Leveraging the Influence of Power Grid Links in Renewable Energy Power Generation

Smart grid construction provides the basic conditions for grid connection of renewable energy power generation. However, the grid connection of large-scale intermittent renewable energy sources increases the complexity of operational control of power systems. With the increase in intermittent renewable energy grid connections, distribution system operators must optimize and integrate these new participants to ensure the flexibility and stability of smart grids. Dividing the smart grid into logical clusters helps to overcome the problems caused by intermittent renewable energy grid connections. In this study, we propose a 2-step modeling approach that includes both link and leverage analyses to detect the network partitioning and assess the stability of the smart grids. Our experimental results of the link analysis show that, despite the identical scores in modularity and Silhouette Coefficients (SC), the total computational time of the linear programming model for linkage (CD1) is 29.8% shorter than that of the quadratic programming model for linkage (CD2) on 7 networks with fewer than 200 nodes, whereas CD2 is 29.5% faster than CD1 on 19 larger networks with more than 200 nodes. The leverage results of benchmark networks indicate that the computational time of each instance with the proposed linear programming model for leverage (ID1) and quadratic programming model for leverage (ID2) was substantially reduced, and the Critical Node Problem (CNP) results of medium- and large-scale networks were better than those reported in the literature, which play a significant role in smart grid optimization.

energy density, and unstable output, increase the complexity 31 of the operation and control of electrical grid systems [2]. 32 In intermittent renewable energy power generation, electric-33 ity is generated by using recycled resources. Its advantages 34 can only be exploited when it is effectively combined with a 35 power grid. 36 Intermittent renewable energy power generation is a green, 37 low-carbon, and less polluting power generation method that 38 can save fossil energy and protect the environment. However, 39 owing to the volatility and uncertainty of renewable energy 40 power generation, the extensive use of renewable energy 41 affects the security and stability of the power grid. The secu-42 rity of a power system is compromised to a certain extent [3]. 43 In combination with relevant smart grid technologies and the 44 grid connection of intermittent renewable sources, the grid 45 division of large-scale renewable energy consumption has 46 A community typically comprises of network nodes with 85 similar functions or properties. Its essence is the regional 86 coupling of the physical, chemical, and social interactions 87 between network nodes. The study of community structure 88 can help analyze the modules, functions, and evolution of 89 an entire network in a divide-and-rule manner. As a typical 90 artificial complex network system, modern power grids have 91 complex systems and network characteristics. The operation, 92 protection, and control of a traditional power grid are signifi-93 cantly different from those of large-scale intermittent renew-94 able energy power generation. Furthermore, the develop-95 ment of large-scale intermittent renewable energy sources has 96 increased the complexity and vulnerability of smart grids [9]. 97 Therefore, the application of research on complex network 98 community structures in power grid analysis is a general 99 trend. 100 The current study focused on two research questions (see 101 Figure 2): (1) How many stable grid partitions do power grids 102 have? (2) How can the vulnerability and critical nodes of the 103 smart grids be assessed? First, we used a linear programming 104 model for linkage (CD1) and quadratic programming model 105 for linkage (CD2) to detect the network structure. Then, CD1 106 and CD2 were employed for leverage analysis to identify 107 the importance and influence of key players in the network. 108 We selected modularity and Silhouette Coefficients (SC) to 109 measure the partitioning quality of link analysis models (CD1 110 and CD2). The Critical Node Problem (CNP) and a measure 111 of fragmentation (F-score) were used to evaluate the perfor-112 mance of the linear programming model for leverage (ID1) 113 and quadratic programming model for leverage (ID2).

FIGURE 2. Framework of smart grids analysis.
In the current study, we propose a comprehensive approach 115 combining link and leverage analyses to detect community 116 structure. We explored the community detection method 117 while considering the influence and disruption of key players 118 in power networks. Existing research mainly focuses on static 119 analysis, whereas the current study includes both static and 120 dynamic power network analyses. We detected the actual 121 grid structure and tracked how it changed over time. In this 122 study, benchmark networks of different sizes are selected 123 to verify the effectiveness of the proposed algorithms. The 124 results show that the proposed algorithm can explore com-125 munity structures more effectively than the other algorithms. 126 The exploration of both link and leverage analyses provides 127 guidance for supervising smart grids. Moreover, the proposed 128 dimensionality reduction algorithm reduces time complexity 129 and significantly shortens computational time. This can play 130 a significant role in smart grid optimization. The remainder 131 of this paper is organized as follows. Section II presents a 132 detailed background and models of link and leverage analyses 133 VOLUME 10, 2022 in smart grid partitions. In Section III, we use three bench-134 mark networks with different sizes to compare the results.  and dynamic power network analyses. We detected the actual 188 grid structure and tracked how it changed over time.

189
As observed in the literature, community detection and 190 key-player discovery are two distinct research topics with 191 many applications. To improve the robustness of a smart 192 grid system, this study focuses on the analysis of smart grid 193 partitioning and the influence of key players. We conducted 194 a leverage analysis on the influence and disruption of social 195 networks. Next, we show a 2-step modeling approach with 196 a common set of variables and constraints on community 197 detection and key-player discovery. Both problems can be 198 formulated as graphical problems.

200
Compared with traditional power grids, smart power grids 201 have self-healing and disaster-resistant abilities. However, the 202 destruction of critical nodes also causes vulnerability to smart 203 grids. Leverage analysis improves the system analysis ability 204 of the algorithm and identifies the importance and influence 205 of the key players (critical nodes) in the network. We define 206 the key players as nodes that generate the largest influence 207 on smart power networks. When such nodes are removed 208 from the network, the stability of the entire power network 209 is significantly disrupted. A leverage analysis of social net-210 works is generally conducted from two aspects: influence and 211 disruption. The influence of key players plays a fundamental 212 role in social networking. Key players in achieving optimal 213 dissemination through the network were identified. The influ-214 ence of key players in leverage analysis is analogous to the 215 ''Key Player Problem/Positive'' (KPP-Pos) [26], in which 216 planners quickly spread information, behaviors, or goods by 217 searching a group of network nodes with the best location. 218 Disruption analysis aims to identify key players by removing 219 critical nodes that disrupt or fragment the network. This 220 aspect of leverage analysis is similar to that of ''Key Player 221 Problem/Negative'' (KPP-Neg) in [26], which depends on the 222 extent to which the network relies on its key players to achieve 223 cohesion.

224
Several studies have examined the identification of the key 225 players in power networks. One way to identify key players is 226 based on centrality measures [27], [28]. Traditional centrality 227 measures for key player problems only consider the network 228 structure and do not consider additional information [29]. 229 A new approach uses entropy measures to detect key player 230 sets within a social network, providing a simple solution 231 to the KPP-Pos problem [30]. Entropy-based measures are 232 employed to identify the key player sets, but the shortcoming 233 of such methods is that they are limited to nondense hetero-234 geneous networks [31], [32], [33].

235
Related studies have aimed to identify a set of key players 236 rather than individual key players. The problem of maximiz-237 ing influence was proposed to identify nodes that initially 238 adopt new products or innovations, such as a greedy-based 239 heuristic with a hill-climbing algorithm [34]. Analysis of dif-240 fusion in social networks has always been a research hotspot 241 [35], [36]. Therefore, several algorithms have been proposed 242 to identify the key players, and each method emphasizes a 243 single object to be examined. However, in complex practical 244 applications, a set of algorithms that can perform well for 245 multiple interesting objects is required [37], [38], [39]. The 246 current study proposes an improved algorithm with multiple 247 objects to detect the actual smart grid structure and track its 248 dynamic change.

250
In this section, we present several models for community 251 detection and measure the influence and disruption of link-252 ages within a network after removing critical nodes.

253
The parameters and variables in the models are presented 254 in Table 1: We use a classical linear programming (LP) model [40] for 256 social network link analysis and community detection (CD): 257 CD1 for LP Model: optimal solution directly because the objective function can-269 not provide the information to assign data points to each 270 cluster.

271
An equivalent quadratic programming (QP) model [41] 272 was employed to address the issues in CD1.

273
CD2 for QP Model: CD2 significantly reduces the number of constraints and 277 directly provides an optimal number of clusters. CD2 is 278 a node-oriented model with fewer variables than the CD1 279 model. The time complexity of CD2 is O(n(c_max)) com-280 pared to that of CD1, which is O(n 3 ). Although CD1 is a 281 linear model and CD2 is a quadratic model, the size difference 282 makes the quadratic model suitable for larger problems in 283 which the computational burden of CD1 hinders its practical 284 application. The main advantage of the CD2 model over the 285 CD1 model is that it can simultaneously determine the opti-286 mal number of communities as well as the optimal allocation 287 of each member of the community.

288
After the optimal number of communities in a social net-289 work is detected, the stability of the formed communities can 290 be assessed by removing the important nodes for leverage 291 analysis. As in CD1, we first introduce an LP model for 292 leverage analysis: influence and disruption (ID). s.t. x ij + y i + y j ≥ 1, ∀(i, j) ∈ E (6) 296 i∈V y i ≤ C (7) 297 In the ID1 model, there are C n 2 variables and C n 3 + 1 299 constraints.

300
ID2 for QP Model: In ID2, the number of variables is nC and the number of 307 constraints is n+nC(n-1)/2+1. ID2 had fewer variables and 308 constraints than ID1. The time complexity of ID1 is O(n 3 ), 309 whereas that of ID2 is only O((C)n 2 ). Both models can be 310 solved using an exact solver, such as Gurobi.    (2) The F-measure is similar to the diversity measure, such 363 as heterogeneity or the Herfifindahl index. The F-score mea-364 sure indicates network fragmentation. The maximum frag-365 mentation occurs when each node is independent, creating 366 the same number of components as the nodes. Measurement 367 of fragmentation (F-score) proposed in [26] as  This study uses the same set of benchmark networks for both 375 the link and leverage analyses. We used the same datasets as 376 in [46], [47], and [48] (see Table 2). The majority of [46] [46] (small-scale network). for partitioning, the output will be a trivial solution. The links 409 and disruptions of the critical nodes are listed in Table 3.

410
From Table 3, we can observe that despite the identical 411 scores in modularity and SC, the computational times are dif-412 ferent. For each problem instance, the optimal computational 413 times are highlighted in bold. The total computational time of 414 CD1 is 29.8% shorter than that of CD2 on 7 networks with 415 fewer than 200 nodes, whereas CD2 is 29.5% faster than CD1 416 on 19 larger networks with more than 200 nodes. Taking the 417 Karate network as an example, computational time of CD1 418 is 0.13 s, which is much smaller than that of CD2 (33 s). 419 However, for large networks such as the BA2500 network, 420 the computational time of CD2 (297 s) is much shorter than 421 that of CD1 (591 s).   We compared the results for the datasets of the Karate Club 426 social network [46], [49].  are represented by edges. The leverage results compared to 434 those in [46] are presented in Table 4.

435
By comparing these three results, it can be observed that which is consistent with the literature [46]. were significantly shorter than those reported in the literature.   is much shorter than that in [48]. Table 8 shows the graphical 491 results of [48]. It shows the changes in the network partition 492 for each benchmark network after critical nodes are removed.

493
The results of the leverage analysis indicate that ID1 and 494 ID2 outperformed the models in the literature in terms of 495 computational time.

497
The current study provides a new framework to solve 498 the problems of smart grid partitioning and critical nodes 499 identification using both link and leverage analyses. It con-500 tributes to the literature by identifying the optimal grid parti-501 tions and critical nodes in the power network to alleviate the 502 vulnerability of the smart grids.

503
Three benchmark networks of different sizes were used 504 to compare the results. Furthermore, linear and quadratic 505 programming models were proposed for link and leverage 506 analyses, and were found to be robust compared with tra-507 ditional approaches. Our experimental results of the link 508 analysis show that, despite the identical scores in modularity 509 and SC, the total computational time of CD1 is 29.8% shorter 510 than that of CD2 on 7 networks with fewer than 200 nodes, 511 whereas CD2 is 29.5% faster than CD1 on 19 larger networks 512 with more than 200 nodes. Therefore, CD1 is more suitable 513 for small-scale networks whereas CD2 is more efficient for 514 large-scale networks with more than 200 nodes. The lever-515 age results of benchmark networks [46], [47], [48] indicate 516 that the computational time of each instance with the new 517 proposed models (ID1 and ID2) was substantially reduced, 518 and the CNP results of medium-and large-scale networks 519 were better than those reported in the literature, which play a 520 significant role in smart grid optimization.

521
The proposed dimensionality reduction algorithm reduces 522 time complexity and significantly shortens computational 523 time. The methods proposed in this study can contribute to the 524 real-time planning and operation of smart grids. The appli-525 cation of partitioning consumers according to their energy 526 demand and supply can be explored. In addition, the link and 527 leverage analyses of smart grids can connect future decen-528 tralized intermittent renewable energy communities to smart 529 grids. By obtaining optimal grid partitions and identifying 530 the critical nodes in the smart grids, we can alleviate the 531 vulnerability of smart grids and improve the security and 532 stability of large-scale power grids.

533
There are some limitations of the current study, and we 534 plan to expand this research further. The effectiveness of 535 the proposed algorithm in an actual power grid is worthy of 536 further study. Meanwhile, additional electrical characteristics 537 of the grid will be considered to detect energy communities in 538 future studies. How to integrate more electrical characteristics 539 into the model and apply the algorithm to larger networks 540 remains to be explored further. In addition, the relationship 541 between the modularization index and the power network par-542 tition process should be studied further, and multiple objec-543 tives should be considered simultaneously using the multi-544 objective method.