Control and Stability of Modular DC Swarm Microgrids

Direct current (dc) microgrids have not yet achieved the promise of true plug-and-play characteristics due to stability issues stemming from power converters. Swarm microgrids, a type of dc microgrids, are aimed at delivering a modular and easy-to-expand infrastructure. In this article, an application-specific control strategy is developed to ensure stable and expandable swarm microgrids. This control strategy makes use of the widely available distributed storage in swarm microgrids and includes active damping techniques at power-drawing units. Utilizing a cascaded state-space system model, experimental validation with dual-active-bridge converters, and eigenvalue robustness analysis, this article demonstrates a high margin of stability for expandable swarm microgrids. The analysis includes variation of the microgrid size and line parameters, among others. The results show that swarm microgrids are stable for up to at least 1000 participating swarm units. As part of the robustness analysis of swarm microgrid stability, specific definitions are provided for the allowed behavior of the power converters used in a swarm microgrid. The developed methodology allows assessing and designing for swarm microgrid stability even without knowledge of the internal structures of the power converters. As such, this article provides a practical framework to support the scaling up of swarm microgrid deployment.

control methods can be used, many of which rely on the 72 availability of energy storage in the microgrid [18]- [24]. 73 In principle, these methods can also be applied to swarm 74 microgrids because of the distributed storage capacity of 75 SHSs [25], [26]. Research on the decentralized control of 76 distributed storage units in dc microgrids has focused on the 77 behavior of these units when supplying power [13], [20]. 78 However, the decentralized control of distributed storage units 79 when drawing power was not addressed to the extend needed 80 for practical application.

81
In regard to stability assessment methods for dc micro-82 grids, two main approaches are described in the literature: 83 large-signal stability analysis and small-signal stability analy-84 sis. A large-signal stability analysis accounts for large dis-85 turbances but is usually only used to analyze single-bus 86 microgrids [27], [28]. In a small-signal stability analysis, 87 a linear model is constructed around a steady-state operating 88 point [8], [12], [29]. The advantage of the small-signal stability 89 analysis is the ability to easily adapt the model to assess 90 different topological configurations of microgrids [25], [30], 91 [31]. However, to date, most studies have been limited in the 92 number of participating units assessed. The number of units 93 is lower than five in [26], [30], and [31] and lower than ten 94 in [9], [18], and [25]. 95 For the modeling of power electronic converters in dc 96 microgrids, two main approaches are discussed in the liter-97 ature. On the one hand, the so-called white-box modeling 98 approach is widely used. This means that a certain internal 99 converter structure is assumed, and the converter behav-100 ior is modeled accordingly [21], [32], [33]. On the other 101 hand, the so-called black-box modeling approach is based on 102 measurements and, thus, real-world behavior [34]- [36]. The 103 black-box modeling approach is an attractive option for the 104 case of swarm microgrids, because the collective behavior 105 of converters and distributed storage units can be reflected. 106 A black-box modeling has been used to design and assess 107 dc microgrids of different scales and with different types of 108 converter topologies [37]- [39]. However, the existing literature 109 lacks measurement-derived models of converters connected to 110 batteries. 111 A systematic analysis that addresses these highlighted short-112 comings of the literature is not yet available. Therefore, this 113 article adds to the existing research with the following key 114 contributions. 115 1) It develops a control strategy specific to the needs of 116 swarm microgrids that ensures stability and makes use 117 of the available distributed storage. 118 2) It demonstrates the stability for the expansion of a swarm 119 microgrid up to 1000 swarm units. 3) It provides practical definitions of allowed and no-go 121 areas for the black-box behavior of battery-connected 122 converters to ensure a stable microgrid. 123 The rest of this article is structured as follows. The archi-124 tecture and attributes of swarm microgrids are discussed 125 in Section II. Section III describes the development of the 126 proposed control strategy. The swarm microgrid model is 127 derived in Section IV. An experimental validation is presented 128 in Section V. A comprehensive stability analysis of swarm 129 microgrids is performed in Section VI. The conclusions are 130 drawn in Section VII.  at any given time. 163 To allow for adequate monitoring and billing, a bidirec-164 tional meter is required. The abovementioned components are 165 connected to the SBB controller, which enables the implemen-166 tation of advanced control schemes. To provide the ability to 167 form meshed network configurations, an SBB has cable ports, 168 which allow for interconnection with other swarm units and 169 can be used to realize zonal protection [40], [41]. These cable 170 ports are joined at the swarm node. The node voltage between 171 the positive and negative rails at swarm node j is denoted 172 by v n, j . The nodal current that is injected at node j is denoted 173 by i n, j . The nodal current i n, j takes negative values when the 174 swarm unit is operating in the charging mode. 176 Based on the architecture presented earlier, swarm 177 microgrids have certain characteristic attributes. Some of 178 them are relevant to stability analysis. These attributes 179 are directly related to the subsequent sections on the 180 control strategy, system modeling, and stability analysis, 181 as follows. 182 First, swarm microgrids are characterized by ad hoc deploy-183 ment and ease of expansion even during operation. This 184 attribute is particularly important in Section III when defining 185 the control strategy, which must cater to this capability of ad 186 hoc deployment. 187 Second, to reach a large number of users consistent with 188 swarm electrification, a swarm microgrid is expected to sup-189 port up to 1000 swarm units. This requirement is important 190 for the overall modeling of the swarm microgrid, as discussed 191 in Section IV, and for the stability analysis undertaken in 192 Section VI.

193
Third, there are two distinct modes in which a swarm unit 194 can operate: the supplying mode and the charging mode. 195 The distinction between these two modes is a key input to 196 the design of the cascaded system model, as discussed in 197 Section IV-A. As shown in Fig. 2, the supplying mode is 198 characterized by the usage of only the step-up converter. The 199 charging mode is characterized by the usage of only the step-200 down converter.

201
Fourth, from a network stability point of view, only the 202 side of the converter that is connected to the swarm node, 203 as indicated in Fig. 2, is of interest. This means that in the 204 supplying mode, only the behavior of the output side of the 205 step-up converter is of interest; in the charging mode, only 206 the behavior of the input side of the step-down converter is of 207 interest. These characteristics are important for the converter 208 modeling in Section IV-C. 209 Fifth, it is assumed that the battery is adequately sized, 210 such that both the power drawn from the loads and the 211 power supplied by the solar PV have negligible effects on the 212 behavior of the converters. These characteristics are important 213 for the modeling of the converter behavior in Section IV-C.

215
In this section, the control strategy for swarm microgrids is 216 developed. Intended to serve as a modular, ad hoc deployable, 217 and highly scalable infrastructure, swarm microgrids are dc 218 microgrids with special requirements in terms of control. 219 As discussed in the first item in Section II-B, rapid deployment 220 and ease of expansion are important attributes of a swarm 221 microgrid. To attain these attributes, four main challenges 222 related to the stability of swarm microgrids need to be 223 addressed. First, the ability to scale the network must be 224 to a negative resistance effect, needs to be mitigated.

230
The key principle underlying the control strategy for swarm  Table I   The reference parameters v 0,ref, j , r dr, j , and i n,ref, j are pro-291 vided from an operational layer. In normal operation, these 292 values change only at the intervals of seconds or minutes. 293 A demonstration of the capacity to cater to changes in these 294 reference parameters is part of the experimental validation in 295 Section V. All discussed control provisions are summarized 296 in Table I.

298
The objective of this section is to derive a mathematical 299 model that allows for the representation and analysis of swarm 300 microgrids of arbitrary sizes. The mathematical model derived 301 here is then used in the subsequent sections to assess the stabil-302 ity of a swarm microgrid. To analyze the small-signal stability 303 of the microgrid, a small-signal system model is required, 304 as developed in this section. In particular, the small-signal 305 model derived here is a linear time-invariant (LTI) model. 306 To represent this system or parts of this system, the transfer 307 function representation and state-space representation are used 308 interchangeably [47]. The following convention is used: x = 309 X +x, where X is the steady-state value, andx is the small-310 signal perturbation.

311
This section is structured as follows. Section IV-A describes 312 the structure of the system model. Section IV-B gives the math-313 ematical representation in terms of subsystems. Section IV-C 314 presents the gray-box modeling approach used to model the 315 behavior of the converters within the different subsystems. 316 Section IV-D introduces the state-space representation of the 317 consolidated microgrid system model by integrating the results 318 from Sections IV-A-IV-C. The overall swarm microgrid is modeled as a multiple-input 321 multiple-output system that consists of multiple subsystems. 322 In turn, each of these subsystems consists of one or multi-323 ple transfer functions. System and subsystem definitions are 324 denoted in the form System [47]. The swarm microgrid system 325  Step-up is an essential part of the model for the swarm    In addition, there is one subsystem type for cables. In summary, the cascaded system model is structured as 368 follows: The different subsystem models are connected through an 372 algebraic coupling κ. This concludes the description of the 373 cascaded system model, as shown in Fig. 3.

375
This section derives the specific attributes of the three 376 types of subsystems and their interactions through algebraic 377 coupling. Mathematically, the algebraic coupling of the sub-378 systems is expressed using the following conventions [47]. 379 The algebraic coupling is represented by the coupling input 380 vectors, the coupling input matrix S, the coupling output 381 vectorz, and the coupling output matrix C z . Building on the 382 overall system model in Fig. 3, the attributes of the subsystems 383 are as shown in Fig. 4 and are discussed, from bottom to top, 384 as follows.

385
Swarm units have internal controllers, which require the 386 input of corresponding reference parameters. These reference 387 parameters are represented by dashed arrows. The required 388 reference parameters are given in Table I. For a swarm unit 389 operating in the supplying mode, the reference parameters 390 are the reference zero-current voltage v 0,ref, j and the droop 391 resistance r dr, j . For a swarm unit operating in the charging 392 mode, there is only one reference parameter, namely, the nodal 393 current reference i n,ref, j . There are no reference parameters for 394 cable subsystem k. 395 The individual subsystems are coupled to each other. The 396 solid arrows in Fig. 4 represent the coupling inputs and outputs 397 of the subsystems. Each subsystem has one coupling input and 398 one coupling output. For swarm unit j , the coupling input 399 parameter is the nodal current i n, j , and the coupling output 400 parameter is the node voltage v n, j . For cable k, the coupling 401 input is the voltage difference v c,k between the two ends of 402 the cable, and the coupling output is the current i c,k through 403 the cable, as shown in Fig. 5. 404 The interconnections between the subsystems through the 405 algebraic coupling κ are defined by the algebraic connection 406 between the coupling input vector and the coupling output vector where the following hold.
The incidence matrix M is further separated into two  As shown in the system model in Fig. 3 and according 442 to (3), an important part of each subsystem model is the 443 representation of the converters of the corresponding swarm 444 unit, as depicted in Fig. 2. The behaviors of both the step-up 445 converter and the step-down converter need to be modeled. 446 As discussed in the third item in Section II-B, a swarm unit 447 can operate in only one of the two modes at any given time. 448 Therefore, a distinct model for the step-up converter behavior 449 in the supplying mode and a distinct model for the step-down 450 converter behavior in the charging mode are developed. 451 To achieve a realistic representation for the stability analysis 452 in Section VI, a gray-box modeling approach is followed.

453
Gray-box modeling refers to an approach in which parts of 454 the system are modeled based on ideal behavior, and other 455 parts are modeled based on measurements and, therefore, real 456 behavior [49]. In the gray-box model derived here, only one 457 component is modeled with ideal behavior. This component 458 is the input capacitor, which is modeled as an ideal capacitor 459 without parasitic resistance. This is done to account for the 460 worst case from a stability perspective, because the parasitic 461 resistance of a capacitor exerts a known stabilizing effect [50]. 462 The other components of the gray-box model are derived from 463 measurements, building on the existing literature on black-box 464 converter modeling [34], [35], [37]. 465 In the black-box converter modeling literature, a converter 466 behavior is typically modeled using an unterminated two-467 port model [37] with three types of components. First, the 468 input admittance and output impedance of the converter are 469 included. Second, two transfer functions, named the audio 470 susceptibility and the back current gain, account for the 471 influence between the input and output quantities. Third, the 472 model may include additional transfer functions related to 473 reference tracking [36].

474
The model derived here is distinctly different, mainly 475 because the converters are not unterminated but rather termi-476 nate at a battery on one side, as shown in Fig. 2. This battery 477 termination allows for the model to be notably simplified by 478 modeling the closed-loop behavior of the combined system of 479 battery and converter. Because the internal behavior is reflected 480 in the closed-loop behavior of this combined system, the 481 internal transfer functions of the converter mentioned earlier, 482 the audio susceptibility and the back current gain, can be 483 omitted. The remainder of the model needs to accurately 484 represent the behavior of the active components in either 485 mode. The active components of a swarm unit operating in 486 the supplying mode are shown in Fig. 6(a), and the active 487 components of a swarm unit operating in the charging mode 488 are shown in Fig. 6(b).

489
As discussed in the fifth item in Section II-B, the influences 490 of the load and solar PV on the battery are regarded as 491 negligible from a network stability point of view. Therefore, 492 the converter models here can focus solely on the behavior 493 of the swarm-node-facing side. As stated in the fourth item 494 in Section II-B, the swarm-node-facing side is the output side 495 of the step-up converter and the input side of the step-down 496 converter.

497
The model boundaries, shown as dashed lines in 498 Fig. 6(a) and (b), encompass both the converter and the 499 battery. The small-signal models for the step-up and step-down 500 converter behaviors are presented subsequently in that order.

501
The reader is reminded that the following convention is used: where X is the steady-state value, andx is the 503 small-signal perturbation. to a change in the nodal currentĩ n, j due to the output In addition, the converter-internal output voltage control (VC) 516 to control the node voltage is included. This is accomplished 517 by including the transfer function T VC (s), which models the 518 response ofṽ VC, j to changes in the reference node voltage The combined small-signal model for the step-up converter 522 behavior is shown in Fig. 6(c) and summarized as follows:

523
Step-up : The behavior due to changes in the nodal currentĩ n, j 531 is modeled through a combination of the input admittance 532 and an explicit model of the input capacitor. As described 533 earlier in this section, when introducing the gray-box modeling 534 approach, an ideal capacitor model is used to account for the 535 worst case scenario from a stability point of view. Following 536 the ideal capacitor model and considering a capacitance C The transfer function T C (s) describes the behavior of the 539 node voltageṽ n, j in response to changes in the capacitor 540 Changes in the capacitor current depend on changes in the 543 nodal currentĩ n, j , inĩ Y,in, j , and inĩ CC, j , as follows: whereĩ Y,in, j represents changes that originate from changes in 546 the node voltageṽ n, j due to the input admittance Y in (s) Furthermore, converter-internal input CC to control the nodal 549 current is modeled. In particular, the response ofĩ CC, j to 550 changes in the nodal current referenceĩ n,ref, j is modeled by 551 the transfer function T CC (s) The combined small-signal model for the step-down converter 554 behavior is shown in Fig. 6(d) and summarized as follows:

3) Transfer Function Identification:
The converter behavior 557 models in (13) and (19) rely on the experimental identifi-558 cation of the black-box behavior transfer functions Z o (s), 559 T VC (s), Y in (s), and T CC (s). These four transfer functions 560 are LTI models that follow a generic structure, consisting 561 of the sum of a first-order subsystem and a second-order 562 subsystem [35]: where the following hold.   The detailed black-box transfer functions and identi-576 fied parameter values are based on experimental data. 577 These data and parameter identification results are provided 578 in Appendix C.

579
It should be noted that if the converter behavior cannot 580 be described by a model with the generic form of (20), then 581 higher-order transfer functions need to be used. Examples for 582 such higher-order transfer functions are discussed in [35].

584
The overall model of the swarm microgrid system SMG , 585 as given in (3), contains three types of subsystem models.

586
As shown in Fig. 3, these three subsystem model types 587 correspond to 1) a swarm unit operating in the supplying 588 mode, s, j ; 2) a swarm unit operating in the charging mode, 589 ch, j ; and 3) a cable, c,k . These models are defined as 590 follows. is shown in Fig. 7 and is defined as follows: 608 s, j :

610
where the following hold. 3) A Zo , B Zo , C Zo , andx Zo, j give the state-space represen-616 tation of Z o (s). 617 4) A s, j is the system matrix, B s, j is the control matrix, 618 C s, j is the output matrix, D s, j is the direct feedthrough 619 matrix, S s, j is the coupling input matrix, and C z,s, j is 620 the coupling output matrix of the subsystem for swarm 621 unit j operating in the supplying mode.  Table I, the subsystem model for a swarm unit operating 624 in the charging mode, ch, j , is also developed in three 625 steps. As for the supplying mode, first, the input and output 626 definitions of Fig. 4 are used. Therefore, the control input is 627 u ch, j =ĩ n,ref, j , the coupling input iss ch, j =ĩ n, j , the subsystem 628 output isỹ ch, j =ṽ n, j , and the coupling output isz ch, j =ṽ n, j . 629 Second, the nodal CC scheme listed in Table I is incorporated. 630 Third, the step-down converter shown in Fig. 2 is represented 631 by an input-side-only model, as discussed in Section IV-C. 632 The input behavior of the step-down converter is modeled 633 according to (19) using the input-side transfer functions 634 T C (s), Y in (s), and T CC (s). As described in Section IV-C, the 635 capacitor is modeled as an ideal capacitor with capacitance C 636 according to (14). The abovementioned steps result in the 637 charging mode swarm unit model, which is shown in Fig. 8 638 and is defined as follows: 639 ch, j : where the following hold.

666
The full model of the subsystem corresponding to cable k with 667 state vectorx c,k is expressed as follows: 668 c,k :

670
where A c,k is the system matrix, B c,k is the control matrix, 671 C c,k is the output matrix, D c,k is the direct feedthrough 672 matrix, S c,k is the coupling input matrix, and C z,c,k is the 673 coupling output matrix of the subsystem for cable k. Note that 674 vector and matrix notation is maintained even for the scalar 675 quantities in (25). This is done to maintain coherence with (21) 676 and (22) and to facilitate clear referencing when combining the 677 subsystems into the system model in Section IV-D4 as follows. 678

4) Combined System Model:
To construct the overall system 679 model, the definitions derived throughout Section IV are com-680 bined. The overall cascaded system model definition according 681 to (3) and Fig. 3 is followed. The individual subsystem 682 definitions for s, j in (21), ch, j in (22), and c,k in (25) 683 are used.

684
In accordance with (3), the individual subsystem definitions 685 are aggregated according to their types: s , ch , and c . This 686 means that all vectors, namely,x s ,x ch ,x c ,ũ s , andũ ch , are 687 combined vectors of the respective quantities corresponding 688 to the individual subsystems. For example,x s is a vector of 689 allx s, j . Furthermore, all obtained matrices, namely, A s , A ch , 690 A c , B s , B ch , C s , C ch , C c , C z,s , C z,ch , C z,c , S s , S ch , and S c , pos-691 sess a diagonal structure consisting of the respective quantities 692 of the individual subsystems. For example, A s is the system 693 matrix for all swarm units operating in the supplying mode and 694 is constructed using the elements of A s, j in a diagonal form. 695 The algebraic coupling definition of (9) is used to connect the 696 submodels to each other. Thus, the overall system model is 697 derived as follows: In this section, the swarm microgrid model developed in 702 Section IV is validated in a laboratory environment. The objec-703 tive of this validation is to demonstrate the accurate modeling 704 of the system. Experimental data are compared with simulation 705 results on the basis of the LTI model. The model validation 706 criteria are threefold. First, the simulation output should match 707 the experimental data. In particular, any mismatches due to 708 ripples should be less than 10% of the average values. Second, 709 the model response should be especially accurate to changes in 710 the reference values with waveforms similar to those observed 711 in the experimental setup. Third, the model must adequately 712 represent spikes and rapid changes, particularly after increases 713 in the power drawn by any swarm unit.

714
To compare the simulation results of the LTI model with 715 the experimental data, the steady-state values need to be 716  the simulation (sim) results for the same quantities obtained 759 using the LTI model derived in (26). In the following, 760 an assessment in accordance with the three validation criteria 761 defined earlier is undertaken.

762
The results of the LTI model generally resemble the 763 real-world behavior well. The highest ripple content is present 764 at unit j = 3 between between t = 0.6 s and t = 0.7 s, when 765 the highest current is drawn from the network. The voltage 766 ripples are less than 2 V in amplitude. This corresponds to 767 a ripple content of less than 3%. The current ripples are less 768 than 0.1 A in amplitude. This corresponds to a ripple content 769 of less than 6%. In both cases, the ripple content is less 770 than 10%. Therefore, the first validation criterion is fulfilled. 771 Furthermore, the behavior in response to changes in the 772 reference values is well modeled, thus also fulfilling the second 773 validation criterion. First, an increase in r dr,1 causes a decrease 774 in v n,1 , as in (1)

783
The waveforms after the corresponding increase and 784 decrease in the power drawn from the network are accurately 785 modeled, particularly the waveforms of the spikes. Therefore, 786 the third validation criterion is also fulfilled.

788
In this section, the stability of a swarm microgrid is 789 analyzed to demonstrate its expandability. Expandability up 790 to n = 1000 is defined as a core requirement for swarm 791 microgrids, as discussed in the second item in Section II-B. 792 Building on the practical validation for a swarm microgrid 793 of size n = 4 in Section V, this section demonstrates the 794 results of increasing the quantity of swarm units to n = 1000. 795 As laboratory validation is no longer feasible at this scale, 796 the LTI model is utilized. The modeling approach derived in 797 Section IV is used to scale the model up to a large microgrid. 798 The generic topology shown in Fig. 11(b) is used. In the first 799 part of this section, a qualitiative assessment of stability is 800 undertaken, followed by a detailed quantitative analysis includ-801 ing single-parameter variation and two-parameter variation.

803
Before the stability of the system is assessed quantitatively, 804 this section first undertakes a brief qualitative assessment. 805 The method is inspired by the assessment of the relationships 806 between species in a given ecosystem [51]. When assess-807 ing such an ecosystem, the interactions between different 808 species are captured with fundamental types of relationships, 809 such as competition and mutualism. Similarly, the relation-810 ships between state variables of a microgrid model can be 811 assessed [51]. For this assessment, the elements a hg ∈ A of the 812 system matrix A defined in (26) Table III.   The direct connection between the capacitance and the input 841 admittance is also apparent from the small-signal model shown 842 in Fig. 8. The relationship between the capacitance and input 843 admittance is further analyzed in the following quantitative 844 stability analysis, in particular, in the section presenting the 845 two-parameter variation. The number of positive links scales 846 proportionally to n, whereas the total number of links scales 847 proportionally to n 2 . Consequently, and as shown in Table II, 848 the discussed robustness indicator improves with an increase 849 in the size of the microgrid. To assess the expandability of the microgrid, the stability of 852 the swarm microgrid system matrices for different microgrid 853 sizes is analyzed based on their eigenvalues. The system matrix 854 A in (26) has eigenvalues of λ{ A}, and the system is stable if 855 the real part λ re of eigenvalue λ is negative for all λ{ A} 856 λ re { A} < 0. (29) 857 In addition, the following performance specifications need to 858 be met. In addition to the validation of the effect of the microgrid 885 size on the stability of the system given in Section VI-B, 886 an assessment of the effect of varying all other modeling 887 parameters is presented in this section. The parameter variation 888 ranges are selected by considering a realistic value range for 889 each individual parameter. The size of the microgrid system 890 is kept constant at n = 1000. An eigenvalue analysis is 891 conducted to assess the impacts of variations in each parameter 892 on the system stability. The parameters varied include the 893 parameters of the transfer functions modeling the dc-dc con-894 verters, as discussed in Section IV. These transfer functions, 895 namely, Z o (s), T VC (s), Y in (s), and T CC (s), follow the generic 896 structure of (20). The parameter identification for these transfer 897 functions is provided in Appendix C. All parameters of these 898 transfer functions except those identified to have trivial values 899 of {1, 0} are considered as part of this analysis. The list of 900 parameters also includes parameters that are less specific to 901 swarm microgrids. For these parameters, value ranges similar 902 to those reported in the literature were chosen [18], [30], [48]. 903 In this section, single-parameter variations are assessed first, 904 followed by an assessment of two-parameter variations.  Table III. For each parameter, the 907 symbol, the description, the reference value, the parameter 908 unit, and the parameter variation range from the smallest to 909 the largest value are provided along with the corresponding 910 eigenvalue results. Positive eigenvalues, which indicate an 911 unstable microgrid, are marked in bold font. In addition to 912 the analysis of the eigenvalues, the performance specifications 913 defined in Section VI-B are also assessed for. The subsequent 914 analysis is divided into three parts, discussing the effects 915 of variations in parameters related to the 1) network; 916 2) supplying mode behavior; and 3) charging mode behavior. 917 For the network parameters, no single-parameter variation 918 can cause the system to become unstable. In particular, neither 919 the cable length l c nor the ratio of cable inductance to cable 920 resistance L c /R c has a detrimental impact on stability. There 921 is no significant impact on the critical eigenvalues. Consid-922 ering (24) and the parameter value ranges of the network 923 parameters presented in Table III, the value range for the cable 924 resistance is between R c = 2 m and R c = 2 . Despite 925 this wide range and the large microgrid size of n = 1000, 926 no detrimental impact on stability is found.

927
Most of the supplying mode parameters do have an impact 928 on the critical eigenvalues, although without making the 929 system unstable. However, variations in the parameters that 930 model the output impedance can result in system instability. 931 The parameters that can lead to an unstable microgrid when 932 varied across their variation ranges are as follows:  2) ζ Z , the damping factor of the second-order subsystem of 936 the output impedance.

954
The identified parameters that can cause unstable microgrid 955 configurations due to single-parameter variations form the

958
This set is further analyzed in Section VI-C2, focusing on 959 two-parameter variations.  the total number of microgrid configurations assessed is 970 N 2 var = 100. Accordingly, the number of microgrid system 971 matrices assessed in this analysis totals 7 · 20 · 100 = 14 000. 972 Due to the high computational effort associated with such 973 a large number of microgrid simulations, the system size 974 is reduced to n = 100. The reference eigenvalue plot for 975 n = 100 is provided in Fig. 12(b).

976
To quantify the impact of each two-parameter variation, the 977 percentage of corresponding microgrid configurations that are 978 unstable is assessed. The two-parameter variation results are 979 compared against a baseline. This baseline is defined as the 980 percentage of unstable microgrids under the single-parameter 981 variation of only the first parameter, a member of K unstable .

982
The baseline analysis is undertaken first. As shown in 983 Table IV, among all parameters in K unstable , K 2,Z is associated 984 with the highest percentage of unstable microgrid configu-985 rations: 50% of the associated microgrid configurations are 986 unstable. For the other parameters in K unstable , the percentage 987 of unstable microgrid configurations under single-parameter 988 variations is between 10% and 30%.

989
The results for the two-parameter variations are provided 990 in Table IV and are discussed as follows, starting with the 991 network parameters, followed by the supplying mode para-992 meters, and concluding with the charging mode parameters. 993 The network parameters do not have a significant impact 994 on the percentage of unstable microgrid configurations in 995 comparison with the baseline. For example, for the baseline of 996  VARIATION RANGES CONSISTENT WITH TABLE III;  THE PERFORMANCE Table I.

1007
The supplying mode parameters give rise to one two-1008 parameter variation that leads to a reduction in the percent-  that when the LPF is deactivated, the range of values for R dr 1026 that result in a stable microgrid configuration is much more 1027 limited.

1028
Supplying mode parameters other than R dr have a detri-1029 mental impact on stability. The most severe impact is exerted 1030 by the parameters related to the output impedance. When any 1031 two of the four parameters modeling the output impedance 1032 are varied simultaneously, the number of unstable microgrid 1033 configurations increases significantly in comparison with the 1034 baseline. However, an even stronger increase in the number 1035 of unstable microgrid configurations is observed when an 1036 output impedance parameter is varied together with an input 1037 admittance parameter. The strongest impact is observed when 1038 varying K 2,Z and K 2,Y together. This two-parameter variation 1039 of K 2,Z and K 2,Y is shown in Fig. 15. To attain a stable 1040 microgrid configuration, the value of K 2,Z needs to be positive, 1041 and K 2,Y should be small. As long as the values are close to 1042 the reference values, the microgrid configurations are stable. 1043 For the charging mode parameters, the results show a 1044 similar pattern. The two-parameter sets that include two input 1045 admittance values lead to an increase in the number of unstable 1046 microgrids. However, the variation of the input capacitance C 1047 has a beneficial effect. For example, when T zero,Y is var-1048 ied together with C, the percentage of unstable microgrids 1049 decreases from 30% to 22%. This direct link between the 1050 model of the input capacitor and the model of the input 1051 admittance was already identified in Section VI-A during the 1052 qualitative assessment. Also for ζ Z , the variation of C has a 1053 positive impact. The two-parameter variation of C and ζ Z is 1054 shown in Fig. 16. A large value of C helps to attain stable 1055 microgrid configurations, even for low values of ζ Z . Fig. 15. Stability analysis for two-parameter variation of K 2,Z and K 2,Y , showing that 56% of the microgrid configurations are unstable; the individual parameter definitions, reference values, and variation ranges are consistent with Table III.  In Fig. 17(a), the characterization of the output impedance 1069 based on the response to a change inĩ n, j is shown.

1070
Allowed and no-go behavior areas are defined. Specific exam-1071 ples are shown for the variation pair of T 2,Z and T zero,Z .

1072
The input admittance is characterized in the same manner in leading to the definition of allowed and no-go areas for 1091 converter behavior. It is important to emphasize that all of 1092 these assessments were possible without knowledge of the 1093 internal structures of the converters. This is particularly rele-1094 vant for organizations implementing swarm microgrids. Such 1095 an organization will most likely not have access to the details 1096 of the internal structures of the power converters.

1097
The following summary aims to enable such organizations 1098 to ensure stable swarm microgrid operation with only a 1099 black-box behavioral assessment using standard laboratory 1100 equipment. The criteria listed in the following can be utilized 1101 early on in a swarm microgrid project, particularly during 1102 planning and procurement activities. 1103 1) All swarm units include a battery, as shown in Fig. 2. 1104 This is essential for the following statements, in partic-1105 ular, for the converter behavior analysis. Details of the 1106 converter behavior models are shown in Fig. 6.

1107
2) The control strategy summarized in Table I  and output parameters is provided in Fig. 4.

1114
3) The network parameters have no significant influence 1115 on stability, as shown in Table III. Neither a variation of 1116 the line inductance nor a variation of parameters deter-1117 mining the line resistance leads to an unstable system. 1118 The relevant sizing and topological layout should be 1119 determined using other methods, such as power flow 1120 optimization.

1121
4) For supplying mode operation using step-up conversion, 1122 the converter must inherently provide output VC. Fur-1123 thermore, the output impedance behavior of the step-up 1124 converter is critical to swarm microgrid stability. A step 1125 response characterization of the output impedance must 1126 fall within the allowed area shaded in light gray in 1127 Fig. 17(a).
1128 5) For charging mode operation using step-down con-1129 version, the converter must inherently provide input 1130 CC. Furthermore, the input admittance behavior of the 1131 step-down converter is critical to swarm microgrid sta-1132 bility. A step response characterization of the input 1133 admittance must fall within the allowed area shaded in 1134 light gray in Fig. 17(b).

1135
All of the presented validations are to be undertaken with 1136 batteries connected to the converters. These batteries should 1137 be of a type and condition as similar as possible to those to 1138 be used in field deployment.

1140
The concept of swarm electrification is based on modu-1141 lar, rapidly deployable, and expandable swarm microgrids. 1142 In this article, a decentralized control strategy that enables 1143 self-stabilization for swarm electrification, even under chang-1144 ing topological conditions, was developed and analyzed. 1145 To validate the performance of the proposed control strategy, 1146 a mathematical model of swarm microgrids of various sizes 1147 was developed. This model captures the core architectural 1148 attributes of swarm microgrids. To achieve a realistic but 1149 highly scalable model, a cascaded system modeling approach 1150 across Bangladesh [54]. An example of swarm microgrid 1165 components in operation is provided in Fig. 18. This figure   1166 shows a tea stall that has an SHS and is connected to a The gray-box model behaviors for both converter topolo-1191 gies derived in Section IV-C are identified on the basis of 1192 input-output step responses, as suggested in [35]. The identi-1193 fication follows the generic LTI function defined in (20). The 1194 identified parameter values are provided in the third column 1195 of Table III. 1196 Laboratory experiments were performed using lead-acid 1197 batteries and dual-active-bridge converters that functioned as 1198 both step-up and step-down converters. The dual-active-bridge 1199 converters were digitally controlled. The digital controller 1200 used was a low-cost 32-bit microcontroller unit. Measurements 1201 were sampled at 100 kHz; control outputs were updated 1202 at 10 kHz.   impedance behavior was analyzed based on an increase inĩ n, j , 1215 as shown in Fig. 19(b). The output impedance was identified 1216 as a second-order system with K 1,Z = 0 1217 Z o (s) = K 2,Z · (1 + T zero,Z · s) 1 + 2 · ζ Z · T 2,Z · s + (T 2,Z · s) 2 .
(A.5) the reference nodal current must be held constant; therefore, 1226ĩ CC = 0. Furthermore, the reciprocal effects between the input 1227 admittance and the capacitor must be accounted for, as shown 1228 in Fig. 6(d). The step response to a change in the input voltage 1229 can be used to identify the parameters of Y in (s) as long as the 1230 behavior of the capacitor is accounted for. The capacitor was 1231 modeled as having ideal behavior according to (14). To assess 1232 the measured step response data, a numerical approximation 1233 of such an ideal capacitor,ĩ C,num, j , was used 1234 i C,num, j = C · ṽ n, j t .

1238
1) The step-down converter was perturbed through a 1239 change inṽ n, j . The sampling time for the measurement 1240 was 1 μs.

1242
3) As the derivative approximation in (A.7) is susceptible 1243 to noise, the data needed to be filtered. For both the mea-1244 sured quantityṽ n, j and the calculated quantityĩ C,num, j , 1245 a 50-μs moving average filter was applied. 1246 4) Then, the parameters ofĩ Y,in, j were identified according 1247 to (A.8).

1248
As shown in Fig. 19(d), the dotted black curve correspond-1249 ing to the capacitor model adequately simulates the response 1250 in the time range from 0 ms to approximately 0.5 ms after 1251 the voltage change. However, after 0.5 ms, the curves devi-1252 ate. Therefore, the admittance needs to model this particular 1253 behavior, with a substantial deviation from the steady-state 1254 value at 2 ms and a return close to the initial steady-state value 1255 at approximately 4 ms. Accordingly, the transfer function for 1256 the admittance Y in (s) was identified as a second-order system 1257 with K 1,Y = 0 1258 Y in (s) = K 2,Y · (1 + T zero,Y · s) 1 + 2 · ζ Y · T 2,Y · s + (T 2,Y · s) 2 . (A.9) share Laboratory Team, Dhaka, Bangladesh, particularly, ments presented in this work.