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Optimal Integration of Phasor Measurement Units in Power Systems Considering Conventional Measurements S. Azizi, Student Member, IEEE, G. B. Gharehpetian, Senior Member, IEEE, and A. Salehi Dobakhshari, Student Member, IEEE

Abstract—This paper presents an integer linear programming (ILP) framework for the optimal placement of phasor measurement units (PMUs), in the presence of conventional measurements. Furthermore, by the proposed method, the power system remains completely observable during all possible single contingencies for lines and measurement devices. In doing so, the potential of circuit equations associated with both PMUs and conventional measurements as well as the network topology are fully utilized by a system of equations to reach the minimum possible numbers of required PMUs. The limitation of communication channels is also taken into account in the proposed ILP-based framework. The method is implemented on several IEEE test systems which have already been equipped with conventional measurements. The comparison between obtained results of the proposed method and those of other methods reveals its superiority in the modeling of robust PMU placement problem (OPP) in the presence of conventional measurements. As such, a smooth transition from the SCADA-based monitoring system to the PMU-dominated WAMS is ensured. Moreover, this method is successfully applied on three large-scale test systems, which demonstrates it can effectively be employed for robust OPP in realistic power systems.

Superscript denoting value of the associated variable or parameter, when flow measurement is out. Indices of bus. Superscript denoting value of the associated variable or parameter, when injection measurement is out. Set of liens. Index of line. Superscript denoting value of the associated variable or parameter, when line is out. Set of measured injection buses. Set produced by removing the th element of set . Superscript denoting value of the associated variable or parameter, when impact of the PMU installed at bus (if any) is neglected. Binary decision variable which is equal to 1 if a PMU is installed at bus , and 0 otherwise. Binary parameter whose value is 0 if is equal to , and 1 otherwise. Binary auxiliary variable indicating the unknown is assigned to the equation state variable obtained by the flow measurement installed at line . Set of injection measurement indices.

Index Terms—Contingency, integer linear programming (ILP), observability, optimal PMU placement (OPP), phasor measurement unit (PMU).

NOMENCLATURE Binary connectivity parameter between buses and . Set of buses. Binary parameter which is equal to 1, if the current for or the voltage phasor phasor of line of bus for , is measured by the PMU installed at bus . Observability function of bus .

Index of injection measurement. Set of flow measurement indices. Index of flow measurement.

Set of measured flow branches. Set produced by removing the th element of set .

Manuscript received February 01, 2012; revised June 01, 2012; accepted July 10, 2012. Paper no. TSG-00049-2012. S. Azizi is with the School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran 14395-515 Iran (e-mail: sadegh. [email protected]). G. B. Gharehpetian is with the Electrical Engineering Department, Amirkabir University of Technology, Tehran 15875-4413 Iran (e-mail: [email protected]). A. S. Dobakhshari is with the Department of Electrical Engineering, Sharif University of Technology, Tehran 11365-11155 Iran (e-mail: [email protected]. edu). Digital Object Identifier 10.1109/TSG.2012.2213279

Installation cost of PMU at bus . Binary auxiliary variable indicating the unknown is assigned to the equation state variable resulting from the injection measurement installed at bus . I. INTRODUCTION EASURING phase angles in different points of a large power system was impractical until recently, due to the lack of a common time reference for synchronizing measurements in distant locations [1]. Advances in the telecommunication technology provided the possibility of synchronized measurements by means of the global positioning system (GPS) signal [2]. Phasor measurement units (PMUs)

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are devices which measure voltage and current synchrophasors of the power system with the accuracy in the order of 1 microsecond. Existing and prospects of PMU applications in various aspects of the power system operation have been addressed in [3] and [4]. The need for a reliable electrical energy and high cost of blackouts in the competitive power markets, necessitate a wide area monitoring, protection, and control system to ensure the electrical power system security while utilizing its assets [5]. PMUs are the most suitable devices for the wide-area monitoring system (WAMS) accomplishment. However, operators may intend to integrate PMUs with the already existing conventional measurements to enhance the power system monitoring performance [5], [6]. Additionally, despite the fact that SCADA can completely be replaced with WAMS in the near future, this replacement is practically out of reach with the present deficiency in the communication infrastructure as well as modern devices for storage and management of the received data [7]. In order to have a completely observable power system, it is not necessary to equip all buses with PMU, because a PMU can make more than one bus observable. Accordingly, optimal PMU placement (OPP) is defined as the problem of finding the least number of required PMUs and their installation locations, in the presence of fixed conventional measurements. In this relation, the network observability is assessed by topological, numerical or hybrid approaches [8]. For a numerically observable network, the gain matrix or the measurement Jacobian matrix is of full rank. Furthermore, if at least one spanning measurement tree of full rank can be constructed from the network, it is said to be topologically observable [9]. As the pioneering work, an optimization model for the OPP problem was developed in [9], using a combination of three intelligent search methods. Afterward, different meta-heuristic approaches have been deployed in [9]–[16] to solve the OPP problem. Despite some advantages, the primordial drawback of methods using evolutionary techniques is that they do not guarantee to find an optimal solution. In recent years, analytical approaches for solving OPP have attracted much attention. Authors in [5] have modeled this problem as an integer quadratic programming, where the presence of other measurements has been considered. Reference [17] has deployed an integer linear programming (ILP)-based method for OPP where taking zero injections into account has made the formulation nonlinear. Nevertheless, this work has been a step forward for the later ILP-based modeling of the OPP problem. Even with the inclusion of injection measurements, the problem still remains linear using the methods presented in [18] and [19]. An algorithm for optimal multistage scheduling of PMU placement has been devised in [20], where the influence of zero injections and the possibility of single PMU outage have been also considered. The methods presented in [17]–[20] solve different equations resulting from conventional measurements and zero injections independently, rather than as a system of equations, and this is their main shortcoming. Authors in [13] have pointed out that zero injection equations may produce a system of linear equations. However, they have only concerned a small number of solvability states of this system of equations. By the bus merging methodology proposed in [21] and [22], zero injection equations are considered as a system of equations. Nonetheless, since all feasible

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merging states are not checked in practice, the obtained result is not ensured to be globally optimal. Reference [23] has proposed an ILP-based approach for robust OPP and has studied different aspects of the problem. Most recently, [24] has formulated the zero injection equations as a system of equations and solved OPP by an equivalent linear formulation for the exhaustive search. It has succeeded in obtaining the optimal solution for the OPP problem by solving a set of linear equations describing the whole system observability. However, the already existing conventional measurements are not dealt with. In this paper, the work of [24] is extended by considering the effect of the existing conventional measurements and their single contingencies while integrating PMUs in the power system for complete observability. By this approach, a smooth transition from the current SCADA-based system monitoring to the future PMU-dominated WAMS is facilitated. The main contribution of this method is that it models the power system equations as a system of equations, and is not confined to merely investigating their individual solvability. Hence, as opposed to the most of other methods, it is capable of reaching the globally optimal solution. This achievement can be verified by the comparison of the obtained results with the existing ones. The rest of the paper is organized as follows. In the next section, power system equations, in relation with the observability issue, are explained. Next, an integer linear framework is proposed to rigorously handle power system equations and OPP in the presence of conventional measurements. Then, in Section III, the proposed linear framework is developed in order to incorporate different types of single contingencies and the PMU channels limitation. Simulation results are presented in Section IV and are compared with those of existing methods to verify the efficiency of the proposed method, even in the case of large-scale power systems. Conclusions are given in the last section. II. OBSERVABILITY AND OPTIMAL PMU PLACEMENT A. Observability Using Power System Equations For an observable power system, all system states, i.e., voltage phasors of all buses, can be specified using gathered data from measurement devices and applying circuit rules. Assume the PMU installed at bus directly measures the voltage phasor of that bus . If this PMU measures current phasor of line as well, an equation is obtained as follows: (1) , and are elements of the transmiswhere sion line matrix modeled by two-port network, when terminal is considered as the sending-end terminal [25]. It is clear that solving of (1) makes bus observable. In the case where a flow at bus side measurement device is installed along branch and measures active and reactive power flow, (1) changes to

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where is the complex conjugate of and is the measured flow of the line . buses and bus is a Assume that the power system has measured injection one. Based on KCL, an equation between the voltage phasors of bus and all its incident buses can be constructed as follows: (3) is the measured injected apparent power and is the where is the th entry of the set of all system buses. In addition, network admittance matrix. If the amount of the injected current to bus is directly measured or this bus is a zero injection one, or 0, respecthe right hand side of (3) should change to tively. If voltage phasors of all buses are directly measured or can be calculated using equations with the general forms of (1)–(3), the power system is said to be numerically observable. It can be easily proved that for an observable power system, if measured and calculated states using PMU data are replaced with their values, other equations of the power system are solvable for the remained states. It must be noted that these equations may be solvable individually or as a system of equations. Without loss of generality, suppose that are the states whose calculation is assigned to independent equations resulting from conventional measurements and zero injections. Substitution of measured values in equations with general forms of (2) or (3) produces a system of equations as follows: .. .

(4)

and indicate real and imaginary parts Let superscripts of the associated variable. Hence, each of complex measurements or state variables can be represented as a 2 1 vector and in rectangular coordinates. Consequently, are the vectors of the th state and measurement, respectively. Accordingly, vectors of measurements and states can be defined as and . Hence, by separating the real and imaginary parts of equations of (4) and linearizing them about operating points, a system of linear equations in unknowns is produced as follows: (5) As state variables of the power system other than the state variables of (5), are supposed to be known by means of installed PMUs, the power system is fully observable if matrix is an invertible matrix. In order to reach the optimal solution of the PMU placement problem, the potential of the system of linear equations (5), should be exploited as much as possible. This means that for a network with independent real equations, all states of solvability of the related system of equations in unknowns have to be investigated. It should be pointed out that this is a necessary and sufficient investigation, although some of unknowns may be redundantly observable through the installed PMUs [24].

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B. Solvability of System of Equations In this paper, it is assumed that the equations resulting from conventional measurements are independent. It means that there is no closed loop of network branches all of which have been equipped with flow measurement device. In addition, no measured injection bus exists so that flows of all branches connected to it are measured by conventional measurements. These assumptions only guarantee that original equations of the system are independent and not the simplified versions of them as in (4). In other words, there is no undertaking for independency of equations of (4), although the same holds for their dependency. It is well-known that any selected equations from a set of linearly independent equations are certainly independent, where can take any integer value between 1 and . As a result, for an matrix of full rank, all of matrices produced by deleting some of its rows should have the rank of . Thus, such produced matrices must have at least non-zero columns as a necessary condition, because the rank of a matrix is lower than or equal to the minimum between the number of its rows and columns. Consider system of linear equations in unknowns (5). Assume that it is possible to set up a one-to-one correspondence between the variables and the equations so that each variable is assigned to an equation in which it appears. It can be easily confirmed that in such a condition, the above-mentioned necessary condition for invertibility holds for matrix . However, to ensure the numerically observability of the power system, the rank of the Jacobian measurement matrix should be checked for each obtained solution of OPP. C. Linear Framework to OPP OPP is defined as finding the minimum number of required PMUs so that their proper placement makes the network completely observable. The more the number of buses which are observable through equations resulting from conventional measurements and zero injections, the less the number of buses which need to be observable using PMUs. Therefore, for a rigorous OPP, it is necessary and sufficient that the whole potential of the network equations is deployed. In spite of the nonlinearity of network equations, OPP is linearly formulated in this section. Assume that is a binary variable, which is equal to 1 if a PMU is installed at bus , and 0 otherwise. Let be th entry of the connectivity matrix defined as: (6) As a result, the observability of bus as a function of locations at which PMUs are installed, is as follows: (7) represents the observability of bus . Accordingly, To specify the states where (5) is a solvable system of equations in complex unknowns, two auxiliary binary variables and are defined. In this way, implies that the unknown variable is assigned to the equation associated with injection measurement , which is an equation as (3). As and the specified equation are both complex, two variables are assigned to two equations, in fact. Similarly, for the conventional



flow measurements, implies that the unknown variable is assigned to the equation associated with the flow measure, which is an equation as (2). This assignment at branch ment can also be interpreted as two assignments in rectangular coordinates. As mentioned before, if all non-assigned variables become observable through PMUs, assigned ones get observable by solving the power system equations. To mathematically represent these assignments, and are defined as sets of buses and lines, respectively. Furthermore, and are sets of measured injection buses and measured flow branches. Therefore, if a flow measurement device is installed at line , then . Accordingly, additional terms should be added to observability function (7) in order to involve observability through conventional measurements. Thus, the observability function should be modified as follows:

presented in [24], all solvable systems of equations in unknowns are enumerated at least one time by such an assignment. Hence, the proposed method investigates all solvability states of the network equations and it is capable of finding the optimal solution among all possible placements of PMUs. In addition, since the formulation is linear, the obtained solution can be ensured to be globally optimal by appropriate adjustment of the solver. This linear formulation is the most comprehensive for OPP problem, achieving full potential of conventional measurements, and thereby is appropriate for power systems whose operators wish a smooth transition from the SCADA-based monitoring to the PMU-dominated WAMS.

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A. Contingency-Constrained OPP Assume that the power system should remain observable during all possible single contingencies. It is obvious that associated constraints, ensuring the observability of the power system, should be added to the problem for each type of contingencies. Nevertheless, objective function still remains the same as (12). Thus, for OPP considering each combination of possible single contingencies, it is sufficient to add the related constraints to the optimization model. Here, observability preservation against single contingency for PMUs and flow measurements are formulated. Associated constraints for the observability robustness against loss of single line and injection measurement are provided in Appendixes A and B, respectively. 1) Loss of Single PMU: In this point, robustness against loss of single PMU is modeled by adding some new constraints to problem constraints. Before OPP, the solution, i.e., candidate buses for PMUs installation, is not known. Hence, it is necessary that in a conservative manner (for each bus), a contingency be defined so that if a PMU is installed at that bus, its influence on the observability is ignored. In this regard, the parameter is defined as follows:

where second and third terms represent the observability through measured injection buses and flow measurement devices, respectively. To ensure the necessary condition for the solvability of (5), auxiliary binary variables and should satisfy following constraints: (9) (10) Constraints (9) and (10) together guarantee that exactly one existing variable in each equation is assigned to it. To ensure that assigned variables are distinct, the following constraint should be appended to the problem constraints: (11) So, the number of complex variables which should be calculated from (5) is equal to , the same as the number of complex equations. Accordingly, the OPP problem in the presence of conventional measurements can be linearly formulated as follows: (12) (13) where is the installation cost of PMU at bus . In relation with observability function given in (8), constraints (9)–(11) should be satisfied as well. Constraint (13) ensures that for all , gets observable by means of installed PMUs; however if PMUs do not make observable directly, (8)–(11) guarantee that gets observable via (4). Therefore, above formulations routinely find the minimum number and locations of required PMUs so that the whole advantage of equations obtained from conventional measurements, zero injection, and PMUs are exploited. It is essential to point out that the proposed formulation sets up a one-to-one correspondence between the equations and variables which are not observed by PMUs. As explained earlier, it means that the associated system of equations can be solved for these unknown variables. On the other hand, based on the proof

III. LINEAR FRAMEWORK TO CONTINGENCY-CONSTRAINED OPP AND COMMUNICATION CHANNELS LIMITATION

if if

.

(14)

Additionally, superscript represents contingency for the PMU installed at bus (if any). Accordingly, related constraints to OPP considering loss of single PMU are as below: (15) where

(16) (17) (18) (19)


and are binary auxiliary variables whose values set up a one-to-one correspondence between unknown state variables and equations, when PMU is out. 2) Single Flow Measurement Failure: Now, the OPP problem should be formulated so that the loss of any single flow measurement device does not cause the loss of observability. Assume that is the set of flow measurement indices. Moreover, is produced by removing the th element of set . Then, if superscript denotes contingency for flow measurement , constraints of OPP considering the loss of single flow measurement are as follows: (20) where

(21) (22) (23) (24) and are binary auxiliary variables as explained before, but when the flow measurement is out. B. OPP Considering Limitation of Measurement Channels For a PMU, measuring currents of all incident branches may not be possible due to the limitation of communication channels [22]. If there is such a limitation on the number of measurements each PMU is allowed to fulfill, a series of constraints should be added to the problem constraints. Suppose is a binary variable, which is equal to 1 if the PMU installed at bus measures the voltage phasor of bus for , or the current phasor of branch for . Therefore, if the maximum number of measurements which the PMU of bus is capable of handling is , then we have: (25) Consequently, the observability constraint related to each bus is as follows:

(26) It is obvious that bus should be equipped with a PMU, if at least one is non-zero. Moreover, if all ’s have the value ’s of zero, it means that no PMU is installed at bus . As all and ’s are binary variables, these relations can be formulated as follows: (27) (28)

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The same approach can be employed to integrate the limitation of communication channels into the contingency-constrained OPP problem. The complete formulation of the proposed method is provided in Appendix C. IV. SIMULATION RESULTS Simulation results are organized in three parts in order to demonstrate the efficiency of the proposed method. First, for several test systems without conventional measurements, and the IEEE 118-bus test system with different configurations of conventional measurements, contingency-constrained OPP is executed and obtained results are compared with those of other methods. Next, the impact of communication channels limitation is studied. Finally, OPP for three large-scale power systems is carried out which demonstrates capability of the presented method in handling OPP for practical power systems. All simulations have been carried out by a 2.66 GHz quadcore processor with 2 GB RAM. The OPP problem has been modeled in GAMS optimization environment [27] and solved using the CPLEX solver [28]. To reach the globally optimal solution in all simulations, the optimality gap in the CPLEX solver has been set to zero. In order to confirm the numerical observability of each case, for all obtained solutions, ranks of measurement Jacobian matrices were calculated using the proposed algorithm in [29]. The results verified that in all cases the system is numerically observable, in addition to its topological observability. Moreover, the costs of PMU installation at different buses were assumed to be the same (1 p.u.) in all simulations. However, if PMU installation and infrastructure costs are different in various buses, the objective function (12) covers differences by applying dissimilar weight factor to each bus regarding the associated costs. Different methods have endeavored to reach the global solution for OPP, so far. By merging every zero injection bus with one of its incident buses, the potential of equations associated with measurement devices is utilized in the individual bus merging method (IBM) [21]. However, the connectivity matrix should be constructed repeatedly to ensure that all possible merging combinations are considered. To enhance the applicability of this approach, reference [6] suggests augmented bus merging (ABM) requiring that the connectivity matrix is built only once. By means of appending nonlinear constraints (ANC), an optimization model is constructed in [17] to solve OPP after eliminating some negligible nonlinear terms from the observability function. Such an approximation does not exist in the appending linear constraints (ALC) method proposed in [20] because the added constraints in relation to circuit rules are all linear. Auxiliary binary variables (ABV) defined in [23] help to linearly formulate the observability function in the presence of zero injection equations. A three-step recursive security (RS) algorithm using a sort of modified depth first search is developed for solving the OPP problem in [16]. A. Contingency-Constrained OPP Considering Conventional Measurements To show the capability of the proposed method in finding the optimal solutions, the results of the proposed OPP method in several IEEE test systems are provided and listed in Table I. For the cases where obtained results are different from that obtained using the method presented in [23] (as the best method reported



TABLE I MINIMUM NUMBER OF PMUS TO MAKE DIFFERENT TEST SYSTEMS OBSERVABLE CONSIDERING SINGLE CONTINGENCIES

TABLE II DIFFERENT CONFIGURATIONS OF

AT

IEEE 118-BUS TEST SYSTEM

COMPARISON

OF

TABLE IV OPP RESULTS FOR IEEE 118-BUS TEST SYSTEM CONSIDERING FLOW MEASUREMENTS

NOT

TABLE V COMPARISON OF OPP RESULTS BY DIFFERENT METHODS CONSIDERING DIFFERENT CONFIGURATIONS OF FLOW MEASUREMENTS

FM: Flow measurement TABLE III MINIMUM NUMBER OF PMUS TO MAKE THE IEEE 118-BUS TEST SYSTEM WITH DIFFERENT CONFIGURATIONS OF FLOW MEASUREMENTS OBSERVABLE

IM: Injection measurement FM: Flow measurement

so far), both results are given. As expected, considering single contingencies increases the number of required PMUs. More PMUs and therefore more total cost is the expense should be paid for having a robust measurement system. For the IEEE 118-bus test system, as described in Table II, three different configurations of flow measurement devices, which have already been used in [6] and [17], are studied. For these configurations, the efficiency of the proposed method to handle OPP in the presence of conventional measurements is discussed. The results of the contingency-constrained OPP

are tabulated in Table III. To examine the impact of injection measurement contingencies on the OPP result, zero injections are dealt with as measured injections so that contingencies can be defined for them. For each configuration, there are at most three types of measurement devices, namely PMU, flow measurement, and injection measurement, which make the network observable. Due to a line outage, a measurement device may be incapable of fulfilling its function. Hence, four types of different contingencies on measurement devices and lines are studied (as listed in rows of Table III). First, it can be deduced that having a robust measurement system against single PMU or line contingencies needs more PMUs than the case of single flow measurement or injection measurement contingencies. It is because of less conventional devices in studied configurations. Additionally, the adverse impact of single PMU loss on the network observability is worse than the impact of single line outage. It can be easily inferred from the comparison of the required number of PMUs in associated cases. For this test system, the comparison between OPP results of this paper and previously reported ones is summarized in Table IV. First, it should be noted that methods considering network equations as a system of equations together have the capability of reaching to the optimal solution. However, as just the linear framework proposed in this paper investigates all solvability states of network equations, it is the only attempt which has reached to the optimal solution in all cases. This superiority is much distinguished in contingency-constrained OPP and in the cases where there are flow measurement devices in the power system. For different flow measurement configurations “ ,” “ ,” and “ ,” the number of required PMUs to make the IEEE 118-bus system observable has been compared with those of different methods in Table V. It is interesting to note that although reference [17] has considered one more flow measurement at branch 4–5 in addition to those of configuration “ ,” its OPP result


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TABLE VI OPP RESULTS FOR DIFFERENT CONFIGURATIONS OF FLOW MEASUREMENT DEVICES IN IEEE 118-BUS TEST SYSTEM

TABLE VII MINIMUM NUMBER OF CAPACITY-RESTRICTED PMUS 118-BUS TEST SYSTEM OBSERVABLE

TO

MAKE IEEE

TABLE VIII RESULTS OF OPP FOR THREE LARGE-SCALE POWER SYSTEMS WITHOUT CONVENTIONAL MEASUREMENTS

power system in winter 1999–2000 and Polish power system in winter 2003–2004 [30]. Characteristics and required PMUs for these networks are tabulated in Table VIII. Here, the OPPs carried out for large-scale power systems aim at demonstrating that thanks to the linearity of the proposed formulation, it reaches the optimal solution within a reasonable time. Therefore, no conventional measurement device is assumed to be present in these test systems. As shown in Table VIII, for these large-scale power systems, the number of PMU-equipped buses for complete system observability is less than 25% of the number of network buses. Also, the optimization does not take longer than 11 s to be executed. Moreover, it is possible to define conventional measurements configuration for each network so that OPP can be solved in such conditions. Nonetheless, as the formulation still remains linear even with the consideration of already existing conventional measurement, the optimal results can be obtained with a low computational burden. From execution time point of view, it can be concluded that the proposed ILP-based method can be applied to any practical large-scale power system. V. CONCLUSIONS

necessitates the installation of 26 PMUs in the power system, which is one PMU more than that of this paper. Moreover, solution of [6] requires at least 27 PMUs for the complete observability in this configuration. In addition to the number of required PMUs, Table VI gives their installation locations obtained using the proposed method. B. Limitation of Communication Channels In this part, OPP is carried out considering communication channels limitation. The number of measurements of each PMU is restricted due to this limitation. OPP is implemented assuming that all PMUs have the same restricted number of channels. The results are presented in Tables VII. In each case, there is one point after which increasing the capability of PMUs measurements has no effect on decreasing the number of required PMUs. Besides, it is obvious that in the cases where PMUs are capable of just one measurement, every bus has to be equipped with a PMU in order to get observable. However, since the IEEE 118-bus test system has ten zero injection buses, it is possible to assign calculation of ten voltage phasors to their associated equations. Hence, as expected, the minimum number of required PMUs decreases by 10, i.e., . C. OPP for Large-Scale Power Systems Finally, the proposed method is applied on some large-scale power systems, namely the IEEE 300-bus test system, Polish

In this paper, it has been demonstrated that the optimal PMU placement problem can be modeled and formulated linearly. All combinations of possible single contingencies, PMU channels limitation, and the presence of conventional measurements have been completely covered by the proposed integer linear framework. Considering the formulation linearity, it can be applied to any practical power system. Unlike previous methods, the potential of solving equations as a system of equations, which is necessary to reach the globally optimal solution, has been addressed and fully achieved in this paper. Therefore, globally optimal results, as well as comprehensiveness and low computational burden are main salient merits of the proposed method which have been verified by the comparison of results with those of reported methods. APPENDIX A A. Single Line Outage For a power system, as many single line contingencies as the number of network lines are definable. It is clear that the connectivity matrix changes in each of these contingencies. Suppose superscript represents outage of line . Thus, parameter is the th entry of the connectivity matrix when line is out. Accordingly, OPP constraints, considering single line outage are as follows: (A-1)



operating condition. The third, fourth, fifth, and sixth sets of constraints ensure the full observability of the power system during single contingencies for PMUs, flow measurements, lines, and injection measurements, respectively.

where

(A-2) (A-11) (A-3) (A-4) (A-12) (A-5)

(A-13) (A-14)

and are binary auxiliary variables as explained before, but when the line is out. (A-15) APPENDIX B A. Single Injection Measurement Loss

(A-16)

Let superscript indicate contingency for injection measurement . In addition, the set of injection measurements indices is denoted by . It should be noted that although zero injections are modeled like injection measurements in the OPP problem, no contingency is defined for them, because they are not measurement devices inherently. Accordingly, constraints of the optimization considering single injection measurement loss are as follows: (A-6)

(A-17) (A-18) (A-19)

(A-20)

where (A-21) (A-7) (A-8) (A-9)

(A-10)

(A-22) (A-23)

(A-24) (A-25)

and are binary auxiliary variables as explained before, but when the injection measurement is out.

(A-26)

APPENDIX C

(A-27)

A. Complete Formulation of OPP Problem The following is the complete formulation of ILP-based framework proposed in this paper for the OPP problem. The first set of constraints, i.e., constraints (A-12) to (A-14), is those in relation to limitation of communication channels of PMUs. The second one represents the observability during the normal

(A-28)

(A-29)


(A-30)

(A-31) (A-32) (A-33)

(A-34) (A-35)

(A-36) (A-37) (A-38)

(A-39)

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