Wireless AMI planning for guaranteed observability of medium voltage distribution grid

Due to the scarcity of measurement devices, distribution systems often suffer from unobservability. The recent expansion of smart meters (SMs) provides the means to enable system observability. To ensure continuous situational awareness of distribution grids, strategic planning of advanced metering infrastructure (AMI) is crucial. However, conventional AMI planning approaches only consider the economically optimal placement of communication devices without considering the need for observability even in case of failures. To address this challenge, this paper proposes a wireless AMI network planning framework that guarantees the observability of distribution grids in the face of potential communication failures. Specifically, given the failure scenario, a closed-form observability criterion is formulated and its observability fortification strategy is derived, based on which the AMI network planning problem is formulated as an integer linear programming (ILP) problem. In addition, a heuristic decomposition technique is applied to the ILP problem in order to address the scalability issues of large-size networks. Finally, case studies demonstrate the robustness and effectiveness of the proposed AMI system planning framework. The findings of this work assist distribution utilities in developing a reliable and economical AMI, while providing guaranteed situational awareness of their


Introduction
Full network observability is the key prerequisite for enabling distribution management applications such as voltage regulation, demand response, and topology reconfiguration in medium voltage distribution grids (MVDGs) [1].As defined in [2], full observability of a distribution system is given if voltage phasors at all buses can be uniquely estimated given a set of measurements.However, except for substations, where voltage and line currents are directly measured, most feeder line buses are not equipped with high-precision sensors [3].This results in limited observability for MVDGs, which was acceptable in the past given the stationary characteristic of traditional consumption patterns [4].However, with the massive integration of renewable energy sources, electric vehicles, and energy storage systems, MVDGs now require realtime observability of system conditions to enable granular and precise management, which cannot be fulfilled by limited observability of the system [5].
To address the limited observability issues, load forecasts from low voltage (LV) networks have been widely used as pseudomeasurements to approximate power injection at unmeasured MVDG buses.However, pseudomeasurements from aggregated load forecasts are inherently inaccurate and outdated, as prior knowledge used for load forecasting models cannot adapt to the real-time variance of system dynamics [6].In addition, load forecasting models typically rely on other input data (e.g.weather) to predict load demands, making the accuracy of pseudomeasurements subject to the availability and uncertainty of other sources of information [7].
To better monitor the MVDG operation conditions, previous works studied the optimal placement of dedicated measurement units such as phasor measurement units (PMUs).For instance, [8] proposes a stochastic meter placement strategy that meets the estimation accuracy requirements with minimum costs under various topology scenarios.The work in [9] proposes a greedy placement strategy of measurement points at MV network nodes to monitor branch voltages with minimum error variance.Similarly, [10] uses a submodular saturation algorithm to minimize the worst-case estimation variance for the state estimation process.By constructing a maximization of determinant problem, [11] formulates a Boolean convex problem to select the optimal subset of measurement units in a candidate set.To ease the computation complexity, a mixed-integer linear programming program is decomposed into two layered subproblems for the optimal placement of micro-PMUs https://doi.org/10.1016/j.apenergy.2024.123598Received 5 January 2024; Received in revised form 12 May 2024; Accepted 28 May 2024 under contingency scenarios in [12].However, all these studies assume full observability of the MVDG and neglect the possibility of failures of measurement devices.
On the other hand, the rapid expansion of SMs in AMI enables close to real-time measurements of load consumption at the LV side.As AMI typically provide measurements from SMs to the distribution system operators (DSOs) at a certain frequency (e.g.every 15 min), the measurements from SMs can be leveraged to improve or replace pseudomeasurements [13].Full utilization of SMs also reduces additional investments in expensive metering devices such as PMUs [14].According to existing studies, SMs can be leveraged for a variety of applications, including load forecasting [15], state estimation [16], topology identification [17], non-technical electricity losses detection [18], and fault detection [19].As demonstrated by the landscape of usage, SMs provide extensive potential to be exploited for smart grid applications.
In practice, however, measurement data from SMs are not fully utilized because of their asynchronous arrival time to the control center.SMs equipped with real-time clocks (RTC) in AMI can coordinate the sampling time with the control center to enable synchronization of measurements [20].Nevertheless, due to the limited capacity of communication bandwidths, the current design of AMI generally cannot accommodate the simultaneous upload of measurement data of all SMs [21].Specifically, the simultaneous upload of measurement data can cause congestion in the communication network, which ultimately leads to packet losses of SM measurements.The limited capacity of AMI hinders the synchronous update of measurement data and prevents the timely observability of MVDGs [22].
Continuous observability of MVDGs is also dependent on the reliability of AMI.Acting as gateways, data aggregation points (DAPs) in AMI are critical components to relay aggregated data from downstream SMs to the control centers [23].Due to quality of service (QoS) constraints, depending on the location, a DAP may not be able to reach downstream SMs that exceed its maximum communication distance, causing a cluster of measurement data to be discarded.For this reason, extending the communication distance for SMs is an approach to alleviate the issues related to the limited coverage area of DAPs.To extend the communication distance of SMs, recent research adopted multi-hop routing techniques to relay the measurement data from one SM to another.For example, [24] proposes a clustering scheme for the DAP placement problem while using multi-hop routing to ensure redundant connectivity to SMs that are not within the same domain.In [25], the evolutionary aggregation algorithm is used to determine the n-hop structure of a wireless smart meter network, whereas [26] uses an iterative clustering approach for multi-hopping enabled SMs to place DAPs such that installation costs, transmission costs together with communication delays are minimized.Using multi-hop routing is able to extend the coverage area of DAPs and avoid excessive planning of additional DAPs.However, SMs that serve as relays in multi-hop routes are susceptible to single-node failure.In fact, as mentioned in [27], the cost-driven setting of multi-hop routing renders the proposed AMI network vulnerable to failures of the intermediate SMs.
To address the single-node failure problem, a few studies consider adding redundancy to enhance AMI communication reliability.For instance, in [28] a theoretical analysis is conducted on a redundant path-searching problem for SMs and theoretically proves the computation complexity of the formulated problem to be NP-hard, i.e., difficult to solve for large networks.On the other hand, [29] proposes a powerdisjoint path for DAPs to increase AMI network reliability.Instead of focusing on infrastructure planning, a middleware is added on top of the physical communication layer to enable redundant connectivity of measurement devices [30].However, the aforementioned works achieve communication redundancy through the addition of SMs and DAPs.In reality, the end-to-end distance between an SM and DAPs on the redundant paths may be longer than the maximum wireless transmission distance of SMs, especially in suburban and rural areas [31].Some remote SMs cannot connect to the neighboring DAPs without the aid of wireless repeaters, making the aforementioned strategies impractical under such geographic situations.
Present research has primarily focused on either reliable AMI network planning or the observability analysis of MVDGs using the existing AMI network.As summarized in Table 1, few papers have studied the interdependence between MVDG observability and AMI network planning in a cohesive manner.Specifically, the impact of AMI network failures on the observability of MVDGs has not been quantified.Furthermore, strategies to fortify the AMI network in order to maintain MVDG observability during failure events are missing.These research gaps highlight the need of a holistic AMI planning framework that can preserve the observability of MVDG under various types of communication failures.To this end, the first objective of our work is to devise an AMI fortification strategy that can withstand multiple communication failures without losing the observability of a MVDG.The second objective is to propose an AMI planning model that incorporates the derived fortification strategy.The main contributions of the paper are outlined as follows: 1. We propose a robust AMI planning model that is capable of handling any single-node failure scenarios by applying a redundancy-driven fortification strategy.Compared with the existing AMI planning paradigm, our robust AMI planning model can withstand disconnections of any single wireless communication path without affecting the observability of the respective MVDG. 2. Considering the non-convex scenario-based observability criterion that cannot be expressed in a closed form explicitly, we reformulate the guaranteed observability constraint into a tractable communication fortification problem using rank multiplication properties.3. Formulated as an integer linear programming (ILP), the conventional AMI planning problem suffers from scalability issues due to the increasing size of the LV grids.Herein, a heuristic decomposition method is proposed to address this issue.The computational burden is reduced such that the large-scale ILP problem can be solved relatively fast while maintaining the accuracy of the solution.
The rest of the paper is structured as follows.Section 2 highlights the critical issues in AMI communication planning with respect to ensuring observability and proposes a general planning framework.In Section 3, we illustrate how to formulate a generalized expression that quantifies the impact of failure scenarios and the corresponding fortification strategies on observability.We then derive a closed-form formulation for a scenario-based observability constraint in Section 4. In Section 5, we present case studies to verify the effectiveness of the proposed framework.Finally, Section 6 summarizes our findings and draws conclusions.

Problem statement
Existing AMI communication networks are designed with data transmission reliability in mind, but in general do not incorporate grid operation requirements, i.e. network observability, into the planning stage.This is primarily due to the fact that smart meters are originally deployed for billing purposes and only later become an integral part of system operation-a function that AMI has been tasked with in recent years.To establish an AMI planning framework that considers network observability, we need to address the following two main challenges.The first challenge is to establish an observability metric that can link the communication failure scenarios with the corresponding fortification strategies.Specifically, a generalized observability constraint should be formulated incorporating possible failure scenarios, ranging from natural disasters to malicious attacks.The second challenge is to include the observability-guaranteed constraint into the AMI planning

Failures of communication devices
Yes [4][5][6] problem.In other words, to integrate the constraint into an optimization problem, a closed-form expression is needed rendering the scenario-based observability criterion tractable.Thus, the proposed AMI planning framework is realized in the following three steps: 1. Observability analysis: formulate a generalized observability criterion that incorporates both failure scenarios and the corresponding fortification strategies.2. Closed-form formulation: transform the intractable scenario-based observability criterion into a closed form formulation that an off-the-shelf solver can solve.3. Communication network planning : design the AMI communication network while minimizing the overall installation costs.
In Section 3.1, we derive a mathematical expression for scenariobased observability for any MVDG.It combines the communication disruption scenarios with the communication network fortification strategies.Section 3.2 then transforms the intractable scenario-based observability criterion into a closed mathematical form, which is later integrated into the AMI planning problem in Section 4.

Observability for MVDG
This section first illustrates how to utilize the AMI network to provide full observability of MVDG.On this basis, the observability criterion that considers both failure scenarios and fortification strategies is formulated and transformed into a closed-form expression.

Observability analysis based on AMI
For a MVDG with +1 buses, the voltage magnitude and angle of the HV/MV substation-connected bus is assumed to be directly measured and is selected as the slack bus for the analysis.Thus, to obtain full observability of the MVDG state variables at the rest of the  buses need to be uniquely determined.Specifically, the states for any MV bus  include both three-phase voltage magnitude  1,2,3 ,  2  ,  3  ]  .Thus, the state vector  for a MVDG can be formulated as: Meanwhile, a measurement vector  can be used to infer the state vector of a fully observable system via the following linearized expression: where  represents the Jacobian measurement matrix.Hence, the Jacobian measurement matrix is based on the system equations that link a measurement vector with the state vector.To construct a Jacobian matrix, we first express both active and reactive power injection measurements at any bus  for phase ℎ ∈ {1, 2, 3} (i.e. ℎ  and  ℎ  ) as a function of its own states and states from all adjacent buses: where  ℎ,  and  ℎ,  are the entries in the admittance matrix Y = G+jB between phase ℎ and phase .In case, the active and reactive power injection at the MV/LV bus is not directly measured, measurements from SMs in the downstream LV networks are aggregated to approximate the power injection to the MV/LV buses for phase ℎ.As shown in Fig. 1, the power injection Eqs.(3a) and (3b) measured at secondary (MV-LV) transformers can be approximated by the sum of all power consumption or production by downstream users at the same phase plus the LV network line losses at this phase.Due to the high correlation between aggregated power consumption and line losses, two factors (i.e.(   ) ℎ , (   ) ℎ ) are used to multiply the aggregated active and reactive power consumption to approximate the total power injection from the LV network into the MV bus, while accounting for errors related to time synchronization.We account for the fact that LV grids can be unbalanced and denote injections from each phases ℎ ∈ {1, 2, 3}.Hence, if the set of all downstream SMs aggregated for a MV bus  at phase ℎ is denoted as N ℎ  , the active power and reactive power injection at bus  for phase ℎ can be approximated by the sum of active and reactive power at each phase: As the most common measurement data from smart meters are active and reactive power consumptions, based on (3a) to (4b), the Jacobian measurement matrix   provided by the AMI network is derived as: Since changes of line parameters or grid topology may affect the Jacobian measurement matrix (5), it is necessary to conduct a pre test on various topologies and line parameters to determine the optimal placement for AMI devices under different scenarios.As a result, the optimal placement of AMI communication devices can be determined at the locations with the most overlap in all topology settings.
In addition to SMs, other measurement devices in the distribution grids can contribute to the Jacobian measurement matrix (5) as well.However, critical nodes require guaranteed observability and cannot tolerate loss of measurements.As a result, planning of AMI on MVDGs is treated independently from other measurement units, in order to enable robust observability.In spite of this, the following observability evaluation can be generalized to other measurement units for MVDG.
Without communication failures, the constructed measurement matrix has a full rank of 6.However, when the AMI communication network experiences disconnections, especially at the sink nodes, the full rank of the MVDG cannot be ensured.For example, when a group of DAPs is disrupted, those aggregated consumption data are lost, which results in some undetermined rows in the Jacobian measurement matrix and eventually this matrix does not have full rank any more.In order to quantify the impacts of disruptive events on the observability of the MVDG, mathematical mappings between the disruptions and the system observability are needed.In addition, the fortification strategy to counteract disruptive events to maintain system observability also needs to be modeled.
Let the communication fortification decisions be defined in a fortification vector  = [ 1 ,  2 , … ,  6 ] for all 6 measurements.Each entry of the fortification vector corresponds to an integer representing the number of communication paths for a measurement.Likewise, any communication failure scenario  can be defined in a disruption vector  = [ 1 ,  2 , … ,  6 ], where each entry corresponds to the vulnerability for the respective measurement.A measurement is considered vulnerable if the estimation error exceeds the allowable threshold after a missing measurement is replaced with its pseudomeausurement.A higher weight is assigned to the vulnerable measurements in the disruption vector .The disruption vector  can also be used to quantify how many communication paths between SMs and DAPs have been compromised after a disruptive event.Together with the measurement matrix, a generalized expression of scenario-based observability is formulated as: where () + leaves the corresponding entry unchanged if it is a positive value and changes it to zero otherwise.The  operator transforms the failure and fortification vectors into matrix form that can be multiplied with the measurement matrix.In this way, a scenario-based mathematical mapping among communication failures, fortification strategy, and system observability is introduced.In spite of various failure scenarios, the scenario-based observability constraint allows impact-driven evaluation and can therefore be extended to consider a wide range of disruptive events.
It should be noted that the mapping expression ( 6) is non-convex, rendering it difficult to integrate it into an optimization problem.In the next section, a closed-form expression is derived to facilitate its integration into the AMI planning problem in Section 4.

Closed-form formulation
The guaranteed observability criterion based on the scenario-based mapping ( 6) can be expressed as: where the failure scenarios are within the defined set of events .Constraint (7) dictates that the lowest rank caused by the worst-case scenario of disruptive events shall be tightly bounded by the number of state variables 6 using any available fortification strategy.We can now apply the following rank properties to derive the closed-form expression.
Theorem 1.If matrix  and matrix  can be multiplied, the rank of the resulting matrix is the minimum rank of either matrix  or matrix  , i.e. (  ) ≤ min(( ), ( )) Lemma 1.Given a full rank square matrix  with size  ×  and a matrix  with size  × , the rank of a product of  and  is equal to the rank of  , namely (  ) = ( ) Proof.Since square matrix  is invertible, we have: According to Theorem 1: Combining inequality (8) and (10) we prove that: (  ) = ( ).□ According to Lemma 1, the equivalent form of the rank expression of ( 7) is derived as: where 1 denotes the indicator function that equals one when (  −   ) > 0 is satisfied.Thus, the guaranteed observability criterion is to ensure that the sum of the indicator functions is larger than the number of required state variables under the worst disruption vector , namely: We can further define the set of measurements to be N  = {1, … , 6}.By introducing ancillary binary variables   to represent the condition of the indicator functions, problem (12) can be reformulated as the following closed-form optimization problem (13): where  in constraint (13b) and (13c) is a positive number whose value can be chosen as: to enforce the inequality within the indicator function in constraint (12), while not introducing infeasible regions for the problem.Here a random small number  is introduced to ensure that the strictly larger requirement is met.By doing so, constraints (13b) and (13c) assess whether all measurements can withstand the disruptions after fortifications.Survived measurements are indicated by   = 1.Constraint (13d) then requires the total survived measurements to be at least equal to the number of states to be estimated.Constraint (13e) enforces the realistic requirement that the number of redundant paths has to be a non-negative integer.Together with the objective function (13a), the full observability constraint ( 12) is reformulated into a closedform expression from which the fortification decisions can be derived accordingly.

Communication network planning
With the fortification decisions derived from problem (13), the next step is to formulate the AMI planning framework that can ensure full observability of a MVDG.For the wireless communication network planning problem, a maximum communication distance constraint is derived based on two quality of service (QoS) requirements and integrated into the corresponding AMI network planning optimization problem.Finally, to address the scalability issues, we propose a heuristic problem decomposition approach to decompose the planning problem into subproblems.

QoS requirement
The first QoS requirement considered in the wireless AMI network is called packet success rate (PSR), which corresponds to the packet error probability.In addition, the packet delivery delay (PDD) should be within an acceptable range.To meet these two requirements, the signal-to-inference and noise ratio (SINR) between device  and  shall be above a threshold SINR ⋆ , which is defined as [32]: where    denotes the transmission power between device  and  and  −  is the distance between device  and  with a path loss factor .The denominator of (15b) represents the sum of Gaussian white noise variance  2 of the communication channel and the cumulative wireless inference in the environment  0 .To avoid fading signals, we further derive the upper bound of the transmission distance under the acceptable SINR by combining (15a) and (15b): Given the minimum SINR ⋆  between two communication devices, the upper bound of the communication distance     can be calculated based on (16).In other words, the maximum coverage distance between device  and device  is parameterized before the planning problem.Hence, by introducing the maximum transmission distance derived from ( 16), we can enforce the placement of devices (i.e.wireless repeaters and DAPs) to satisfy both PSR and PDD in the following wireless AMI planning problem.

Conventional planning problem
Given a set of N  = {1, … , } MV buses, let N  = ∪ ∈N  N   be the set of all DAP candidate positions, where N   is the subset of DAP positions located in the LV network connecting to MV bus .Likewise, let N  = ∪ ∈N  N   be the set of all repeater candidate positions.
denotes the set of SM indexes and here we assume every household in the LV networks is equipped with a SM.   ∈ {0, 1} and    ∈ {0, 1} are the decision variables for placing DAP and wireless repeaters at candidate location  ∈ N  and  ∈ N  , respectively.For brevity, ∀ ∈ N  , ∀ ∈ N  and ∀ ∈ N  are simplified as ∀, ∀, ∀ in the following problem formulation.
A core objective of the AMI planning problem is to minimize the total installation costs for DAPs and wireless repeaters and the communication costs associated with the redundant paths.In addition, to minimize the routing delays, the objective function should also penalize any unnecessary long-routing paths.In other words, direct connections between SMs and DAPs are prioritized to reduce the hopping delay that indirect connections between SMs and DAPs may cause.On the other hand, to realize fortification, the planning problem aims to build vertexdisjoint paths such that a path failure does not affect the functionality of the other path.Hence, the planning problem is formulated as follows: ∈ {0, 1},   ∈ {0, 1} ∶ ∀, ,  (17o) where   ,   ,   represents the link between SM  and DAP , SM  and repeater  and repeater  and DAP , respectively.  denotes the longrouting path between SM  and DAP  through repeater .Parameters   and   denote the unit price of a repeater and DAP, respectively.  is the unit cost for the long-routing path.  in constraint (17i) counts the number of disjoint paths leaving SM , which has to be greater than the fortification decisions derived from problem (13).Constraints (17f) to (17h) limit the maximum coverage distance of SMs, repeaters, and DAPs to satisfy their minimum acceptable SINR requirements.Constraints (17k) to (17l) specify the maximum load capacity that a DAP and a repeater can handle.In (17m), the vertex disjoint path is enforced between the direct connection and indirect long-routing connection starting from a SM .
Since both constraints (17d) and (17e) include the product of binary variables which make the constraints non-convex, the following linearization of these two constraints is used.
Hence, the bilinear terms associated with   and   are linearized into multiple linear constraints, rendering constraints convex.
With these linearized constraints, the conventional AMI planning problem is formulated as the following ILP problem: Hereafter we will refer to this ILP problem as the conventional planning problem which couples the placement of repeaters and DAPs to represent the communication paths.

Decomposed planning problem
Due to the fact that redundant path variables   combine decisions from both repeaters and DAPs, the total number of decision variables increases greatly for larger LV networks.This is because the decision variable of a redundant path considers the entire candidate set of repeaters and DAPs in a MVDG.As a result, for large LV networks, scalability issues can occur due to the NP-hard nature of an ILP problem.To address the scalability issue, a two-step heuristic divide-and-conquer strategy is applied to decompose the conventional planning problem.The first step is to formulate the AMI placement subproblem that ensures that the basic communication requirements are satisfied.Subsequently, the AMI planning that considers the fortification decisions is solved based on the results obtained from the first subproblem.
In practice, for power consumption billing purposes, at least one communication route from SMs to DAPs is needed for each SM.In addition, objective function (17a) penalizes the use of long routes, which incentivizes the optimizer to select a direct connection path instead of a long-routing route.In this case, we can first optimize subproblem 1 with a single communication path for all LV networks as basic connections.Based on these results, the second subproblem is solved separately on every single MV bus where at least one more redundant path is needed, i.e.,   ≥ 2.
First, subproblem 1 for the basic connection is formulated as: where only DAPs are considered to establish basic connections to all the downstream SMs.The solution of subproblem 1 provides a subset N 1 ⊆ N  of DAPs which denotes preselected DAP positions that are fixed in the following subproblem 2.Here we denote the DAP capacity used for subproblem 1 as ( ∑      ) 1 .To build redundant communication routes for critical MV buses that requires   ≥ 2, an iterative repeater selection process is applied.In each iteration, given a fixed number of repeaters, a capacity-constrained clustering approach [33] is applied to pre-select a subset N 1 ⊆ N  of repeaters that can cover all SMs from critical buses while satisfying the repeater capacity constraint (17l).The number of repeaters, same as the number of clusters, is incrementally added until both the maximum communication distance between SMs and repeaters (17g) is within the acceptable range.On this basis, subproblem 2 is to derive the optimal placement strategy for DAPs and repeaters to satisfy the disjoint path requirement (17j): where   in constraint (20i) denotes the size of clustered SMs belonging to repeater .Constraint (20g) ensures that the repeaters do not connect to the DAP that is physically connected to the same LV network from bus  to meet the disjoint path requirement.In this case, Fig. 2 shows the overall problem decomposition procedure.

Numerical validations
Under extreme disruptive events (e.g., malicious attacks, natural disasters, etc.), the substation metering devices may be compromised and generate untrustworthy measurement data.In the worst-case scenarios, DSOs may have to completely rely on the AMI network to provide full observability of the distribution system.The reliability of the AMI network and overall situational awareness is however endangered, when the network is subjected to communication failures.For this reason and taking state estimation (SE) as an example application, we first study the impact of a single DAP failure scenario.By doing this, we are able to evaluate the benefit of observability provided by fortification strategies taking into account various communication failure scenarios.Then, we present the AMI planning layout corresponding to a single DAP failure scenario to verify the realization and effectiveness of its corresponding fortification strategy.Additionally, the effectiveness of the proposed decomposed method is compared with the conventional planning approach with respect to the optimality of planning decisions and the usage of computational efforts.

Simulation settings
Numerical assessments are conducted using a medium-voltage (MV) test system based on the IEEE 33 bus distribution grid.It is assumed that a radial LV network is connected to each MV bus.Each node of the LV network represents a household or building that is equipped with at least one SM.In order to define geographical positions for the SMs, SM positions are sampled using a Monte-Carlo simulation within a square area of 1km × 1km centered around the MV buses as shown in Fig. 3.By randomly repositioning the SMs, we can study the robustness of the proposed method under different communication topologies for the LV networks.The candidate positions of DAPs are limited to the locations of the MV/LV transformers.Wireless repeaters are selected from candidate positions that maximize the coverage area from downstream SMs to redundant DAPs while meeting installation requirements.The relevant parameters for communication network components are given in Table 2.In the following tests, an actual load profile is obtained from the Global Energy Forecasting Competition 2012 (GEFCom2012) [34].On the other hand, the SM measurement data are generated from using this load profile with the addition of simulated measurement errors according to the American National Standard Institute (ANSI) C12.20-1.0class for electricity SMs.The proposed approach is implemented in Python 3.9.10environment using Gurobi optimizer version 10.0.0.All numerical evaluations are executed using a workstation equipped with an Intel i7-8700K computing processor and 32 GB of RAM.

Evaluation metrics
Two metrics called relative absolute voltage magnitude error (RME) and relative absolute voltage angle error (RAE) are used to evaluate the accuracy of the estimated voltage magnitude and angle   and   at bus  over three phases: where the true voltage magnitudes and angles are given as  ℎ  and  ℎ  at bus  and phase ℎ.
The thresholds of acceptable voltage magnitude deviations (AMD) and acceptable voltage angle deviation (AAD) are set according to Ref. [35] to 1% and 5%, respectively.Hence, unacceptable SE results are identified by the number of over-RME (ORME) buses and over-RAE (ORAE) buses, i.e.: and the sum thereof as:

Simulation results
If a MV bus becomes unobservable due to AMI network failure, missing aggregated active and reactive measurements from SMs are typically replaced by pseudo-measurements.
Here the pseudo-measurements are taken from a typical short-term load forecasts with mean square error of 5% [7].Therefore, by comparing the SE results obtained from the AMI network with fortified communication to those communication failure cases that have to rely on pseudomeasurements, we can quantify the benefits of the fortification strategy and its ability to ensure guaranteed observability.In this case, the SE errors associated with the single-point failure of each DAP are presented in Fig. 4.
Except for the slack bus (i.e.bus 1), Fig. 4 shows the RME and RAE of SE results under the single-point failure scenarios of DAPs from bus  2 to bus 33, with a case when the AMI network is completely fortified.
To further assess the extent of estimation errors, two red dashed lines in each subfigure indicate the upper limit AMD for RME and AAD for RAE, respectively.Note that scenarios that do not violate limits are plotted in grey.
As can be seen in Fig. 4(a), RME increases with increasing distance from the substation.This is expected given that the feeder line losses associated with the shunt capacitance of MVDG are not incorporated into (4a) and (4b).This error is accumulated as the line becomes longer.For this reason, the RME reaches a peak at the end bus (bus 18) of the longest feeder, violating the AMD limit.The same is true for bus 33, which is the end bus of the second longest feeder line.On the other hand, the magnitude of the connected load is found to affect the accuracy of the estimated phase angle.In particular, failures at heavily loaded buses 24 and 25 result in AAD violations as shown in Fig. 4(b).Here, because of the large loads connected to these buses, the high equivalent impedance from the LV network has a similar impact as those end buses on the feeder.In addition, when both physical distance and loads from LV networks are significant, the impact of line losses is exacerbated exceeding both AMD and AAE limits, e.g. for bus 30.This is also reflected in a larger TUE value.
From the simulated 32 failure scenarios, the estimation of the phase angle is more sensitive than its magnitude counterpart.This is expected given the small angle differences between buses due to the geographical distance and the line R/X ratio.In other words, the angle estimation is more sensitive to measurement accuracy than the magnitude estimation.We can argue that the vulnerable buses, i.e. the buses that experience violations, are those that have a large equivalent impedance with respect to the feeder, which typically involves MV buses connected to large loads and located physically far from the substation (see Tables 3 and 4).
Since vulnerable buses are sensitive to the inaccuracy of measurement data, it is imperative to fortify the AMI networks to ensure reliable measurement data uploading paths.Fig. 5(a) shows an example of how the fortification strategy is implemented on the identified vulnerable buses to satisfy the N-1 communication network reliability.Specifically, the fortification routes associated with the LV network connected to MV bus 30 are shown in an enlarged plot in Fig. 5(b).Two aspects are important: there are two types of vertex disjoint paths connecting every SMs within the fortified LV network to the DAPs on MV buses; one is through a direct connection from SM to DAP, and the other is through redundant paths to neighboring DAPs.This allows downstream SMs to have a backup route to upload their data.Second, wireless repeaters are commonly installed in a diagonal direction geometrically in the LV networks.This is manifested in Fig. 5(b) that the repeaters for both bus 30 and 33 are placed on the diagonal direction with respect to the DAPs in their LV networks.With a minimum number of repeaters installed, this placement strategy leads to a maximum coverage area of LV networks.In addition, the long route penalty in the objective function eliminates the placement of any unnecessary repeaters.These properties for the placement of the repeaters with respect to the topology of a feeder can be observed in all fortified LV feeders.To verify the scalability of the proposed method, we conducted AMI planning on both the IEEE 69 distribution system and a large-scale 300 buses MVDG.Similar observations with respect to the repeater placement locations can be made in these two large scale MV network test case, as shown in Figs. 6 and 7. Lastly, to verify the computational effectiveness of the decomposed method, we compare the difference in computation time and the RAM usage between the conventional planning problem and the proposed planning framework with the model decomposition.Overall, the decomposed method is not sensitive to the size of the LV networks, i.e., computing time is constant irrespective of the network size.In contrast, the computation time of the conventional problem increases non-linearly when the LV network size increases as shown in Fig. 8.Such increasing trend verifies the NP-complete nature of the conventional problem.As the number of SMs reaches 150 in each LV network, the memory usage of the Gurobi solver surpasses 32 GB RAM.The high demand for memory space for the conventional planning problem is due to the fact that the ILP is solved by the branch-and-bound method.During the optimization process, the intermediate branch results need to be stored in RAM.As the number of integer variables increases, the intermediate branch results grow exponentially, running out of memory.On the other hand, the proposed decomposition method decouples the size of the LV system from the DAP/repeater placement problem, which effectively reduces the number of integer decision variables to a minimum and hence renders the decomposed problem scalable to large-scale problems.The efficiency of the decomposition method is further verified in the large scale MV network with 300 buses, shown in Fig. 7.For this 300 bus MV network test case, the decomposition method requires around 6 min, while the conventional method exhausts all memory and fails to plan the AMI.
To evaluate the optimization performance of the decomposed method, Fig. 9 illustrates the difference in the installation cost between the conventional method and the benchmarked decomposed approach (red dashed line).Note that the price of DAP and repeater are based on [26].The difference is observed to gradually decrease as the LV network size increases.This is because the decomposed problem prioritizes the placement of DAPs and assigns SMs to each repeater only based on their connection capacity constraints, resulting in the decomposed problem overestimating the demand for DAPs and repeaters.This optimality gap reduces significantly when the number of SMs in the LV network exceeds 25.That is, the marginal cost of building a repeater-based redundant route exceeds the marginal cost of building a direct route with an additional DAP installed.Therefore, the placement of DAPs becomes more economical and thus is favored in problems with an increasing size of LV networks.For this reason, the LV network size of the conventional problem gradually approximates the upper bound set by the decomposed method.This result suggests that the decomposed method is suitable for AMI planning in large LV networks, and is especially advantageous when ad-hoc communication networks need to be planned in the event of an emergency.

Conclusions
In this paper, we present a generalized AMI planning framework to ensure the observability of MVDGs using downstream SMs in the LV networks.From simulation results, we observe that communication failure scenarios associated with MV buses that have a high equivalent impedance can lead to unacceptable state estimation errors.On this basis, a fortification strategy is proposed to strengthen the reliability of the AMI network.The most economical layout for the placement of repeaters to enable a fortification path typically lies in the diagonal direction of the respective MV buses.In this case, the installation cost and the communication latency are both minimized.By comparing the computational time and optimality of the results against the conventional planning method, we demonstrate the scalability and accuracy of the proposed decomposed planning method for large LV networks.The findings also imply that the reliability achieved by the AMI network remove the need to install additional measurement devices in the substations to maintain full situational awareness during grid emergencies.This could lead to significant cost saving for power utilities in terms of upgrade investments to strengthen monitoring resilience.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 1 .
Fig. 1.Aggregation of SM measurements to ensure the observability of every MVDG bus.

Fig. 3 .
Fig. 3. Overview of the IEEE 33 bus distribution grid (MV-LV) topology.The dashed box illustrates the connectivity between MV and LV networks.Note that SMs are installed in LV networks.

Fig. 4 .
Fig. 4. Summary of state estimation errors (dots along the vertical axis) incurred at all 33 buses due to various DAP failure scenarios compared with the fortified AMI case: (a) relative absolute voltage magnitude error (RME), and (b) relative absolute voltage angle error (RAE).Note that the red dash-lines refer to their respective thresholds, i.e., AMD and AAD, respectively.The colored dots indicate DAP failure scenarios with estimation errors at any of the 33 buses exceeding the threshold, while the grey dots indicate DAP failure scenarios with estimation errors at all of the 33 buses below the threshold.

Fig. 5 .
Fig. 5. Placement of DAPs and repeaters with fortification strategy for 7 most vulnerable failure scenarios with (a) an overall view on IEEE 33 buses, and (b) a zoom-in view between bus 30 and bus 33.

Fig. 6 .
Fig. 6.Placement of DAPs and repeaters with fortification strategy on IEEE 69 bus distribution system: (a) overall view and (b) zoom-in view around the central area.

Fig. 7 .
Fig. 7. Placement of DAPs and repeaters with fortification strategy on a MVDG with 300 MV buses: (a) overall view and (b) zoom-in view around MV buses 55, 61 and 76.

Fig. 8 .
Fig. 8.Comparison of computing time and RAM usage between the conventional planning method and the proposed fortification strategy with model decomposition.

Fig. 9 .
Fig. 9. Comparison of installation cost with respect to both DAP and repeater between the conventional planning method and the proposed fortification strategy with model decomposition.

Table 1
Summary of existing AMI planning models.

Table 2
Parameters for simulation.

Table 3
Highest aggregated active and reactive power consumption in the MVDG.

Table 4
Summary of DAP failures resulting in violation of AMD and AAD thresholds.