Generalized optimal placement of PMUs considering power system observability, communication infrastructure, and quality of service requirements

This paper presents a generalized optimal placement of Phasor Measurement Units (PMUs) considering power system observability, reliability, Communication Infrastructure (CI), and latency time associated with this CI. Moreover, the economic study for additional new data transmission paths is considered as well as the availability of predefined locations of some PMUs and the preexisting communication devices (CDs) in some buses. Two cases for the location of the Control Center Base Station (CCBS) are considered; predefined case and free selected case. The PMUs placement and their required communication network topology and channel capacity are 
co-optimized simultaneously. In this study, two different approaches are applied to optimize the objective function; the first approach is combined from Binary Particle Swarm Optimization-Gravitational Search Algorithm (BPSOGSA) and the Minimum Spanning Tree (MST) algorithm, while the second approach is based only on BPSOGSA. The feasibility of the proposed approaches are examined by applying it to IEEE 14-bus and IEEE 118-bus systems.


OBSERVABILITY CONSTRAINT
In general, given a PMU with unlimited number of channels at a bus, bus voltage phasor and all current phasors along lines connected to that bus will be available. As shown in (1) presents observability constraint in general form as introduced in [10,16] for complete observability with a required degree of redundancy (Without Conventional Measurements -With Conventional Measurements), one depth of unobservability (Without Zero Injection Measurement -With Zero Injection Measurements)and,two depth of unobservability (Without Zero Injection Measurement -With Zero Injection Measurement ).
Observability constraint: TX ≥ B (1) T and B are a matrix and vector depend on each case [10,16]. X is the PMUs placement variables X = [x 1 x 2 … x n ], if No PMU at bus i, The minimum number of PMUs (PMU Smin ) can be formulated as a problem of Integer Linear Programming [29] as shown in the following equation.
Subject to: Observability constraint (2) where N is the number of buses

BINARY OPTIMIZATION USING HYBRID PSO AND GSA
The PSOGSA is a hybrid optimization algorithm, combining strengths of both PSO and GSA. This algorithm performs both PSO and GSA in terms of improved exploration and exploitation [30]. This technique has the nature of meta-heuristic optimization techniques. One of the main advantage of these techniques is that they do not need a function formulation, but rather need a fitness function only or any other way for distinguishing the results. As a result, the black box problem can be solved using these techniques. The BPSOGSA algorithm is a binary version of hybrid PSOGSA. Readers may refer to [31] for more detailsabout this algorithm. For the above-mentioned reasons, this version will be used in this study.

MINIMUM SPANNING TREE
The vertices (nodes) of the CI in a power grid correspond to PMUs, CDs, and CCBS, while the edges correspond to high-voltage lines [32] or a new data transmission paths. Dijkstra's algorithm, conceived by Dutch computer scientist Edsger Dijkstra in 1959 [33] is a graph search algorithm that solves the shortest path problem for a graph with nonnegative edge [34]. The Dijkstra's algorithm is used to search the short path in the MST algorithm. The complete pseudocode for MST algorithm is shown in Figure 1 Step 3 in this algorithm could be modified to start with a highest short path and end with the lowest short path. This modification is preferable when small propagation time delay is required, where this modification shrinks the network and reduces the maximum propagation time delay of the farthest site. In step 4, the node is connected with tree through switch.

QUALITY OF SERVICE
The Wide Area Measurement System (WAMS) is a distributed communication network. The QoS in the WAMS depends on the latency time, data losses and reliability of the system. The latency time performance is very important especially in protection and dynamic control applications [35]. The tree network is a common methodology in order to design the networks [36,37]. The network architecture consists of several PMUs, CDs, and Phasor Data Concentrators (PDCs) as shown in Figure 2. PDC collects the data generated by these PMUs over a shared communication network. Additionally, it performs quality checks on phasor data and interprets and inserts the missing data at their appropriate position [38,39]. Typically, many PMUs located at various substations gather data and send it in real time to a PDC. Many PDCs can be connected to a common central PDC, in order to provide an interconnection wide snapshot of the power grid measurements. In large systems, they may contain more than one PDC, where each PDC is placed in a subarea. For simplicity in this study, the power system is assumed as one area, and only one PDC is used in the control center. The measurements are made at specific time instances and physical distant locations. They are then transmitted to a common location for use by wide area applications.
The latency time experienced by data between PMU and the destination node (CCBS) is a combination of PMU reporting delay, the network propagation delays, queuing, routing delays, and PDCs delay [40]. A representative latency time of the data network is shown in Figure 3. PMU reporting delay is defined as the maximum time interval between the data report time as indicated by the data timestamp, and the time when the data becomes available at the PMU output. This delay includes many factors, such as the window over which data is gathered to make a measurement, measurement filtering, and the PMU processing time. PDC delay is defined as the maximum time interval between the data input time as indicated by the data timestamp, and the time when the data becomes available at the PDC output. This delay includes many factors such as processing, and alignment received data from PMUs/PDCs. The PDC aligns received data and places that data in a packet. In addition, the PDC data processing may include filtering, reporting rate conversion, interpolation, extrapolation, phase and magnitude adjustment, etc. Most of the time the data frame is transmitted continuously from the PMU or PDC at the designated reporting rate.  Queuing and transitions delays ( ) are caused by the amount of data that has to be transported across the medium and the data rate of the medium. Considering an M/M/1 model, can be expressed as follows [41]: where μ is the average packet length in bits, and , respectively, represent the capacity and the flow of the link l in bps.
The propagation delay ( ) is dependent on the medium and thus is a function of both the medium and the physical distance separating the individual components of WAMS. In the fiber optic, the propagation delay can be considered as in the following equation [42].
where S is the speed of the light in a vacuum. L is the length of the communication link N is the group index of the material≈1. 5 Consider the network is connected using backbone switches. We can conclude the above-mentioned facts and summarize the total communication latency time as in the following equation: where T is the total latency, is the number of links between PMU and CCBS SW n is the number of switches between PMU and CCBS t pdc is PDC delay Based on the typical values shown in Table C.2 in [43], and with assuming PDC uses direct forward mode t pdc ≈ 2ms, and t pmu ≈ 25ms. The (5) will be as follows: Queuing packet losses mostly occur because of the finite queue capacity of packet switching networks. To compute the average loss rate at each switch, each node is modeled with the M/M/1/k queuing system. On the topic of [44], the total number of packet losses is estimated as a function of the link flow and the capacity of buffers and links. The form of average packet loss will be: where ρ e = f e y e h d = Traffic volume for all PMUs and CDs b e = Buffer capacity of link e , y e = Capacity of link e , f e = Traffic flow of link e.
For a system, which contains 500 PMUs and 500 CDs with each data flow 128 kbps and 500 links with flow link ratio equal to 0.8, if we use a buffer memory equal to one mega, average packet loss ≈ 0. Therefore, the buffer memory can be assumed with enough value and the cost of this memory relative to other component in CI can be neglected. This leads to reducing variables in the optimization problem and reducing the run time.
The reliability of the WAMS depend on the reliability of media channel and nodes elements. Based on the concept in [45] we can describe the relation which assess the reliability of connection between any Required-node (Rnode) and CCBS as follows: where R s =series element reliability, R p =parallel element reliability, s = Number of series elements in a path p = Number of parallel elements in a paths. For each bus reliability calculation, there are two cases: Case1) Complete observability without redundancy. In this case, there is only one path. If the PMU is located at Rnode the series components are only communication components (i.e. switches, communication links (cl), and PDCs) as shown in Figure 4(a). If the PMU is located at Neighbor-node (Nnode), the series components are communication components plus the transmission line (T.L.) as shown in Figure 4(b) (Case2) Complete observability with redundancy. In this case, there are series path and parallel paths as shown in Figure 5 and  In this study, the reliability of each switch is assumed as 0.99 and the reliability of OPGW and transmission lines is calculated as follows: Reliability cl or T.L. = R L/BL (10) where L is the length per km of the OPGW link or transmission line BL is the base length R is the reliability of the base length In this study, the base length and the reliability of the base length is assumed 20 km, and 0.99 respectively

COST CALCULATION
The WAMS cost depends on CI cost, PMUs cost, and CCBs cost. The cost of a CI is mainly composed of two major costs including the cost of passive components and the cost of active devices. In fiber optic networks, the price of passive components mainly depends on OPGW length and capacity. On the other hand, the cost of active devices mainly depends on the number of switches, which are installed at backbone nodes [46]. As a result, the cost of CI correspond directly to the number of switches, and data transmission medium (i.e. OPGW) price and installation cost as in the following equation where =number of the links L i = L crp i + L in i =link capacity rate price factor (Depend on the link capacity) =link installation cost factor =length of the link SWc i ≈ switch crp i + switch in i switch crp =switch capacity rate price factor switch in =switch installation cost Subscript i indicate link or node i Actually, however, the model should be such that the price of channel capacity can take only discrete values. In addition to the CI cost, there are the cost of PMUs, which equal to the total price of the PMUs and its installation cost. In the case of adding new data transmission paths between buses, the economic study has considered the establishment of new towers; the cost of the total new towers will depend on the link length. The total link cost can be calculated as follows: where =cost for direct link i Lavcrp=Capacity rate price factor of the new data transmission paths =link installation cost factor =towers cost= α=tower cost factor = + =actual distance for link i = virtual direct distance for link i β=direct connection factor The minimum number of the PMUs required could be calculated using (2). In addition, the number of CDs (Ncds) are known, so that the capacity of the new data transmission line C av ) could be approximated as follows: where PMU dataflow is rate of the pmu data flow (kbps) CD dataflow is rate of the CD data flow (kbps) For each existing economic study between two buses, calculate virtual distance from (14) for all possible new data transmission paths. After calculating virtual distance there are two-distance matrices: distance matrix corresponding to transmission lines distance matrix ( ) and distance matrix from virtual calculating ( ). Merge the two matrixes in one matrix as follows: -For direct connected buses, compare the link distance in D power with and take the as link distance if it is less than in D power . -For not direct connected buses, take the virtual length as link distance. -Then modify (11) as follows: where from 1 to lp are the links from power system network from +1 to are the links from new added paths Finally, the total cost will be as following

PROBLEM FORMULATION AND IMPLEMENTATION
The PMUs optimal placement problem can be considered as nondeterministic polynomial complete problem [47]. For a system with N buses, the search space is 2 without considering CI topology, channel capacity allocation, and number of PMU channels. Therefore, the PMUs optimal problem is considered as a combinatorial optimization problem [48]. Meta heuristic algorithm population based methods, such as BPSOGSA, are candidate for solving such problems. In the following, two approaches are presented to minimize the total cost with considering the observability and CI. In these approaches, the optimization problem is defined as follows: Prob. ∶ The following considerations are made in these approaches: -Some CDs are existed, and will be connected with the communication network.
-Two Cases are considered for location of the CCBS; predefined and free selected.
-Some PMUs locations are predefined and included in the cost calculation.
-Conventional measurements and ZIB with a required degree of observability and required redundancy are considered in the observability constraint such as in Section 2 -The buffer to store the packet in the switching node has a fixed enough value and not considered in the optimization problem. -Based on fairness grade of service, the link capacity is allocated to minimize the maximum latency time.
The maximum Latency time for any PMU( ) is less than 0.04 Sec. -PMU and CD dataflow are assumed 128 kbps.
-The reliability of the observability for any node (rreq) is greater than 0. 8 In the following sections, the IEEE 14 and IEEE 118 systems with given data in below is investigated for full observability condition using PC with Intel Core i5-430M @ 2.27 GHz and Matlab 2016. In addition, the results of these approaches are compared with the method, which was presented in [18] with fixed channel capacity (ten times actual data flow) and without adding new data transmission paths. Distance matrix of each IEEE test network, we have assumed that all transmission lines have the same conductors. Thus, the relative distances between system buses can be extracted from system admittance matrix. In addition, the distances of the new data transmission paths are assumed as listed in Table 1 and Table 2 for IEEE 118 bus and for IEEE 14 bus. Table 3 and Table 4 show CCBS per unit cost at each bus for IEEE 14 bus and IEEE 118 bus systems respectively. Table 5 shows the rate of the data flow for PMU and CD. Table 6 presents cost factors values, which are used to calculate the total cost.

Using BPSOGSA Combined with MST
In this approach, the optimization is divided into three loops as shown in Figure 7. The first loop, the main loop, the BPSOGSA in Section 3 is tseb uh secrao uoe desu location of the CCBS and PMUs that achieve the observability constraint as shown in Section 2. If the observability condition is not met, the inner loops are not required. Therefore, the following equation is used as the cost function in the outer loop.
Total Cost = C 1 + OBS_Penalty (19) where OBS Penalty = C 2 * ineqdsum 1 , 2 are constants with large value ineqdsum = summation of all postive elements in OBS d vector OBS d = observability right hand side − observability left hand side Based on Dmerged, the MST or MMST in Section 4 to connect all PMUs, CDs, and CCBS is used in the second loop. The third loop, BPSOGSA is used to allocate the links capacity of the connected network. Cost of the connected network according to (17 This loop return the total) with considering the (20) and (21) as a weighted penalty. max (T pmu ) < t req (20) min (R Node ) < r req (21) where T pmu is a vector of latency time for all PMUs according to (6) R Node is a vector of latency time for all nodes according to (9), (10).  Table 7 lists the predefined locations of the PMUs and CDs. In addition, Table 8 shows the results of the proposed approach, and the network topology is shown in Figure 8. The results of the used method in [18] for this case are: cost= 33.048 per unit, maximum latency 0.027844 Sec., and minimum reliability =0.87146. b. Predefined selection of the CCBS (CCBS=14) Table 9 lists the Predefined locations of the PMUs, CDs, and CCBS. In addition, Table 10 shows the results of the proposed approach, and the network topology is shown in Figure 9. The results of the used method in [18] for this case are: cost= 47.561 per unit, maximum latency 0.02826 Sec., and minimum reliability =0.83678.

. IEEE 118 bus Case study (Free selection of the CCB)
The Predefined locations of the PMUs, CDs, and the results of the proposed approach are listed in Table 11. In addition, the network topology is shown in Figure 10. The results of the used method in [18] for this case are: cost= 54 per unit, maximum latency 0.029074 Sec., and minimum reliability =0.76225.  Figure 10. Network topology (118 bus free selection case)

Using BPSOGSA
In this approach, the optimization is divided into three loops.
-The first loop, the main loop, is treated as explained in the Section 7.1 -The second loop, the BPSOGSA with particles dimension equal length of is used to search the low cost network connection topology, which connect all CCBS, PMUs, and CDs. The value of the fitness function for this loop is evaluated using the connectivity algorithm, which is shown in Figure 11.

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-The third loop is treated as explained in the Section 7.1 complete the flowchart of the approach is shown in Figure 12.
The main difference between the approach in this section and the approach in Section 1.7 is that the connection topology is not depend on the length of the network, but the topology connection is depend on the CI cost.  Table 7 lists the Predefined locations of the PMUs and CDs. In addition, Table 12 shows the results of the proposed approach, and the network topology is shown in Figure 13.  b. Predefined selection of the CCBS (CCBS=14) Table 9 lists the Predefined locations of the PMUs and CDs. In addition, Table 13 shows the results of the proposed approach, and the network topology is shown in Figure 14.  Table 11 lists the Predefined locations of the PMUs and CDs. In addition, Table 14 shows the results of the proposed approach, and the network topology is shown in Figure 15.  Figure 15. Network topology (118 bus free selection case)

RESULTS DISCUSSION
The following results were observed from simulation results. -The run time of second approach is longer than the first approach -The second approach is more efficient than the first approach, especially if the difference in price resulting from the change in channel capacity is significant. -Indeed, the methods presented in [18] was unsuccessful to achieve the global solution. Since it used MST algorithm to find the network topology and did not take into account the channel capacity allocation. Also due to, the multi-loop is not used in this method, the runtime is large.

CONCLUSION
In this study, optimal placement of PMUs and their required CI for power systems are co-optimally designed. Two approaches have been presented. The first approach (i.e. BPSOGSA Combined with MST) and the second approach (i.e. BPSOGSA) to find the optimum placement of PMUs and their CI are investigated using IEEE 14 buses and IEEE 118 buses. The simulation results indicate that the second approach is cost effective. Moreover, the second approach, due to using BPSOGSA in all loops, may succeed converge to the global solution. In contrast, the first approach due to using MST for links topology can take less run time but it may not converge to the global solution. The cost of the CI in this study is not depend on the accumulative length of the OPGW only. However, it considered the switches and the link capacity in the objective function. In addition, the quality of service such as latency time and the reliability of the communication network and the degree of the observability are considered. Also, the partially optimization problem (predefined locations of some PMUs and CDs), and the economic study for additional new data paths are considered in the proposed approaches.