Analysis of Ill-conditioning of hybrid state estimators in the presence of PMU measurements

It has been observed by the practicing engineers that the inclusion of measurements from phasor measurement units (PMUs), current measurements in particular, into the conventional state estimators (SEs), may deteriorate the numerical performance of the SEs in terms of convergence and accuracy. This paper analyzes the reason for such possible ill-conditioning of the hybrid SEs that use both conventional and PMU measurements. It presents simpliﬁed analytical approaches to identify what causes such ill-conditioning when PMU measurements are used. The condition number of SE gain matrix may vary depending on the number of the PMUs and its placement in the network. It has been identiﬁed that the large difference in weights that are allocated to the traditional measurements and PMU measurements is the main reason of the ill-conditioning of the hybrid SEs.


INTRODUCTION
The loss of estimation accuracy and difficulty in convergence are some of the critical problems that are sometimes encountered by power system state estimators (SEs). One of the main reasons for such performance issues in SEs is the numerical illconditioning of the state estimation problem.
For weighted least squares (WLS) state estimation, condition number of the gain matrix is a popular means of assessing the numerical conditioning of the estimation problem [1,2]. The conventional SE based on supervisory control and data acquisition (SCADA) system uses measurements such as real and reactive power flows and injections, and RMS values of bus voltages and line currents. It has been observed in practice that the SCADA based SE may encounter convergence problem for certain measurement configurations, such as large number of injection measurements, or current magnitude (ampere) measurements [3][4][5]. Also, the gain matrix tends to become ill-conditioned as the system size increases. To overcome the numerical problems associated with SCADA based SEs, several techniques, like Peters and Wilkinson method [6], orthogonal decomposition [7,8], Levenberg-Marquardt algorithm [9], linear programming [10], and regularization method [11]  sion for the condition number of the gain matrix was formulated in [6] and [12], and it was found that the injection measurements caused degradation in the numerical performance of the SE.
Phasor measurement units (PMUs) are being increasingly used in power systems due to their advantages such as high refresh-rate (sub-second), direct measurement of phase angles, and time-stamping of the measurements using the global positioning system (GPS). When placed at a bus, a PMU can provide measurement of the voltage phasor at that bus, frequency, and the rate of change of frequency. It can also measure current phasors through the lines connected to that bus (depending on the available number of current measurement channels) [13]. Due to the relatively high cost of PMUs and the associated communication infrastructure, they are being incrementally placed in power systems [14,15]. Most of the existing measurements in power systems are conventional asynchronous ones. One immediate choice at this moment for SE is to combine the PMU measurements with the estimated states of the existing conventional state estimator, and come up with the estimate of the states of the system [16]. As the number of PMUs increases in power system, which is being envisaged to be the case soon, it will be wise to adopt a state estimator that can handle conventional and PMU measurements simultaneously. In the analysis presented in this paper, it is assumed that the hybrid SE (HSE) uses this type of pre-processing algorithm [14].
The basic objective of the SE is to estimate the bus voltage phasors. Inclusion of the voltage phasor measurements by PMUs in an HSE is, therefore, straightforward. Care, however, needs to be taken while including PMU current measurements in the HSE, since current measurements usually degrades the numerical characteristics of the SEs [3]. A number of relevant publications have reported the inclusion of PMU current phasor measurements in an HSE [13,14]. In this paper, the PMU current measurements are assumed to be in rectangular form, as this leads to most desirable performance of the SE in terms of convergence characteristic and estimation accuracy, as compared to its inclusion in polar form, or in pseudo-voltage measurement form [14].
The numerical behavior of the estimation process depends on the type of measurements, placement of the measurements, measurement redundancy, measurement weights, and the parameters of the network [4,9]. Differential evolution based algorithms [17] have been used to handle the illconditioning problem in hybrid estimators [18]. However, This paper explores the factors behind numerical problems of the HSEs when PMU measurements are considered, which is an extended version of the work reported in [19]. Approximate expressions for the condition number of the gain matrix are developed for the following measurement configurations for a radial system: the PMU measurements only (either voltage phasor or current phasor), and combination of the PMU current measurements with different types of conventional measurements. This analysis is also extended for several IEEE test systems.
The main contributions of the paper are as follows.
• It has been observed by researchers that assignment of differential weights to measurements tends to cause numerical ill-conditioning of the SE. However, so far there has not been formal proof that differential weights are the factors for such ill-conditioning. The main contribution of this paper is that it analytically confirms and validates this observation. • The assignment of significantly different weights to the PMU and the conventional measurements, due to their different accuracy levels, is established as the main reason for such numerical degradation of SE performance. Such analysis was not done earlier considering PMU measurements. • The paper also proposes an analytical method to compute approximate condition number of the SE gain matrix for radial systems. • The condition number analysis is also extended for standard non-radial IEEE test systems.
The remainder of the paper is arranged as follows. In Section 2, the SE model is explained, along with the assumptions for analysis of its numerical ill-conditioning problem. Section 3 determines approximate condition numbers for the gain matrix of the SE problem, for a radial system, having only one type of measurement. In Section 4, for a radial system, the effect of PMU current measurements in the presence of conventional measurements is investigated. The numerical analysis is performed for IEEE test systems in the next section, that is, Section 5. The conclusions are drawn in Section 6.

BASIC WLS ALGORITHM AND ASSUMPTIONS
For state estimation, the commonly used WLS formulation is as follows: where J (x) is the objective function; x is the vector of voltage angles and magnitudes; z is the measurement vector; h(x) is the vector of measurement functions relating z and x; R is the error covariance matrix of the measurements. The state estimation problem is solved by carrying out the following iterations: where H = h(x)∕ x is the Jacobian matrix of the measurement; G = H T WH is the so-called GM; x (k) denotes vector of voltage angles and magnitudes at the kth iteration. One needs to invert the matrix G to solve (2). However, to minimize the requirement of computational resource and numerical errors, direct gain matrix inversion is rarely performed. Numerically efficient methods, such as, decomposition, factorization, and regularization are typically used to carry out the inversion process. Numerical ill-conditioning of the estimation problem is frequently encountered in state estimation. For an ill-conditioned state estimation problem, the inversion of the gain matrix may introduce significant numerical errors, which can subsequently lead to convergence problem in the estimation algorithm. A commonly used way of quantifying the numerical characteristics of a matrix is finding its condition number [20]. The condition number of a non-singular square matrix A may be given by [6,20]: where ‖.‖ represents the matrix norm. When the condition number is high, it usually implies numerical ill-conditioning problem. In the context of state estimation, convergence characteristics of the SE are often reflected by the condition number of the estimation gain matrix [6]. For the state estimation problem, usually the second order norm is used. Higher order norms are not considered because of increased computational effort [21]. In this paper, for simplified analysis of the condition number, first order norm of the gain matrix is used. The condition number for a number of measurement configurations is investigated. The simplifying assumptions that are taken are listed below [6]: Numerical ill-conditioning of the SE problem in the presence of conventional measurements is well-explored in the literature. The paper focuses on finding the root cause of numerical illconditioning when PMU measurements are considered for state estimation. Section 3 presents the condition number analysis for different but uniform type of measurements in the system. Section 4 analyses the problem for hybrid measurement configuration including PMU and conventional measurements.

CONDITION NUMBER ANALYSIS OF RADIAL NETWORK FOR DIFFERENT TYPES OF MEASUREMENTS
In this section, the effect of different types of measurements, such as line flow, injection, current, and voltage, acting alone, on the gain matrix is investigated. An (n + 1)-bus radially oriented system is considered as the representative system, as shown in Figure 1. Four different measurement configurations considered are as follows: Case I: Flow measurements at all of the n lines. Case II: Injection measurements at all of the (n + 1) buses. Case III: Voltage measurements by PMU at all (n + 1) buses. Case IV: Current measurements by PMUs at all n lines For Case I, having n flow measurements in the radial network, [6] discusses a method of finding the approximate value of the condition number of the gain matrix. Assuming R c to be the variance in the conventional measurements, such as flow and injection measurements, the first order norm of the SE gain matrix and its inverse are computed as 4y 2 ∕R c and n(n + 1)R c ∕(2y 2 ), respectively, resulting in the condition number of 2n(n + 1).
For Case II, that is, for injection measurements at all buses, the first order norm of the gain matrix and its inverse are computed as 16y 2 ∕R c and n(n + 1) 2 (n + 2)Rc∕(24y 2 ), and the condition number is found to be 2n(n + 1) 2 (n + 2)∕3 [6].
For Case III, the voltage phasors are observed at all buses of the n + 1 bus radial system. The corresponding measurement Jacobian may be found as: where the sub-matrices H and H V correspond to measurements of bus voltage angle and magnitude, respectively. I n is an unity matrix of size n × n. Using (4), and assuming R p to be the variance of PMU measurements, the gain matrix and its inverse can be computed as: The condition number of the gain matrix in this case is found to be unity by using (3), (5), and (6).
Approximate condition number of the gain matrix for Case IV, that is, in the presence of PMU-measured current phasors for all lines, is derived below. The (n + 1)-bus radial system is made observable by n number of PMU-measured current phasors. The ith branch current from bus i + 1 to i, in rectangular form, can be written as, where I i,real and I i,imag are the real and reactive parts of the i th current phasor obtained from PMU;V i = V i ∠ i is the voltage phasor of the i th bus. Assuming 1 pu voltage magnitude for all buses, the elements of the measurement Jacobian corresponding to the real and imaginary parts of the current phasors measured by PMUs can be expressed as, With the above formulation, the measurement Jacobian matrix can be written as: where all the four sub-matrices are non-zero, unlike the ones for injection and flow measurements [6].
From (9)-(13), it can be observed that in Case IV, the following holds.
From (13) and (14), the gain matrix can be written as, where G caseIV and G VcaseIV are the sub-matrices for voltage angle and magnitude, respectively.
: : 0 0 0 : : : : : : : : : : : : : : : : The sub matrix G caseIV for the current phasor measurements shown in (16) is the same as in the case of flow measurements [6]. The inverse of the sub matrix can be expressed as: For Case IV, the first order norm for the sub matrix G caseIV and its inverse are calculated from (16) and (17), and are found to be (4y 2 ∕R p ) and n(n + 1)R p ∕(2y 2 ), respectively. Thus, the condition number of the gain matrix is equal to that of the submatrix G caseIV , that is, 2n(n + 1). From the above discussion, it is evident that the presence of current phasor measurement alone does not pose any numerical problems to state estimation. Table 1 shows the norm of the gain matrix, its inverse, and the condition number for the four test cases discussed above. For Case III, that is, for measurements consisting of voltage phasors only, the condition number of the gain matrix is independent of the number of buses. For the three other cases, as the number of buses increases, the condition number also increases. The highest condition number is observed for Case II, that is, for measurements consisting of injections only. This conforms to the general experience that the presence of large number of injection measurements deteriorates the numerical conditioning of the state estimation problem [6].
In actual power systems, however, we hardly have only one type of measurement, as assumed in the four cases described above. Our focus, in this paper, is on hybrid state estimation. We will, therefore, consider a measurement configuration with conventional SCADA measurements, as well as measurements from PMUs. The next section discusses the effect of inclusion of PMU measurements in SE, along with the conventional measurements, on the numerical characteristics of the state estimation problem. We will also attempt at identifying the root cause of the ill-conditioning for hybrid SEs.

ANALYSIS OF SE WITH HYBRID MEASUREMENTS
In the preceding section, the condition numbers of the gain matrix for individual measurement types are formulated. As evident from Table 1, these condition numbers do not depend on the measurement variances. However, when two or more different types of measurements are considered, the variance of the measurements, or rather, the ratio of the variances plays a major role in determining the condition number. The following simplified analysis shows that, as the ratio of measurement weights increases, the estimation problem becomes ill-conditioned. The analysis is done for a radial system, for which the number of buses is arbitrarily chosen as 9, as shown in Figure 2

Conventional power flow measurement and PMU
The measurement system for the 9-bus radial system considered here consists of eight flow measurements. Three PMUs are placed at bus 2, 5, and 8 (arbitrarily selected) to develop the approximate expression for the condition number. However, this expression may be different for different number of PMUs and their placement. There are six PMU current measurements (located between buses 2-1, 2-3, 5-4, 5-6, 8-7, and 2n(n + 1)

FIGURE 2
Nine-bus radial system (eight flow measurements with three PMUs) 8-9) and three voltage phasor measurements at bus 2, 5, and 8, corresponding to the three PMUs placed. For the measurement of the power flow in the ith branch, the power flow meter is assumed to be placed near the (i + 1) th bus. The variance in the power flow and PMU current measurements are denoted as R c and R p , respectively. The respective weights for WLS formulation are taken as W c and W p , respectively.
The number of PMU current measurements and their placement, the size of the system, individual measurement weight etc., are important factors that can affect the numerical conditioning of the problem. Some of the arbitrarily chosen measurement configurations are investigated in the following.

Measurement configuration: Six PMU current measurements with eight power flow measurements
The measurement configuration is shown in Figure 2. The voltage phasors obtained from PMUs are ignored to study the effect of PMU current measurements along with the flow measurements on the condition number of the gain matrix.
For the placement of power flow meter and PMU current measurements in Figure 2, the first order norm of gain matrix and its inverse are found to be From (18) and (19), the condition number of the gain matrix can be written as where w = W p ∕W c .

Measurement configuration: Three PMU voltage measurements and eight power flow measurements
Let there be three PMU voltage measurements in the network shown in Figure 2, along with eight power flow measurements. The PMU current phasor measurements are ignored to study the effect of PMU voltage measurement on the condition number.
The first order norm of gain matrix and its inverse are found to be, From (21) and (22), the condition number of the gain matrix can be written as PMU measurements are typically much more accurate than the conventional measurements. Therefore, the ratio w is usually high. Equations (20) and (23) indicate that the condition number increases with increasing value of w. From these equations, the impact of current measurement on the condition number is very high as compared to the voltage measurement. If the network is completely covered by either flow measurements or current measurements in a (n + 1) bus radial system, the condition number is 2n(n + 1), that is, the condition number is 144 for the 9-bus system. The condition number for the current and voltage phasor measurement in presence of eight flow measurements are higher, and found to be approximately as (36w + 144) and (4 + w), respectively.

Conventional power injection measurement and PMU
In this subsection, the effect of the presence of PMU current phasor measurement along with the conventional injection measurements is investigated. For the 9-bus radial system shown in Figure 2, nine injection measurements are considered, one at each bus. The analysis is done for three PMUs placed at buses 2, 5, and 8. The condition number of the gain matrix in the absence of PMUs, that is, for nine injection measurements only, is found to be 4320 for the 9-bus radial system using Table 1. A number of arbitrarily chosen measurement configurations are investigated below.

Measurement configuration: Six PMU current measurements and nine power injection measurements
For this case, the 9-bus system is assumed to have six PMU current measurements (located to measure the flow between buses 2-1, 2-3, 5-4, 5-6, 8-7, and 8-9), along with 9 injection measurements placed at each bus. The voltage phasors measured by PMUs are ignored, to study the effect of PMU current measurement on the condition number. The first order norm of the gain matrix and its inverse, as well as its condition number can be evaluated as Cond(G) = (16 + 4w)(9w 3 + 135w 2 + 405w + 270) (2w 3 + 8w 2 + 7w + 1)

Measurement configuration: Three PMU voltage measurements and nine power injection measurements
Let there be three PMU voltage measurements, as shown in Figure 2, along with nine power injection measurements. The PMU current phasor measurements are ignored to study the effect of PMU voltage measurement on the condition number. The first order norm of the gain matrix and its inverse, as well as its condition number for this placement of measurement, can be evaluated as Due to the high accuracy of the PMU measurements compared to the conventional injection measurements, the weight ratio, w, can be very high. For w > 1000, the condition numbers in (26) and (29) can be expressed approximately as (18w + 72) and (16 + w), respectively, which clearly indicates the deterioration of the numerical condition of the gain matrix in the presence of PMU current phasor measurements.

Conventional power injection and power flow measurement along with PMU
In this subsection, the effect of combination of power flow and power injection measurements with PMU measurements on the condition number is analyzed with the 9-bus radial test system. In this case, eight flow measurements (refer to Figure 2) through each branch and nine injection measurement, located at each bus, are the conventional measurements. The condition number is analyzed for different numbers and the corresponding all possible locations of PMUs, along with the conventional measurements.
As discussed in [14], measurement variances may be assumed proportional to the square of the maximum specified error in measurement in the data-sheet of the device. For a usual error of 3% in SCADA-based conventional measurements [22] and 0.03% for the current measured by PMU [23], the weight ratio becomes (0.03∕0.0003) 2 ≈ 10 4 . Similarly, for 3% error for the conventional measurement [22] and 0.01 o for the PMU voltage angle measurement [23], the weight ratio becomes ((0.03 * 180)∕(0.01 * )) 2 ≈ 2.9545 × 10 4 . In this calculation of weight ratio, the effect of the meter reading is not taken into consideration.
For these weight ratios, and for a given number of PMUs, the maximum and the minimum values of the condition number is listed for all combinations of PMU placements, considering either voltage phasor or current phasor or both, in Tables 2, 3, and 4, respectively. For all the combinations, one PMU is placed at slack bus 1. In case of 5 PMUs being paced in the network, there are 70 = ( 8 C 4 ) combinations of PMU placements. For this case, the condition number for the combination of PMU voltage phasor measurement and the conventional measurement is maximum, when PMUs are placed at buses 1, 2, 3, 4, 5, and minimum, when placed at buses 1, 3, 5, 7, 9. Similarly, for 5 PMUs, the condition number for the combination of PMU current phasor measurements and the conventional measurements is maximum when PMUs are placed at buses 1, 6, 7, 8, 9, and minimum, when placed at buses 1, 2, 4, 6, 8. The condition number for the combination of PMU voltage and current phasor measurements  along with the conventional measurement is maximum when PMUs are placed at 1, 2, 3, 4, 5, and minimum, when placed at 1, 3, 5, 7, 9. From Tables 2-4, it is evident that the condition number varies in presence of PMU voltage phasor or current phasor or both, depending on the number of PMUs and their placement in the radial network. The numerical ill-conditioning is more prominent with less number of PMUs. The analysis presented in Sections 4.1, 4.2 and 4.3 tries to identify the cause of the numerical problems in hybrid state estimation. It is evident that the ratio of the weights allocated  to the PMU measurements to the conventional ones plays a very significant role in degrading the numerical conditioning of the state estimation problem. From the approximate expressions of the condition numbers shown in this section it can be seen that the ratio of the measurement weights, w, plays a very significant role. For an already large value of w, its 2 nd , 3 rd , or higher powers become very large to handle; and the gain matrix becomes numerically ill-conditioned. The condition number depends on the number of PMUs in the network and their location. In case of small number of PMUs, the condition number is observed to be high in presence of PMU measurements. It is to be noted that, even for the simple 9-bus radial system, expressing the condition number of the gain matrix as a function of the measurement weights is significantly involved. For actual power systems with large number of buses, such task will be extremely difficult, if not impossible. In the next section, therefore, for test power systems, instead of expressing the condition numbers as a function of the measurement weights, their numerical values are observed after assigning typical values to the measurement weights.

CASE STUDIES ON IEEE TEST SYSTEMS
In Sections 3 and 4, the condition number of the gain matrix is investigated for a radial system. However, practical power systems are not necessarily radial. In this section, the analysis is extended to test power systems, viz., the IEEE 14-bus and IEEE 118-bus systems [24]. With the given IEEE test systems, the assumptions (2-4) described at the end of Section 2 are relaxed in the analysis presented in this section. The results shown in this section confirms the fact that the presence of PMU measurements can make a hybrid SE ill-conditioned.
In this section, SCADA measurements are combined with a number of PMU measurements. Typical values of the maximum measurement uncertainties (specified by the manufacturers) for different types of conventional measurements [22] as well as the PMUs [23] are listed in Table 5.
The variance in the i th measurement is determined with the assumption that the measurement error has a uniform probability distribution for the range of maximum uncertainty that is specified by the manufacturer [14,25]. where a is the % maximum measurement uncertainty in the measurement.

IEEE 14-bus system
The system is assumed to have two power injection measurements (bus 8 and 12), and 20 power flow measurements (at each line). In this test system, bus 7 is zero injection bus and is included as a measurement with variance 10 −6 . Two PMUs are assumed to be placed at buses 1 and 6, which measure the voltage phasors at buses 1 and 6, and current phasors at lines 1-2, 1-5, 6-5, 6-11, 6-12, and 6-13. With the given maximum measurement uncertainties in Table 5, two test Cases I and II are considered for the condition number analysis for the different measurement configurations. In Case I, variance of the measurements are obtained considering only the relative accuracy assumed for the measurements (refer Section 4), whereas in Case II, the variances are evaluated using (30) and (31), considering the meter reading and its corresponding maximum measurement uncertainty. For the two methods of calculating measurement variances, the condition number of the gain matrix is listed in Table 6 for different type of measurement configurations. For the hybrid measurement configuration (conventional as well as PMU), the condition number is higher when only PMU current phasors are considered. The value of the condition number may vary depending on the placement of the PMUs. For a combination of the conventional measurements along with two PMUs (only current phasor considered), the maximum and minimum value of the condition number is 2.6353 × 10 9 (PMU at buses 1 and 10) and 4.6089 × 10 6 (PMU at buses 1 and 7).
To study the effect of variation in the relative weights of PMU and conventional measurements, three more test cases, III, IV, and V are considered for the IEEE 14-bus test system. In these  cases, PMU voltage and current phasors, both are considered for the condition number analysis with the variance defined for Case II. In Cases III and IV, the maximum measurement uncertainty for the conventional flow and injection measurement is taken as 2% and 1%, respectively, while for PMUs, it is same as listed in Table 5. In Case V, the maximum measurement uncertainty in the voltage magnitude, current magnitude and the phase angle are 0.01%, 0.02% and 0.005, respectively, while it is 3% for the conventional measurement (flow and injection). If the accuracy in the conventional measurement is improved, then the ratio between the weights for conventional and PMU measurements is reduced, and the value of the condition number is reduced, as evident from Cases III and IV in Table 7. In case of improvement in the measurement accuracy of PMU (refer Case V in Table 7), the condition number is higher.

IEEE 118-bus system
The condition number analysis is extended for IEEE 118-bus test system with 3 injection measurements (at buses 47, 95, and 112), and 186 flow measurements (through each line). The measurement set also consists of 10 zero injection measurements at buses 5, 9, 30, 37, 38, 63, 64, 68, 71, and 81 with a variance of 10 −6 . There are 5 voltage phasor and 20 current phasor measurements corresponding to 5 PMUs placed in the network at buses 17, 34, 69, 98, and 113. With the given maximum measurement uncertainties in Table 5, two test Cases, I and II, are considered for the condition number analysis for the different measurement configurations. Both the Cases I and II are defined in the same manner as for the IEEE 14-bus system. From Table 8, the numerical ill-conditioning is experienced in case of inclusion of PMU phasor currents only along with the conventional measurements. Like the study for the 14-bus system, to study the effect of varying measurement accuracy for the conventional as well as PMU measurements, three more test cases are generated. If the accuracy in the conventional measurement is improved, the ratio between the weights for conventional and PMU measurements is reduced, and the value of the condition number is also reduced, as evident from Cases III and IV in Table 9. In case of improvement in the measurement accuracy of PMU (refer Case V in Table 9), the condition number is higher.
From the preceding results, it is observed that the effect of PMU current phasor measurement on the condition number is more detrimental in comparison to the PMU voltage phasor measurements. The increase in the accuracy of the conventional measurement will make the condition number of the system better, as evident from Cases III and IV from Tables 7 and 9. The increase in the accuracy of the PMU measurements compared to conventional measurements results in more ill-conditioning, as clear from Case V for both the test systems.
To resolve the convergence issues associated with illconditioned gain matrix, several methods have been reported [6][7][8][9]. Reference [11] by the present authors proposes a regularization method using L-curves to address the ill-conditioned state estimation problems that may be difficult to solve by the traditional WLS methods.

CONCLUSION
Numerical ill-conditioning of the state estimation problem may lead to loss of estimation accuracy and convergence problems. Condition number of the state estimation gain matrix is a commonly used means of evaluating the numerical conditioning of the estimation problem. In practice, it has been observed that the inclusion of PMU measurements often creates numerical ill-conditioning in the so-called hybrid state estimator that uses both PMU and conventional measurements. This paper attempts to identify the root cause of such ill-conditioning of the state estimation problem. It analytically finds the approximate condition number of the gain matrix for simple systems considering hybrid measurement sets. It is found that the large difference in the weights allocated to the PMU measurements compared to the conventional ones is the main cause of such numerical ill-conditioning. In other words, higher relative accuracy of PMU measurements compared to the conventional measurements is the main reason for degrading the numerical condition of the state estimation problem. The condition number of the gain matrix may vary over a wide range depending on the number of PMUs and their locations. The condition num-ber analysis is also validated for large non-radial test systems with appropriately chosen weight ratio of the two types of measurements. The condition number of the gain matrix is high if only PMU current measurements are considered. It is, therefore, suggested that numerically robust solution techniques are used to address the ill-conditioning problem, when PMU measurements are included along with conventional SCADA based measurements in a state estimator.