyond 3-4 pi-sections. This observation is important because it meant that it was not necessary to compute sigmas for end-points that were far away as their combination would not have significant influence on the net variance. This theory was further tested for the IEEE-300 bus system and the results obtained are shown in Table I. Fig. 6. Growth of sigmas when only one reliable measurement is present in the systems; the dashed line shows the pre-defined threshold (T 0 ) of pisection TABLE I OPTIMAL LOCATIONS OF GOOD MEASUREMENTS FOR DIFFERENT PI-SECTION LENGTHS OF IEEE-300 BUS SYSTEM All pi- 4 pi- 5 pi- 6 pi- 7 pi- section section section section section Table I compares the optimal locations obtained when sigmas for all pi-section lengths are considered, and when sigmas up to 4 to 8 pi-section lengths are considered. An entry i j in Table I means that the good measurement is placed on the i end of line i j. The location of the initial good measurements are shown in red. They were so chosen because the i th node of those lines was connected to only one other node of the network (the j th node). Therefore, in all practicality, they could be assumed to be the tie-lines that join the test system with its neighbors, with the i th nodes becoming the tie-line buses. The numbers in the disconnected row of Table I indicate the value of T min obtained for the location set given in the corresponding column. From the table it is realized that the numbers and locations are identical for all pisection and 8 pi-section lengths. This meant that sigmas computed up to a length of 8 pi-sections was sufficient for finding optimal locations where new good measurements can be added. This result is particularly important for large systems where it is computationally complex to consider all possible pi-section lengths. In the final set of simulations, the optimal locations for adding good quality measurements for all the three test systems were computed using the proposed approach. Table II shows the results obtained with the disconnected row listing the T min values for the three systems. The results of Table II are also validated by Fig. 7. In Fig. 7, the dots correspond to the sigmas while the dashed line depicts T 0. From the figure it becomes clear that by using the proposed approach the growth in sigmas has been contained for all the three test systems.

8 8 TABLE II OPTIMAL LOCATIONS OF GOOD MEASUREMENTS FOR TEST SYSTEMS 8-Bus 300-Bus System System 383-Bus Polish System VI. DISCUSSION This section presents some clarifications regarding the results obtained using the proposed methodology. The good quality measurements described in this paper are obtained from revenue quality meters present in the network. The instrument transformers that are present on the tie-line buses (that connect one utility to another) have such meters already placed on them. The number of these high quality meters is limited in a network because of the high cost associated with them. However, since market price changes occur at the tieline buses, utilities prefer placing the revenue quality meters at those locations. Using this knowledge in the proposed methodology, such good measurements were identified to be present by-default in the system, and were used as the starting set for calibrating voltage instrument transformers. However, contrary to [3], it was not assumed that those better quality measurements were free from errors. Instead, it was proved that by adding optimal number of such good quality measurements to other locations inside the network, all the sigmas could be kept below a pre-defined threshold. Fig. 7. All the sigmas for the three test systems are below the pre-defined threshold (T 0 ) where T 0 = 0.0

9 9 A robust formulation of (7) is essential to get valid results. In the simulations, this was ensured by considering sets of measurements for each of the 000 runs with each set of measurement occurring at a different instant in time. Each individual set of measurement had an equation of the form shown in the first two rows of (7). Therefore, for every pisection, the A 3 matrix had size 4 3. For the three test systems, the sets of measurements were made by mirroring the morning load pick-up. The loads and generations were increased over a period of one hour, and measurements were made every five minutes of that hour. More details about the simulation set-up for the morning load pick-up can be found in [8]. As the measurements were made at different instants in time of the morning load pick-up, they corresponded to different operating conditions. The diverse operating conditions thus created ensured that A 3 had full-rank. In real systems, the measurements can be made during the morning load pick-up or even over the period of an entire day. As an example of the latter, if one day is reserved by a utility for calibration purposes and they make a measurement every hour of that day, then N will be equal to 4 (higher the value of N, the better the results). Now, since the ratio-errors do not change over a period of one day, the results obtained by solving (7) will remain consistent over that time-period. However, the ratio-errors do change over a period of months. So as long as the measurements made over a period of one day are repeated every few months, the proposed approach will be able to track the change in ratio errors that occur. The growth of the sigmas depicted in Fig. 6 depended on two factors: (a) it was proportional to the number of pisections that separated the location of the reliable measurement from the location whose sigma was being computed, and (b) it was a function of the line parameters of the pi-sections that lay in between. It was the combination of these two factors that made the growth non-uniform as was observed in Fig. 6. It is also important to note that different number of pi-sections must be traversed to reach all possible locations from the location of the single reliable measurement. As an example, for the IEEE-8 bus system, if the single reliable measurement was placed at bus 0, then 7 pi-sections must be traversed to reach all possible locations. However, if the single reliable measurement was at bus 6, then at most 5 pi-sections must be traversed in order to reach all possible locations. It is for this reason that sigmas shown in Fig. 6 got higher and sparser as the number of pi-sections increased. Regarding the T min values listed in the disconnected row of Tables I and II, it is to be realized that the solutions obtained by BIP, although always optimal, are seldom unique. In order to address this feature of BIP, the value of the variable T was systematically decreased from its initial value of T 0 as is described in the flowchart. This was done to find the smallest possible value of T, denoted by T min, for a given system (thereby making T min a system dependent quantity) for which the total number of good measurements required would be the same as that required for keeping all the sigmas below T 0, but the locations would be different. That is, if S T0 and S Tmin denote the location sets corresponding to T 0 and T min, respectively, then S T0 = S Tmin, but S T0 S Tmin. This manipulation of the value of T ensured that with the smallest number of good measurements, the best results were obtained. For the methodology developed in this paper, the line parameters were assumed to be known accurately. Research is currently being done on developing a formulation that will use the concept of adding optimal number of good measurements for estimating line parameters along with the calibration of voltage and current instrument transformers. However, due to the level of mathematics involved in that formulation, it will be the subject of a future publication. VII. CONCLUSIONS This paper presents a methodology for calibrating voltage transformers without having to take them off-line. Presence of PMU errors in the measurements is accounted for along with the absence of a pre-calibrated measurement. 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Thomas, MATPOWER steady-state operations, planning and analysis tools for power systems research and education, IEEE Trans. Power Syst., vol. 6, no., pp. -9, Feb. 0. [6] The MathWorks Inc., MATLAB version 7.0.0, (computer software), Natick, MA, 00. [7] GUROBI Optimization, Inc., GUROBI Optimizer Reference Manual, Version 6.0., Houston, Texas, Feb. 05. [Online]. Available: [8] F. Gao, J. S. Thorp, A. Pal, and S. Gao, Dynamic state prediction based on auto-regressive (AR) model using PMU data, in Proc. IEEE Power and Energy Conference at Illinois, Champaign, IL, pp. -5, Feb. 0. Tata Steel Ltd. (TSL), Jamshedpur, India. He was also a summer intern at Electric Power Group, LLC in Pasadena, California in 0 and 03. His research interests include power system modeling, transient and dynamic stability analysis, and wide area measurements-based protection, monitoring, and control. Paroma Chatterjee (S 4) is currently pursuing her Ph.D. degree in the Bradley Department of Electrical and Computer Engineering at Virginia Tech, Blacksburg (USA). She received her Bachelor of Engineering (B.E.) degree in Power Engineering from Jadavpur University, Kolkata (India) in 0. She worked as a Trainee Engineer in the Power Transmission and Distribution section in MECON Ltd., Ranchi (India) for a year (0-03). Her research interests include wide-area measurement based monitoring and protection of power systems. James S. Thorp (LF 00) was the Hugh P. and Ethel C. Kelley Professor of Electrical and Computer Engineering and Department Head of the Bradley Department of Electrical and Computer Engineering at Virginia Tech from 004 to 009. He was the Charles N. Mellowes Professor in Engineering at Cornell University from He was the Director of the Cornell School of Electrical and Computer Engineering from 994 to 00, a Faculty Intern, American Electric Power Service Corporation in and an Overseas Fellow, Churchill College, Cambridge University in 988. He was an Alfred P. Sloan Foundation National Scholar and was elected a Fellow of the IEEE in 989 and a Member of the National Academy of Engineering in 996. He received the 00 Power Engineering Society Career Service award, the 006 IEEE Outstanding Power Engineering Educator Award, and shared the 007 Benjamin Franklin Medal with A.G. Phadke. Virgilio A. Centeno (M 9 SM 06) received the M.S. and Ph.D. degrees in electrical engineering from Virginia Polytechnic Institute and State University (Virginia Tech), Blacksburg, in 988 and 995, respectively. He worked as a Project Engineer at Macrodyne, Inc., Clifton Park, NY, in the development of phasor measurement units from 99 to997. He joined the faculty of Virginia Tech as a Visiting Professor in the fall of 997and became an Associate Professor in 007. His area of interest is wide area measurement and its applications to power system protection and control. X. BIOGRAPHIES Anamitra Pal (S 0 M 5) received his Bachelor of Engineering (B.E.) degree (summa) from Birla Institute of Technology, Mesra, Ranchi (India) in 008 and his M.S. and Ph.D. degree from Virginia Tech, Blacksburg in 0 and 04, respectively. He is now an Applied Electrical and Computer Scientist in the Network Dynamics and Simulation Science Laboratory at the Virginia Bioinformatics Institute (VBI). From , he worked as a Manager in