# Optimal PMU Placement in the Presence of Conventional Measurements

Authors: Zhaoyang Jin; Yueting Hou; Yue Yu; Lei Ding
Journal: International Transactions on Electrical Energy Systems
 (2022)

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## Abstract

This article presents a novel method for optimal phasor measurement unit placement (OPP) for full power system observability in the presence of conventional measurements. The method considers eligible PMU placements that may be omitted by existing OPP methods and hence may provide a better solution and can guarantee that the PMU placement can restore network observability. This is achieved by creating a new observability criterion and a network transformation scheme. The new observability criterion uses injection and zero injection measurements to improve the solution and the network transformation scheme reduces the size of the OPP problem and is a prerequisite for applying the proposed observability criterion. The OPP problem is solved using binary integer linear programming (BILP) or binary integer linear programming (BILP). Therefore, the proposed OPP method can be easily combined with other OPP methods, considering other constraints/scenarios using BILP, such as OPP considering PMU current channel limits. The new method is tested using simulations of the IEEE 14-, 118-, and 2736-bus test systems. These results show that the proposed new method is feasible in terms of execution time, is free of solutions that fail to make the system fully observable, and is capable of identifying optimal placement solutions that may be overlooked by existing OPP methods.


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## Body

## 1. Introduction

State estimation is a critical application in electric power systems. The Energy Management System (EMS) must be equipped with an online state estimator so that the database input to all other online applications is reliable and accurate. In order to make sure that a unique estimate can be obtained for all states of the system, i.e., the voltage magnitudes and angles at every bus, a sufficient number of measurements must be placed at the proper places. A network is deemed to be observable if a unique estimate of the system states can be obtained with the measurements available. If a PMU is installed at a bus, then this bus and the direct neighbouring buses will be observable, provided that the PMU monitors the current flow on the branch between the two buses. Therefore, if PMUs are placed at strategic locations, then network observability can be delivered at a lower overall cost. In this article, only the observability requirements for state estimation are considered.Optimal PMU placement for observability is the subject of abundant literature that considers different algorithms [1, 2], scenarios (e.g., in the presence of redundant measurements [3]), and constraints. Integer linear programming (including binary integer linear programming) [3–5] is the optimisation algorithm that has received the most attention due to its high accuracy, flexibility, and ease of implementation. Taxonomies of PMU placement methodologies [1, 2] classified the existing OPP methods into six different formulations separated according to the factors they consider: the effects of single branch outage contingency [5–8], PMU outage contingency [6–10], the contingency of single line outage or single PMU loss [6, 8, 10–13], the effects of PMU channel capacity (i.e., how many branch currents a PMU can measure) [6, 14–16], the effects of zero injection measurements [5–7, 9, 14, 17, 18], the effects of conventional measurements [8, 9, 11, 19–24], the effects of DC lines in the network [25], OPP considering probabilistic and stochastic observabilities [26, 27], OPP considering fault [28], OPP for applications and purposes such as controlled islanding [29], small-signal stability [30, 31], and state estimation [32, 33]. In recent times solving the OPP problem for observability while considering the presence of conventional measurements has become an important subset of this work. This is mainly because of two reasons: first, hybrid state estimators require the use of a mixture of PMU and conventional measurements (a promising application of data from wide area monitoring systems [34]; second, zero injection measurements can be assumed for many bus stations that are known not to possess any load or generation.Most algorithms, such as [8, 9, 19–22], are based on Kirchhoff’s Current Law (KCL). According to KCL, n buses that have an injection measurement or that neighbour a bus with an injection measurement will be observable if any n − 1 of them are observed by PMUs. However, KCL-based methods are local observability criteria that can decide whether or not a specific part of the network is observable. It can only ensure optimal placement when there are only injection measurements in the network and no injection measurements are located at neighbouring buses, which are rather unrealistic in a power system. Some other algorithms, such as [11, 24], are based on traditional observability analysis. The observable islands [25] identified through traditional observability analysis are regarded as the new buses in the reduced new network. It is assumed that there are no conventional measurements in the new network. However, this assumption is not always valid because of the existence of some special injection measurements referred to as irrelevant injection measurements [35]. They are named “irrelevant injection measurements” since they can be eliminated without affecting the identification result of the observable islands. However, irrelevant injections do contribute to the rank of the gain matrix, and more PMUs will be placed if they are neglected when finding the optimal solution to restore the full observability of the network (optimality problem).In this article, a novel method for OPP using BILP is proposed that overcomes the abovementioned problems using an innovative network transformation scheme and a new global observability criterion that can directly decide the observability of the whole network. BILP is chosen for solving the OPP problem because there have been abundant OPP methods that consider different constraints and scenarios using BILP. The function of the proposed method can be easily extended by combining it with them if it is solved by BILP. The network transformation scheme creates a network model that is reduced in size and contains only injection measurements. It is based on the traditional observability analysis while also taking the impact of the irrelevant injection measurements into consideration. The network transformation scheme is necessary for two reasons: first, it is difficult to find the global observability criterion when the network contains multiple types of conventional measurements (e.g., injection measurements and flow measurements); second, even if such a global observability criterion can be found, it might be too complicated to implement with BILP. Thus, the presented global observability criterion was designed specifically for the transformed networks. It considers the observability of the whole network rather than the local observability of parts of the network. The proposed algorithm can offer an improved optimal placement solution and can guarantee 100% valid PMU placement that satisfies the observability requirement, while the KCL method might have PMU placements that cannot make the network fully observable (validity problem). This is because it removes the assumption of nonadjacent injection measurements and is mathematically proved to be valid. Furthermore, it has reduced execution time compared to the observability analysis-based method. This is because of the reduction in the network size after the network transformation and the consequent reduction in the number of elements in the state vector.The article is structured as follows. Section2 introduces the new network transformation scheme and how it reduces the system to the observable islands and irrelevant injections. Section 3 describes the new observability criterion and provides a proof of its validity. Section 4 provides an overview of the new methods execution and its ability to avoid the erroneous placements (placement that fails to make the network fully observable) that KCL-based methods are vulnerable to is demonstrated in Section 5. Section 6 provides examples of the improved optimal placement found by the new method for the IEEE 14- and 57-bus systems and discusses the relative execution time of the method. Finally, Section 7 concludes the article and identifies opportunities for future developments.


## 2. Observability Analysis and the Network Transformation Scheme

The first step of the proposed method is to perform observability analysis and apply the proposed network transformation scheme, which will be detailed in this section.Observability analysis is the procedure for determining whether or not the available measurements are sufficient to find a unique solution to the state estimation problem and, if this condition is not satisfied, identify suitable corrective actions [25, 36]. Examples of corrective actions are as follows: placing additional measurements into the network to restore the system observability or identifying the observable islands so that state estimation can still be performed for these subnetworks. In this article, identifying the observable islands [35, 37–40] serves as the initial part of the network transformation scheme. The method proposed in [40] is used for identifying observable islands since it guarantees 100% correct identification of observable islands with fast execution speed. For the sake of brevity, the algorithms for identifying observable islands will not be discussed in this article, whose details can be found in [35, 37–39]. The network transformation is proposed as follows:(a)
Perform observability analysis for the unobservable network; identify the observable islands and irrelevant injection measurements.(b)
Transform the network into a reduced form, where the observable islands are the new buses and the injection measurements deemed irrelevant in a) become the only injection measurements.(c)
Form the bus–placement incident matrix,A.The purpose of the network transformation scheme is to convert the network to be analysed that comprises various types of conventional measurements in a network with only power injection measurements. This allows the network observability criterion detailed in Section3 to be applied for optimal PMU placement in any network.


### 2.1. Example

In the IEEE 14-bus test system with the measurement placement illustrated in Figure1, five observable islands are identified by the observability analysis: {1, 2, 3, 4, 5}, {6, 12, 13, 14}, {7, 8}, {9, 10}, and {11}. In addition, the injection measurements incident to Bus 4, Bus 9, and Bus 11 are found to be irrelevant injection measurements. The irrelevant injection measurements are defined as the power injection measurements which are incident to more than one observable island [35]. The transformed network is given in Figure 2.Figure 1
IEEE 14-bus test system and the measurement placement.Figure 2
transformed 5-bus network, the bus numbers correspond to the observable islands in the full network.Since each bus in the transformed network represents an observable island, the PMU at a certain bus might not be able to measure the current phasors of the neighbouring buses when the full network is considered (e.g., a single PMU at Bus 1 of Island 1 of the original network can only measure the current of the branch connecting to Island 2). Here, aplacement is defined as a PMU placed at a given bus measuring current phasors of given branches. Thus, it is possible to have different placements at the same bus. The bus–placement incidence matrix of the transformed network and the definition of its elements are given in (1) and (2), respectively:(1)A=1110110001110101100011001001111100100011,(2)Aij=0,if Busiis not measured by PMU with Placementj,1,if Busiis measured by PMU with Placementj.Eight placements are available for the transformed network, as summarized in Table1. The available placements are found following the following procedure: finding a bus in the original network and representing it as a placement in the transformed network from the bus corresponding to the observable island where it is located to the buses corresponding to the observable islands it connects. For example, the PMU placed at Bus 5 and Bus 4 of Observable Island 1 in the original network are represented as Placements 1 and 2, the two placements with the PMUs are placed at Bus 1, and the current phasors of the branches are measured connecting to Bus 2 and Buses 3 and 4, respectively. The injection measurements at Buses 4, 9, and 11 are identified as the irrelevant injection measurements in the original network. According to the network transformation scheme, these irrelevant injection measurements in the original network are regarded as the injection measurements in the corresponding placements in the transformed network, Placements 2, 7, and 8, as shown in Table 1. The placements incident with injection measurements is treated differently according to the proposed observability criterion, which will be introduced in the next section.Table 1
Placements found in the transformed network.
At bus #To bus #With injection measurement?Placement 112NoPlacement 213, 4YesPlacement 321, 5NoPlacement 424NoPlacement 534NoPlacement 641, 2, 3NoPlacement 745YesPlacement 852, 4Yes


## 2.1. Example

In the IEEE 14-bus test system with the measurement placement illustrated in Figure1, five observable islands are identified by the observability analysis: {1, 2, 3, 4, 5}, {6, 12, 13, 14}, {7, 8}, {9, 10}, and {11}. In addition, the injection measurements incident to Bus 4, Bus 9, and Bus 11 are found to be irrelevant injection measurements. The irrelevant injection measurements are defined as the power injection measurements which are incident to more than one observable island [35]. The transformed network is given in Figure 2.Figure 1
IEEE 14-bus test system and the measurement placement.Figure 2
transformed 5-bus network, the bus numbers correspond to the observable islands in the full network.Since each bus in the transformed network represents an observable island, the PMU at a certain bus might not be able to measure the current phasors of the neighbouring buses when the full network is considered (e.g., a single PMU at Bus 1 of Island 1 of the original network can only measure the current of the branch connecting to Island 2). Here, aplacement is defined as a PMU placed at a given bus measuring current phasors of given branches. Thus, it is possible to have different placements at the same bus. The bus–placement incidence matrix of the transformed network and the definition of its elements are given in (1) and (2), respectively:(1)A=1110110001110101100011001001111100100011,(2)Aij=0,if Busiis not measured by PMU with Placementj,1,if Busiis measured by PMU with Placementj.Eight placements are available for the transformed network, as summarized in Table1. The available placements are found following the following procedure: finding a bus in the original network and representing it as a placement in the transformed network from the bus corresponding to the observable island where it is located to the buses corresponding to the observable islands it connects. For example, the PMU placed at Bus 5 and Bus 4 of Observable Island 1 in the original network are represented as Placements 1 and 2, the two placements with the PMUs are placed at Bus 1, and the current phasors of the branches are measured connecting to Bus 2 and Buses 3 and 4, respectively. The injection measurements at Buses 4, 9, and 11 are identified as the irrelevant injection measurements in the original network. According to the network transformation scheme, these irrelevant injection measurements in the original network are regarded as the injection measurements in the corresponding placements in the transformed network, Placements 2, 7, and 8, as shown in Table 1. The placements incident with injection measurements is treated differently according to the proposed observability criterion, which will be introduced in the next section.Table 1
Placements found in the transformed network.
At bus #To bus #With injection measurement?Placement 112NoPlacement 213, 4YesPlacement 321, 5NoPlacement 424NoPlacement 534NoPlacement 641, 2, 3NoPlacement 745YesPlacement 852, 4Yes


## 3. Network Observability Criterion

After the original network is transformed into a reduced network, the OPP algorithm should obtain the optimal placement of the PMUs according to the network observability criterion as follows:A network is defined to be observable if a unique estimate of the states can be obtained with the measurements available. In other words, the gain matrix of the following normal equation should be invertible:(3)G=HTWH,where G is the gain matrix, H is the measurement Jacobian matrix, and W is the weighting matrix. For classical state estimators, the gain matrix is only one rank less than the full rank if the system is fully observable. This is because of the necessity to set up a reference bus. For hybrid state estimators, the gain matrix is of full rank if the system is fully observable, as the angles of the voltage and current phasors measured by PMUs have already been calculated on a common reference using the synchronising GPS signals. Since W is a nonsingular diagonal matrix, it is derived that the rank of G is equal to the rank of H:(4)rankH=rankHTH=rankHTWH=rankG.Hence, the general observability criterion for a hybrid state estimator is summarized in the following:(5)rankH=Nbus,where Nbus represents the number of buses in the network. However, this observability criterion requires an algorithm to check the rank of the measurement Jacobian matrix, which is impractical to be implemented with BILP. Therefore, a more direct observability criterion should be derived from this criterion, which is presented in Part B of this section.


### 3.1. DC Decoupled State Estimation Model

The DC state estimation model is a simplified model typically used in numerical state estimation. It is known that there are strong correlations between active power and voltage angle and between reactive power and voltage magnitude. Furthermore, if the equations describing active power are observable, then it follows that the equations describing reactive power will also be observable. Hence, only the active power and the voltage angle need to be considered when formulating the measurement Jacobian matrix. In addition, it is assumed that the network has DC power flow and unity branch reactance as expressed in the following:(6)Pik=θi−θk,where θi and θk are the voltage angles of Bus i and Bus k and Pik is the active power flow between the two buses. These assumptions are valid since the network observability is independent of the system states.


### 3.2. Proposed Observability Criterion

For a network with only injection measurements (including zero injection measurements), the following observability criterion can be applied for optimal PMU placement.Theorem 1.
A network withq buses incident with injection measurements will be fully observable if the remaining p buses are all observed by PMUs.The proposed observability criterion can be easily implemented with BILP as follows:(7)mincTx,(8)s.t.Ax≥b,where c is the cost coefficient vector for the placements, x is the placement vector whose elements are either 0 or 1, A is the bus–placement incidence matrix, and b is the linear equality constraint vector. In b, the elements corresponding to placements with injection measurements are assigned to be 0 and the other elements are assigned to be 1. Since the elements of A and x can only be nonnegative numbers, the zero constraints are effectively released constraints. Thus, the constraints corresponding to the placements with injection measurements could be neglected without affecting the final placement result.Proof for Theorem 1:Define matrixB as the network connection matrix. The entries of B are defined as follows:(9)Bik=1,Buskis the sending bus of Branchi,−1,Buskis the receiving bus of Branchi,0,otherwise.Define matrixBq as the matrix constituted by the columns corresponding to the q buses incident with injection measurements. The measurement Jacobian matrix corresponding to the measurement placement with the q injection measurements, Hq, can be represented as follows:(10)Hq=BqTB.After proper matrix permutations and row eliminations, the measurement Jacobian matrix,H, corresponding to the measurement placement with the q injection measurements and the p PMU observed buses, can be expressed as follows:(11)H=Ip00BqTBq.It is obvious that the statement thatH is of full rank is equivalent to the statement that Bq is of full rank since for any vector or matrix, x, rank(x) = rank(xTx). After proper matrix permutations and row eliminations, any B can be represented as follows:(12)B=1−1⋱⋱1−1R,where R is the redundant part of B. rank(B) = rank(Bp+q−1) = p + q − 1 because the sum of each row of B is 0 and an identity matrix of size p + q − 1 can be found after deleting one column of B and performing Gaussian elimination. Since Bp+q−1 has full rank, the matrix constituted by any columns of Bp+q−1 also has full rank. Therefore, Bq and hence H are of full rank. So, the network will be full observable since the necessary and sufficient condition of the network to be fully observable is that the measurement Jacobian matrix H has full rank.End of the proof.


### 3.3. Optimal PMU Placement considering Current Channel Limits

To demonstrate the ability of the proposed method to combine with the other OPP methods using BILP, this subsection introduces how the proposed OPP method is combined with the OPP method proposed in [41], which considers the current channel limits of the PMUs. If the PMUs have a current channel limit of N, while bus i is connected to M buses with M > N, then there are C(M, N) = M!/(N!(M − N)!) different placements with different current channel connections at bus i. Instead of using only 1 element in the placement vector, x, to represent the PMU placement situation at bus i, a total of C(M, N) elements are used in x to show how the current channels are connected. For example, the corresponding row of bus i, in the bus–placement incident matrix A, is given by Ai shown in (13). If the PMU has a current channel limit of 2, then Ai should be rewritten as in (14):(13)Ai=1011,(14)Ai=100110100011.If different types of PMUs that have different current channel limits,M1, …, Mk, are available, then all ∑l=1kCMl,N possible placements should be considered.


## 3.1. DC Decoupled State Estimation Model

The DC state estimation model is a simplified model typically used in numerical state estimation. It is known that there are strong correlations between active power and voltage angle and between reactive power and voltage magnitude. Furthermore, if the equations describing active power are observable, then it follows that the equations describing reactive power will also be observable. Hence, only the active power and the voltage angle need to be considered when formulating the measurement Jacobian matrix. In addition, it is assumed that the network has DC power flow and unity branch reactance as expressed in the following:(6)Pik=θi−θk,where θi and θk are the voltage angles of Bus i and Bus k and Pik is the active power flow between the two buses. These assumptions are valid since the network observability is independent of the system states.


## 3.2. Proposed Observability Criterion

For a network with only injection measurements (including zero injection measurements), the following observability criterion can be applied for optimal PMU placement.Theorem 1.
A network withq buses incident with injection measurements will be fully observable if the remaining p buses are all observed by PMUs.The proposed observability criterion can be easily implemented with BILP as follows:(7)mincTx,(8)s.t.Ax≥b,where c is the cost coefficient vector for the placements, x is the placement vector whose elements are either 0 or 1, A is the bus–placement incidence matrix, and b is the linear equality constraint vector. In b, the elements corresponding to placements with injection measurements are assigned to be 0 and the other elements are assigned to be 1. Since the elements of A and x can only be nonnegative numbers, the zero constraints are effectively released constraints. Thus, the constraints corresponding to the placements with injection measurements could be neglected without affecting the final placement result.Proof for Theorem 1:Define matrixB as the network connection matrix. The entries of B are defined as follows:(9)Bik=1,Buskis the sending bus of Branchi,−1,Buskis the receiving bus of Branchi,0,otherwise.Define matrixBq as the matrix constituted by the columns corresponding to the q buses incident with injection measurements. The measurement Jacobian matrix corresponding to the measurement placement with the q injection measurements, Hq, can be represented as follows:(10)Hq=BqTB.After proper matrix permutations and row eliminations, the measurement Jacobian matrix,H, corresponding to the measurement placement with the q injection measurements and the p PMU observed buses, can be expressed as follows:(11)H=Ip00BqTBq.It is obvious that the statement thatH is of full rank is equivalent to the statement that Bq is of full rank since for any vector or matrix, x, rank(x) = rank(xTx). After proper matrix permutations and row eliminations, any B can be represented as follows:(12)B=1−1⋱⋱1−1R,where R is the redundant part of B. rank(B) = rank(Bp+q−1) = p + q − 1 because the sum of each row of B is 0 and an identity matrix of size p + q − 1 can be found after deleting one column of B and performing Gaussian elimination. Since Bp+q−1 has full rank, the matrix constituted by any columns of Bp+q−1 also has full rank. Therefore, Bq and hence H are of full rank. So, the network will be full observable since the necessary and sufficient condition of the network to be fully observable is that the measurement Jacobian matrix H has full rank.End of the proof.


## 3.3. Optimal PMU Placement considering Current Channel Limits

To demonstrate the ability of the proposed method to combine with the other OPP methods using BILP, this subsection introduces how the proposed OPP method is combined with the OPP method proposed in [41], which considers the current channel limits of the PMUs. If the PMUs have a current channel limit of N, while bus i is connected to M buses with M > N, then there are C(M, N) = M!/(N!(M − N)!) different placements with different current channel connections at bus i. Instead of using only 1 element in the placement vector, x, to represent the PMU placement situation at bus i, a total of C(M, N) elements are used in x to show how the current channels are connected. For example, the corresponding row of bus i, in the bus–placement incident matrix A, is given by Ai shown in (13). If the PMU has a current channel limit of 2, then Ai should be rewritten as in (14):(13)Ai=1011,(14)Ai=100110100011.If different types of PMUs that have different current channel limits,M1, …, Mk, are available, then all ∑l=1kCMl,N possible placements should be considered.


## 4. Proposed Algorithm

Based on the new network transformation scheme (Section2) and the new observability criterion (Section 3), a new algorithm for optimal PMU placement in the presence of conventional measurement is proposed here as follows:Step 1: perform the network transformation scheme following the steps given in Section2.Step 2: find the optimal PMU placement in the transformed network using BILP with the optimisation function and constraints given in (7) and (8), respectively.Step 3: find the optimal PMU placement in the original network.The steps given above are summarized in the flowchart in Figure3.Figure 3
Block diagram of the proposed OPP algorithm.


## 5. Advantages of the Proposed Method over Existing OPP Methods

To demonstrate the validity of the proposed observability criterion, the advantages of the proposed method are compared with the two types of existing OPP methods in terms of validity and optimality.


### 5.1. KCL-Based Methods: Failure to Identify a Valid Placement Solution

The most severe problem of the KCL-based methods is that the optimal PMU placement it provides can fail to make the system fully observable. For context, the fundamental aspects of a KCL-based method are briefly described as follows.If a bus is incident with a PMU or directly neighbouring a bus that is incident with a PMU, this bus is said to bereached by the PMU. Define yi as the number of times of Bus i reached by PMUs. For buses not associated with conventional measurements, yi should be greater than or equal to 1, while for the n buses associated with a certain conventional measurement (i.e., incident to or neighbouring a bus that is incident to a conventional measurement), the following constraint is imposed:(15)yi1+yi2+⋯+yin≥n−1.According to KCL,n buses that have an injection measurement or that neighbour a bus with an injection measurement will be observable if any n − 1 of them are observed by PMUs. This condition is not guaranteed by the constraint given by (15) since some of the yis might be greater than 1. For example, suppose that yi1 = 2, yi2 = 0, yi3 = 0, and the other yis are equal to 1, the constraint given by (15) is still satisfied, but only n − 2 of the n buses are observed by PMUs. In these cases, the PMU placement given by the KCL method might fail to restore full system observability.Simulations have been performed in the IEEE 14-bus test system to investigate the validity of both the KCL-based methods and the proposed method in this article. All 2^14 possible injection measurement placements without other types of conventional measurements are tested in the simulations. The simulation results are summarized in Tables2 and 3.Table 2
Valid solutions found.
KCL-based method (%)Proposed method (%)Valid solutions found96.71100Table 3
Examples of failed PMU placement by the KCL-based method.
Injection measurements placed at busesPMU placement at busesCase 13, 115, 7, 13Case 23, 56, 7, 9Case 32, 115, 7, 13


### 5.2. KCL-Based Methods and the Observability Analysis-Based Methods: Suboptimal Solutions

KCL-based methods might not give the placement with the minimum number of PMUs needed to provide the system with full observability. An example is as follows:Consider a simple 6-bus network with the measurement placement illustrated in Figure4.Figure 4
Six-bus network with measurement placement.According to Theorem1, placing a PMU at either Bus 1, Bus 2, Bus 5, or Bus 6 can ensure that the network will be fully observable. The validity of the placement result can easily be validated by checking the rank of the measurement Jacobian matrix with both the conventional measurements and the additional PMU measurement. The constraints of the proposed method are given in the following:(16)y1≥1,y6≥1.In comparison, the KCL-based method requires the following constraints to be satisfied:(17)y1+y2+y3≥2,y2+y3+y4≥2,y3+y4+y5≥2,y4+y5+y6≥2.It can be seen that it is impossible to satisfy all of the four constraints given in (17) with only a single PMU since one PMU at the network given in 0 can only reach at most 3 buses. Thus, two PMUs are suggested by the KCL-based method to restore system observability.As for the observability analysis-based method, its obvious flaw in finding the optimal PMU placement is that the contribution of the irrelevant injection measurements to the system observability is neglected. In the example in Section2, the constraints of the observability analysis-based method and the proposed method in the transformed network are given in the following, respectively:(18)x1+x2+x3+x5+x6≥1,x2+x3+x4+x6+x8≥1,x1+x5+x6≥1,x1+x4+x5+x6+x7+x8≥1,x3+x7+x8≥1,(19)x2+x3+x4+x6+x8≥1,x1+x5+x6≥1,where xi is the element of the placement decision vector, x, xi = 1 means that a PMU is placed at the corresponding placement, and xi = 0 means that it is not. It can be seen that at least two PMUs are required to satisfy the constraints given by (18), while only one PMU at Placement 6, which corresponds to Bus 9 in the original network, is required to meet constraints given by (19).


## 5.1. KCL-Based Methods: Failure to Identify a Valid Placement Solution

The most severe problem of the KCL-based methods is that the optimal PMU placement it provides can fail to make the system fully observable. For context, the fundamental aspects of a KCL-based method are briefly described as follows.If a bus is incident with a PMU or directly neighbouring a bus that is incident with a PMU, this bus is said to bereached by the PMU. Define yi as the number of times of Bus i reached by PMUs. For buses not associated with conventional measurements, yi should be greater than or equal to 1, while for the n buses associated with a certain conventional measurement (i.e., incident to or neighbouring a bus that is incident to a conventional measurement), the following constraint is imposed:(15)yi1+yi2+⋯+yin≥n−1.According to KCL,n buses that have an injection measurement or that neighbour a bus with an injection measurement will be observable if any n − 1 of them are observed by PMUs. This condition is not guaranteed by the constraint given by (15) since some of the yis might be greater than 1. For example, suppose that yi1 = 2, yi2 = 0, yi3 = 0, and the other yis are equal to 1, the constraint given by (15) is still satisfied, but only n − 2 of the n buses are observed by PMUs. In these cases, the PMU placement given by the KCL method might fail to restore full system observability.Simulations have been performed in the IEEE 14-bus test system to investigate the validity of both the KCL-based methods and the proposed method in this article. All 2^14 possible injection measurement placements without other types of conventional measurements are tested in the simulations. The simulation results are summarized in Tables2 and 3.Table 2
Valid solutions found.
KCL-based method (%)Proposed method (%)Valid solutions found96.71100Table 3
Examples of failed PMU placement by the KCL-based method.
Injection measurements placed at busesPMU placement at busesCase 13, 115, 7, 13Case 23, 56, 7, 9Case 32, 115, 7, 13


## 5.2. KCL-Based Methods and the Observability Analysis-Based Methods: Suboptimal Solutions

KCL-based methods might not give the placement with the minimum number of PMUs needed to provide the system with full observability. An example is as follows:Consider a simple 6-bus network with the measurement placement illustrated in Figure4.Figure 4
Six-bus network with measurement placement.According to Theorem1, placing a PMU at either Bus 1, Bus 2, Bus 5, or Bus 6 can ensure that the network will be fully observable. The validity of the placement result can easily be validated by checking the rank of the measurement Jacobian matrix with both the conventional measurements and the additional PMU measurement. The constraints of the proposed method are given in the following:(16)y1≥1,y6≥1.In comparison, the KCL-based method requires the following constraints to be satisfied:(17)y1+y2+y3≥2,y2+y3+y4≥2,y3+y4+y5≥2,y4+y5+y6≥2.It can be seen that it is impossible to satisfy all of the four constraints given in (17) with only a single PMU since one PMU at the network given in 0 can only reach at most 3 buses. Thus, two PMUs are suggested by the KCL-based method to restore system observability.As for the observability analysis-based method, its obvious flaw in finding the optimal PMU placement is that the contribution of the irrelevant injection measurements to the system observability is neglected. In the example in Section2, the constraints of the observability analysis-based method and the proposed method in the transformed network are given in the following, respectively:(18)x1+x2+x3+x5+x6≥1,x2+x3+x4+x6+x8≥1,x1+x5+x6≥1,x1+x4+x5+x6+x7+x8≥1,x3+x7+x8≥1,(19)x2+x3+x4+x6+x8≥1,x1+x5+x6≥1,where xi is the element of the placement decision vector, x, xi = 1 means that a PMU is placed at the corresponding placement, and xi = 0 means that it is not. It can be seen that at least two PMUs are required to satisfy the constraints given by (18), while only one PMU at Placement 6, which corresponds to Bus 9 in the original network, is required to meet constraints given by (19).


## 6. Simulated Examples

The results of four simulations are provided in this section. Performance comparisons of the proposed method and two existing types of methods in the IEEE 14-, 118-, and 2736-bus test systems [42] are given in Sections 6.1, 6.2, and 6.4, respectively. Note that in these three simulations, the costs for placing the PMUs at different buses are assumed to be the same. An additional simulation in IEEE 14-bus test system considering PMU current channel limits is presented in Section 6.4. The function “intlinprog” in MATLAB is used to solve the BILP problem since MATLAB has used it to replace the function “bintprog” since the version MATLAB 2014. The simulations are conducted using Intel(R) Core(TM) i9-10980HK CPU with 64 GB of RAM. The average execution time of the OPP methods given in the following subsections is obtained after 100 Monte Carlo simulations.


### 6.1. IEEE 14-Bus Test System

In this simulation, the IEEE 14-bus test system with a number of existing conventional measurements is considered. Details of the conventional measurement placement are given in Table4 and the IEEE 14-bus test system is illustrated in Figure 1.Table 4
Placement of conventional measurements in IEEE 14-bus test system.
Injection measurement at bus#1, 6, 7, 8, 9, 11, 13Flow measurements at branch#7, 13, 18As shown in Table5, all of the OPP methods give different PMU placements: the proposed method suggests to place only two PMUs, while the KCL-based method (KCL method) and the observability analysis-based method (OA method) give PMU placements that include 3 PMUs. Table 6 shows that the rank of the gain matrix (rank(G)) after PMU is placed into the network are all equal to 14, the full rank. This indicates that PMU placements given by all three methods can make the network fully observable, which demonstrates the optimality of the proposed method.Table 5
PMU placements in IEEE 14-bus test system.
Proposed method4, 13KCL method3, 6, 9OA method2, 6, 9Table 6
Performance of the OPP methods in IEEE 118-bus test system in seconds.
Proposed methodKCL methodOA methodNumber of PMUs233Rank (G)141414Execution time (ms)13.7811.7613.83It is also shown in Table6 that the proposed method and OA method have a similar execution time of about 13.8 ms, which is 17% longer than that of the KCL method. This is because the proposed method and the OA method need extra time to perform observability analysis. The slight speed advantage of the proposed method over the OA method comes from the reduced size of the transformed network, which reduces the dimension of the OPP problem.


### 6.2. IEEE 118-Bus Test System

This subsection presents the PMU placements in the IEEE 118-bus test system with the 36 injection measurements and 49 power flow measurements, as detailed in Table7.Table 7
Placement of conventional measurements in IEEE 118-bus test system.
Injection measurement at bus#2, 8, 16, 18, 19, 20, 21, 26, 31, 37, 40, 41, 46, 48, 54, 55, 56, 58, 60, 63, 64, 77, 78, 82, 85, 87, 91, 93, 98, 99, 101, 105, 107, 109, 113, 115Flow measurement at branch #1, 2, 3, 4, 7, 8, 14, 15, 16, 18, 20, 25, 28, 33, 34, 38, 40, 42, 43, 58, 72, 78, 84, 90, 91, 96, 101, 104, 105, 109, 112, 122, 129, 130, 133, 134, 137, 139, 142, 146, 149, 157, 162, 169, 170, 171, 172, 174, 178Table8 shows very different PMU placement results by the three compared OPP methods: the proposed method gives a PMU placement with 17 PMUs and the KCL method suggests allocating 16 PMUs, while the OA method requires 21 PMUs to be placed into the network. Although the number of PMUs suggested to be placed by the KCL method is the least, this PMU placement cannot make the network fully observably since rank(GKCL) = 109 < 118, as shown in Table 9. In comparison, the ranks of the gain matrices of both the proposed method and the OA method are of full rank. Therefore, the proposed method has better performance over the KCL method in terms of placement validity and better performance over the OA method in terms of placement optimality.Table 8
PMU placements in IEEE 118-bus test system.
Proposed method5, 10, 12, 17, 20, 32, 37, 49, 59, 66, 70, 77, 85, 94, 100, 108KCL method6, 9, 12, 32, 34, 37, 45, 53, 57, 67, 68, 73, 74, 76, 96, 103, 110OA method6, 9, 12, 17, 25, 34, 37, 45, 49, 52, 56, 63, 68, 73, 74, 86, 96, 100, 110, 115, 118Table 9
Performance of the OPP methods in IEEE 118-bus test system in seconds.
Proposed methodKCL methodOA methodNumber of PMUs171621Rank(G)118109118Execution time (s)0.22720.05810.2278In terms of execution time, the ranks among the three methods remain the same. However, the ratio in execution time between the the other two methods and the KCL method grows from about 1.17 to 3.91. This is because the time taken for observability analysis increases much more than that taken for solving the BILP problem. Again, a tiny execution time advantage can be seen from the proposed method compared to the OA method due to the reduced size of the transformed network.


### 6.3. IEEE 273-Bus Test System

To demonstrate the validity and efficacy of the proposed method, the three OPP methods are compared with simulations in a large network of IEEE 2736-bus test system.Due to the large data size, the placement of the conventional measurements and the placement results of the three OPP methods are not given for brevity. Their performances are summarized in Table10. In this case, the proposed method suggests 661 PMUs to be placed and the OA method suggests a PMU placement of 697 PMUs, while the placement by the KCL method also gives a very small number of PMU of 337. However, the PMU placement by the KCL method can hardly make the network fully observable: rank(GKCL) = 1872, which is far less than the full rank of 2736. Therefore, it can be seen that as the size of the network grows, the advantages of the proposed method over the existing methods in terms of both validity and optimality also increase significantly. This is because the KCL method is based on the assumption that only a single power injection is considered or that no injection measurements are located at neighbouring buses. When the number of the network and the number of the measurements increase, this assumption becomes invalid for a higher proportion of the injection measurements, leading to invalid PMU placements. As for the OA method, since it does utilize the irrelevant injection measurement to improve the network observability, its PMU placement result becomes less optimal when the number of irrelevant injection measurements increases.Table 10
Performance of the OPP methods in IEEE 2736-bus test system in seconds.
Proposed methodKCL methodOA methodNumber of PMUs661337697Rank(G)273618722736Execution time (s)182.6916.41189.75As shown in Table10, the execution time rank among the three OPP methods is still the same: the KCL method is the fastest, and the proposed method is faster than the OA method. The reason for this is still the same. That is, the KCL method does not need to identify observable islands, and the proposed method solves BILP problems with smaller dimension than those solved by the OA method. Although the proposed method is slower than the KCL method, it is still acceptable considering the size of the network and the fact that OPP is usually done offline for planning the placement of PMUs.


### 6.4. PMU Placement considering PMU Channel Limits

To demonstrate the capability of the proposed method to optimally place PMUs under current channel limits, some more simulations are conducted with PMUs that have different current channel limits. As listed in Table11, three types of PMUs are considered, and it is assumed that the PMUs having more current channels are more expensive. In the first three cases, only one of the three types of PMUs is available for PMU placement, while in the fourth case, all of the three types of PMUs are available. As shown in Table 12, the proposed OPP method is able to optimally place the PMUs with different types of PMUs and a mixture of them. The results show that if the type of PMU has more current channels, fewer PMUs are needed, but the overall cost is higher. The cost becomes even lower if a mixture of different types of PMUs is used.Table 11
PMU types with different costs and channel number limits in IEEE 14-bus test system.
PMU typesCostChannel number limitType 111Type 21.52Type 33No limitTable 12
PMU placement results in IEEE 14-bus test system with different types of PMU available.
PMU types availablePMU placementCostType 1Bus 2 connected to Bus 44Bus 11 connected to Bus 10Bus 13 connected to Bus 12Bus 13 connected Bus 14Type 2Bus 3 connected to Bus 2 and Bus 44.5Bus 11 connected to Bus 6 and Bus 10Bus 13 connected to Bus 12 and Bus 14Type 3Bus 4 and Bus 136Type 1, type 2, and type 3Type 1 at Bus 2 connected to Bus 53.5Type 1 at Bus 4 connected to Bus 3Type 2 at Bus 13 connected to Bus 12 and Bus 14


## 6.1. IEEE 14-Bus Test System

In this simulation, the IEEE 14-bus test system with a number of existing conventional measurements is considered. Details of the conventional measurement placement are given in Table4 and the IEEE 14-bus test system is illustrated in Figure 1.Table 4
Placement of conventional measurements in IEEE 14-bus test system.
Injection measurement at bus#1, 6, 7, 8, 9, 11, 13Flow measurements at branch#7, 13, 18As shown in Table5, all of the OPP methods give different PMU placements: the proposed method suggests to place only two PMUs, while the KCL-based method (KCL method) and the observability analysis-based method (OA method) give PMU placements that include 3 PMUs. Table 6 shows that the rank of the gain matrix (rank(G)) after PMU is placed into the network are all equal to 14, the full rank. This indicates that PMU placements given by all three methods can make the network fully observable, which demonstrates the optimality of the proposed method.Table 5
PMU placements in IEEE 14-bus test system.
Proposed method4, 13KCL method3, 6, 9OA method2, 6, 9Table 6
Performance of the OPP methods in IEEE 118-bus test system in seconds.
Proposed methodKCL methodOA methodNumber of PMUs233Rank (G)141414Execution time (ms)13.7811.7613.83It is also shown in Table6 that the proposed method and OA method have a similar execution time of about 13.8 ms, which is 17% longer than that of the KCL method. This is because the proposed method and the OA method need extra time to perform observability analysis. The slight speed advantage of the proposed method over the OA method comes from the reduced size of the transformed network, which reduces the dimension of the OPP problem.


## 6.2. IEEE 118-Bus Test System

This subsection presents the PMU placements in the IEEE 118-bus test system with the 36 injection measurements and 49 power flow measurements, as detailed in Table7.Table 7
Placement of conventional measurements in IEEE 118-bus test system.
Injection measurement at bus#2, 8, 16, 18, 19, 20, 21, 26, 31, 37, 40, 41, 46, 48, 54, 55, 56, 58, 60, 63, 64, 77, 78, 82, 85, 87, 91, 93, 98, 99, 101, 105, 107, 109, 113, 115Flow measurement at branch #1, 2, 3, 4, 7, 8, 14, 15, 16, 18, 20, 25, 28, 33, 34, 38, 40, 42, 43, 58, 72, 78, 84, 90, 91, 96, 101, 104, 105, 109, 112, 122, 129, 130, 133, 134, 137, 139, 142, 146, 149, 157, 162, 169, 170, 171, 172, 174, 178Table8 shows very different PMU placement results by the three compared OPP methods: the proposed method gives a PMU placement with 17 PMUs and the KCL method suggests allocating 16 PMUs, while the OA method requires 21 PMUs to be placed into the network. Although the number of PMUs suggested to be placed by the KCL method is the least, this PMU placement cannot make the network fully observably since rank(GKCL) = 109 < 118, as shown in Table 9. In comparison, the ranks of the gain matrices of both the proposed method and the OA method are of full rank. Therefore, the proposed method has better performance over the KCL method in terms of placement validity and better performance over the OA method in terms of placement optimality.Table 8
PMU placements in IEEE 118-bus test system.
Proposed method5, 10, 12, 17, 20, 32, 37, 49, 59, 66, 70, 77, 85, 94, 100, 108KCL method6, 9, 12, 32, 34, 37, 45, 53, 57, 67, 68, 73, 74, 76, 96, 103, 110OA method6, 9, 12, 17, 25, 34, 37, 45, 49, 52, 56, 63, 68, 73, 74, 86, 96, 100, 110, 115, 118Table 9
Performance of the OPP methods in IEEE 118-bus test system in seconds.
Proposed methodKCL methodOA methodNumber of PMUs171621Rank(G)118109118Execution time (s)0.22720.05810.2278In terms of execution time, the ranks among the three methods remain the same. However, the ratio in execution time between the the other two methods and the KCL method grows from about 1.17 to 3.91. This is because the time taken for observability analysis increases much more than that taken for solving the BILP problem. Again, a tiny execution time advantage can be seen from the proposed method compared to the OA method due to the reduced size of the transformed network.


## 6.3. IEEE 273-Bus Test System

To demonstrate the validity and efficacy of the proposed method, the three OPP methods are compared with simulations in a large network of IEEE 2736-bus test system.Due to the large data size, the placement of the conventional measurements and the placement results of the three OPP methods are not given for brevity. Their performances are summarized in Table10. In this case, the proposed method suggests 661 PMUs to be placed and the OA method suggests a PMU placement of 697 PMUs, while the placement by the KCL method also gives a very small number of PMU of 337. However, the PMU placement by the KCL method can hardly make the network fully observable: rank(GKCL) = 1872, which is far less than the full rank of 2736. Therefore, it can be seen that as the size of the network grows, the advantages of the proposed method over the existing methods in terms of both validity and optimality also increase significantly. This is because the KCL method is based on the assumption that only a single power injection is considered or that no injection measurements are located at neighbouring buses. When the number of the network and the number of the measurements increase, this assumption becomes invalid for a higher proportion of the injection measurements, leading to invalid PMU placements. As for the OA method, since it does utilize the irrelevant injection measurement to improve the network observability, its PMU placement result becomes less optimal when the number of irrelevant injection measurements increases.Table 10
Performance of the OPP methods in IEEE 2736-bus test system in seconds.
Proposed methodKCL methodOA methodNumber of PMUs661337697Rank(G)273618722736Execution time (s)182.6916.41189.75As shown in Table10, the execution time rank among the three OPP methods is still the same: the KCL method is the fastest, and the proposed method is faster than the OA method. The reason for this is still the same. That is, the KCL method does not need to identify observable islands, and the proposed method solves BILP problems with smaller dimension than those solved by the OA method. Although the proposed method is slower than the KCL method, it is still acceptable considering the size of the network and the fact that OPP is usually done offline for planning the placement of PMUs.


## 6.4. PMU Placement considering PMU Channel Limits

To demonstrate the capability of the proposed method to optimally place PMUs under current channel limits, some more simulations are conducted with PMUs that have different current channel limits. As listed in Table11, three types of PMUs are considered, and it is assumed that the PMUs having more current channels are more expensive. In the first three cases, only one of the three types of PMUs is available for PMU placement, while in the fourth case, all of the three types of PMUs are available. As shown in Table 12, the proposed OPP method is able to optimally place the PMUs with different types of PMUs and a mixture of them. The results show that if the type of PMU has more current channels, fewer PMUs are needed, but the overall cost is higher. The cost becomes even lower if a mixture of different types of PMUs is used.Table 11
PMU types with different costs and channel number limits in IEEE 14-bus test system.
PMU typesCostChannel number limitType 111Type 21.52Type 33No limitTable 12
PMU placement results in IEEE 14-bus test system with different types of PMU available.
PMU types availablePMU placementCostType 1Bus 2 connected to Bus 44Bus 11 connected to Bus 10Bus 13 connected to Bus 12Bus 13 connected Bus 14Type 2Bus 3 connected to Bus 2 and Bus 44.5Bus 11 connected to Bus 6 and Bus 10Bus 13 connected to Bus 12 and Bus 14Type 3Bus 4 and Bus 136Type 1, type 2, and type 3Type 1 at Bus 2 connected to Bus 53.5Type 1 at Bus 4 connected to Bus 3Type 2 at Bus 13 connected to Bus 12 and Bus 14


## 7. Conclusion

This article presents a new method for optimal PMU placement (OPP) in the presence of conventional measurements using binary integer linear programming (BILP). It has been shown that the two types of existing methods, KCL-based methods and the observability analysis-based methods, might fail to provide the least number of PMUs to ensure system observability (optimality problem). In addition, the KCL-based method is likely to have PMU placements that cannot achieve network observability (validity problem). The optimality problem of the observability analysis-based method is solved by the proposed network transformation scheme considering the effect of irrelevant injection measurements. The optimality and validity problem of the KCL-based method is solved by the proposed global observability criterion that considers the eligible PMU placements, which might be omitted by the KCL-based methods that use the local observability criterion. The validity of the proposed global observability criterion is mathematically proven and tested with computer simulations by considering all possible injection placements in the IEEE 14-bus system. However, a drawback of this criterion is that it might as well neglect certain eligible PMU placements. Since the proposed method applies the most widely used BILP to solve the OPP problem, it can be easily combined with other OPP methods that consider other constraints and scenarios, such as the limit of PMU current channel limits and the contingency of loss of a branch or a PMU. Simulations in IEEE 14-, 118-, and 2736-bus test systems clearly demonstrate the feasible execution time of the proposed OPP method and its better performance in terms of both optimality and validity compared to the existing two types of OPP methods in the presence of conventional measurements. More work will be done in the future to further improve or eliminate the suboptimal problem of the proposed observability criterion.



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*Source: 8078010-2022-02-22.xml*