Sensitivity analysis-based optimal PMU placement for fault observability

This paper proposes an algorithm that ﬁnds the optimal sets of phasor measurement units (PMUs) to achieve a fault observable system, while it addresses the multi-estimation issue. The optimal PMU placement (OPP) problem here is to ﬁnd a set of PMU locations with minimum number of members that enables fault observability in a system and satisﬁes a deﬁned minimum sensitivity requirement in measurements. The proposed algorithm gen-eralizes the impedance method in fault analysis and optimizes PMU utilization to maintain a required minimum sensitivity in each set of measurements, given the required fault detection accuracy. Also, the set of measurements is unique and distinctive for each fault scenario, preventing multi-estimation. A fault is referred to as a set of affected faulty line, fault location, and fault impedance. A sensitivity analysis is performed and sensitivity indices are derived to evaluate measurements quality to detect changes in fault line, location, or resistance. The algorithm is executed on IEEE 7-bus, 14-bus, and 30-bus test systems. Subsequently, artiﬁcial neural networks (ANN) are employed to build fault locators through ofﬂine training. ANN use an optimal PMU set obtained by the proposed algorithm to uniquely map between the corresponding measurements set and the faults.


INTRODUCTION
The roles of synchronized phasor measurement units (PMU) in power systems monitoring, control, and protection are prominent and constantly developing [1][2][3], This enables the operator to take advantage of wide area monitoring, protection and control (WAMPAC) [4]. Some of the applications of phasor measurement devices include event detection [5], normal [6] and fault observabilities [7], state estimation [8], and postcontingency analysis [9]. Since PMUs can measure voltage phasor of buses as well as current phasors of branches connected to the buses, it is feasible to observe the network with optimal number of devices. Therefore, optimal PMU placement (OPP) techniques have been introduced to determine a minimal set of PMUs to make the entire network observable. Many techniques and algorithms have been proposed in recent years to find OPP solution in power systems targeting system's normal observability. Numerical and topological algorithms are two common methods for solving the OPP problem. System's normal observability is guaranteed using numerical methods if the rank of the measurement matrix is com-This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2020 The Authors. IET Generation, Transmission & Distribution published by John Wiley & Sons Ltd on behalf of The Institution of Engineering and Technology plete, i.e. it is equal to the number of system state variables [10]. In topological methods, graph theory is employed and normal observability is ensured if it is possible to have an observable spanning tree [11]. Authors in [12] propose an optimization algorithm named as modified binary cuckoo optimization algorithm (MBCOA) to solve OPP problem by using topological rules. A binary particle swarm optimization technique for the optimal allocation of PMUs for the full observability of a connected power network is proposed in [13]. In [14] several optimization methods such as binary particle swarm optimization method, greedy algorithm, flower pollination algorithm, single vertex, and binary integer linear programming are compared in solving OPP and achieving full system observability. The idea of optimal placement of PMUs to ensure full power system observability in the presence of non-synchronous conventional measurements is studied in [15].
A power network is normally observable when all of its bus voltage phasors are known using available measurements under normal operation [16]. However, a network is fault observable if under a fault scenario occurring at any point of the system, voltage and current phasors at both ends of all lines are determinable. It should be mentioned that normal observability does not guarantee fault observability [16]. Thus, a normal observable power system may not be fully observable during fault condition since fault alters the system structure. While many approaches are proposed to solve OPP problem for power system normal observability (under normal operating condition), there are a very limited number of studies that target OPP for fault observability. Optimal PMU placement for fault observability is investigated in [17] and [18]. Authors in [17] employ the popular one-bus-spaced strategy to find the OPP by genetic algorithm and using only PMU voltage measurements. The topic is expanded in [18] by considering zero injection buses (that reduce the system size) using both PMU voltage and current measurements followed by integer linear programming (ILP) methodology. Authors in [19] study deterministic and stochastic algorithms for placing minimum number of PMUs in a power system in order to locate any fault in the power system.
Though the available approaches take advantage of various algorithms to impose observability constraints, the important issue of measurement sensitivity (quality) and its impact on OPP set and fault location is considered in very few literatures. In [20] the branch sensitivity indicator is developed and introduced into the PMU placement by giving higher priority to terminals pertaining to the higher sensitive lines. A pattern search algorithm (PSA) is proposed in [21] to minimize and optimally allocate PMUs considering nonlinear sensitive constraints of buses. However, in these publications the sensitivity analysis is considered for normal observability (sensitivity of either line current or voltage to the load change) rather than fault observability (sensitivity of both voltage and current to fault impedance and location). Authors of [22] utilize a minimization algorithm to reduce the number of sensors. This effort was followed by considering the measurement precision in the fault location problem [23] given the sensor locations; however, the precision has not been used in the measurement optimal placement. The effect of the measurement precision in PMUs placement is of paramount importance and adds additional constraints to the available methods while this has not been given enough attention in OPP solution methods. This paper considers PMU direct measurements with adequate channel availability for voltage and current measurements. A more comprehensive definition of fault observable system than [16][17][18]24] is adopted in this paper. If location and impedance of all faults of interest in a power system can be determined with predefined accuracy through a set of voltage and current measurements, the system is considered fault observable. A unique function mapping between measurements and faults is discussed and obtained in a systematic manner for the first time to the authors' best knowledge. The contributions of this research include: 1. Introducing sensitivity analysis in OPP problem for networks fault observability. The quality of measurements is assessed at PMU locations using the proposed sensitivity indices. Thus, through sets of faults, one can judge whether a specific network bus is a proper measurement location for PMU placement. Using the proposed sensitivity analysis, measurement precision or inaccuracy instigated by the current transformers (CTs), potential transformers (PTs), and PMUs can be incorporated into the OPP problem. Considering measurement quality is also vital for other system analyses such as voltage stability, contingency studies, etc., which are mostly fault related. 2. Formulating minimal PMU placement and finding pertinent optimal PMU sets for fault observability and fault location. The proposed algorithm based on the defined accuracy of PMUs finds the optimal PMU sets such that using minimum number of PMUs the faults are located uniquely, avoiding multi-estimation. Multi-estimation is a condition where different faults result in similar measurements in a selected PMU set. 3. Developing a fault locator by utilizing the obtained optimal PMU set along with artificial neural networks (ANNs).
Since there is assumed to be a function that relates the fault location and impedance to the voltage and current measurements, the function approximation property of the ANNs is employed to map between the faults and the measurements of the optimal PMU set.
Machine learning algorithms have already revolutionized many fields. Machine learning techniques such as reinforcement learning and extreme learning have been employed in power systems due to their generalization performance [25][26][27]. The reinforcement learning in the power system literature has been used to tune the learning mechanism based on a critic response or a target objective while enables the design of model-free approaches. On the other hand, the applications of learning methods such as ANNs is growing in power systems due to the ability of ANNs to learn complex non-linear relationships [28]. In this application, we found ANNs relatively easier than other learning mechanisms given the availability of the data and certainty of the system topology and parameters. Although deep learning mechanisms are great solutions to systems with large number of variables and uncertainties, they are generally time consuming and require large data, whereas the ANNs used in this paper are simpler and with the multistep training method proposed here, it does not impose much computation burden.
The remainder of this paper is organized in the following order: Section 2 presents the proposed sensitivity analysis and introduces the sensitivity indices. In Section 3, the sensitivity and multi-estimation criteria are presented, followed by the proposed algorithm for solving OPP problem in Section 4. Section 5 includes simulation results obtained from applying the proposed method on the IEEE 7-bus, IEEE 14-bus, and IEEE 30-bus test systems followed by artificial neural network fault locator results. Finally, concluding remarks are provided in Section 6.

SENSITIVITY ANALYSIS
The approach presented in this paper is built upon the classical fault analysis and is considered for symmetrical threephase systems. However, the approach can be generalized to Steps for Z bus modification: Z 0 through Z 3 are changing steps in Z bus single-phase and unsymmetrical networks as well [29,30]. A fault in power system changes the structure of the network, while its location and impedance are unknown. Therefore, previously known system states, impedance matrix Z 0 , and admittance matrix Y 0 should be altered to accommodate the fault analysis (see Figure 1) [30]. This study considers faults on power system lines (note that faults on grid buses is a special case). That is, an extra bus p = N + 1 is designated at the point of fault, where N is the network total number of buses. Figure 1 shows the procedure of adding a fault to the system. The unfaulty power system with known impedance matrix Z 0 , voltages, and currents is depicted in Figure 1a. Figure 1d depicts the faulty system with a fault on one of the network lines, where the impedance matrix is Z 3 (fault not included). The line exposed to the fault is located between system buses l and k that are unknown due to the random nature of fault with unknown fault distance D and fault resistance R f .

Definitions.
The following terms are used frequently in this paper.
Normal value: The value of a bus voltage or a line current in an unfaulty power system is called normal value. from Figure 1), and R f ∈ R f = [0, R max ] is the fault lineto-ground resistance in the single-phase equivalent circuit with R max being the maximum fault resistance of interest. If R max is selected very small (short circuit), the loads can be ignored in the proposed method. Otherwise, the load information may be needed to locate the fault accurately. Multi-estimation: Multi-estimation is a condition where different faults cause similar measured values in an observant set.
Four steps are required to modify Z 0 and obtain Z 4 (dashed elements in Figure 1b imply faulty line removal from Z bus ). Each of these steps results in a new system with impedance matrix subscripted by the step number as shown in Figure 1 [24,30]. By using the standard fault analysis, the voltage changes at observant bus h (when fault F occurs at bus p) can be described in Equation (1): where Z 3 (h, p) is the (h, p) entree of Z 3 , Z 3 (p, p) is the system Thevenin impedance seen from imaginary bus p, and V pre f is the prefault voltage at the point of fault in the system. With the assumption of linear voltage drop along the transmission lines between buses and by ignoring line capacitances to avoid complexity, V pre f can be calculated in Equation (2): For more accurate calculation in long transmission lines, hyperbolic voltage drop can be considered [24]. From the previous discussion, voltage and current changes in all buses of the system can be calculated by using original impedance matrix Z 0 along with D and R f , as will be explained next.

Voltage sensitivity indices
If deviation from the initial value of voltage at the observant bus h after fault F = (l f , D, R f ) is represented by ΔV h,F , then the voltage sensitivity indices are defined as derivatives of fault distance D and impedance R f with respect to ΔV h,F as shown in Equation (3): respectively. One can use derivatives of ΔV h,F with respect to D and R f instead and use the inverse function to achieve volt- h,F can be found in a similar manner. The derivation of indices (3) are given in the appendix.

Current sensitivity indices
An installed PMU on any grid bus measures the phasor of the bus voltage as well as that of the currents of all the connected lines. The number of line currents that a PMU is capable of measuring depends on the PMU's available channels that are assumed to be large enough in this study. This assumption is widely acceptable due to recent developments in PMU technology and the need for acquiring the line current data for various purposes [17,18]. Similar to voltage sensitivity indices, current sensitivity indices can be defined for each line connected between observant bus h and an adjacent bus u where u ∈ U h and U h is the set of all connected buses to observant bus h. These indices are defined in Equation (4): The number of current sensitivity indices derived for each observant bus h in (4) is equal to the number of lines connected to bus h. Figure 2 illustrates an example of a line current in the state of fault.
At any observant bus h within the network, line current changes can be expressed as in Equation (5): where Z hu is the impedance of line hu, Z hu = (−Y 4 (h, u)) −1 , and Y 4 (h, u) is the (h, u) entree of the admittance matrix that corresponds to Z 4 according to Figures 1 and 2. Note that manipulating the admittance matrix is much easier than that of the impedance matrix and it is not detailed here. Admittance matrix modification procedure results in obtaining Y 4 elements, many of them are not functions of D or R f . Derivation of the current sensitivity indices are also given in the appendix.

Definition. Consider an observant set
In order to demonstrate the concept through some numerical results, IEEE 7-bus test system is utilized for exemplary sensitivity analyses. Figure

Sensitivity requirements
Low values of the defined sensitivity indices (3) and (4) where h c is the number of connected buses (lines) to observant bus h.
Consider line l f , observant bus h, and adjacent buses u ∈ U h . Measurement sensitive range sets are defined in Equation (6): and  (1) and (3) where terms indicate desired sensitivity thresholds. That is, for example, set Θ DV h,F (l f ) contains all faults on line l f for which voltage at observant bus h is sensitive to the fault distance (D).   Figure 5a illustrates that not all possible faults on line 8 in terms of fault location (D) will cause distinguishable measurements on observant bus 2. That is, the red region, which correlates to a set of possible faults, needs line current measurements or other buses coverage so that we achieve fault detection coverage on line 8 for full observability. Figure 5b illustrates that all sensitivity indices pass the threshold criteria, which shows the capability of bus 2 voltage measurement in detecting line 8 fault resistances up to 0.1 p.u. In a similar manner In practice, realization of such condition may be difficult, especially for high values of fault impedance. Thus, a slightly simpler (and probably more conservative) approach is proposed here. One of the objectives in this paper is to benefit the provided fault location results and select observant buses that are able to locate at least 90% of all possible faults in the region D × R f for each faulty line l f ∈ L f . This practical criterion adds some flexibility in observant bus selection. By adding the 90% coverage criterion one has the option to lower the number of required PMUs. This also affects the implementation cost due to the lower number of required PMUs. The case with 100% coverage is also executed in this paper and the OPP results are compared with the ones of 90% coverage. The fault observability (neural network accuracy) of 90% and 100% coverage are almost the same; however, the number of PMUs in the latter case is higher. Due to the piecewise continuity of the defined sets in (6), an observant bus h is chosen if for ∃l f ∈ L f = {1, 2, … L} condition (7a) or (7b) is satisfied: implies that observant bus h is sensitive to the distance of 90% of the faults, indicated by the region D × R f , on the line l f . Similarly, SVIR f implies that observant bus h is sensitive to the impedance of 90% of the faults indicated by region D × R f on the line l f . Subsequently as in Equation (8), Binary value SVIDR f indicates whether the observant bus h is capable of illustrating (using its measurements) the changes in distance and/or impedance of a vast majority of the faults of interest that occur on line l f with the desired precisions indicated by (6). Condition (8) will be checked for all the power system lines to find observant bus h's domain of fault coverage. This step will reduce the number of required observant buses in obtaining fault observability in the entire system. In practice, one observant bus may not cover the faults on all power system lines and thus other observant buses must be exploited so that faulty lines that are not observed by one observant bus are observed by others. Thus, the above process is repeated for all the power system's buses to lay out an initial mapping between the faults of interest and the power system buses as potential observant buses. A group of observant buses (i.e. an observant set, if one exists), that satisfies condition (8) for all l f ∈ L f provides a solution to the fault location problem and thus renders the power system fault observable. This is equivalent to an observant set whose measurements (measurement set) are sensitive to 90% of distances or impedances of the faults on all power system lines.

Uniqueness and multi-estimation
After finding sensitive bus locations for measurement placement, multi-estimation is a necessary criterion to check to ensure that a measurement set is capable of locating all possible faults in the power system uniquely. The ability of precisely locating a fault in the system depends on distinguishable measurements for any two different faults in the system. Multi-estimation exists if for an observant set H ⊆ {1, 2, … , N} and two faults F 1 = (l f 1 , D 1 , R f 1 ) and F 2 = (l f 2 , D 2 , R f 2 ), where F 1 ≠ F 2 all corresponding measurements from the observant set H are the same; i.e., M H F 1 = M H F 2 (see Section 2). Analytically, for any pair of faulty lines l f 1 , l f 2 ∈ L f and observant bus h ∈ H , this results in the nonlinear equalities, shown in Equation (9), in terms of D 1 , R f 1 , D 2 , and R f 2 for ∀u ∈ U h :  (1) and (5) that lead to nonlinear equations that can be solved numerically.
This approach in the simplest form can be represented as an optimization problem in the form of min H W T H under constrains (7) and (9), where H is an N × 1 vector with its elements (0 or 1) represents selection of an observant bus, and W = [w 1 , w 2 , … , w N ] T is a weight matrix that reflects practical or operational priorities in selecting observant buses with 0 ≤ w i ≤ 1. The optimization problem can be developed further to include other constraints such as contingencies, etc., but this is not the objective of this paper and neither discussed further here. Thus, an exhaustive search is used in this paper to solve the OPP problem.

PROPOSED ALGORITHM FOR OPP AND ARTIFICIAL NEURAL NETWORK FAULT LOCATOR
Previous works consider optimal PMU placement with much emphasis on the PMU cost as a weight vector in the optimization problem [10][11][12][13][14][15]. However, measurement precision and bus suitability for fault observability are mostly neglected in assigning PMU locations. PMU fault location capability is a function of its location in the system. Measurement from a PMU installed in an improper location may cause significant inaccuracy in fault location. The proposed formulation and algorithm in this paper aims to thoroughly consider this issue. Power system buses should be checked for conditions (7) and (9) to obtain the most proper observant set H . These conditions can be translated as sensitivity and uniqueness conditions required for fault observability and location, and are evaluated for all grid buses so that a set of appropriate observant buses are selected. Numerical solutions can be sought to evaluate observant buses which are explained next.
Before we proceed, the following discussions need to be conducted.
Remark 1. (Measurement precision): IEEE C57.13 standard for instrumentation transformers suggests 0.3% error for current and voltage transformer [31,32]. Since PMU measurement precision is usually higher than that of the instrumentation, precisions of 0.1% for current and 0.1% for voltage measurements total vector error are considered in this study, denoted by TVE V and TVE I , respectively, where TVE x = | It is worth mentioning that accurate phasor estimation can be made during fault transients [33][34][35]. Nevertheless, in this study fault duration is considered to be 0.1 second, which is 6 cycles at 60 Hz and is equal to the operating time of the circuit breakers. Since the transients caused by the faults are generally damped within two cycles [36], an installed PMU has enough time to measure the steady-state fault phasors. In case, a severe fault occurs at a PMU location, the amplitude of the measured voltage or current phasors can be very inaccurate; however, the proposed method exploits multiple measurements across the grid to assure that enough accurate measurements are taken.
Fault location precision: Define TP D as "target precision for fault distance D". Also, define TP R f as "target precision for fault resistance R f ". Note that fault location range is 0 ≤ D ≤ 1 on a power line and thus for a given TP D ≤ 1, fault can be located on one of 1 TP D + 1 equally spaced points on any power lines. Also, if fault resistance range of interest is 0 ≤ R f ≤ R max , for the given TP R f the fault resistance can be any of R max TP R f + 1 equally spaced resistances between 0 and R max . Remark 2. From the above discussion, the desired upper limits for sensitivity indices (Equations (3) and (4)) can be calculated in Equation (10):

Proposed algorithm
The algorithm to find optimal PMU sets is explained as follows: 1. Enter the algorithm inputs: TP D , TP R f , TVE V , TVE I and S DR . Calculate the sensitivity thresholds DV , R f V , DI and R f I using (10). 2. Select an observant bus h and a faulty line l f ∈ L f and obtain the sensitivity indices (3) and (4)  That is, at least one measurement from the measurement set must pass the introduced sensitivity criteria for a given fault from the set of faults of interest; otherwise, the set is dismissed. Once an observant set that satisfies (8) is found for all faults of interest, it is retained and must be checked against multi-estimation. 8. Check against multi-estimation criterion using exhaustive search. The observant set obtained in Step 7 must be checked against multi-estimation criterion. Select a pair of faulty lines l f 1 , l f 2 ∈ L f . Equations (9) must yield no solutions but the trivial solution F 1 = F 2 , for selected lines l f 1 , l f 2 and measurements of selected observant set. Discard the measurement set and go back to Step 7 for another measurement set selection if Equations (9) yield non-trivial solutions; i.e. two faults yield similar measurement sets in the selected observant set. Otherwise, go to Step 9. Then, go to Step 10. Otherwise, go back to Step 7 for another measurement set selection. 10. Collect all the observant sets that pass Step 9. Typically, the observant set obtained in Step 9 is not unique. Thus, search can be continued to obtain more qualified observant sets that pass Step 9 with the number of observant busses equal to that of the last obtained observant set. This in turn determines multiple optimal PMU locations. Among the optimal sets, the set with the minimum number of measurements (voltages and currents) outperforms and is chosen.
The flowchart of the proposed algorithm is depicted in Figure 6. The proposed algorithm may include other constraints for practical applications such as excluding inaccessible measurement points (out of reach) and considering topological changes due to operational requirements (especially in micro grids). Thus, certain measurements can be removed from observant sets. For topological changes, the algorithm should be applied to the new topologies.

ANN fault locator
Once the optimal observant set is obtained, it is ensured that the set can locate all faults of interest uniquely without multiestimation. Thus, a one-to-one map exists between the corresponding measurement set and the faults of interest including the faulty line, the fault distance, and impedance. Consequently, artificial neural networks (ANNs) are capable of and used to map the measurement set from the optimal observant set to corresponding faults comprising faulty line l f , distance D, and resistance R f . It is worth mentioning that, in this paper, the ANN is used as a fault locator rather than fault detector or identifier. Since the fault location is a matter of fault place and intensity, there is no associated time component. That is, the fault location strategy is the same at all times. In addition, fault location in the proposed method is performed in a very short time after the observant set is obtained. Therefore, the complications of spatiotemporal  [37,38]. can be avoided, leaving temporal fault analysis unnecessary.
ANNs are intelligent mechanisms that can approximate complex nonlinear functions through employing a set of input and output data [39]. The function approximation property of ANNs is used here to estimate the function that maps the measurement set as the input and corresponding fault as the output. ANNs are trained offline and weights and bias values are obtained in MATLAB using the Levenberg-Marquardt optimization method [39] as an efficient method in training of feedforward ANNs. The ANNs here have one hidden layer and one output layer with sigmoid and linear activation functions, respectively.
In this study, instead of using one large neural network, a structure of networks is employed to have a more precise fault locator. That is, faulty line l f is found in the first neural network using input data from the measurement set. Then, based on the detected faulty line, a pertinent neural network is activated to determine fault distance D, and resistance R f , as shown in Figure 7. Input vector X of the first ANN is the measurement set's (corresponding to the obtained OPP) voltage and current magnitudes and angles. Output vector Y 1 is the faulty line l f . That is, where W 1 is the output layer weight matrix, Φ is the Sigmoid activation function, and V 1 is the hidden layer weight matrix. Next, a second ANN is selected based on the resultant faulty line from the first ANN. In the second ANN, the input vector is X as explained and output vector Y 2 = [D R f ] T is the location and resistance of the fault located on faulty line l f . That is, where W l f is the output layer weight matrix, Φ is the Sigmoid activation function, and V l f is the hidden layer weight matrix corresponding to faulty line l f .
The individual ANNs are trained separately using relevant generated fault data. All ANNs utilize one hidden layer whose number of neurons vary with the size of the grid (e.g. 20-40 neurons for 7-bus and 35-65 neurons for 30-bus grid) where higher number of neurons are used for higher precision scenarios (lower TP D and TP R f ). Approximately 20% of the generated fault data is separated and used to test the trained neural networks. Neural network fault locator results presented in the next section are the percentage of the correct estimations for this portion of data. ANN design data is also provided in the tables in the next section.

RESULTS AND DISCUSSION
The proposed algorithm is applied to the IEEE 7-bus (Figure 3), IEEE 14-bus, and IEEE 30-bus [40] test systems to assess the performance of the algorithm and the obtained optimal PMU set in fault location. The test systems consist of 3, 2, and 6 generators as well as 10, 21, and 42 transmission lines, respectively [40]. The IEEE 14-bus and 30-bus test cases oneline diagram are provided in the appendix. Once the proposed algorithm finds the optimal observant set(s) for each power system, artificial neural networks are utilized to obtain a fault locator using the observant set. It should be mentioned that despite ignoring the network capacitance in developed methodology (Equation 2), the shunt capacitance of transmission lines is available during the algorithm execution to include their effect  on OPP solutions and fault location accuracy. The algorithm is also executed with zero shunt capacitance and results compared with those considering the lines' capacitance showing negligible differences. The ANNs are trained by known fault data that are measured by the optimal PMU set (observant set) and create a one-to-one map between the measurement set and the corresponding fault; i.e. fault line, distance, and impedance. After the training is completed, the ANN fault locator is tested by new fault data and accuracy of fault location is examined. Fault impedance is considered to be purely resistive in this study [24]. The maximum fault resistance of the interest is considered to be R max = 17.4 for all test cases, i.e. 0.1 p.u in 132 kV base voltage. By increasing the maximum fault resistance of interest the number of PMUs in the set may increase since the measurement set is demanded to cover higher resistance faults. Various voltage and current measurement precisions are used to solve the OPP problem to include various PT and CT precisions.

5.1
Proposed algorithm results Table 1 presents OPP results for the IEEE 7-bus system. Two cases are performed in the simulation: with target precisions of 1% for fault distance D and resistance R f , and with target precisions of 5% for D and R f . These precisions are the desired fault location accuracies and thus they are used to generate faults for training and testing the ANN fault locator. The first two columns of Table 1 show voltage and current measurement precisions. These precisions are used in solving the OPP in the proposed algorithm where sensitivity indices are utilized. Columns 3 and 4 represent the minimum number of required PMUs and the optimal observant set(s) suggested by the proposed algorithm. The ANN fault locator is trained by employing the optimal observant set shown in bold. For the case with 1% target precision, 11,000 fault scenarios are generated throughout the system for ANN training, and 2,200 fault data are used for test and validation. Similarly, for the case with 5% target precision, 600 fault scenarios are used for training and 120 fault scenarios are used for validation. The remaining columns show the accuracy of fault location using the trained ANN fault locator. The fault locator dedicates an artificial neural network for each line in the second stage when the faulty line is found, as shown in Figure 7. The top and bottom percentage values in the last two columns show the average and minimum estimation accuracies, respectively, across all network lines. One can observe that by using current and voltage precision of 10 −2 (1%) only two optimal observant sets with 2 PMUs in each set are found by the proposed algorithm. By only improving the voltage precision to 10 −3 , the minimum number of PMUs and the optimal observant sets remain the same. However, if the current measurement precision is also improved to 10 −3 (0.1%), one PMU would be enough for the system to be fault observable. On the other hand, by reducing the preferred precision for the fault location to 5%, only one PMU is adequate for observing all system faults. Tables 2 and 3 present the results of the proposed OPP and ANN locators for IEEE 14-bus and 30-bus systems, respectively. The bus numbers given in [40] are adopted in this paper. It is observed that higher current measurement precision is more effective than that of voltage in reducing the number of required PMUs. Overall, these results show the impact of measurement precision on OPP solutions. In addition, provided results illustrate a significant improvement over the conventional one-busspaced method where approximately 50% of buses are required for the system fault observability [16,18]. For example the number of suggested PMUs for one-bus-spaced method is 17 [18] for the IEEE 30-bus system as opposed to 13 PMUs obtained here based on 10 −2 measurement precision. Moreover, [18] proposes 14 PMUs for the IEEE 30-bus system when considering  (2,5,7,9,12,13,14)- (2,5,8,9,12,13,14)- (2,6,7,9,12,13,14)- (2,6,8,9,12,13,14)- (2,7,9,11,12,13,14)- (2,8,9,11,12,13,14) 99    (2,3,6,7,10,11,12,13,15,17,19,22,24,26,27,28,29) 17 (2,3,6,7,8,10,11,12,13,15,17,19,22,24,26,27,29) six zero-injection buses (that reduce grid size), and [16] proposes eight PMUs using 15 additional flow measurements to achieve fault observability. By contrast, the proposed algorithm suggests 13 PMUs based on 10 −2 measurement precision and two PMUs based on 10 −3 measurement precision with the desired fault location accuracy of 1% for both fault distance and impedance. Table 4 summarizes the results of references [16] and [18] that employ ILP in the context of onebus-spaced strategy for full system fault observability. Note that the measurement precision is not considered and elaborated in these works, whereas the precision plays an important role in the number of required measurement units. That is, higher fault location and/or impedance precision need larger number of PMUs.

CONCLUSION
A new algorithm has been introduced for power system optimal PMU location using sensitivity analysis, where the fault location accuracy is specifically taken into account. With the proposed sensitivity analysis, appropriate indices are defined that can be used to qualify the measurements' locations in the network in detecting fault location and impedance. Also, multi-estimation is introduced and checked in the proposed algorithm to guarantee a unique mapping between a PMU measurement set and all faults of interest in the system. The proposed algorithm finds the minimum number of PMUs required for system fault observability. By using obtained optimal PMU sets, an ANNbased fault locator is developed that map the measurements of the optimal PMU set on the system faults.  (1) is discussed. From the transition from impedance matrices Z 1 to Z 2 [24], one concludes that for any fault, term Z 2 (p, p) is the only D-dependent variable in Z 2 shown as Z 2 (p, p) = Z 1 (k, k) + (1 − D) × Z lk . Subsequently, the transition from impedance matrices Z 2 to Z 3 , resulting from the addition of impedance DZ lk between buses p and l , leads to

ORCID
Thus, the derivative of Z 3 (h, p) with respect to D is Similarly, the derivative of Z 3 (p, p) with respect to D is Note that derivatives of Z 3 (h, p) and Z 3 (p, p) with respect to R f are zero.
Current sensitivity indices: Five elements that are D dependent and one element that is R f dependent are obtained, for which )−1 .

FIGURE 9 IEEE 30-bus benchmark
It should be mentioned that for cases where fault is on the line whose current is measured, S DI hp,F and S R f I hp,F are calculated with p = N + 1 due to an additional bus at fault location. IEEE 14-bus and IEEE 30-bus test cases are shown in Figures 8 and 9, respectively. Line numbers are provided in parenthesis.