An Improved OPP Problem Formulation for Distribution Grid Observability

Phasor measurement units (PMUs) are becoming popular and populating the power system grids rapidly due to their wide range of benefits and applications. This research paper proposes a comprehensive, effective, and revised formulation of the optimal PMU placement (OPP) problem with a view to minimizing the required number of PMU and ensuring the maximum number of measurement redundancy subjected to the full observability of the distribution grids. The proposed formulation also incorporates the presence of passive measurements/zero injection buses (ZIB) and the channel availability of the installed PMU. Additionally, the formulation is extended to various contingency cases i.e., the single line outage and single PMU loss cases. This paper solves the proposed OPP formulation employing a heuristic technique called backtracking search algorithm (BSA) and tests its effectiveness through different IEEE standard distribution feeders. Additionally, this study compares the obtained results with the mixed integer linear programming (MILP)-based approach and the referenced works. The obtained results demonstrate the superiority of the proposed formulation and solution methodology compared to other methodologies.


Introduction
The rapid increase in electricity demand and the integration of renewable energy resources into the power generation mix along with the deregulation of the electricity markets are pushing the electric grids to be operated with their highest capacity and closer to their stability limits. These modifications are also creating a situation of perfect uncertainty along with potential opportunities for the power system operators and load aggregators. Consequently, the real-time estimation of power system states is gaining huge momentum with a view to ensuring secure and reliable power system operation and to optimizing the potential opportunities. In response, the wide area measurement systems by placing PMUs throughout the electricity grids for the full grid observability could be a promising solution [1][2][3][4]. The PMU is considered as the most prominent measuring devices because of their abilities in providing time synchronized voltage and current measurements with promising accuracy. The recent hybridization of the global positioning system (GPS) technology with the electrical measurement devices paves the way for commercial production of the PMU [4][5][6].

Improved OPP Problem Formulation for Distribution Grids
The numerical and topological approaches are two widely used observability analyses techniques for the electricity grids. However, most of the researchers are prone to use graph theory based topological approaches due to the drawbacks of numerical approaches including the computational burden and singularity issue of the Jacobian matrix [18]. Additionally, the topological approaches can provide the same solution of their numerical counterparts without being subjected to any major difficulty [34]. According to the topological approach, the OPP problem of an electric distribution grid with NB number of nodes can be formulated as [7,14,15]: Cy p (1) Subject to: where 1 i f p = q 1 i f there nodes p and q are connected 0 otherwise The topological observability rules considering the ZIB of the electrical transmission grids observability analysis are well established. In this paper, similar rules will be followed for the distribution grids observability analysis. The details about the rules can be found in [9], that are summarized by considering unlimited channels of the PMU as: Rule 1. The voltage phasors of the PMU installed node along with current phasors of all connected branches can be measured by a single PMU. Eventually, Kirchhoff's law using the measured current phasors and branch parameters can calculate the voltage phasors of all adjacent nodes. Consequently, the PMU installed node along with all adjacent nodes can be observable by a single PMU.

Rule 2.
If a single passive measurement/ZIB along with its adjacent node form a set of 'n' numbers of nodes, the observability constraint of the whole set of nodes can be stated by the following inequality: Rule 3. If 'm' number of ZIB are connected to each other directly or through a single non-ZIB and if the ZIB along with adjacent nodes form a set of 'n' numbers of nodes, the observability constraint of the whole set of nodes can be stated by the following inequality: Rule 4. If any node is included to more than one sets of ZIB then the observability constraint (f q ) associated to that node can be kept unchanged for one set and replaced by zero for others.

Rule 5.
If a node is not adjacent to any ZIB, the f q associated to that node should be kept unchanged.
Now, the abovementioned rules modify the 'f ' vector by modifying the node connectivity matrix 'A'. The detail explanation with example can be found in [9], how the node connectivity matrix is modified in the presence of ZIB. So far, the formulated OPP problem considers the PMU with an infinite number of channels and can measure branch currents of all adjacent branches but as mentioned earlier, the PMU is manufactured with a limited number of channels and their costs vary accordingly. Consequently, the above formulation should be modified further considering the channel limits of the PMU. Let's assume a PMU with L number of channels installed at node k that is connected to Nk number of nodes. If L ≥ Nk, meaning if the number of channels is more than adjacent nodes; the installed PMU makes observable the node and all adjacent nodes by following Rule 1 and the row associated with node k of the connectivity matrix A should be kept unchanged. Otherwise, L combinations of Nk will replace the respective row. Consequently, the number of rows (Rk) for nodes where Nk > L can be obtained from the following equation and further details are available in [9]: The formulated problem is well defined and can be solved employing MILP approach. However, the MILP approach always leads towards only one optimal solution by satisfying the constraints though other combination of solutions can be achieved. Conversely, the heuristic search algorithms can find all viable solutions that require a slight modification of the formulated objective function as: where the P p is set to a very high value, for instance, P p = CN B as the aim of the OPP problem is to ensure full grid observability. However, to ensure the maximum number of measurement redundancy employing the heuristic search approach the objective function can be further modified as: where the Qp is set to a low value, for instance, Qp = 1/CNB as the incentive should not be as high as the PMU cost. Finally, the proposed OPP formulation also incorporates the contingency cases i.e., the single PMU loss and single line outage of the electric distribution grids. For instance, during a single line outage case, each node of an electric grid must be observed by at least two PMU, hence, the associated element of the vector b as given in Equation (2) will be '2'. However, the nodes located at the termination point of the radial lines are linked to the distribution grids through a single line only. The associated elements of the vector b will be '1' for those nodes as they need to be observed by only one PMU and the outage of that line will not affect the observability of the remaining part of the grid [7][8][9]. Similarly, the complete observability of an electric distribution grid will be guaranteed during a single PMU loss case, if each node of the grid is observed by at least two PMU. Consequently, all the elements of the vector b should be replaced by '2' for such a contingency [7][8][9].

Mixed Integer Linear Programming
The MILP is an optimization technique that is employed to solve many complex planning and control problems. Though the technique is not new, the advent of faster computers and improved software packages has made it popular among researchers. Additionally, the technique is effective not only for mixed problems, but also for the pure-integer and pure-binary problems [35].

Backtracking Search Algorithm
The BSA [36], a recently developed heuristic optimization technique, has already been employed successfully to solve many engineering problems [37][38][39][40]. It is comprised of the following steps: Step 1: Initialization The BSA generates a set of initial population randomly using Equation (8) and evaluates the fitness of the individuals. Then, it stores the best individual as the global best solution: where, i = 1, 2, 3, . . . , N pop and j = 1, 2, 3, . . . , D m .
Step 2: Selection I The Selection I generates/updates Q his to find the search track through a random selection of the individuals from P pop and Q his employing the following rule. Then, it reshuffles the positions of the individuals of the updated Q his : if r 1 < r 2 then Q his(i,#) = P pop(i,#) else Q his(i,#) = Q his(i,#) ∀ i (10) Step 3: Mutation This step generates T mutant using Equation (12) where the F cp controls the magnitude of search direction matrix (Q his -P pop ). In each generation, the BSA updates the F cp using Equation (13) where the values of F cp and α are chosen as '3'and '0.99', respectively, through a systematic trial and error process while solving the formulated OPP problem: It is very likely that few parameters of a few individuals go beyond their specified ranges after this mutation operation. The BSA selects random values of the violating parameters from the search space using Equation (14): if T mutant(i,j) < low j or T mutant(i,j) >up j , then T mutant(i,j) = U low j , up j ∀ i & j (14) Step 4: Crossover The crossover is considered as the most complex part of the BSA that generates T final through the generation of a binary integer-valued matrix (B map ) of size N pop × D m randomly that contains '0' and '1'. Then it updates the elements of the trial population from T mutant and P pop using Equation (15).
Step 5: Selection-II This step evaluates the fitness of the updated trial population (T final ). Then it updates P pop from T final and P pop based on their relative fitness. Finally, it updates the global best solution if it is relatively better than the previously stored one.
Step 6: Stopping criteria The BSA terminates its operation if the global best solution does not change for a pre-specified number of generations or it reaches the maximum number of pre-specified generations. Figure 1 illustrates the complete flowchart of the discussed BSA technique.  It is worth mentioning that, the operators of the BSA technique shares the similar name of other evolutionary algorithms including the genetic algorithm and differential evolution. However, the selection, mutation, and crossover operations of the BSA are different from other techniques. The BSA creates more effective and diversified population in each generation through these operators. Besides, it controls the amplitude of search direction in a balanced way as required for the global and local searches. Moreover, the historical population plays a dynamic role in attaining diversity even in the advanced generations [36].

Results and Discussion
This research tested the effectiveness of the proposed OPP formulation on four different IEEE standard test distribution feeders including IEEE 13-node, 34-node, 37-node and 128-node feeders employing the BSA and MILP approaches. The objective function of the employed optimization approaches was to minimize the overall PMU installation cost subjected to full grid observability. It is worth mentioning that, the operators of the BSA technique shares the similar name of other evolutionary algorithms including the genetic algorithm and differential evolution. However, the selection, mutation, and crossover operations of the BSA are different from other techniques. The BSA creates more effective and diversified population in each generation through these operators. Besides, it controls the amplitude of search direction in a balanced way as required for the global and local searches. Moreover, the historical population plays a dynamic role in attaining diversity even in the advanced generations [36].

Results and Discussion
This research tested the effectiveness of the proposed OPP formulation on four different IEEE standard test distribution feeders including IEEE 13-node, 34-node, 37-node and 128-node feeders employing the BSA and MILP approaches. The objective function of the employed optimization approaches was to minimize the overall PMU installation cost subjected to full grid observability. The BSA approach was initiated with a population of 100 individuals in size and terminated after 300 generations where the proposed OPP formulation was executed in a personal computer having a 3.50 GHz Core-i5 processor with an installed Random Access Memory of 8 GB. However, the required information regarding the standard test feeders are summarized in Table 1 and the details about them can be found in [41]. However, the IEEE 123-node test feeder is connected to five different incomers (power supplies). Hence, the authors considered the incomers as extra nodes and modified the 123-node test feeder to 128-node feeder and kept unchanged other test feeders. It is worth mentioning that, there are two types of loads in distribution test feeders namely the spot and the distributed loads. The spot loads are connected to nodes whereas the distributed loads are distributed between two nodes. In this paper, the nodes associated with any types of loads or generations are considered as non-ZIB and the others are assumed as ZIB. Following simulation cases have been carried out on the selected test distribution feeders with a view to minimizing the required number of PMU and ensuring the maximum number of measurement redundancy subjected to:   Table 2 presents the optimum number and locations of the PMU along with the required and the achieved measurement redundancies subjected to the full grid observability of the selected test distribution feeders during the normal operating mode. The summation of the elements of the final 'b' vector of the proposed OPP formulation gives the required number of measurement redundancy whereas the summation of the elements of the production of final 'A' and 'Y' matrices gives the achieved measurement redundancy. As can be seen, both BSA and MILP approaches ensured full grid observability of the test distribution feeders with an equal number of PMU installation. Besides, the BSA approach achieved more measurement redundancy over the MILP approach by rearranging the positions of the PMU among the nodes. For instance, Figure 2 presents the optimal locations of PMU and respective measurement redundancy employing the MILP and BSA approaches for the IEEE 13-node feeder. As can be observed, both approaches selected the node 684 for PMU installation that can make observable a total of four nodes (611, 652, 671, and 684). Hence, the MILP and BSA approaches achieved 14 and 23 measurement redundancy, respectively. Likewise, the MILP approach achieved 35, 43 and 132 whereas the BSA achieved 42, 47 and 169 measurement redundancies for the IEEE 34-node, 37-node, and 128-node test feeders, respectively. Consequently, the presented results illustrate the superiority of the BSA approach over the MILP approach in obtaining the global best solutions. Table 2 also presents the execution times of the MILP and BSA approaches. As can be observed, the BSA approach required relatively larger computational time compared to the MILP approach for the tested cases. However, the computational burden of the BSA approach can be ignored as the OPP formulation are solved offline before the installation of PMU and this research aimed to achieve the maximum number of measurement redundancy.  Table 2 also presents the execution times of the MILP and BSA approaches. As can be observed, the BSA approach required relatively larger computational time compared to the MILP approach for the tested cases. However, the computational burden of the BSA approach can be ignored as the OPP formulation are solved offline before the installation of PMU and this research aimed to achieve the maximum number of measurement redundancy.  Table 3 compares the obtained results of the proposed OPP formulation with the results of the referenced works without considering the presence of ZIB and channel limitation of the available PMU. The optimal number of PMU required through the proposed formulation is equal to the minimum number of PMU recommended by the ILP method of Ref. [29] and less than to the minimum number of PMU recommended by the two-stage method of the same literature. As mentioned earlier, this research modified IEEE 123-node test feeder to 128-node feeder and still required the same or a smaller number of PMU installation to ensure full grid observability. Consequently, it can be concluded that the proposed BSA approach is more promising over the  Table 3 compares the obtained results of the proposed OPP formulation with the results of the referenced works without considering the presence of ZIB and channel limitation of the available PMU. The optimal number of PMU required through the proposed formulation is equal to the minimum number of PMU recommended by the ILP method of Ref. [29] and less than to the minimum number of PMU recommended by the two-stage method of the same literature. As mentioned earlier, this research modified IEEE 123-node test feeder to 128-node feeder and still required the same or a smaller number of PMU installation to ensure full grid observability. Consequently, it can be concluded that the proposed BSA approach is more promising over the reported approaches in solving the formulated OPP problem. Additionally, the proposed approach ensured the maximum number of measurement redundancy whereas the reported methodologies did not consider the measurement redundancy issue. Besides, the reported OPP formulations for the distribution grids did not consider any contingency case, the inclusion of passive measurements and channel limitations of the PMU. Tables 4 and 5 present the optimum number and locations of the PMU along with the required and the achieved measurement redundancies subjected to full grid observability of the selected test distribution feeders during single line outage and single PMU loss cases, respectively. Both approaches observed the test distribution feeders completely and the BSA achieved more or at least an equal number of measurement redundancy with an equal number of PMU installation. Consequently, the obtained results confirmed the superiority of the BSA approach over the MILP approach in terms of achieving measurement redundancy even for the contingency cases. However, this paper refrained from presenting the execution times in Tables 4-10 as they are similar to the values of Table 2.       Table 6 presents the optimum numbers and locations of the PMU along with the required and the achieved measurement redundancy subjected to the full observability of the selected distribution test feeders for normal operating mode after incorporation of passive measurements/ZIB. As can be observed, both solution methodologies, the BSA and MILP approaches confirmed grid observability through the installation of an equal number of PMU. The incorporation of ZIB on the selected test feeders reduced the required number of PMU significantly hence the installation and configuration costs can be compared from Tables 2 and 6. For instance, the ratio of the required number of PMU to the total number of nodes (ψ) was reduced from 46.15% to 30.77%, 35.29% to 29.41%, 32.43% to 21.62% and 37.50% to 21.88% for normal operating mode of the selected IEEE 13-node, 34-node, 37-node and 128-node test distribution feeders, respectively. Additionally, the BSA approach achieved more measurement redundancy over the MILP approach for all the test distribution feeders. Similarly, Tables 7 and 8 present the optimum number and locations of the PMU along with the required and achieved measurement redundancies subjected to the full observability of the selected test distribution feeders during single line outage and single PMU loss cases, respectively.

Full Grid Observability without Considering ZIB and Channel Limits
The incorporation of the passive measurements on the selected test feeders reduced the required number of PMU significantly hence the installation and configuration costs, as can be compared from Tables 4 and 7. For instance, the ratio, ψ got reduced from 53.58% to 46.15%, 55.88% to 44.12%, 48.65% to 37.84% and 57.03% to 37.50% for the single line outage case for the selected IEEE 13-node, 34-node, 37-node, and 128-node distribution feeders, respectively. However, both solution methodologies observed the test distribution feeders completely and the BSA achieved more or at least equal number of measurement redundancy through the installation of an equal number of PMU. Therefore, it can be concluded that the BSA approach was superior in figuring out the global optimal solution while solving the proposed OPP formulation over the MILP approach.

Full Grid Observability Considering Both ZIB and Channel Limits
This section considers the channel limitations of the available PMU along with the incorporation of ZIB while ensuring full grid observability of the selected test distribution feeders during both normal and contingency cases. Table 9 presents the required number of PMU and their locations along with the required and achieved numbers of measurement redundancies during a single PMU loss case considering single channel of the available PMU and the incorporation of the passive measurements.
As can be observed, some nodes required several PMU due to the availability of the least number of PMU channel. Additionally, the obtained results show both solution methodologies achieved full grid observability with an exactly same number of PMU installation. Besides, both approaches achieved only the required number of measurement redundancy as Table 9 presents the most severe contingency case with the minimum number of PMU channel. However, for brevity's sake and to avoid repetition of similar tables and discussions, this paper refrained from presenting results of other operating cases by varying the channel limitations of the PMU, instead, all the cases have been summarized in Table 10. The locations of the PMU and the number of measurement redundancies for the presented simulated scenarios are omitted for brevity's sake. As can be observed from Table 10, the BSA approach required an equal or a smaller number of PMU installation over the MILP approach to achieve the full grid observability for both normal and contingency cases. For instance, the BSA approach achieved full grid observability with a total of 46 PMU installation whereas the MILP approach required a total of 48 PMU installation having a single channel for the normal operating situation of IEEE 128-node distribution feeder. In addition, the BSA approach achieved more measurement redundancy over the MILP approach for many cases and at least an equal number of measurement redundancy for other cases with the same number of PMU installation. Hence, the obtained results confirmed the superiority of the BSA approach over the MILP approach in terms of achieving more measurement redundancy.
It can also be observed that the required number of PMU was reduced with the increase of available PMU channels to ensure full grid observability of the selected distribution test feeders. It is worth mentioning that the proposed OPP formulation and the solution methodologies required the same number of PMU installation to ensure full grid observability with the PMU having four or more channels. Hence, it is neither necessary nor economically viable to install PMU with more than four channels for the tested distribution feeders. The PMU manufacturers, as well as the decision-makers of power system monitoring, may find this conclusion useful while designing the wide area measurement and monitoring scheme problem considering channel limits of the available PMU.

Conclusions
This paper formulated a revised, effective and comprehensive OPP problem for the distribution grids through achievement of the maximum number of measurement redundancy with the installation of a minimal number of PMU. The formulation was extended considering the channel limitations of available PMU and presence of the passive measurement under various contingency cases related to electric distribution feeders, i.e., the single line outage and single PMU loss. This research solved the proposed OPP formulation employing an effective heuristic optimization approach called BSA. The results exhibited that the heuristic approach can ensure full grid observability with lower or at least an equal number of PMU installation than the MILP approach and other referenced works. Additionally, the results confirmed the superiority of the BSA approach over the MILP approach in terms of measurement redundancy. However, it is worth mentioning that the employment of PMU with more than four channels did not minimize the required number of PMU for the tested distribution feeders rather increased the investment cost.
Author Contributions: All authors contributed to this research work through collaboration. Besides, all of them revised and approved the manuscript for publication.
Funding: This work was funded by the King Abdulaziz City for Science and Technology through the Science and Technology Unit, King Fahd University of Petroleum & Minerals, under Project 14-ENE265-04, as a part of the National Science, Technology and Innovation Plan.

Conflicts of Interest:
The authors declare no conflict of interest.

Nomenclature α
Multiplying factor for the control parameter F cp .