A General Optimal Substation Coverage Algorithm for Phasor Measurement Unit Placement in Practical Systems

Transcription

1 1 A General Optimal Substation Coverage Algorithm for Phasor Measurement Unit Placement in Practical Systems Anamitra Pal 1,*, Chetan Mishra 2, Anil Kumar S. Vullikanti 1, and S. S. Ravi 3 1 Network Dynamics and Simulation Science Laboratory, Virginia Bioinformatics Institute, Virginia Tech, Blacksburg, Virginia-24061, USA. 2 Virginia Electric & Power Company (d/b/a Dominion Virginia Power), Richmond, VA-23219, USA. 3 Computer Science Department at University at Albany - State University of New York, Albany, NY , USA. * Corresponding Author. Address: 1880 Pratt Drive (0477), Blacksburg, Virginia-24061, USA. anam86@vbi.vt.edu Abstract The primary objective of the conventional optimal phasor measurement unit (PMU) placement problem is the minimization of the number of PMU devices that, when placed in a power system, measures all bus voltages. However, due to advancements made in the field of relay technology, digital relays can now act as PMUs. This has significantly reduced device costs. Moreover, although the goal is to observe all the buses, the devices themselves can only be placed in substations, whose upgrade costs are much higher than those of the devices. Considering these factors, the approach proposed here optimizes the number of substations where traditional bus type PMUs, as well as the more modern dualuse line relay PMUs, can be placed. Called the general optimal substation coverage (GOSC) algorithm, it is also able to incorporate practical challenges such as redundancy in measurement of critical elements of the system, and estimate tap ratios of the transformers present. The results indicate that the proposed GOSC algorithm has significant techno-economic benefits. Keywords Criticality, integer linear programming (ILP), observability, optimal substation coverage, phasor measurement unit (PMU) placement, redundancy, tap setting estimation.

2 2 1. INTRODUCTION Phasor measurement units (PMUs) are devices that provide real-time voltage and current phasor measurements at those locations of a power system network where they are placed. When present at a sufficiently large number of locations inside the grid, these devices are capable of creating a power system state estimator that is completely linear and non-iterative [1]. Called a linear state estimator (LSE), it has numerous advantages over the classical state estimator in terms of speed, accuracy, and reliability [2]. However, PMUs cannot be placed at random inside the power system. This is primarily because of the associated communication infrastructure costs, as well as the costs incurred in upgrading substations. A US Department of Energy (DOE) report published in October 2014 identified the communication infrastructure cost as the major portion of a PMU installation cost [3]. The report mentions that, in the absence of adequate existing communications and/or upgrades to communication infrastructure, the cost of installing PMUs could increase by a factor of seven. However, it also states that once a high-speed backbone telecommunications network is installed, the cost of installing additional PMUs drops considerably. Labor and substation outage costs were identified as the next most significant cost drivers. The report concluded by identifying that the PMU hardware cost was typically less than 5% of the total installed synchrophasor system cost. In light of this report, it is clear that the objective of the traditional optimal PMU placement (OPP) problem of minimizing the number of devices that must be added to the system for its complete observability is no longer valid. An alternate formulation called general optimal substation coverage (GOSC) is proposed in this paper to minimize the overall synchrophasor installation cost while simultaneously catering to practical constraints of realistic systems. The rest of the paper is structured as follows. An overview of the state-of-the-art with respect to the OPP problem is presented in Section 2. Section 3 explains the GOSC formulation and associated constraints that it can address. Its application to a multi-voltage power system model is also discussed in that section. The results obtained by applying the GOSC algorithm to standard IEEE systems as well as a 2383-bus Polish system are described in Section 4. The conclusions are provided in Section 5.

3 3 2. TRADITIONAL OPP FORMULATION The traditional OPP problem is an optimization challenge which tries to place the minimum number of devices in a power system network so that all the buses of that network are directly or indirectly observed. Observability is defined as the ability to measure the complex voltages (known as states) of a power network. Direct observability of a bus implies that the PMU is placed on that bus, while indirect observability implies that the indirectly observable bus is connected to a bus that has PMU on it by transmission lines whose parameters are assumed to be known. From a graph-theoretic perspective, where buses are nodes and transmission lines are edges, the traditional OPP problem is a variation of the minimum dominating set problem [4]. Over the past two decades engineers, computer scientists, and mathematicians have shown considerable interest in optimally solving the PMU placement problem. A search of the phrase PMU placement in popular forums yields more than 400 publications, with the bulk of them occurring in the last five years. The reason for the continued interest in this apparently theoretical problem is the accelerated growth of applications that rely on phasor data for their reliable functioning. In terms of applications, the PMU placement papers can be grouped into the following categories: a. Monitoring (through hybrid/linear state estimation) [5]-[6] b. Control [7]-[8] c. Protection [9]-[10] A variety of mathematical techniques have also been applied to solve for OPP. These include genetic algorithms [11], linear programming [12], semi-definite programming [13], particle swarm optimization [14], Tabu search [15], etc. However, integer linear programming (ILP) has emerged as the most popular choice for solving OPP problems [16]. The reason for this is that unlike meta-heuristic approaches, ILP always gives an optimal solution. Due to its inherently heavy computational burden, the application of ILP to large systems was a concern [17]. However, with the emergence of efficient optimizers such as GUROBI and CPLEX, the computational time required for finding optimal locations for PMU placement even for large systems is no longer substantial.

4 4 In terms of practical constraints, popular topics have been the presence of zero injection buses [18], incorporation of conventional measurements [19], consideration of communication infrastructure costs [20], provisions for redundancy [21], and accounting for measurement channel capacity [22]. Most of the papers published on the topic of OPP have tried to address one or more of these constraints in their formulations. For a more detailed description of PMUs and their placement methodologies, we refer the reader to [23]. 3. THE GENERAL OPTIMAL SUBSTATION COVERAGE (GOSC) ALGORITHM All papers on PMU placement summarized above considered the following formulation: an optimal placement must satisfy all the specified constraints and require the least number of PMUs for doing so. This is because the cost of the device was assumed to be the primary reason for not placing PMUs at all buses. However, in a real system, there are multiple voltage levels (buses) at a particular substation and the tap settings between the different voltages are not usually known. Thus, different voltage levels are decoupled from the point of view of observability. Secondly, although the buses must be observed by PMUs, the PMUs themselves can only be placed inside the substations. Therefore, a distinction must be made between buses and substations when choosing optimal locations for PMU placement. It has already been shown in [3], [14] that minimizing the number of PMUs does not necessarily result in minimizing the cost of PMU deployment. Likewise, during an actual implementation at Dominion Virginia Power (DVP), a US-based utility, it was observed that the majority cost associated with synchrophasor deployment was not due to the devices but rather due to substation outage and infrastructure/labor costs. For this reason, the aim of the problem considered here is the minimization of the total number of substation locations where installations are performed so as to observe all the buses of the network. We refer to this formulation as optimal substation coverage. In contrast to the optimal substation coverage algorithm proposed in [24], the GOSC algorithm developed here optimizes placement of two types of PMUs a traditional bus-type PMU (TPMU) and a dual-use line relay branch-type PMU (DULRP). The key features of the GOSC formulation are summarized below:

5 5 1. A bus at a particular voltage level that is monitored inside a substation does not imply that buses at other voltage levels are also monitored inside the same substation. This is because in practice, for most utility companies, real-time values of the transformer tap settings are not known. 2. Different voltage levels are connected differently. This implies that a multi-voltage power network can be decomposed into multiple sub-networks, each of which is at a different voltage level. 3. In a double bus-bar substation, if the buses at the same voltage level are connected by normally open switches (or isolating transformers), then they are to be treated as separate buses. 4. Since considerable investment is made when placing a PMU at a substation [3], once a substation is chosen for PMU placement, all lines present inside the substation as well as the ones that connect it to other substations will be monitored by the PMUs placed inside the chosen substation. A mathematical formulation of GOSC is developed in the following two sub-sections and is illustrated using an example in Section Terminology Used Let the power network be represented by an undirected graph G(V, E), where V is the set of nodes (buses) and E is the set of edges (transmission lines or transformers). Further, the node set V is partitioned into k 2 blocks B 1, B 2, B k where each block represents a substation. Let r i denote B i for 1 i k. Now, each node of V can be denoted by a pair of integers (x, y) where x {1, 2, k} is the block number and y {1, 2, r x } is an index number within block B x. Next, a lexicographic ordering amongst the nodes is introduced: Given two nodes v 1 = (x 1, y 1 ) and v 2 = (x 2, y 2 ) from two different blocks B x1 and B x2, we define v 1 v 2 if x 1 < x 2. For two nodes v 1 = (x 1, y 1 ) and v 2 = (x 1, y 2 ) lying within the same substation, v 1 v 2 if y 1 < y 2. An edge e = {v 1, v 2 } E can either join two nodes inside the same block, or two nodes lying in two different blocks. In the former case, v 1 = (x 1, y 1 ) and v 2 = (x 1, y 2 ), while in the latter case, v 1 = (x 1, y 1 ) and v 2 = (x 2, y 2 ). When specifying an edge e as {v 1, v 2 }, it will be assumed that v 1 v 2 ; thus, v 1 and v 2 can be referred to as the low end and the high end of edge e, respectively. For any node v V, the neighborhood of v, denoted by N v, contains the node v itself and all

6 6 nodes that are adjacent to v. As a simple extension of this definition, the neighborhood of a set of nodes X, denoted by N X, is defined by N X = v X N v. As mentioned earlier, two types of PMUs are considered in this formulation: a traditional PMU (TPMU), and a dual-use-line-relay PMU (DULRP). When placed at a node v 1 = (x 1, y 1 ), a TPMU is expected to observe the nodes in N v1 while causing a disruption of block x 1. However, the observability of the nodes in N v1 {v 1 } is subject to the number of measurement channels available to the TPMU placed at v 1. A DULRP may be placed at either end of an edge. When a DULRP is placed on an edge e = {v 1, v 2 }, it can observe both the nodes v 1 and v 2. If it is placed on the v 1 = (x 1, y 1 ) end, which is the low end of e, then the block x 1 must be disrupted. Likewise, if it is placed on the v 2 = (x 2, y 2 ) end, which is the high end of e, then the block x 2 must be disrupted. If x 1 = x 2, that is both the nodes lie inside the same block, then only that block will be disrupted. As one DULRP monitors only one edge, the issue of measurement channel limitation does not arise. Finally, considering measurement channel limitations of a TPMU, and by definition for a DULRP, from an observability perspective, both these devices can be thought of as placed on one (in case of a DULRP) or on many (in case of a TPMU) edges of a network. A mathematical formulation based on the terminology described here is developed below Mathematical Formulation The basic version of the optimization assumes that the given system does not have any TPMU or DULRP pre-installed. Its goal is to place PMUs inside the network so that the following conditions are satisfied: (a) all the buses are observed; and (b) the number of affected substations is minimized. An ILPbased formulation of this problem is developed as follows. In the formulation described below, unless specified explicitly, the word PMU refers to both TPMU and DULRP. For each substation B i for 1 i k, there is a binary valued variable y i such that y i = { 1 0 if substation B i is disrupted otherwise (1) For each edge e, there are two binary valued variables w e l and w e h such that

7 7 w e l = { 1 0 w e h = { 1 0 if PMU observes low end of edge e otherwise if PMU observes high end of edge e otherwise (2) (3) Using (1)-(3), the basic objective function can be defined as k Minimize ( y i ) (4) i=1 Three sets of constraints that must be imposed on this basic objective function are defined as follows. For any node v, let L v denote the set of edges incident on v such that for each edge in L v, v is the low end of that edge. Similarly, for any node v, let H v denote the set of edges incident on v such that for each edge in H v, v is the high end of that edge. Let E v = L v H v denote the set of all edges incident on node v. Then, for each node v V {w h e + w l e } 1 (5) e E v Eq. (5) ensures that each node v is observed by PMUs placed at any of the edges that are incident to v. For each node v = (i, j), 1 i k and 1 j r i, and each edge e L v y i w e l (6) Eq. (6) ensures that if a PMU observes an edge for which v = (i, j) is the low end, then B i must be disrupted. Similarly, each node v = (i, j), 1 i k and 1 j r i, and edge e H v h y i w e (7) Eq. (7) ensures that if a PMU observes an edge for which v = (i, j) is the high end, then B i must be disrupted. This completes the formulation of the basic version of the optimization. Practical constraints further imposed on the basic objective function are defined as follows: Redundancy to critical buses: Measurement redundancy under N 1 contingency has been a common theme in many PMU placement papers ([11]-[13], [18]). However, a closer inspection of results reveal that a large number of substations (> 50%) must be disrupted in order to provide redundancy to all phasor measurements. A more practical scheme is to provide redundancy in

8 8 phasor measurements of only the most important buses of the system. A methodology to identify such critical buses has already been proposed [21]. To provide redundancy in measurement for only the critical buses of the network, the basic GOSC formulation is modified as follows. Depending on the type of PMU device (TPMU or DULRP), when the device fails it loses the ability to observe edges (and thereby, nodes) that are incident to it. When a TPMU on node v fails, it loses observability of the edges it was monitoring. When a DULR on an edge e fails, it cannot observe any end point of e. Consider a given set C B V of critical buses and an integer t 1 that represents the maximum number of edges that will lose observability (due to TPMU or DULRP failures). Then, the goal is to ensure that each node v C B is observed by at least one PMU even when any subset of t or fewer edges lose observability. Moreover, since the failures can be due to the device itself and/or outage of the line which that device observes, this constraint ensures observability under N t contingency. In the proposed ILP formulation, this requirement is accommodated by ensuring that each critical bus is observed by at least t + 1 PMUs. This is done by replacing the constraint specified by (5) with the one shown in (8). {w h e + w l e } t + 1 v C B (8) e E v Ensuring observability of important lines: Similar to the concept of critical buses described in Item 1 above, there are some lines in the network which are more important than others. Typically, these are the high-voltage (HV) lines present inside the system or the tie-lines that join one system to another. To the best of our knowledge, [24] was the first paper that placed PMUs while considering the fact that some lines needed to be observed by them. In this paper, the idea is extended further by proposing two levels of line criticality: (a) Critical Lines those lines which must be observed by PMUs under normal conditions; and (b) Super Critical Lines those lines which must be observed by PMUs under N 1 contingency conditions. Thus, if C L E is the set of critical lines present in a given system, their observability is ensured by (9a).

9 9 {w e h + w e l } = 1 (9a) e C L Similarly, if SC L E is the set of super critical lines present in a given system, their observability under N 1 contingency is ensured by (9b). {w e h + w e l } = 2 (9b) e SC L Handling prohibited substations: In the field, there exist some substations where synchrophasor installations cannot be made in the specified planning horizon. This is a practical constraint and is addressed as follows: Suppose S {B 1, B 2, B k } is the set of substations that cannot be disrupted. Then, for each substation B i S, the constraint y i = 0 must be added to the ILP formulation described in (4)-(7) Handling pre-installed PMUs: Similar to Item 3 above, some substations may already have installed PMUs. For instance, this can happen due to a synchrophasor deployment phase that might have happened previously. Existing PMU locations can be integrated into the proposed formulation in the following manner. Let P V be the set of nodes at which PMUs have already been placed. Then, PMUs placed at the nodes in P observe all the nodes in N P. In other words, the placement of new PMUs needs to observe only the nodes in V N P. Therefore, to account for P, the constraint specified by (5) in the ILP formulation must be applied to each node v V N P Considering measurement channel limitations: As mentioned previously, the problem of measurement channel limitation occurs only in the case of TPMUs. This is because DULRPs only monitor the currents flowing through the edge on which they are placed, and the voltages of the two nodes that lie on either end of that edge. Let the number of channels for measuring threephase currents present on the j th TPMU be c j. Now, when a substation is selected for PMU placement, all the lines present inside as well as outside that substation are made observable by PMUs. Then, if the k th substation which has l k lines to be monitored is selected for PMU

10 10 placement, the break-up of the number of TPMUs and DULRPs required for that substation is done according to the following logic: Number of TPMUs required = Quotient(l k, c j ) Number of DULRPs required = Remainder(l k, c j ) (10) In (10), Quotient(l k, c j ) denotes the (integer) quotient when l k is divided by c j ; while Remainder(l k, c j ) denotes the remainder when l k is divided by c j Transformer tap ratio estimation: In a real-system, transformers are often under local control, and so, the tap position is not communicated to the control center. An erroneous tap measurement or the presence of an unmeasured tap can lead to a high error in the state estimator [25]. Thus, utilities often wish to estimate tap settings using phasor measurements. In [26], an OPP problem with the additional goal of estimating all transformer tap settings was introduced. However, it was done by minimizing the number of devices/buses instead of substations. In the proposed GOSC algorithm, since all lines inside a substation selected for PMU placement are monitored, tap ratios of the transformers present in those substations can also be determined. We note that our algorithm for GOSC does not guarantee observability of tap settings of all the transformers present in the system. This completes the ILP formulation of GOSC. Its application to a simple test system is described in the following sub-section Illustration using a Model System A typical visualization of a practical system with three voltage levels is shown in Fig. 1. It shows a 17- bus system with one or more buses present inside a substation. The black circles represent substations and the colored dots are buses at different voltage levels. Following the lexicographic ordering proposed in Section 3.1, each bus is shown as an ordered pair of integers. Thus, from the figure it becomes clear that the 17-bus system is equivalent to a 12-substation system (denoted by the prefix S). Considering the most practical scenario, it is assumed that every substation has a non-zero net injection and that the transformer

11 11 tap settings between different voltage levels are unknown. The latter is depicted in Fig. 1 by joining the colored dots inside the black circles using dotted black lines. Fig. 1. Example system illustrating the proposed GOSC algorithm Table 1 shows the basic observability results obtained for this system. The proposed technique is compared with two OPP formulations, both of which minimize the number of PMU devices required for complete observability. The first formulation finds the minimum number of buses where TPMUs must be placed so that the given system becomes observable. This was done using the ILP technique developed in [27]. Ref. [27] assumed that each TPMU has an unlimited number of measurement channels and hence the number of TPMUs required was equal to the number of buses where PMUs were to be placed. Thus,

12 12 that formulation gave the absolute minimum number of TPMU devices that a system needed for its complete observability. The second formulation found the minimum number of DULRPs required for complete observability using the integer programming technique developed in [28]. The costs given in Table 1 were computed based on the information from [3] and [29]. Ref. [3] mentions that the cost of a PMU is 1/20 th of the cost of upgrading a substation for synchrophasor deployment, while [29] indicates that the cost of a TPMU with six three-phase current channels and two three-phase voltage channels is approximately 5 times that of a DULRP that has one three-phase current and voltage channel. Ref. [29] also indicates that for each j, a suitable value of c j for the TPMUs is 6. Thus, c j = 6 was used in the rest of the simulations. The entry (i, j) in Table 1 implies that the TPMU is placed at the bus having an index number of j inside substation i. Similarly, the entry (i, j) (k, l) implies that the DULRP is placed on the (i, j) th end of the line between (i, j) and (k, l). From Table 1 it becomes clear that for the same technical constraints (no redundancy, no pre-installed PMUs, etc.) the proposed algorithm provides a more economical solution than either of the OPP formulations that minimized the number of devices. Moreover, even in its most basic form, the proposed approach is able to estimate the tap ratios of the transformers in substations S1, S2, and S7, which the other two methods do not do. In the next section, results from more detailed simulations on standard IEEE systems as well as a 2383-bus Polish system are presented to show the wide-spread applicability of the GOSC algorithm. 4. RESULTS A variety of test systems were used to analyze the ability of the proposed GOSC algorithm to handle practical constraints. In the first set of simulations, its application to a modified IEEE 30-bus system is demonstrated. The modified IEEE 30-bus system is represented by a 26-substation system as shown in Fig. 2. In that figure, numbers with a prefix of S denote substations, while those without are buses. Substations S4, S6, and S24 are multi-bus substations with transformers in them (having unknown tap ratios), while the other substations are one-bus substations. The system has three voltage levels depicted by the colors red, blue, and green, respectively. The results obtained using the proposed method are also

13 13 shown in Fig. 2. There, the lines which must be observed by TPMUs or DULRPs in accordance with the proposed scheme have blue dots on them. From the figure, it can be seen that by default, the GOSC algorithm is able to identify the tap ratios of the transformers in substations S4 and S6. The basic observability results for this system are shown in Table 2. In that table, the results obtained using the proposed algorithm are compared with those obtained using the techniques developed in [26]-[28]. From the table it becomes clear that the GOSC algorithm provides the most cost-optimal results. TABLE 1: Cost comparison of GOSC algorithm with traditional OPP formulations for the model system Method TPMU Locations DULRP Locations Substations affected Total Cost* Ref. [27] (1,2);(2,1);(4,1); (5,2);(7,2);(11,1) N/A S1;S2;S4; S5;S7;S11 150x Ref. [28] N/A (1,1)-(2,1);(1,2)-(6,1);(1,2)-(7,1);(1,3)- (11,1);(4,1)-(2,2);(4,1)-(5,1);(7,2)-(8,1);(9,1)- (11,1);(10,1)-(5,2);(10,1)-(12,1) S1;S4;S7; S9;S10 110x S1{(1,1)-(1,2);(1,2)- (1,1)-(2,1);(1,3)-(11,1);(2,1)-(1,1);(2,1)- GOSC Algorithm (1,3);(1,3)-(1,1);(1,2)- (5,1);(1,2)-(6,1);(1,2)- (3,1);(2,2)-(4,1);(7,1)-(1,2);(7,2)-(8,1);(7,2)- (12,1);(10,1)-(5,2);(10,1)-(8,1);(10,1)- S1;S2;S7; S10 99x (7,1)} (9,1);(10,1)-(12,1);(2,1)-(2,2);(7,1)-(7,2) * x is the cost of a single DULRP; 1 TPMU = 5x [29]; Outage cost of 1 Substation = 20x [3] In the next set of simulations, the following constraints are applied to the modified IEEE 30-bus system: (a) Substation S2 is unsuitable for PMU placement; (b) Bus 28 inside Substation S24 is a critical bus and needs N 1 redundancy; (c) The line between buses 23 and 24 is critical and must be observed under normal operating conditions; (d) The line between buses 12 and 13 is super critical and must be observed with N 1 redundancy; and (e) Substation S26 has a pre-installed TPMU that monitors the voltage of bus 30 and currents flowing in lines and Section 3.2 explains how the proposed

14 14 GOSC algorithm is able to handle these constraints. As an example, the N 1 redundancy in observability of bus 28 inside substation S24 is ensured by setting t = 1 in (8). Fig. 2. Modified IEEE 30-bus system with different colors indicating different voltage levels; the blue dots indicate the lines which must be monitored by PMUs in accordance with the GOSC algorithm

15 15 The results obtained are shown in Fig. 3 where the red dots indicate the lines observed by the preinstalled PMU while the blue dots indicate additional lines which must be observed by TPMUs or DULRPs in accordance with the proposed scheme. Now, if x is the cost of one DULRP, the total cost for this synchrophasor installation set-up comes out to be 216x (=total cost of 4 TPMUs, 16 DULRPs and 9 Substation disruptions). The results indicate that the GOSC algorithm is able to incorporate a wide variety of practical constraints and yield results that are techno-economically beneficial. TABLE 2: Cost comparison of GOSC algorithm with other OPP formulations for the IEEE 30-bus system Method # TPMUs # DULRPs # Substations affected Transformer Tap Ratios Observed Total Cost* Ref. [26] 11 N/A 7 S4, S6, S24 195x Ref. [27] 10 N/A 10 None 250x Ref. [28] N/A None 256x GOSC Algorithm S4, S6 175x * x is the cost of a single DULRP; 1 TPMU = 5x [29]; Outage cost of 1 Substation = 20x [3] The proposed GOSC algorithm is now applied to some of the largest systems that have been analyzed for OPP studies. These are the IEEE 118-bus system, IEEE 300-bus system, and a 2383-bus Polish system. The data for the test systems was obtained from the MATPOWER [30] toolbox of MATLAB [31], while the ILP optimization for these systems was performed using GUROBI [32]. By combining buses into substations (based on the locations of transformers), the three test systems became a 109-substation, a 235-substation, and a 2218-substation system, respectively. Table 3 shows results obtained when basic observability for the three systems were analyzed using different approaches. From the results it becomes clear that the most cost-optimal results are obtained using the proposed GOSC approach. Moreover, although the methodology developed here was not meant to observe all tap ratios, it did observe most of them at a significantly lower total cost. This proves that the proposed method provides a good balance of costs incurred versus benefits gained.

16 16 Fig. 3. Modified PMU placement-based edge observability for the IEEE 30-bus system in presence of practical constraints; red dots indicate lines that were observed by pre-installed PMU at Bus 30

17 17 TABLE 3: Cost comparison of GOSC algorithm with other OPP formulations for larger systems Test System Method # TPMUs # DULRPs # Substations affected # Transformer Tap Ratios Observed Total Cost* Ref. [26] 44 N/A x IEEE 118-bus system Ref. [27] 33 N/A 31 None 785x Ref. [28] N/A None 1263x GOSC Algo x Ref. [26] 198 N/A x IEEE 300-bus system Ref. [27] 129 N/A 89 None 2425x Ref. [28] N/A None 2731x GOSC Algo x Ref. [26] 887 N/A x 2383-bus Polish system Ref. [27] 775 N/A 728 None 18435x Ref. [28] N/A None 24421x GOSC Algo x * x is the cost of a single DULRP; 1 TPMU = 5x [29]; Outage cost of 1 Substation = 20x [3] In the last set of simulations, the high voltage networks of the three test systems were assumed to be very critical. This translated to the highest voltage buses of these systems becoming critical buses, while the high voltage transmission lines became super critical lines that, from a PMU placement perspective, required observability under N 1 contingency conditions. The results obtained for this scenario are summarized in Table 4. The table compares the results obtained using the proposed approach with the one developed in [21]. From the table it is clear that the GOSC algorithm is able to address practical constraints associated with OPP at a lower total cost.

18 18 TABLE 4: Cost comparison of GOSC algorithm with other OPP formulations for larger systems when the HV system is considered critical Test System Method # TPMUs # DULRPs # Substations affected # Transformer Tap Ratios Observed Total Cost* IEEE 118-bus system IEEE 300-bus system 2383-bus Polish system Ref. [21] 47 N/A 42 None 1075x GOSC Algo x Ref. [21] 138 N/A 92 None 2530x GOSC Algo x Ref. [21] 809 N/A 748 None 19005x GOSC Algo x * x is the cost of a single DULRP; 1 TPMU = 5x [29]; Outage cost of 1 Substation = 20x [3] 5. CONCLUSIONS This paper describes a PMU placement procedure to optimize the number of substations where installations must be made to observe all the buses when traditional bus-type PMUs (TPMUs) as well as dual-use line relay branch-type PMUs (DULRPs) can be added into the network. The proposed approach does so with the additional constraint that all tap settings are unknown to start with. The fact that many of the transformer tap settings become known in the process of reaching the final solution is an added benefit of the proposed approach. Similarly, since all substations have a minimum internal power consumption that should not be neglected, zero-injection substations were not considered as part of the simulation. Also, the proposed formulation was developed in order to cater to aid utilities like DVP that wish to create a linear state estimator for their whole system. Therefore, conventional measurements obtained from the SCADA network were not incorporated in the proposed framework. The constraints that were considered in the proposed approach are: providing redundancy in measurements of critical elements of the system, acknowledging presence of prohibited substations and substations with pre-installed PMUs, and accounting for measurement channel limitations. The results obtained based on these constraints indicate that the GOSC algorithm developed in this paper provides a cost-optimal solution while

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