Ecological Robustness-Oriented Grid Network Design for Resilience Against Multiple Hazards

Power systems are critical infrastructure for reliable and secure electric energy delivery. Incidents are increasing, as unexpected multiple hazards ranging from natural disasters to cyberattacks threaten the security and functionality of society. Inspired by resilient ecosystems, this article presents a resilient network design approach with an ecological robustness (RECO)-oriented optimization to improve power systems' ability to maintain a secure operating state throughout unknown hazards. The approach uses RECO, a surprisal-based metric that captures key features of an ecosystem's resilient structure, as an objective to strategically design the electrical network. The approach enables solvability and practicality by introducing a stochastic-based candidate branch creation algorithm and a Taylor series expansion for relaxation of the RECO formulation. Finally, studies are conducted on the RECO-oriented approach using the IEEE 24 Bus RTS and the ACTIVSg200 systems. Results demonstrate improvement of the system's reliability under multiple hazards, network properties of robust structure and equally distributed power flows, and survivability against cascading failures. From the analysis, we observe that a more redundant network structure with equally distributed power flows benefits its resilience.


I. INTRODUCTION
Power systems deliver the electric energy that ensures the functionality of modern society.However, the infrastructure is aging and remains vulnerable to physical disturbances and natural disasters [1], such as the Winter Storm Uri in Texas in 2021.The integration of communication networks into critical infrastructure enables improved functionality but also increases the risk of cyber-originated and combined cyberphysical attacks to cause unexpected outages [2], [3].The design of a resilient grid network is thus an essential foundation for its inherent abilities to withstand such hazards.
Resilience is a property of systems that represents their ability to recover from adverse conditions.From a regional transmission operator perspective, Chen et al. emphasize the necessity of constructing a robust grid to allow operators to address various contingencies on any given day [4].In [5], Gholami et al. list different areas of resilience enhancement regarding system planning and operations, where long-term planning resilience enhancements lay the foundation for shortterm operational resilience enhancements.Both [6], [7] find that redundant and robust network structures are effective for improving power system resilience under extreme conditions.These works highlight the importance of network design for enhancing power system resilience, while motivating the need to better understand and characterize the effective use of design against extreme events.Inspired by resilient ecosystems, this work develops a resilience-oriented design approach for large scale power systems that improves their inherently ability to absorb the disturbances from multiple hazards.The novelty of this work is to introduce an optimization based resilient design approach that is realistic, scalable, and extensible to translate the long-term resilient trait of ecosystems, ecological robustness (R ECO ), into power network design with the consideration of power system constraints, including the power balance, power flow equations, and operational limits.The goal of the proposed R ECO Oriented Power Network Design Problem is to strategically add redundancy to power networks and satisfy the constraints of power systems for improving the system's resilience.The main contributions of this paper are as follows: • This paper presents a resilience-oriented approach to improve power systems' inherent ability to tolerate unexpected high-impact disturbances and maintain functionality securely.A quantitative resilience metric, R ECO , is formulated as an objective for network optimization considering the power system's constraints to guide resilient power network design, ahead-of-time without intelligence of the threat.• A stochastic-based algorithm to create candidate branches and a Taylor Series Expansion of the logarithm function in the formulation are proposed to scalably solve the optimization, and the R ECO -oriented design problem is solved for 24-and 200-bus systems under different scenarios.• The R ECO -oriented power networks are examined under different levels of N-x contingencies, and their network properties and power flow distributions are analyzed.The analyses show that a more redundant power network structure with more equally distributed power flows contributes to a more resilient power system.• R ECO is shown to be an effective metric to help measure and improve the inherent resilience of power networks.
The formulation can guide design of power network structures considering power flows and ecosystems' resilient traits to achieve power systems' long-term resilience.
Section II reviews other resilient power network design approaches and introduces the research objective of this work.Section III presents related work on unexpected critical multihazards in power systems and the background of R ECO .Section IV introduces the proposed R ECO -oriented approach for improving power system resilience through resilient network design.Section V applies the R ECO -oriented approach to a 24and a 200-bus system, respectively.Section VI analyzes optimized networks regarding the system reliability under different levels of N-x contingencies and network properties.More discussions are in Section VII, and Section VIII concludes the paper.

II. BACKGROUND AND RESEARCH OBJECTIVES
Recently, several works have been proposed to optimally design and plan transmission and distribution systems to improve a system's resilience against natural disasters.In [8], Ma et al. present a two-stage stochastic mixed-integer linear program to optimally design the network with minimum investment and minimum expected loss of load during climate hazards.In [9], a framework is proposed to analyze the investment in power network enhancements with the evaluation of system resilience under natural disasters.In [10], a trilevel planning approach is proposed to expand and harden the coupled power distribution and transportation systems for improved resilience under random natural disasters with minimum investment.In [11], Garifi et al. propose a method to harden the power grid structure with minimum investment.The investment decision will improve the grid's recovery against natural disasters.All above works consider the adverse impact of natural disasters with stochastic models and formulate the resilient network design problem from the cost-effectiveness perspective.The improvement of resilience is observed and validated with less loss of load under the adverse scenarios.These works address resilience through different economic incentives for targeted hazards.However, there is a lack of an accepted and unified resilience objective that captures the inherent property of resilience considering the power network structure.By comparison, the research question addressed in this paper is how to design a resilient power network structure that can enhance power systems' inherent ability to tolerate disturbances and maintain functionality securely regardless of the source of threats.R ECO captures the inherent property of resilience regarding the network design and power flow distribution, and it represents the inherent ability to absorb disturbances regardless of their sources or causes.
As presented in [12], design, preparedness, and planning have been recognized as the top three needs to enhance grid resilience; importantly, design and construction standards for higher performance are required.The research gap addressed in this paper is to integrate the property of resilience into power network design for enhanced inherent resiliency.This paper presents a resilience-driven approach for power network design with its inspiration from naturally resilient ecosystems.The proposed approach translates ecosystems' survivability and resilience traits to power grids under the guidance of a quantitative resilience objective.
The concept of resilience that we adopt dates back to the 1970s when C.S. Holling defined the resilience in ecology as "a measure of the ability to absorb changes of variables and parameters in systems [13]."Over millions of years' growth and development, ecosystems have survived from various large-scale and unexpected disturbances, showing the ability to absorb sudden changes in the system and maintain their state.This long-term resilience contributes to an ecosystem's unique network structure, and it results in a novel and practical benchmark for robust, sustainable, and resilient human networks design.This benchmark is quantified as ecological robustness (R ECO ) [14], [15] that adopts a surprisal model from information theory [16].By modeling ecosystems as directional graph representations of energy transfer, the optimal R ECO recognizes a balance of the pathway efficiency and redundancy in resilient ecosystems.Based on the similarity between ecosystems and power systems, [17], [18] introduce the potential of R ECO to guide power network design for improved reliability.In [19], the authors propose a R ECO -oriented optimal power flow to improve power systems' survivability against unexpected contingencies.All above works show the potential of applying R ECO into power systems to improve resilience.However, the approach in [17] would not be practical to implement.For a 14-bus power grid, it is unrealistic to construct 80 branches to improve its resilience.Besides, [17], [18] are limited to smallscale power systems due to the mathematical formulation of the R ECO and its optimization, while [19] only optimizes the power flow dispatch.Therefore, this paper introduces a comprehensive R ECO -oriented resilient power network design approach that facilitates scalability and practicality for largescale power systems.
Three challenges previously impeded the application of R ECO for large-scale power network design.First, the nature of network design is a mixed-integer problem, which is a typical class of NP hard problem.With the increase of case size, the search domain expands exponentially, which adversely limits the efficiency for finding a global optimal.Second, the optimized networks in [17] directly connect buses in different voltage levels, which are impractical in power systems.Third, the formulation of R ECO involves several layers of logarithm functions which need the input variables to be positive, namely the power flow direction needs to be the same during the solving process.However, power flow direction changes are prevalent in large-scale power systems, and it makes the mixed-integer nonlinear programming (MINLP) problem in [18] invalid for larger power systems.To deal with above challenges, this paper first introduces a stochastic based algorithm to create candidate branches with realistic electric parameters for large-scale power systems.It greatly reduces the search domain for the optimal structure and keeps the realism of the network structure.Then, we relax the formulation of R ECO with a Taylor series expansion for the logarithm functions.The change of flow direction during the solving process will not cause the problem to be invalid.This improves the solvability and efficiency of the network design problem and ensures the practicality of the optimized resilient network design.

A. Unexpected Multi-Hazard Scenarios
N-1 reliability is the basic requirement for modern power systems planning and operation [20].However, the integration of communication networks and the increasing of system size expose power systems to more threats from both cyber and physical domains.Thus, the abruptness of contingencies is increasingly harder to predict [21].In [22], [23], authors have utilized Line Outage Distribution Factors (LODFs) and Group Betweenness Centrality (GBC) to identify sets of critical elements in large scale synthetic grids [24].These sets of critical elements consist of multiple (3 to 8) branches across the wide area, which are statistically unexpected and can adversely disrupt power systems' operation and security.In [25], those unexpected multi-hazards have been achieved through Man-in-the-Middle attacks (MiTM) in a high fidelity cyberphysical power system testbed.Those incidents make the system experience operational stress, threatening grid security and resilience.The above multi-hazard scenarios provide a touchstone for measuring power system resilience against unexpected cyberattacks and natural disasters.Under unexpected multi-hazards, the system's inherent ability to absorb disturbances can be measured by its resulting operational violations as an indicator of its resilience.

B. Background of Ecological Robustness (R ECO )
Ulanowicz et al. and Fath et al. utilize a model of surprisal from Information Theory [16] to quantify the resilience of ecosystems as R ECO .It considers the network structure and the transitions of energy and material among all species over the network [14], [15], [26].Its formulation represents a given network's robustness as a function of its energy flow pathway's redundancy and efficiency.
Surprisal is defined with the following expression, where s i is one's "surprisal" at observing an event i that occurs with probability p i , and k is a positive scalar constant [27].
The indeterminacy (h i ) of an event i is then formulated as the product of the presence of an event p i and its absence s i : It measures how likely the event i will change for a given event i, if we know the probability of event i will occur (p i >> 0) and the surprisal of event i that the system is doing something else most of the time (s i >> 0).It can be interpreted as follows: for a given system, those low probability events can cause high impacts to the system, because they happen so rarely that the system doesn't expect; high probability events possess a low impact because they occur often and the system adapts to them [28].
With the above models of surprisal and indeterminacy, R ECO is formulated with the following metrics.
The Total System Throughput (TSTp) is the sum of all flows within the system, which represents the system size [29], where T ij is the entry in Ecological Flow Matrix (EFM) [T].Following the ecologists' modeling of food webs, the EFM is constructed with a system boundary.that exchange energy based on a prey-predator relationship are within the system boundary, and the energy providers, energy consumers, and energy dissipation are placed outside of the system boundary [29].Thus, [T] is a square (N +3) × (N +3) matrix containing flow magnitudes of transferred energy.N is the number of actors inside the system boundary, and the extra three rows/columns represent the system inputs, useful system exports, and dissipation or system exports [30].It captures the energy interactions within and across the system boundary.
Fig. 1: The conversion of a hypothetical ecosystem into Ecological Flow Matrix.Replicated from [17].
The Ascendency (ASC) measures the scaled mutual constraint for system size and flow organization that describes the process of ecosystems' growth and development [31] with following expression, where Tij T ST p is recognized as the probability of an event that interrupts T ij with respect to all flows circulated in the system.

Tij T ST p TiTj
measures the conditional probability of joint event i and j with knowledge of source node (i) and end node (j), where T nj .With the model of indeterminacy (Eqn.2), the sum of Tij T ST p log 2 ( Tij T ST p TiTj ) multiplies with TSTp, giving a dimensional version of network uncertainty.For the same size systems, a higher value of ASC means that a network has fewer options of pathways for flows moving from any one actor to another, resulting in a lower level of uncertainty.
The Development Capacity (DC) is the upper bound of ASC as the development and growth of ecosystems are limited [32], Similar to ASC, DC is also the aggregate uncertainty, but without considering the source and end nodes.It captures the aggregated impacts (uncertainty) from all events (surprisals).
Then, R ECO is then formulated as follows: The ratio of ASC and DC reflects the pathway efficiency for a given network, while its natural logarithm shows the network's pathway redundancy [14].Thus, R ECO is a function of these two opposing but complementary attributes, where their balance achieves the optimal R ECO that directly affects a system's long-term survival [14].Multi-element contingency analyses in systems controlled for optimal R ECO [19] have shown the ability for R ECO to account for the presence of unknown events, or interruptions, that can happen in the system.

IV. ECOLOGICAL ROBUSTNESS-ORIENTED APPROACH
FOR RESILIENT POWER NETWORKS Modeling a power system analogous to an ecosystem enables construction of [T] with real power flows which enables R ECO optimization and analysis [17], [18].The analogy adopted between power grids and food webs sets the food web actors as generators and buses, the system inputs as energy supplied to generators from outside the system boundaries, the useful exports as loads (demand), and the dissipation as real power losses.With [T] constructed from real power flows, DC estimates the aggregated impacts of all events as the maximum power flow changes that can happen in the system.ASC estimates the dependence between events, and R ECO estimates the robustness of the system.Then, by including R ECO as an objective to guide network design, the optimized networks can better inherently absorb disturbances while maintaining functionality securely, thus improving their resilience.

A. Mixed-Integer Optimization Model
The R ECO -Oriented Power Network Design Problem is built upon the Transmission Network Expansion Planning (TNEP) problem and implemented using PowerModels.jlwith the objective of achieving optimal R ECO .The problem is formulated through Equation ( 7)-( 17) with the direct current (DC) power flow model.The novelty of this model is integrating knowledge of this resilient property from ecosystems with the physics in power systems for resilient power network design. Objective: Subject to: where B is the set of existing branches, N B is the set of candidates of new branches, M is the set of buses, and G is the set of generators; P l ij and P u ij are the lower and upper bound of branch limit, respectively; P l geni and P u geni are the lower and upper bound of generator output, respectively.The TNEP problem is formulated as a mixed-integer optimization problem where each candidate branch has a binary decision variable α ij for the candidate branch from bus i to bus j.The initial value of α ij equals to zero if the corresponding branch does not exist in the original network.If α ij after optimization equals one, the branch is built to reach a maximum R ECO .
The calculation of R ECO depends on [T] as expressed in Eqn. 8 with generator real power output P geni of each generator, real power flow P ij and P neij from existing branches and candidate branches, power consumption at each load P loadi , and binary decision variables α ij for candidate branches.Fig. 2 illustrates the detailed formulation of [T] using the above variables.Power flow dispatch (P ij and P neij ) depends on the real (P i ) and reactive power (Q i ) injection at each bus, bus voltage (voltage magnitude V i , voltage angle θ i ), and the network structure (α ij ) [33].In this formulation, a DC power flow model is used, so voltage magnitude V i is one, reactive power Q i and real power losses P lossi are zero.Therefore, the decision variables of the R ECO -Oriented Power Network Design Problem include each generator's real power input P geni , voltage angle θ i of each bus, and the binary decision variable α ij for candidate branches.In this way, the proposed R ECO -oriented approach will optimize the network structure (α ij ) and power flow dispatch (P geni and θ i ) to maximize R ECO .Eqn. ( 9)-( 12) are the calculation of R ECO using [T] through several layers of logarithm functions.Eqn. ( 13)-( 17) are the power flow constraints for operating limits and power balance.This MINLP problem is thus a nonlinear non-convex optimization problem.

B. Stochastic Based Candidate Branches Creation
In the proposed MINLP problem in Section IV-A, rather than considering all potential branches to build, a set of candidate branches is considered.This represents that in reality, the planners have some a priori information about new lines to consider.To represent the impact of the variability of such a set on the formulation and its solution, assuming here that we do not know and cannot control what lines they will choose, and to serve as a proxy for this set, we implement the hypothetical scenario where this set is chosen randomly.This selection mechanism can be considered as a worst case scenario, which is suitable to study, as the true planners may be able to choose a better set than a random selection.Hence, by demonstrating algorithm's effectiveness even when it is assumed that no information is given about the candidate branch locations, it shows the potential of the approach to be this good or better in reality.This introduces opportunity for future study.Since the test cases do not include a candidate branch set, Algorithm 1 is used, with suitable electric parameters for each branch based on the existing grid information.This fills the gap of the lacked information for candidate branches to expand power networks.Unlike [17], [34] that use heuristics or pre-screening methods to find an optimal network structure, Algorithm 1 is a stochastic approach to select candidate lines that supports direct inclusion of R ECO with power system constraints to optimize power network structure for inherent resilience.
The input for Algorithm 1 is the bus and branch information of a given power network, including identifier information, voltage levels, and branches' electric parameters.Algorithm 1 first classifies the existing branches into different voltage levels.The normal distribution is then used to represent the real-valued random variables.Thus, we take branches' electric parameters, including series resistance (R), series reactance (X), and shunt conductance (C), and capacity (MVA limit), as real-valued random variables following the normal distribution.Based on the case information, Algorithm 1 generates normal distributions for different electric parameters ) ) end for end for Select the from bus and to bus at the same voltage level using a Uniform Distribution (U(0, M)) Insert the parameter for the candidate branch from the Normal Distribution (N (µ, σ 2 ) ). end while of branches at each voltage level.Algorithm 1 takes a 40% confidence interval to create valid and different electric parameters of R, X, and C in per unit for candidate branches in our case studies.The candidate branches' capacity are twice the average capacity of existing cases' branches.From ecologists' perspective, power networks are more efficient than redundant.Each network has a corresponding value of R ECO , and any new branch could contribute to the improvement of R ECO .In selecting the initial candidate branches, we hypothesize that all network structures have approximately the same probability of being the most resilient network based on R ECO ; hence, all branches are assumed to have the same probability to be selected using the uniform distribution.Algorithm 1 will select M candidate branches from all possible branches with the uniform distribution to reduce the searching domain.With a specified number of candidate branches (M), the probability of selecting candidate branches is ( 1 M ) Additional information such as geographic location, cost, and government policies can further improve the realism for choosing candidate branches and validating the costeffectiveness for the network construction.Such information can help stakeholders determine the candidate branches instead of using Algorithm 1.The material, electrical parameters, and construction cost of candidate branches can also then be practically and accurately estimated.
One potential issue that may arise when adding branches is the so-called Braess paradox where adding one or more roads can cause congestion and slow down the traffic [35].A similar situation has been observed in power systems where added branches induced congestion in the system [36], [37].The Braess paradox is avoided in the proposed R ECOoriented power network design in Section IV-A because the optimization model will reject the branches that can cause congestion in the system.The results and analyses from the case studies also show this.

C. Relaxation of the Ecological Robustness Formulation
The formulation of R ECO involves with several layers of logarithm functions, whose hard constraint is that its inputs must remain positive during the solving process for the proposed optimization problem using state-of-art MINLP solvers.However, the inputs for calculating R ECO are the power flows, and their directions can be reversed during the solving process.In [18], its formulation fails to capture the feasible space for even small scale power systems, since the inputs are not constantly positive for the logarithm functions during the solving process.This creates a problem for large cases, where flow direction changes are more prevalent during the solving process.A Taylor Series Expansion of the natural logarithm function is thus used here to relax the formulation of R ECO to ensure the feasibility of the proposed R ECO -oriented power network design problem.
Considering the domain for the expansion, this paper utilizes the following relaxation, with x > 0 [38]: The logarithm function has a base of 2 in Equations ( 5) and (4).Using a property of logarithm functions, the Taylor Series Expansion of log 2 (x) can be expanded: ) By adapting the first order Taylor Series Expansion of Equations (18) and (20) into Equation ( 9) - (11), the formulation of R ECO can keep valid even with flow direction changes during the optimization process.The above formulation requires the input x not equal to -1.The x for the logarithm function in Equations ( 9)-( 11 is not equal to -1 for power systems.Then, the relaxed R ECO in the proposed approach can thus be solved with large power grid networks.

V. CASE STUDIES
This section applies the R ECO -oriented approach for two power system cases: the IEEE 24 Bus RTS [39] and the 200bus synthetic grid from [24], to improve their inherent ability to tolerate disturbances and maintain functionality securely.Algorithm 1 created 50, 100, 150, and 200 candidate branches for each case, respectively.Each case has a unique set of candidate branches, and each set of candidate branches does not belong to the others.For example, the set with 100 candidate branches does not include the set with 50 candidate branches.These candidate branches constitute 2 50 , 2 100 , 2 150 , and 2 200 different network structures to find the optimal R ECOoriented structure through solving the proposed R ECO -oriented design problem.
The candidate branches are selected with the highest voltage rating for each case, since the highest voltage transmission lines are the backbone of the system for power transfer.The proposed approach (Equation 7-17) not only solves the network structure (α ij ), it also solves the optimal power flow dispatch with an output vector of generator real power and bus voltage setpoints (P geni and θ i ).The resultant network design is analyzed with both the optimized network structure and the optimized network structure with the output vector, respectively.Thus, there two types of optimized networks analyzed for each scenario under each case study.The naming convention used for each network follows the pattern of Original Case Name-Number of Candidate Branches-Structure/Str-OPF.For the -Structure cases, they are the optimized network structure with the selected branches (α ij ) from the solution to analyze the optimized resilient network structure under original operating points; while for the -Str-OPF cases, they are the optimized network structure with the operating points of each generator's output and bus voltage (α ij , P geni and θ i ).The detailed case information have been made publicly available at [40].
The solver for the MINLP problem uses Ipopt [41], Juniper [42], and Cbc [43].Since the MINLP in Section IV-A is a nonlinear non-convex problem, the solver can only find the local optimal point.All the problems were solved using a laptop with a 2.4 GHz processor and 8 GB memory.The value of the Optimal R ECO from the solver is 0.3431.It is the mathematical optimal value of R ECO with the Taylor Series Expansion.The results in Table I and II show the Achieved R ECO , the Operational Cost, the Number of Added Branches, the Real Power Losses, the Reactive Power Losses, and the Computation Time for the IEEE 24 Bus RTS system and ACTIVSg200 system, respectively.The Achieved R ECO is based on the optimized network structure with/without the output vector of generator real power output and bus voltage after solving the power flows of optimized case with the alternative current (AC) power flow model.The Operational Cost is based on the marginal cost (C i ) $/MWhr and generator's output (P i ) MW with Eq. ( 21), so the unit is $/hr.

A. IEEE 24 Bus Reliability Test System (RTS)
The IEEE 24 Bus Reliability Test System (RTS) [39] has 24 buses and 37 branches.With 24 buses, there are 276 links that can be selected as candidate branches to expand the network structure.Table I shows the results of all four scenarios for the IEEE 24 Bus RTS cases.The results of Achieved R ECO show that the optimized networks have a higher value of R ECO than the original case, and the -Str-OPF networks have a higher value of R ECO than the -Structure networks (except for the 100 candidate branch scenario).The value of optimized R ECO is close to the 'Window of Vitality' (0.3469-0.3679), which is the unique range of R ECO for the resilient ecosystems [44].
With more branches constructed, the system has fewer real power losses but more reactive power losses, and the apparent power losses (MVA) are increasing as shown in Table I (except IEEE 24 Bus RTS-200-Structure and -Str-OPF).However, the extra losses from the new branches do not incur extra operational cost.When we compared the -Structure cases to the original case, the operational cost is reduced.On the other hand, the operational cost of all -Str-OPF cases increases with a slightly higher R ECO values (except IEEE 24 Bus RTS-100-Str-OPF).With the optimized output vectors, P geni and θ i , the generators are also more equally contributing to the power supply for improving the R ECO .Some expensive generators are generating more power, while some cheaper generators are producing less.It shows that the operational cost will not change much if only the network structure is more robust.As mentioned in Section IV-B, each set of candidate branches is unique.With the increasing numbers of candidate branches, the number of added branches does not increase.The added redundancy does not necessarily depend on the number of candidate branches.This confirms that R ECO can strategically construct the network structure and operate power systems to improve the system's resilience and maintain power system constraints.

B. ACTIVSg200
The ACTIVSg200 case [45] has 200 buses and 246 branches.With 200 buses, there are 19900 links that can be selected as candidate branches, which contains 2 19900 different network structures to be explored.4 shows the R ECO optimized for the ACTIVSg200 system with 50 candidate branches (2 50 different network structures), and 26 branches are added after the optimization.All four scenarios are successfully solved and the results are shown in Table II.Compared to the IEEE 24 Bus RTS, the Achieved R ECO values are much smaller in ACTIVSg200 cases.The original synthetic power grids are highly close to the real U.S power grids, which are quite sparse and efficient.Considering there are 2 19900 different structures that can be explored, the created candidate branches may not have the exact optimal structure.Thus, the R ECO for this synthetic grid is not improved as much as the IEEE 24 Bus RTS system.
For the ACTIVSg200 cases, all -Str-OPF networks have higher R ECO than their corresponding -Structure networks.The new built branches incur extra whole power losses.Similar to the IEEE 24 RTS case, the real power losses also decrease while the reactive power losses increase.The operational cost of the ACTIVSg200-50-Structure and ACTIVSg200-150-Structure cases are less than the original operational cost even though there are extra branches and losses.The operational cost of other R ECO -oriented cases increase slightly compared to the original case.The number of built branches does not increase with the increasing of candidate branches.This also demonstrates the R ECO is strategically constructing the network structure and operating power systems.

VI. NETWORK ANALYSES
The optimized networks are analyzed and compared with their original network for their reliability under multi-hazard scenarios and network properties regarding their structure and power flow distribution.All analyses are performed using AC power flow model.

A. Network Reliability Analysis
The multi-hazard contingencies are applied as different levels of N-x contingencies for each case.For x=1, they are planned contingencies; for x>1, they are unexpected contingencies.Under the contingencies, if there is one branch's power flow is over the limit or the voltage magnitude is out of the required limit, it is counted as one violation.If the power flow cannot be solved, then the contingency is marked as unsolved.
As for different case studies, the generation of N-x contingencies are different since the IEEE 24 Bus RTS system is relatively small compared with the ACTIVSg200 system.For the IEEE 24 Bus RTS cases, comprehensive N-1, N-2 and N-3 contingency analyses are performed for all power system components, including branch, bus and generator.The loss of any bus can cause more elements to be disconnected simultaneously.Thus, the N-3 bus contingencies can cause multiple components (generators and branches) disconnected.This can have a similar impact on generator unavailability like the Texas Winter Storm [46].For the ACTIVSg200 cases, a comprehensive N-2 and N-3 contingency analysis is difficult to complete, due to the large number of components.The N-1 contingency analysis is done for the branch, bus, and substations, respectively.Since all generators in ACTIVSg200 case are connected through transformers, the N-1 branch contingencies include all N-1 generator contingencies.The loss of one bus and one substation can catastrophically impact the entire system with multiple components (N-x) disconnected.It provides validation of the redesigned system's ability to tolerate disturbances and maintain functionality securely.For the ACTIVSg200 cases, the unexpected critical multi-hazard contingencies from [22], [23] are also considered.As mentioned in Section III-A, such critical N-x contingencies (x ranges from 3 to 8) are selected through LODFs and GBC as multiple branches widely spread in the system, whose loss may cause catastrophic impact to the system.Such critical contingencies are both geographically wide spread and statistically rare, which make them a touchstone to study resilience in large-scale systems.All the contingency analyses investigated here are performed without remedial actions.The basic control mechanism, such as automatic generation control (AGC) and automatic voltage regulator (AVR), are retained at their original settings.This provides a fair study about each system's inherent ability to tolerate unexpected multihazard disturbances and maintain functionality securely, thus justifying the improvement of resilience.
With more branches built after the optimization, there are more N-1, N-2 and N-3 contingencies than the original case, especially for the IEEE 24 Bus RTS case.To fairly compare the reliability, we then normalize the number of violations with the total number of N-x contingencies.Fig. 5 shows the normalized violations (total violations/total number of contingencies), and Fig. 6 shows the unsolved N-2 and N-3 contingencies for all variations of the IEEE 24 Bus RTS cases.Overall, the R ECO -oriented network structure and operation schemes are more reliable than the original case with far fewer normalized violations and unsolved contingencies.With the proposed R ECO -oriented approach, the unsolved N-2 contingencies are completely resolved and the number of unsolved N-3 contingencies reduced from 148 to less than 20.This ensures the observability of the system during disturbances and shows an outstanding improvement of resilience.The IEEE 24 Bus RTS-100-OPF case has the best performance among all cases.Even though its achieved R ECO is smaller than the corresponding -Structure case, they have the same network structure.The redundant network structure contributes to the improved resilience.II) the R ECO -oriented networks are more reliable with much fewer violations and unsolved situations.The ACTIVSg200-200-OPF has the best performance out of all the optimized networks with minimum violations and unsolved contingencies with the highest achieved R ECO .

B. Network Properties Analysis
An entropy based network robustness metric (R CF ) is used to identify cascading failures in power systems [47].The analysis of R CF can capture how likely it is for the network to experience cascading failures.With a higher value of R CF , the network is more robust and less likely to have a cascading failure [47].The calculation of R CF follows, where α i is the ratio between the maximum capacity and the load of corresponding line i; p i is the normalized flow values on the out-going links; P i is the total power distributed by node i and N is the number of nodes in the network.All network structures are analyzed for typical complex network properties, including the average node degree ( d), clustering coefficient (c), average betweenness centrality measures ( b) and average shortest path ( l) [48], where e is the edge and n is the node in graph; n is the total number of nodes in the graph; A is the adjacency matrix of the graph; σ(s, t) represents the number of shortest paths in the graph between s and t; σ(s, t|e) is the number of shortest paths in the graph between s and t that contain edge e.
The power flow distribution is also investigated by calculating the Mean and Standard Deviation (STD) of all branches' real power flow (pf), reactive power flow (rf), and the line percentage of MVA limit (MVA%) using Eqn.(26).For the power flow, the x i are all branches' pf and rf, respectively.For the line percentage, the x i are all branches' MVA%.The N is the total number of branches.
Table III shows the network properties for all network structures and the corresponding optimal power flow.The R ECO -oriented networks have better network properties than their original counterparts.All the R ECO -oriented networks have higher R CF , showing that the R ECO corresponds to an improved R CF against cascading failures.Increasing R CF is found to highly correlate with increasing R ECO , except the optimized results of the IEEE 24 Bus RTS with 100 candidate branches.Although formulations of both R ECO and R CF are based on an entropy model, their modeling details are different.R CF is based on branch flow limits, while R ECO is based on network structure, flow magnitudes, and flow directions.There can be some discrepancies between these two metrics.
All the R ECO -oriented networks have larger d and c, and reduced b and l.It shows these networks are more robust, reducing the significance of nodes (buses) and paths (branches) in the system, which spreads out the system's risks, from both perspectives of severity and probability.For actual networks, the d is in the range of (2.58, 2.61), the c is in the range of (0.032, 0.058), the b is in the range of (0.083, 0.40), and the l is in the range of (14.2, 29.2) [45].The results show that the optimized ACTIVSg200 networks' d and c are close to the actual systems, but the b and l are not.These can be explained by the way candidate branches were selected at their highest voltage level for each case, whose distance is shorter than branches between different voltage levels.
The Mean and STD of all the branches' real power flow (pf), reactive power flow (rf), and line percentage of MVA limits (MVA%) show that the R ECO -oriented networks distribute power flow more equally than the original network with reduced values in those measures.The Mean (pf) and STD (pf) in each -Str-OPF network are smaller than the -Structure network showing a more equally distributed real power flows, while the -Structure networks more equally distribute reactive power flows than -Str-OPF networks with smaller value of Mean (rf) and STD (rf).These facts could explain that even though the real power flows of the IEEE 24 Bus RTS-150-Str-OPF are more equally distributed than IEEE 24 Bus RTS-150-Structure, its R CF is smaller.The α i for R CF (Eqn.23) is the ratio between maximum line capacity considering real and reactive power, while the line loading in its calculation is only real power.With less equally distributed reactive power, its R CF can be reduced.Similarly, since [T] and R ECO only consider the real power flows, the reactive power flows can be distributed less equally to support the new built branches.Thus, the -Str-OPF cases may less equally distribute the power flows regarding the loading capacity, with higher value of the Mean (MVA%) and STD (MVA%) than the corresponding -Structure cases.In the optimized network, the reduced STD (pf), STD (rf), and STD (MVA%), compared to their original distributions, show that the power flows are closer to each other, and the newly built branches do not cause power flow increases on other branches.This also shows the proposed approach does not cause Braess paradox.

VII. DISCUSSION
The proposed R ECO oriented approach for resilient power networks is a typical NP-hard problem.Although the cases are different, the total number of topologies that the proposed approach explored is the same, which are 2 50 , 2 100 , 2 150 , and 2 200 .With the case size increasing, the computation time increases from 1.7 seconds to 84.09 seconds because of more power system variables (P i and θ i ) and more complicated network structures.Thus, the computation time and complexity of the proposed approach depend on the number of power system variables and network structure.
Unlike a traditional network expansion problem using the AC power flow model [49], [50], [34], this paper does not consider auxiliary equipment for new branches in the formulation.The proposed R ECO -oriented power network design problem is based on the DC power flow model.The optimized network's reliability and network properties are then analyzed through solving the AC power flow model.From the analyses, the optimized power network structures with more equally distributed power flows have a greatly improved inherent ability to tolerate disturbances and maintain functionality securely.The improved resilience is shown by fewer operational violations and unsolved contingencies under the conventional N-1 and unexpected multi-hazard contingencies.The candidate branches are created in Algorithm 1 without construction cost data.Thus, we are not able to perform as detailed a cost effectiveness analysis as in [8], [9], [10], [11].
Fig 8 shows the comparison of R ECO for eight power grids and a set of 38 food webs.The smaller power grids (5-to 14-bus cases) are optimized by a heuristic method in [17] and the larger power grids are optimized by the proposed approach in this paper.The R ECO of the food webs fall into the range of 'Window of Vitality', while the R ECO of the original power grids fall outside this range, especially the large and sparselyconnected power grids.After the network optimization, their R ECO is improved, as well as their inherent ability to absorb disturbances.However, the R ECO is not within the 'Window of Vitality' for the cases in this paper.Two possible reasons for this are: (1) the desired 'Window of Vitality' values may be different for power systems compared to food webs, and (2) the sets of candidate branches do not include all network structures, so it is possible that the solution is not the exact optimal structure recognized by R ECO .Compared to the heuristic method in [17] whose optimized cases are within 'Window of Vitality,' the proposed approach in this paper is more realistic with far fewer branches built.The approach in [17] is limited to a 14-bus system, thus we cannot directly compare both methods.The 14-bus case constructs 60 branches in [17] with a global heuristic search, while the proposed approach builds 51 branches for the ACTIVSg200 case.It shows that the proposed R ECO -oriented approach with power flow constraints and limited search domain can realistically and strategically guide the power network design.Although the added branches slightly increase the operational cost for some scenarios, the improvement of reliability under different levels of N-x contingencies and their network properties justifies this increased cost.In [19], R ECO was used to optimize the power flow distribution.This paper uses R ECO to guide the power network design to further enhance its inherent capability to tolerate disturbances and maintain functionality securely.By strategically adding branches, the R ECO -oriented power networks are more resilient and survivable against multi-hazard contingencies, with much fewer violations and unsolved contingencies.From Table I, Figure 5 and 6, the optimal R ECO -oriented IEEE 24 Bus RTS system reduces 70% violations and 96% unsolved contingencies with 25 added branches.From Table II and Figure 7 the optimal R ECO -oriented ACTIVSg200 case reduces 98% violations and unsolved contingencies with 51 added branches.This level of resilience enhancement was not achieved in [19].It shows that R ECO can be an accepted and unified metric that captures power networks' inherent property of resilience.
The correlation among R ECO , R CF , complex network properties and power flow distribution shows that the R ECO -oriented power network structure is more resilient against multi-hazard and cascading failures due to the redundant network structure with equally distributed power flows.It is worth noting that the reactive power losses are predominant in transmission network as observed in Table I and II.With more branches built, the optimized systems have more reactive power losses.There should be some auxiliary equipment along with the new branches for reactive power compensation as in [34].However, to investigate the influence of network structure to resilience, Fig. 8: R ECO curve for eight power grids and their R ECOoriented versions, as well as a set of 38 food webs (Data source: [17]).
all systems keep their original real and reactive power capacity.Thus, the improvement of resilience solely comes from the R ECO -oriented network structure.With extra auxiliary devices for reactive power support, the optimized systems can be more reliable and resilient under the contingencies.All above analyses demonstrate the effectiveness of using R ECO as a guidance to strategically design and operate power grids to improve its ability to absorb sudden and big disturbances in the system while maintaining their functions securely, thereby enhancing their resilience.

VIII. CONCLUSION
This work addresses a power system's need to withstand distributed threats arising from natural, accidental, and intentional causes that can create multi-hazard scenarios of x elements across a wide area with severe impact.To achieve this, a power system resilient design approach is presented, inspired from long-term resilient ecosystems.The resilience-oriented power grid network design problem is formulated and solved, with the goal to improve power systems' inherent ability to tolerate disturbances and maintain functionality securely.The R ECO -oriented power networks are analyzed under N-x contingencies, network properties, and operational cost.Results show the R ECO -oriented networks have fewer operational violations and unsolved contingencies with more redundant network structure and more equally distributed power flows.The R ECO -oriented optimization is generalizable as a resilient network design approach that improves a network's ability to withstand unknown threats.
Future work can extend upon this methodology from the following two aspects.On the one hand, the impact of reactive power for the calculation and optimization of R ECO in power networks can be investigated for reactive power planning for better resiliency.On the other hand, the economic factors, such as construction fee, electricity price, and penalty of unserved load, can be integrated with the proposed model to better understand the trade-offs between inherent resiliency and economics.Further, the projection of load growth and renewable energies integration can be taken into consideration for future resilient and economic power network design.
Fig 1 shows a hypothetical ecosystem and its conversion to [T].The actors (species) Fig 2 shows an exemplar [T] for a grid with n generators and m buses.
are guaranteed within (-1,1) and Tij T ST p TiTj Fig 3 shows the R ECO optimized network for the IEEE 24 Bus RTS system with 50 candidate branches (2 50 different network structures), and 21 branches are added after the optimization.

Fig. 5 :
Fig. 5: Normalized Violations Comparison of R ECO -Oriented Power Network for all variations of IEEE 24 Bus RTS cases (TableI)

Gen 1 . . . Gen n Bus 1 . . . Bus m Output Dissipation
The generators are treated as lossless with no dissipation.The P loadi and P lossi are the real power consumption and real power loss, respectively, at Bus i.The P ij and P neij are the real power flows on the corresponding existing branch and candidate branch, respectively.Entries with zero values mean there is no power flow interaction among buses and generators.
Fig.2: An exemplar of Ecological Flow Matrix [T] for a grid with n generators and m buses.The entries of [T] are P geni , P ij and P neij , P loadi , P lossi .The P geni is the real power output from generator i, which locates at the input row and the flow between generator and corresponding bus.
Algorithm 1 Stochastic Based Realistic Candidate Branches Selection and Creation Input = All branches' information from the case, the total number of candidate branches (M ) Classify branches based on the voltage level while The number of candidate branches < M do for Each Voltage Level do Collect the branch information for all parameters for Each Parameter do Compute the mean (µ) and variance (σ 2 ) Generate a Normal Distribution (N (µ, σ 2

TABLE I :
Results of R ECO -Oriented Power Network Design for IEEE 24 Bus RTS Fig. 4: R ECO -oriented ACTVISg200 network topology with 50 candidate branches (26 branches are constructed)TABLE II: Results of R ECO -Oriented Power Network Design for ACTIVSg200

TABLE III :
Network properties for all variations of the IEEE 24 Bus RTS and ACTIVSg200 systems.