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Improving dynamic performance in dc microgrids using trajectory control Bianchi, Marco Andrés 2017

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Improving Dynvmixezrformvnxz in DC bixrogriysjsing irvjzxtory ControlbyMarco Andre´s BianchiIng., Universidad Nacional del Comahue, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Electrical & Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2017c© Marco Andre´s Bianchi 2017AwstrvxtDirect-current (dc) microgrids interconnect dc loads, distributed renewable energy sources,and energy storage elements within networks that can operate independently from the maingrid. Due to their high efficiency, increasing technological viability and resilience to natu-ral disturbances, they are set to gain popularity. When load-side converters in a microgridtightly regulate their output voltages, they are seen as constant power loads (CPLs) fromthe standpoint of the source-end converters. CPLs can cause instability within the network,including large voltage drops or oscillations in the dc bus during load transients, which canlead to dc bus voltage collapse. Traditionally, the stability of CPL-loaded dc microgridsrelies on the addition of passive elements, usually leading to dc-bus capacitance increase. Inthis scenarios, source-end converters controllers are usually linear dual proportional-integral(PI) compensators. The limited dynamic response of these controllers exacerbates the CPLbehavior, which leads to the use of larger passive elements. Recent contributions focus onimplementing control modifications on the source-end converter in order to improve the sys-tem performance under CPLs. Particularly, the use of state-plane based controllers has beenstudied for the case of a single dc-dc power converter loaded by a CPL, showing fast androbust transient performance. However, the microgrid problem, where these faster convert-ers interface with others of a slower response has not been studied thoroughly. This workproposes the use of a fast state-plane controller to replace one of the systems source-endconverters controllers in order to improve three aspects of the microgrid operation: resiliencyunder CPL’s steps, load transient voltage regulation, and voltage transient recovery time.iiAwstrvxtSince the converter is operating within a microgrid, the controller incorporates a traditionaldroop rule to enable current sharing with the rest of the converters of the network. Thesmall-signal stability improvement of the whole system obtained by the addition of a sin-gle faster controller is analyzed for a linear model, and a parametric analysis demonstratesthe improvements in a detailed model. Simulations and experimental results of a microgridwith three converters feeding a CPL prove the effectiveness of the technique for large-signaltransients.iiiavy hummvryDirect-Current (dc) microgrids are electrical grids that connect electrical loads and low powerelectrical energy sources through a dc bus. Power converters enable the power flow from theenergy sources to the dc bus and from the dc bus to the loads. The speed at which a powerconverter reacts to source or load perturbations depends on its hardware and the controllerin its firmware. When the combined speed of the load-side converters is similar or largerthan that of the source-end converters the first one is seen as a constant power load (CPL)during perturbations. The CPL behavior is often mitigated with oversized hardware, whichis a sub-optimal solution. This work proposes to replace the conventional controller in onlyone of the source converters with a faster trajectory controller. The new controller, whichrequires only firmware modification, mitigates the CPL behavior and expands the microgridsstable operating area.iverzfvxzThis work is based on research performed at the Electrical and Computer Engineering depart-ment of the University of British Columbia by Marco Andre´s Bianchi, under the supervisionof Dr. Martin Ordonez.A first version of this work was presented in IEEE 7th International Symposium on PowerElectronics for Distributed Generation Systems (PEDG), 2016, [1], and later extended andin preparation for submission to IEEE Transactions on Power Electronics.As first author of the above-mentioned publication, the author of this thesis developedthe theoretical concepts and wrote the manuscripts, receiving advice and technical supportfrom Dr. Martin Ordonez, and developed simulation and experimental platforms, receivingcontributions from Dr. Ordonez’s research team.vivwlz of ContzntsAwstrvxt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiavy hummvry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iverzfvxz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vivwlz of Contznts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viaist of ivwlzs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixaist of Figurzs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAxknowlzygzmznts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiDzyixvtion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivF Introyuxtion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 Passive and Active Damping Using Additional Circuitry . . . . . . . 41.2.2 Load-End Converters Stabilizers . . . . . . . . . . . . . . . . . . . . 51.2.3 Source-End Converters Stabilizers . . . . . . . . . . . . . . . . . . . 61.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Contribution of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9viivwle of Contents1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Constvnt eowzr aovy Wzhvvior Any bvximum eowzr htzp . . . . . . . 142.1 Single Buck Converter Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Multiple Buck Converters in a DC Microgrid . . . . . . . . . . . . . . . . . 192.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Fvstzr hourxzBEny Convzrtzr Controllzrs in v DC bixrogriy . . . . . . . 223.1 Traditional Controllers: Nested Dual Loop PI . . . . . . . . . . . . . . . . . 233.2 Parallel Converters Linear Model Derivation and Small-Signal Stability . . . 263.3 Tradeoff Between Droop Dynamics and CPL Transient Dynamics Robustness 333.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34I Cirxulvr hwitxhing hurfvxz Controllzr jnyzr Droop avw . . . . . . . . . 354.1 Derivation of CSS for a Buck Converter Under Droop Control . . . . . . . . 364.2 Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Control Law Operation: Single Converter and Microgrid Scenarios. . . . . . 384.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 himulvtions vny evrvmztrix Anvlysis . . . . . . . . . . . . . . . . . . . . . . 445.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.2 Parametric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 Expzrimzntvl gzsults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54L Conxlusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62viiivwle of ContentsWiwliogrvphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64viiiaist of ivwlzs3.1 Root Locus Analysis Normalized Parameters . . . . . . . . . . . . . . . . . . 295.1 Simulations Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.1 Experimental Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55ixaist of Figurzs1.1 Diagram of a microgrid showing the three categories of techniques used tomitigate the CPL negative behavior of the system. . . . . . . . . . . . . . . 41.2 Dynamic performance of a dc microgrid for the proposed approach vs. thetraditional approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 One buck converter, as the ones found in microgrids, connected to a constantpower load (CPL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Behavior of a buck converter under constant power loads steps . . . . . . . . 162.3 Microgrid with parallel converters and equivalent circuit . . . . . . . . . . . 193.1 Nested dual loop control scheme . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Normalized Bode plots of the control loops for a buck converter in light loadcondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 Linearized model for M parallel buck converters controlled using nested dualPI compensators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4 Root locus for M parallel buck converters when VPB2 is varied, and resultingconverter 2 Bode plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.5 Dominant poles of a microgrid for load power sweep . . . . . . . . . . . . . . 323.6 Tradeoff between droop dynamics and dc voltage dynamic regulation when afaster converter is introduced in the microgrid . . . . . . . . . . . . . . . . . 33xaist of Figures4.1 State-plane and time domain representations of transient responses of a buckconverter with a CSS controller connected to a constant power load . . . . . 394.2 State-plane and time domain representations of start-up response for threeparallel converters connected to a constant power load, with one of the con-verters using CSS control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3 State-plane and time domain representations of a load step-up transient ofthree parallel converters connected to a constant power load, with one of theconverters using CSS control . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.1 Simulation of a start-up transient for the proposed CSS (A) and traditionalPI (B) approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2 Simulation of a load step-up transient for the proposed CSS and traditionalPI approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.3 Simulation of a load step-down transient for the proposed CSS and traditionalPI approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.4 Microgrid dc bus voltage response for different power step-up values . . . . . 485.5 Maximum power step vs. compensators’ bandwidths for a microgrid compris-ing three converters, which are controlled using four different approaches. . . 505.6 Voltage drop within a load step-up transient vs. power step for different com-pensators’ bandwidths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.1 Photograph of the experimental setup . . . . . . . . . . . . . . . . . . . . . . 546.2 Experimental results of a start-up transient: proposed vs. traditional approach 576.3 Experimental result of a 800 W CPL step-up transient: proposed vs. tradi-tional approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.4 Experimental result of a 800 W CPL step-down transient: proposed vs. tra-ditional approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59xiaist of Figures6.5 Experimental result showing maximum CPL step-up that the system can with-stand for both approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60xiiAxknowlzygzmzntsI would like to thank my supervisor Dr. Martin Ordonez for recruiting me to be part of hisresearch team. I am very grateful for his great enthusiasm, and his continuous support andcommitment that made this learning experience so rich and fascinating.I would also like to acknowledge my lab mates. Working with them along my programwas a privilege and a beautiful adventure. Special thanks to my closest friends in Canadafor encouraging and helping me when I needed it, and to my friends in Argentina for beingin touch and making me feel they are close.Finally, I would like to thank my parents Claudia and Roberto and my sister Mara, fortheir unconditional love and their faith in me.xiiiho my fumily unx friynxs.xivChvptzr FIntroyuxtionFCF botivvtionMicrogrids are electrical networks that interconnect power energy sources with storage andconsumption elements, and that can operate as a single controllable system. Their powerrange can vary from around 0-10 kW [2] up to 0-100 kW [3]. Their proliferation is a conse-quence of the increasing technological viability for integrating distributed renewable energyresources efficiently and reliably at a rapidly reducing cost. The nature of microgrids of beingable to operate independently from the main electrical grid, makes them an appealing solu-tion to supply the customer unmet demand in developing countries, which is currently of 1.2billion people that do not have access to electricity [2]. Moreover, its resilience to natural dis-asters and other disruptions, and its capability of incorporating more carbon-efficient waysof producing and distributing energy have motivated an increased number of installationsaround the world.It is expected that the rapid growth of dc-native loads will lead to the popularization of dcand hybrid ac/dc microgrids [4]. DC microgrids show high efficiency and enable the simpleinterconnection of renewable energy sources with loads and energy storage systems using asmall number of power conversion stages. At a given instant a microgrid can be analyzedas a set of converters supplying energy from the sources to a dc bus, and another set ofconverters transferring energy from the bus to the loads. Because of the small scale storageelements, microgrids dynamics are faster than that of the traditional ac grid, which is sub-1FCFC botivvtionjected to the inertia of massive rotating electrical machines. Microgrids fast dynamics makethese networks especially vulnerable to sudden changes in the operating conditions of thepower converters interconnected within them. Particularly, when load-end converters tightlyregulate their load voltage they behave as constant power loads (CPL), which dynamic effectemulates a variable negative resistance and challenges the microgrid’s stability. A microgridincluding a CPL load can be modeled as a set of parallel converters supplying energy to dcbus where a single CPL is connected that simulates the combination of all the electronic loads[5–7]. The most critical-scenario, from the dc bus stability standpoint, happens when an in-stantaneous power mismatch between sources and loads leads to a sudden voltage change inthe bus capacitors. During sudden transients, which could be the result of a load step or aconverter turnoff, the effects of both fast and slow system dynamics can be identified in thedc voltage signal[8]. While fast dynamics can be directly linked to the converters’ voltagecontrollers, the slower transients are imposed by the droop controllers and the interactionamong them. Several contributions have focused on improving the droop dynamics, ana-lyzing the system’s long-term behavior and neglecting the fast dynamics during transients[9, 10]. However, the mitigation of the negative effects of CPL transients, such as voltagedrop and oscillation, requires the fast dynamics of the system to be addressed. A number ofstrategies have been studied in the literature that aim to mitigate the CPL behavior. Fromthe wide range of possibilities, including passive damping, and the application of advancedcontrol techniques, the use of state-plane based control schemes in source-end converters canresult in very fast and robust performance during sudden transients, which is of particularinterest to increase the resilience of microgrids. Although, these techniques have been im-plemented in power converters connected to CPL’s, the effect of their use in the source-endconverter of a microgrid have not been studied extensively. Moreover, some contributions inthe literature looked to improve the microgrid resilience under transients by implementingfaster controllers in some of the converters of the microgrid. These works, where dissimilar2FC2C aitervture geviewspeed converters can be found, are mostly restricted to hybrid energy storage applications.Furthermore, the converters’ inner control techniques employed are conventional linear tech-niques, which are known for having a limited dynamic response. Applying a faster controltechnique in only some of the converters of a microgrid to increase the system resilience is anappealing concept when the challenge of improving the pre-existing infrastructure’s dynamicperformance is needed. This work proposes to replace one of the conventional controllers inone of the power converters of a microgrid with the faster state-based one in order to improvethe system resilience and its dynamic performance under sudden CPL transients.FC2 aitzrvturz gzvizwThe behavior of CPL’s was extensively discussed in the literature and many strategies havebeen studied that aim to mitigate its negative effect on the stability of the system. Whilesome of the proposed techniques are designed for its application in dc microgrids, other worksfocus on systems comprised of a single-converter connected to a CPL. The adaptation of thetechniques employed to improve the dynamic performance of stand-alone converters for theiruse in microgrids is often possible, and can potentially motivate new ideas. The universe oftechniques that tackle the CPL challenge can be grouped in three main categories, dependingon the element of the microgrid where these are applied, and are shown in Fig. 1.1. Firstly,external stabilizers (Ê in Fig. 1.1) add new devices to the network, usually passive elements,with the sole purpose of stabilizing the system. Secondly, load-end converters stabilizers (Ëin Fig. 1.1) implement the stabilization as part of the load converter control law. Finally,a third group of techniques implement the stabilization techniques whithin the source-endconverter’s controllers (Ì in Fig. 1.1). The next three sub-sections will cover the mentionedmain contributions categories that are found in the literature.3FC2C aitervture geviewini2i1vocontrolPloadsource-end convertersload-end converters(grouped)CPL (tight regulation)v1control231Figure 1.1: Diagram of a microgrid showing the three categories of techniques used to mitigatethe CPL negative behavior of the system. The stabilization techniques are grouped accordingto the element of the microgrid where they are applied: addition of new elements to thenetwork (Ê); load-side converters’ controllers (Ë), source-end converters’ controllers (Ì).FC2CF evssivz vny Axtivz Dvmping jsing Ayyitionvl CirxuitryThe techniques that stabilize the CPL negative behavior using circuitry that is external oradditional to the source-end and load-end converters (Ê in Fig. 1.1) can be separated inactive and passive damping stabilization. External active damping relies on the use of addi-tional power electronics to increase the damping of the circuit. A bidirectional buck-boostconverter in parallel to the dc bus is usually employed, whose current controller is shaped toemulate the behavior of a resistive load or an RC branch connected to the bus [11–14]. Whilesome of the proposals re-circulate the absorbed energy to a neighboring dc bus comprised of4FC2C aitervture geviewbatteries or super-capacitors, other topologies need to deal with zero net energy since theyonly count with a regular capacitor to store energy.On the other hand, passive damping mitigates the CPL behavior with the addition of net-works of passive elements or the re-sizing of the source-converters output filters . The externalpassive dampers include RL and RC arrays that connect to the converters output filters orthe dc bus [15]. Alternatively, when re-sizing converters LC output filters, two main ap-proaches are possible: decreasing the filter series inductor values or increasing the outputcapacitor [16]. Because a decrease in the inductance deteriorates the converter’s electromag-netic compatibility, the over-sizing of the output capacitors is usually preferable and is acommon practice to improve the system stability.In general, in both active and passive methods, the damping impedance is determined bythe application of Routh-Hurwitz’s or Middlebrook’s criterion [17]. The stability criterionrequires a linear model of the system to be obtained, which is usually done considering theopen-loop transfer functions of the source-end power converters and the linear model of thenon-linear load under certain load conditions.The use of active damping techniques exhibits a larger level of complexity than that of pas-sive stabilization. However, its application could be desirable in high power systems wherehaving non-dissipative dampers could signify a large reduction in energy consumption. Onthe other hand, passive stabilization particularly by resizing the dc bus capacitance can bean appealing solution because of their simplicity but it also increases the size and cost of themicrogrid building blocks.FC2C2 aovyBEny Convzrtzrs htvwilizzrsThese stabilizers constitute a family of solutions that increase the system stability by im-plementing control strategies embedded in the load-end converters (Ë in Fig. 1.1). Theircontrol law aims to limit the maximum rate at which the load-end converter input current5FC2C aitervture geviewcan change. Consequently, the CPL behavior is mitigated at the expense of a slower loadvoltage dynamic regulation. In [18], this method is implemented by emulating a virtual ca-pacitance of the dc bus for a single CPL, and in [19] this is done using stabilizing agentsfor a number of parallel load-end converters. The stabilizers decrease the converters inputpower reference as a function of the high-frequency components of the square of the inputvoltage signal. The objective is for the load-side converter its CPL behavior during tran-sients. On the other hand, [20] proposes to implement a virtual impedance emulated at theinput of the CPL converter. The impedance, which can be emulated in series or in parallel,is mathematically defined using a piecewise function, that gives the impedance a differentbehavior depending on the frequency. In this way, the impedance stabilizes the system atthe frequencies where the Middlebrook’s criterion is not originality met, but his behavior isnull outside of this frequency scope. A different approach is studied in [21], where the CPLcontroller maximum current saturation values are theoretically determined as a function ofthe source-end converters limitation. The paper establishes a simple rule that should be fol-lowed when configuring the CPL controller’s operating limits in order to guarantee stabilityin start-up and step-up transients.Load-end converters stabilizers are an interesting approach that tackles the challenge of theCPL behavior by decreasing the fast dynamic response of the load-end converters. However,this is expected to be detrimental to the load voltage regulation and is undesirable for manycases.FC2C3 hourxzBEny Convzrtzrs htvwilizzrsThe described methods so far mitigate the negative behavior seen in CPL-loaded microgridswith methods that do not require any modification of the control law of the controller ofthe source-end converter. In these cases, source-end converters are traditionally controlledusing conventional linear control techniques. Particularly popular is the use of current-mode6FC2C aitervture geviewdual proportional-integral compensators (PI)[22, 23]. In these control schemes, the existenceof a current-loop simplifies the voltage control and provides safety benefits, but imposesthe bandwidth reduction of the outer voltage loop. Consequently, dual PI ruled convertersdynamic performance tends to be slow, which exacerbates the negative effects of the CPLsin the microgrid.On the other hand, the application of source-end controllers specifically designed to mitigatethe CPL behavior (Ì in Fig. 1.1) covers a wide range of options: linear, mixed and non-linear techniques. Linear approaches focus on stabilizing the small-signal closed-loop transferfunction using linear feedback control. In [5, 24–26] this is done by the application of activedamping techniques, while in [16] it is done using passivity-based control. Alternatively, alinear modification of the current sharing controller in the source-end converters of hybridenergy storage systems (HESS) was studied recently to improve the dynamic performance ofdc microgrids [27–29]. The inner control law of the source-end converters is a conventionaldual loop PI compensator. But, the external current sharing controller is modified to producea transient response that is coherent with the discharge time of the energy storage technologythat the converter is interfacing. With this strategy, converters connected to super-capacitorswill respond faster than those connected to batteries, providing a higher frequency currentpulse that should extinguish after the transient is over. Although its application is specificallydesigned for HESS scenarios, the concept of having a few converters ruled by a faster controllaw with the objective of improving the microgrid stability for a more general applicationwas not studied extensively in the literature.A number of controllers combine the application of non-linear operations with linear con-trol techniques. These controllers achieve better transient performance while keeping thesimple implementation of classic linear control theory. In [30] and [31] an outer voltage loopis used in combination with a hysteresis current control and peak current control respectively.Another hybrid controller is introduced for different converter topologies in [32–34], where7FC2C aitervture geviewa non-linear geometric loop regulates the voltage and a conventional inner PI controls theinductor current, although in these cases the converter is not loaded with a CPL. Moreover,adaptive controllers like in [35] also enable the application of linear techniques by estimatingthe parameters of the system and modifying the linear compensator’s gains in real time.In [36, 37] a non-linear state feedback loop is employed in order to obtain a large-signallinearized plant that is then compensated applying conventional linear techniques. Otherstudies apply less traditional but faster controllers after a nonlinear feedback transforma-tion is performed, at the cost of increased mathematical complexity. In [38], the nonlineartransformation is followed by a proportional state feedback which gain array is obtained bythe Ackerman’s formula for poles allocation; while in [39], the nonlinear feedback takes thesystem to a canonical form that enables the application of a backstepping method to obtainthe control law. Both methods implement state variables observers in order to increase thecontrol immunity to parameters mismatch. Among non-linear control techniques, boundarycontrollers have demonstrated very good performance while ruling the behavior of the con-verter during sudden load changes. Since these controllers do not experience the bandwidthlimitation of conventional linear, and hybrid controllers, they can make the converter workcloser to the theoretical limit, studied in [40], and its implementation does not require com-plex mathematical operations. First order sliding mode controllers were studied in [41, 42]for their use in CPL loaded converters, and the natural trajectories derived in [43, 44] wereapplied in the control laws of a power converter with a CPL in [45]. While state-planecontrollers display rapid dynamic response, the work described above focuses on one con-verter only (stand-alone), rather than the multiple converters interacting in a dc microgrid;the interaction of a fast state-plane ruled converter with other conventional converters in amicrogrid is yet to be studied.8FC3C Contriwution of the lorkFC2CI hummvryThe presence of CPLs in dc microgrids imposes an instability challenge that has been tack-led using three main approaches: use of additional stabilizing circuitry, modification of theload-side converter’ controller, and modification of the source-end converter’ controller. Thepublished work succeeds on improving the dc microgrid performance under CPL behaviorbut there are still areas that can be explored. In particular, the use of faster state-based con-trollers embedded in the source-end converters of a microgrid while interacting with otherconventional slower controllers in a microgrid is lacking in the literature. The use of dis-similar speed controllers in a dc microgrid is mostly limited to the specific case of hybridenergy storage systems, and its study for a more general case is yet to be done. These tech-nical challenges are addressed in this thesis and a new dc microgrid stabilization approach isproposed.FC3 Contriwution of thz lorkThis work presents a novel stabilization approach that mitigates the negative effects of CPLsin a dc microgrid by implementing a faster controller in one of the converters and studies theimplications of having a faster controller in the system. The contributions of this thesis aresummarized below:• A state-trajectory controller with an embedded droop law is designed and assessed forits implementation in one of the source-end converters of a dc microgrid. The proposedapproach improves three critical aspects of the microgrid operation: 1) resiliency underlarge CPL’s steps; 2) load transient voltage regulation; 3) voltage transient recoverytime. The strategy does not require to update all the controllers of the microgrid, andthe improvements in the performance are obtained by only implementing the controllerin a single source-end converter. Figure 1.2 shows a conceptual comparison between the9FC3C Contriwution of the lorkproposed and traditional strategies. The proposed controller (case A) leads to dynamicperformance improvements under the sudden load change (CPL) when compared totraditional dual PI current-mode controllers (case B).• The implications of having a faster controller in one of the source-end converters ofthe dc microgrid are studied. It is shown that having faster controllers will improvethe small-signal stability of the system. Moreover, the existence of a tradeoff betweendroop dynamics and dc bus dynamic regulation is observed when one of the parallelconverters is faster.10FC3C Contriwution of the lorkCPLBuckConvertervcplio,cplio1ionioxDroop PIDDroopDroopPIDCSSPIDTraditionalProposedABvcpl CPL Steptntntnio,cplio,xLess recovery timeLess voltage dropDroopdynamicsBABMore voltage dropMore recovery timeLess resilient to large CPL steps32A5%BA current sharingconvergence Resilient to larger CPL steps1Figure 1.2: A dc microgrid comprising several parallel converters connected to a CPL. Itsdynamic performance is depicted for the traditional approach (B) versus the proposed ap-proach (A), where a traditional linear control is replaced with a CSS controller in one of thesource-end converters). Case A improves three aspects of the microgrid operation: resiliencyto CPL steps À, transient voltage regulation Á, transient recovery time Â. In case A, tvBxtakes a larger share of tvBcws during the transient, what results in a faster dynamic regulationof vcws. The current imbalance after the transient for case A slows down the droops dynamics,however, the current sharing capability is maintained.11FCIC ihesis dutlineFCI ihzsis dutlinzThe present work is organized as follows;• In Chapter 2, the behavior of a CPL when connected to a power converter with an LCoutput characteristic is explained. The ON state trajectories of a normalized versionof this system are obtained for a range of initial conditions and CPL steps, and themaximum stable power step is found. Then, the results obtained for a single powerstructure are extended for the case of parallel source-end converters in a microgrid.• In Chapter 3, the small-signal stability of a linearized model of the system is studiedwhen one of the converters is ruled by a faster controller. And the tradeoff for theproposed strategy between current sharing dynamics and dynamic voltage regulationis also explained. The different speed controllers are modeled using dual PI compen-sators, and the practical limitations of expanding the bandwidth of these convertersare discussed.• Chapter 4 derives and explains the control law to be used, and the proposed strategyexplained for the stand-alone and microgrid cases in the state-plane domain.• In Chapter 5 the concept is assessed using a microgrid simulation model. And a morecomprehensive description of the system performance of the proposed approach ver-sus that of the traditional approach is obtained through a parametric analysis for anormalized microgrid.• Experimental results of a microgrid composed of three power converters feeding a CPLprove the validity of the concept in Chapter 6. When one of the controllers rulingone of the power converters is replaced with a CSS controller with embedded droop,the dynamic performance of the dc microgrid improves, showing less voltage drop, andfaster responses during load transients.12FCIC ihesis dutline• Finally, in Chapter 7 a summary and conclusions of this work are presented, along withsome details of future research ideas.13Chvptzr 2Constvnt eowzr aovy Wzhvvior Anybvximum eowzr htzpIn this chapter, the behavior of a system comprising power converters loaded with a constantpower load is studied. The buck topology is a common structure found in dc microgrids andwill be used as source-end converter. It is observed that for a given initial condition, there isa theoretical maximum load power step-up (∆Pcrpt) that the system can withstand before thedc voltage collapses. Stable power steps that are close to ∆Pcrpt are possible if the control lawruling the buck converter reacts immediately after the step occurs. First, the value ∆Pcrpt isobtained for a normalized system and sets a maximum limit against which the performanceof any control law can be compared to characterize its large signal behavior. The analysis islater extended from the case of a single converter to several power converters in a microgrid.2CF hinglz Wuxk Convzrtzr CvszIn order to add generality to the present work, the analysis will be performed in a normalizeddomain when possible. The normalization base values are given by the filter’s characteris-tic impedance and time constant, and the power supply input voltage of the buck converter.When the system includes more than one converter, the base values will be obtained from theparameters of one of them. This procedure makes the study independent of the specific filterand voltage settings. In Figure 2.1 a schematic of a normalized buck converter connected to142CFC hingle Wuxk Converter CvseCPLVccn vonLn =12piCn =12piiLnuionFigure 2.1: One buck converter, as the ones found in microgrids, connected to a constantpower load (CPL). Variables and parameters are normalized using the power supply inputvoltage (acc) and the LC output filter resonant frequency (F0) and impedance (e0) as nor-malization base values. This is indicated using the subscript “n”.a CPL is depicted. The parameters and variables names have the subscript “n” to indicatethat the respective quantities have been normalized. The normalization for values of voltageax, current Tx, impedance ex, or time Tx is performed as follows:axn = ax=arlm F Txn = Tx · e0=arlm F exn = ex=e0F Txn = Tx=T0Fwhere the normalization base quantities are given by:arlm = acce0 =√L=CT0 = 1=F0 = 2√LC:In particular, the normalized values of the inductance and capacitance are obtained fromthe normalization of their respective reactances using the impedance base quantity e0. Theequations are:Ln = L1T0e0=12(2.1)Cn = Ce0T0=12: (2.2)Using the normalized values, the current through the ideal CPL in Fig. 2.1 is given by:tvn =PvnvvnF (2.3)152CFC hingle Wuxk Converter Cvse(a)von(0)iLn(0)iLnvonPCritnP0=0P2P3P4P5P1Vccntnvonvon(0)tniLniLn(0)(b)0.2 0.4 0.6 = 0.80 p.u.0.1 0.2 0.3 0.4 tnPonPon(0)PCritnP0P1P2P3P4P5Pon(0) = von(0) • iLn(0)Pi>PCritnleads to voncollapseif Pi<PCritnconverter canrecover from transient max. stable Pon  forinitial condition Pon(0)ΔPCritnΔPCritn=Pon(0)-PCritnΔPCritn0.1Figure 2.2: Behavior of a buck converter under constant power loads steps. (a) ON trajecto-ries in both state-plane and time domain for a buck converter loaded with a CPL. For any setof initial conditions, there is a maximum stable power step value ∆PCrptn = PCrptn − Pvn(0),where PCrptn is the CPL power after the step and Pvn(0) is the initial power. If the powerafter the step is smaller than PCrptn the system can recover from the transient keeping theON a move to the new target point by adopting right switching actions. For values of powerlarger than the critical the system is unstable independently on the switching actions taken.(b) Normalized maximum stable power step ∆PCrptn vs. converter initial power Pvn(0). Thevalue of ∆PCrptn when Pvn(0) = 0 is approximately 0.3 p.u. for a normalized buck converter.where Pvn is the normalized load power. The dynamic behavior of the CPL can be understoodfrom its small-signal resistance:dvvndtvn= − v2vnPvn= −rcwsn: (2.4)162CFC hingle Wuxk Converter CvseThe negative incremental resistance introduces instability into the system, which may leadto dc voltage collapse if not compensated on time. Its destabilizing effect can be understoodduring a load step-up, which is the most challenging condition for a converter feeding a CPLoccurs. At that moment, rcwsn decreases instantaneously. The consequent sudden increaseof tvn tends to discharge the dc bus capacitor, leading to a decrease of vvn. This furtherdecreases rcwsn and the process continues. If the inductor current tLn does not grow fastenough to compensate for the rapidly increasing tvn before the dc-voltage is too small, thetransient could lead to a dc-voltage collapse. Therefore, the converter supplying the CPLshould be fast enough for vvn to recover before rcws becomes too small. The maximum responsespeed that the converter can achieve is given by its reactive components, and at the sametime limited by the effectiveness of the control technique ruling the converter.Right after a CPL step-up, an optimal control law would make the buck converter adoptits ON structure (u = 1) since that would lead to an immediate increase of tLn. Figure 2.2(a)shows the converter’s ON trajectories in the state-plane and the respective time domainresponses for different power steps. For a given initial power [vvn(0); tLn(0)], the converterfollows different trajectories that vary with the power value after the step, which is calledPp with t = 0F 1F 2F :::. If the power after the step is lower than a critical value PCrptn, thetrajectories move the operating point closer to the new target point, which will be at vvn(0)but at a different current tLn that depends on the power step. In these transients, marked inthe figure as P0, P1, P2, and P3, if the correct switching actions are taken, the converter canrecover from the transient and reach the target value. While for the trajectories with finalpower P1, P2, and P3 the load increases its power after the step, the ON trajectory markedas P0 corresponds to a step-down. A negative power step is less challenging because rcwsnincreases instantaneously after the step and so does the dc voltage.It is observed that, the larger the power step, the larger the voltage drop on vvn and thelonger it takes for the converter to recover. For each initial condition, there is a maximum172CFC hingle Wuxk Converter Cvsepower step final value that the converter can withstand, marked as PCrptn in the figure. If itis exceeded, the voltage collapses irreconcilably as shown for the cases P4 and P5 in whichthe trajectories reach the vertical axis (vvn = 0). This value defines a hardware limitationfor the maximum power step that can be applied to the system. For lower Pp values, the useof an effective control technique can lead to an stable operating condition after the load steptransient.The value of PCrptn was calculated numerically for a number of initial conditions usingMATLAB. The range of current values tLn(0) was selected from 0 p.u. to 0.8 p.u while thevoltage initial condition was set to vvn = 0:8 p.u., which will be the operating point for thecases studied in this work. The code implemented solves numerically the ON differentialequations of the normalized buck converter (see (4.2) and (4.3)) for each set of initial condi-tions and for a given CPL step initialization value. Then, the resulting trajectory is markedas ‘unstable’ or ‘stable’, depending on whether or not it leads to a voltage what leads to thecalculation of a new CPL power value. The process is iterated for different power values andPCrptn is found using successive approximations.The results are summarized in Figure 2.2(b), as a function of the initial power Pvn(0) =0:8 · tLn(0) and the power step value ∆PCrptn = PCrptn−Pvn(0). The chart shows that a buckconverter can ride through a maximum ∆P of around 0:3 p.u. when initialized at zero Pvn,and its capability decreases as Pvn(0) increases.It is expected that a fast compensator will bring the converter close to its physical limit,allowing the system to withstand maximum load steps that are closer to ∆PCrptn. The resultssummarized in Fig. 2.2(b) can be used as a benchmark to compare the performance of anycontroller with the theoretical limit when the converter is subjected to CPL load step-ups.As the values in the curve correspond to a normalized buck converter, they can be extendedto any converter after its parameters have been normalized.182C2C bultiple Wuxk Converters in v DC bixrogriyLVcc,11C 1iL,1CPLLVcc,22C 2iL,2LVcc,MMC MiL,M(a)LVcc,eq1C 1 C 2 C ML2LMCPLvo(b)LeqCeqFigure 2.3: (a) M parallel buck converters in a microgrid supplying a CPL. (b) Microgridequivalent single converter model for synchronized ON or OFF actions.2C2 bultiplz Wuxk Convzrtzrs in v DC bixrogriyIn a dc microgrid, the number of buck converters supplying the CPL is larger and the systemcan be represented as in Fig. 2.3(a). In order to extend the analysis to account for multipleconverters, the parallel buck converters can be combined into a single equivalent unit. Themaximum possible power step that the system can withstand, is obtained for the case inwhich theM converters have synchronized switching actions. For this scenario, the equivalentinductance Llx and capacitance Clx can be obtained by paralleling the individual elementsof each converter.Llx = LA ‖ LI ‖ ::: ‖ LT (2.5)Clx = CA + CI + :::+ CT (2.6)192C2C bultiple Wuxk Converters in v DC bixrogriyTo obtain the input voltage of the equivalent circuit some extra steps are needed. The differ-ential equations of the inductor current for the m-th converter and for the equivalent circuitare:dtLBtdt=1Lt(accBt − vv) (2.7)dtLBlxdt=1Llx(accBlx − vv) : (2.8)If the total current through the equivalent inductor is tLBlx =∑Tt=1 tLBt, then (2.8) can berewritten as:T∑t=1dtLBtdt=1Llx(accBlx − vv) : (2.9)Finally, substituting (2.7) in (2.9) and solving for accBlx:accBlx = LlxT∑t=1accBtLt: (2.10)The equivalent circuit is depicted in Fig. 2.3(b), where Llx and Clx are the result of parallelingthe converters’ inductances and capacitances. Now, using the equivalent buck converter, thetheoretical maximum power step that the system could withstand (∆PCrptn) can be calculatedfrom the results obtained for a single buck converter in Fig. 2.2(b). The normalizing powerof the equivalent circuit is obtained from the normalizing base values arlm = alx and ev =√Llx=Clx as shown below:Prlm =a 2rlmev= a 2lx ·√ClxLlx(2.11)Then, the critical power step-up for a given initial power Pvn(0) is defined by:∆PCrpt = ∆PCrptn · Prlm F (2.12)202C3C hummvrywhere ∆PCrptn is obtained from the curve in Fig. 2.2(b). The value of ∆PCrptn obtainedfor a microgrid with parallel buck converters loaded by a CPL gives a theoretical limit ofthe maximum power step-up that can be applied in the system. As in the case of a singleconverter, it can be used to contrast the performance of the system against the ideal case,and measure the margin for improvement.2C3 hummvryIn this chapter, the incremental resistance of a CPL was derived and its implications on sta-bility were discussed for a buck converter loaded by a CPL. The large-signal behavior of thesystem during load step-up transients was analyzed using the state-plane domain represen-tation. Moreover, the maximum power step that the system can recover from (∆PCrpt) wasobtained for a range of initial power conditions for a single buck converter, and the resultswere extended for the microgrid case. ∆PCrpt can be used as a benchmark to observe howclose the maximum stable power step of a given system is to the theoretical maximum, andmeasure the transient performance of the source-end converters controllers. The analysis wasdone in the normalized domain, and the normalization process was introduced. This allowsthe results to be extended to any problem that shares the same circuit topologies.21Chvptzr 3Fvstzr hourxzBEny ConvzrtzrControllzrs in v DC bixrogriyThe use of faster response control techniques in power converters with constant power loadsshould lead to an improvement in the system stability. Since CPL behavior is a consequence ofthe load-end converter being faster than the source-end one, an increased transient responseof the latter should mitigate the negative CPL effects. Moreover, if the faster controllerreplaces one of the regular compensators in a microgrid with parallel converters loaded witha CPL, the stability of the system as a whole could be potentially improved. The inclusion ofa faster compensator in the system increases the speed at which the combined set of parallelsources supplies power to the dc bus, and decreases the difference in the transient responsebetween source converters and CPL.When the converters are controlled using linear compensators, the dynamic responsecan be enhanced if the compensator’s bandwidth is increased. However, this requires aproportional increase in switching frequency to prevent non-attenuated switching harmonicsto be amplified by the compensator. Usually, an increase in switching frequency is undesirablesince it moves the hardware out from the operating point it was designed for. Althoughtraditional linear compensators exhibit a very limited response for a given switching frequencywhen compared with other non-linear strategies as state-plane based controllers, they can beused to model the microgrid behavior effectively. In particular, the effects of having a fastersource-end controller in a microgrid is of interest in the present work.223CFC irvyitionvl ControllersO cestey Duvl aoop eIPIv PI iiLn*Gid Gvivon* voniLnGi,CLGv,CLGv,OLGi,OLFigure 3.1: Nested dual loop control scheme: an inner PI loop controls the converter inductorcurrent (tLn), while an outer PI loop controls the capacitor voltage (vvn).This chapter first reviews some of the features of a traditional dual loop PI controller,which will be used later to model the source-end converters controllers in a microgrid. Sec-ondly, a state-space model approximation of a system comprising a number of converters inparallel is derived and the small-signal stability of the model is then analyzed. The analysis isperformed for the case in which one of the converters is controlled using a faster compensatorthan the others. Finally, section 3.3 assesses the large-signal behavior for a three parallelconverter system with dissimilar controller speeds.3CF irvyitionvl ControllzrsO czstzy Duvl aoop eIA traditional control scheme for power converters involves using a nested dual loop PI com-pensator to control both current and voltage, as depicted in Fig. 3.1. The outer voltage loopis tuned to have a slower response than the inner current loop, which can be observed whencomparing the bandwidths between both closed-loop Bode plots. Figure. 3.2 (b), depicts thevoltage and current closed-loop frequency responses for a normalized buck converter com-pensated with a nested dual PI. The current compensator was tuned to achieve a crossoverfrequency fp approximately 10 times lower than the switching frequency fsw. The distancebetween fp and fsw gives enough room for the switching harmonics to be attenuated by233CFC irvyitionvl ControllersO cestey Duvl aoop eIPhase (deg) -15-10-50Magnitude (dB)10-310-210-1100101102f [p.u.]f v -3dB-180-90090nf i f Closed loop|G      |v,CL∠Gv,CL|G      |i,CLGi,CL∠(a)(b)≈10x ≈10x10-310-210-1100101102-180-90090Phase (deg)f [p.u.]n-100-50050100Magnitude (dB)Open loop|G      |v,OL∠Gv,OL|G      |i,OLGi,OL∠ *ideal current source approximation∠Gv,OL*swn v,OL   |G      |*Figure 3.2: Normalized Bode plots of the control loops for a buck converter in light loadcondition. (a) shows the open loop Bode plots for both current (RpBOL), and voltage (RvBOL)loops. RvBOL∗ is the open loop frequency response when the current closed-loop is approxi-mated with a constant current source. The approximation is accurate for a range that exceedsthe closed-loop cut-off frequency. (b) Shows the closed-loop Bode plots and the separationrequired for cross-over frequencies, and with respect to the switching frequency.243CFC irvyitionvl ControllersO cestey Duvl aoop eIconverter 1converter 2CMiL,MPIMIo^Rcplrrrvo,M^^Rd,MVspconverter Mvcpl^Figure 3.3: Linearized model for M parallel buck converters controlled using nested dual PIcompensators. Since the dynamics of the inner current closed-loop are very fast in comparisonwith the voltage loop, the current loop behavior can be emulated using a controlled currentsource.the compensator’s low pass filter response, preventing the switching ripple to be amplifiedthrough the feedback loop [46, 47]. On the other hand, the voltage compensator closed-loopcrossover frequency fv is close to 10 times lower than that of the current compensator [47].The separation between fp and fv makes the inner loop fast from the point of view of thevoltage loop. Consequently, the dynamics of the current loop can be neglected when ana-lyzing the behavior of the whole system, what leads to a lower order representation of thesystem. In Fig. 3.1, this is done by replacing the current closed-loop transfer function RpBCLwith a unity gain, which is equivalent to replacing the current loop with a controlled currentsource. In Fig. 3.2, it is observed that the open loop Bode for the voltage compensator andplant when the current source approximation is used (RvBOL∗), matches that of the completesystem representation (RvBOL) for the frequency range of interest. The approximation RvBOL∗deviates from RvBOL for frequencies where the response attenuation is already large enough,around -20 dB for the example in Fig. 3.2(a).The described compensator constitutes a traditional control approach and will be em-ployed in this chapter to model the behavior of a microgrid when one of the source-end253C2C evrvllel Converters ainevr boyel Derivvtion vny hmvllBhignvl htvwilitycontrollers has increased speed. In the following chapters, it will be used to compare themicrogrid dynamic performance of the proposed approach with that of the traditional ap-proach.3C2 evrvllzl Convzrtzrs ainzvr boyzl Dzrivvtion vnyhmvllBhignvl htvwilityThe effects on stability of having a single faster controller in one of the source-end convertersof a microgrid can be studied modeling the compensators dynamics with a traditional nesteddual loop PI controller. Because of the bandwidth separation between inner and outer closed-loops, the inductor dynamics in each converter can be neglected to obtain a reduced ordersystem representation. The inner closed-loop can be modeled with a current source whichoperating point is set by the PI voltage compensator. Moreover, since the converters willshare the load current in a microgrid the target voltage vv in each of them will be set usinga conventional droop law [48]:vv = asw − tv ·Rd (3.1)where asw is the reference voltage at no load, tv is the converter output current, and Rd isthe droop resistance. If a set of M buck converters with the mentioned characteristic arestacked in parallel, the small-signal equivalent circuit can be represented as in Fig. 3.3. Inthis model, the output voltage dynamics for each converter are given by the capacitor voltageequation:dvˆvBtdt=1Ct(tˆLBt +vˆvBt − vˆcwsr): (3.2)The subscript m means that the variable belongs to the m-th converter, and the hat symbol“∧” indicates that the small-signal is being considered. The resistances of the lines that263C2C evrvllel Converters ainevr boyel Derivvtion vny hmvllBhignvl htvwilityconnect the capacitors and the CPL are named as r, they are considered equal for the sakeof simplicity. Since the inductor current tLBt is set by the voltage PI compensator and thedroop law (3.1), as observed in Fig. 3.3, its dynamics are defined by:tˆLBt = VPBt · (aˆsw − vˆvBt − tˆLBt ·RdBt) +VPBt∫(aˆsw − vˆvBt − tˆLBt ·RdBt)dtF (3.3)where Vw and VP are the compensator’s proportional and integral gains. In this case, thedroop law is implemented by using the inductor current tLBt instead of the actual outputcurrent tvBt. Since tLBt is equal to tvBt in steady-state this substitution still guaranteesbalanced current sharing and is a common practice in the literature. The CPL voltage, vcwsin (3.2) is:vˆcws =Rcwsr +M ·Rcws(T∑tvvBt − Tˆv)F (3.4)where Tv is the output current operating point.In order to analyze the stability using the small-signal equations, the system equationscan be expressed in the form tx = Ax+Wu. Then, the system’s stability can be analyzedfrom the eigenvalues of A. First, the following change of variable is done.dwˆtdt= aˆsw − vˆvBt − tˆLBt ·RdBt (3.5)Now, equations (3.2) and (3.5) can be written using the state variables and system inputsonly. In addition, if we do u = 0 then:264w˙v˙375 = A ·264wv375 F (3.6)273C2C evrvllel Converters ainevr boyel Derivvtion vny hmvllBhignvl htvwilitywhere w = [wˆ1wˆ2 · · · wˆt · · · wˆT ]T and v = [vˆ1vˆ2 · · · vˆt · · · vˆT ]T . The matrix A and its entriesare specified in equations (3.7)-(3.12) for the case of 3 parallel converters, and can be easilygeneralized to M converters.A =266666640 0 0 a1 m1 m10 0 0 m2 a2 m20 0 0 m3 m3 a3e1 0 0 n1 d1 d10 e2 0 d2 n2 n20 0 e3 d3 d3 n337777775 (3.7)at =RdBtr(Rcpl(r + 3Rcpl)− 1)− 1 (3.8)mt =RcplRdBtr (r + 3Rcpl)(3.9)nt =atVPBtCt− (r + 2Rcpl)Ct r (r + 3Rcpl)(3.10)dt =Rcpl (VPBtRdBt + 1)Ct r (r + 3Rcpl)(3.11)et =VIBtCt(3.12)Now, the matrix A is obtained for three parallel buck converters of different power rate. Thevariables and parameters of the system are normalized with respect to the base referencevalues of the middle power converter 2, and are summarized in Table 3.1. The eigenvalues ofmatrix A can be plotted in the complex plane in order to study the impact of a single fastcontroller on the stability of the system. The root locus analysis is done sweeping two differentparameters: the gain of converter 2 (VPB2), and the CPL power (Pcws), which translates in achange of Rcws.The dominant poles of A when VwB2 and VPB2, are varied are observed in Fig. 3.4(a). Fromthe figure, the effect that the increase in bandwidth of single converter has on the stabilityof the system can be analyzed. It is observed that two of its poles, originally located in theunstable region, become stable when the middle power converter’s bandwidth is sufficientlylarge. The root locus when the VPB2 is equal to VP0B2, 3 VP0B2, 10 VP0B2, and 50 VP0B2 aremarked, where VP0B2 corresponds to a compensator that is tuned for fswn ≈ 80 as indicatedin Table 3.1. Figure 3.4(b) shows the voltage closed-loop magnitude Bode plots of converter283C2C evrvllel Converters ainevr boyel Derivvtion vny hmvllBhignvl htvwilityTable 3.1: Root Locus Analysis Normalized ParametersParameter Formula Conv. 1 Conv. 2 Conv. 3ZnBX ZrlmBX=ZrlmB2 = 1=Z0nBX 3=2 1 1=2Z0nBX Z0BX=Z0B2 2=3 1 2P0nBX P0BX=P0B2 F:8 1 1:2VccnBX C 1 1 1VnBX12a0n;XM0n;X5121256CnBX121a0n;XM0n;X121516524rnBX rn=Z0B2 F:F1 F:F1 F:F1RdnBX F:4 · Z0nBX F:27 F:4 F:8fswnBX 8F · P0n 64 8F 96KP0BX C 1 1 1KP0BX F:F1 · fswnBX ·KP0BX F:64 F:8 F:962 for the mentioned values of VPB2. The plots depict that the increase in bandwidth shouldbe followed by an increase in the switching frequency, which sometimes is not possible and itcan bring other disadvantages. For this particular case, VPB2 should be increased three timesfor the system to be marginally stable. The consequent increase in the bandwidth is of twotimes, what requires the fsw to be increased proportionally.In Fig. 3.5, the impact that a fast controller has on the maximum stable load the systemcan operate at is studied. It is observed that the increase of VPB2 in 10 times expands themaximum CPL power Pcwsn at which the system is stable from 0.43 p.u. to 0.7 p.u. Notethat the normalization of the power value is done as detailed in the equation:Pcwsn =PcwsPrlm=v2cwsRcws· eva 2lxF (3.13)where ev and alx are the characteristic impedance and the equivalent voltage obtained in(2.11). Then, Pcwsn is obtained for the case vcwsn = 0:8, where alx is used as normalizing base.293C2C evrvllel Converters ainevr boyel Derivvtion vny hmvllBhignvl htvwilityFinally, according to the Bode plots in Fig. 3.4(b), an increase of 10 times in VPB2 wouldrequire fsw to be increased 6 times in order to keep the proper separation with the closedloop crossover frequency.303C2C evrvllel Converters ainevr boyel Derivvtion vny hmvllBhignvl htvwilityf [p.u.]-60-40-20020Magnitude (dB)10-210-11001011021032x6x 20x2x 6x 20xKP03KP050KP010KP0fsw needs to be increased to keep fsw/fv ≈ 100 n|G      |v,CL≈100x-3 -2 -1 0 1 2-2.5-2-1.5- Regionf v0 f sw0 f sw1 f sw2 f sw3 f v1 f v2 f v3 3KP010KP050KP0KP,2 increasesKP0a)b)Pcpln=Pcpl/Pref  = 0.57 p.u.(at vcpln  = 0.8 p.u.)Figure 3.4: a) Dominant poles of a linearized model forM parallel buck converters controlledusing nested dual PI compensators. The poles originally located on the RHP move to theLHP when the proportional gain of one of of the controllers is large enough. b) Bode plotsthat correspond to 4 different VPB2 values of the parameter sweep. The increase of VPB2 leadsto increasing bandwidth that should be followed by a proportional increase of fswn, which isoften not desirable. The power value Pcwsn is defined for a given Rcws using (3.13).313C2C evrvllel Converters ainevr boyel Derivvtion vny hmvllBhignvl htvwility-4 -2-2-10120.43 p.u.0 2-4 -2 0 2b)a)-2-1012Unstable RegionKP,2 = KP0PcplincreasesPcpln = 0.43 p.u.max. stable load increases for faster com-pensatorUnstable RegionPcplincreases(regular compensator)(faster       compensator,requires higher fsw)KP,2 = 10 KP0Pcpln = 0.7 p.u. 0.12 p.u. Pcpln = Pcpl/Pref (at vcpln = 0.8 p.u.)Figure 3.5: Dominant poles of a microgrid for load power sweep. The analysis is done for twodifferent values of the proportional gain of the middle power converter (VPB2). In b) VPB2 is10 times larger than in a), increasing the maximum CPL power at which the system is stablein 60% (0.7 p.u. vs. 0.43 p.u.). The power value Pcwsn is defined for a given Rcws using (3.13).323C3C irvyeoff Wetween Droop Dynvmixs vny Cea irvnsient Dynvmixs gowustness3C3 irvyzoff Wztwzzn Droop Dynvmixs vny Ceairvnsiznt Dynvmixs gowustnzssThe study of the system during CPL steps becomes necessary to understand its large signalbehavior. In Fig. 3.5 the CPL step-up response for a model with three buck converters isshown when the bandwidth of one of the converters controllers is varied. The parameters arethe same as the specified in Table 3.1. The results show that an increase in VPB2 leads to atransitory imbalance in the current sharing that allows an improvement in the dc bus voltagedynamic regulation. The tradeoff that exists between dc bus voltage dynamic regulation and0.50.70 1 500.20.450 1 5 0 1 5KP,2 = KP0,2ion,3ion,1ion,2ion,3ion,1ion,2ion,3ion,1ion,2Improved voltage dynamic regulationFaster current sharing equalizationFaster controller 2vcplnKP,2 = 3KP0,2 KP,2 = 10KP0,2vcplnvcplntn tn tnFigure 3.6: Tradeoff between droop dynamics and dc voltage dynamic regulation when afaster converter is introduced in the microgrid. A faster controller in one of the parallelconverters of a microgrid improves the dc voltage regulation during CPL transients. Thedissimilar speed in the converters leads to an transitory imbalance in the output currentsright after the transient, what increases the currents settling time.333CIC hummvrydroop dynamics is depicted in the figure. Since the current of the faster controller growsfaster during the transient, it can take a larger share of the load during the transient. Whenthe voltage starts recovering the larger imbalance in the currents increases the settling time ofcurrents and voltage. However, after the transient is over the current sharing is guaranteed.An unequal, but yet transitory, current sharing is necessary if the dynamics of the dc voltageare to be improved using a single faster converter.3CI hummvryThis chapter introduced the concept of increasing the stability of a microgrid by using a fastercontroller in one of the source-end converters. The small signal stability of a linear model ofa microgrid with a CPL was analyzed when using conventional dual PI compensators. Forthat purpose, the conventional dual PI compensator was first introduced and its bandwidthlimitations were discussed. Then, the linear model of the microgrid was derived and itsstability was analyzed from its root locus. The system dominant poles were obtained whenvarying the bandwidth of the middle power converter, and the CPL power. The analysisshowed that having a faster controller in one of the converters of a microgrid increases thestability for a given CPL power and expands the maximum power at which the system canoperate. Finally, the time domain large signal behavior of a microgrid was analyzed for thecase of a CPL step-up, and the tradeoff between dc bus voltage dynamic regulation anddroop dynamics was introduced and discussed.34Chvptzr ICirxulvr hwitxhing hurfvxz Controllzrjnyzr Droop avwThe use of a single faster controller in one of the source-end converters of a microgrid can leadto a general improvement of the stability and dc voltage dynamic regulation of the system. Inchapter 3, it was noted that conventional dual-loop PI compensators have a limited dynamicresponse due to their bandwidth limitation. Their speed can only be improved at the costof a larger switching frequency and the subsequent hardware specifications re-design. Theuse of other control schemes that can achieve fast response without increasing the steadystate switching frequency is of interest to tackle this challenge. In particular, the CircularSwitching Surface controller (CSS) features a very fast dynamic response. Since its controllaw is based on the knowledge of the possible trajectories that the converter describes in thestate-plane, it is particularly effective during large-signal transients. Because of its fasterdynamic response for a given hardware, the CSS controller will be chosen to replace oneof the conventional dual PI controllers of the system. In this section, the CSS control lawis derived including a modification of its target voltage that enables the converter to workunder a droop control scenario. The behavior of the controller is first studied for the singleconverter case under a droop scheme and later compared with its performance in a microgrid.35ICFC Derivvtion of Chh for v Wuxk Converter Unyer Droop ControlICF Dzrivvtion of Chh for v Wuxk Convzrtzr jnyzrDroop ControlSince the converter being controlled is operating within a dc microgrid, its target voltage(vvn) must follow a voltage droop law:vvn = aswn −Rdn · tvnF (4.1)where, aswn is the target voltage at no load, and Rd is the virtual droop resistance. Then,the target point of the CSS controller, given by (vvn; tvn), is not fixed but can be located atany point of the droop line. The CSS controller uses a geometrical approximation of the ONand OFF state-plane trajectories of the converter to base its control law. These trajectoriescan be obtained from the differential equations that describe the dynamics of the powerconverter. For the buck converter depicted in Fig. 2.1, the differential equations are given by(4.2) and (4.3). The time domain solutions are (4.4) and (4.5).12dtLndtn= uaccn − vvn (4.2)12dvvndtn= tLn − tvn (4.3)vvn = [vvn(0)− uaccn] cos(2tn) + [tLn(0)− tvn] sin(2tn) + uaccn (4.4)tLn = [tLn(0)− tvn] cos(2tn) + [uvvn(0)− accn] sin(2tn) + tvn (4.5)Then, combining the expressions in (4.4) and (4.5), the time parameter tn can be eliminatedand the state-plane trajectories of the converter are obtained as in (4.6). : (vvn − uaccn)2 + (tLn − tvn)2 − [vvn(0)− uaccn]2 − [tLn(0)− tvn]2 = 0 (4.6)36IC2C Control avwThe initial conditions [vvn(0); tLn(0)] are given by any point in the state-plane that is con-tained in the converter’s trajectory. Then, since the control law is based on the trajectoriesthat passes through the operating point, (4.6) becomes: : (vvn − uaccn)2 + (tLn − tvn)2 − (vvn − uaccn)2 = 0 (4.7)Equation (4.7) can be rewritten in the following form: : (vvn − Cvvn)2 + (tLn − CpLn)2 −R2 = 0F (4.8)which is the equation of a circumference of radius R centered on (Cvvn;CpLn). For smallvariations in tvn, the radius R can be approximated as constant. Then, when the controlsignal u is equal to 1, the converter ON trajectory can be described as a circular curvecentered on (accn; tvn). On the other hand, if the converter adopts the OFF structure, theconverter trajectory describes a circumference centered on (0; tvn).IC2 Control avwThe control law of the CSS controller is described using the two trajectories that result from(4.7) when vvn is defined by the droop law given in (4.1):Wusy IN (tpLn I tvn)if (1 I 0), thyn u = 0, ylsy u = 1Wusy IIN (tpLn G tvn)if (2 I 0), thyn u = 1, ylsy u = 0with8>><>>:1 = v2vn + (tLn − tvn)2 − (aswn −Rdn · tvn)22 = (vvn − accn)2 + (tLn − tvn)2 − (aswn −Rdn · tvn − accn)2:37IC3C Control avw dpervtionO hingle Converter vny bixrogriy hxenvriosCAs can be observed, the control law employes two circular surfaces, 1 I 0 and 2 I 0, todefine the state of the switches. The switching surfaces are obtained from the ON and OFFcircular trajectories that pass over the target operating point (aswn −Rdn · tvn; tvn). The twotrajectories are obtained by replacing u with 1 or 0 in (4.7).IC3 Control avw dpzrvtionO hinglz Convzrtzr vnybixrogriy hxznvriosCFigure 4.1 illustrates the start-up and step-up responses of a buck converter connected to aCPL that is being controlled by a CSS control law, which target voltage is defined by a drooplaw. It depicts the converter trajectories in both the state-plane and the time domain, andgives insight into the operation of the control law. During start-up, the converter’s operatingpoint corresponds to Wusy I of the control law. The compensator sets the converter in itsON structure until the operating point crosses the curve described by 1 = 0. At thatmoment, 1 I 0 and the control signal changes to u = 0. As tvn is constant during thestart-up process, it can be observed that the converter’s OFF trajectory matches the circulartrajectory described by 1 = 0.Once in Ì, the converter is subjected to a sudden power step-up. The rapid increase incurrent puts the converter under Wusy II and its control signal is set to 1 until it crosses1 = 0. Since the CPL current varies as a function of its voltage (2.3), the radius of theideal circle changes along the trajectory. The difference between the approximated circulartrajectory (1 = 0) and the trajectory (Í-Î) can be observed. Since the converter’s operatingpoint does not reach the target point with a single switching action, an extra switching actionis required (Î). It is important to notice that, unlike in other applications, in this case, thetarget of the controller is not a point but the droop line.38IC3C Control avw dpervtionO hingle Converter vny bixrogriy hxenvriosCVspn von14523iLn6Droop LineSwitch ON actionSwitch OFF action123 Target op. pointfor Pcpln = 045 Switch ON actionSwitch OFF action6 Target op. pointfor Pcpln = 0.27 p.u.(σ1= 0)Pcpln=0λu=1λu=0Vccnion1(σ1= 0)Pcpln=0.27 p.u.tnvon1tniLntnPcpln0.27 p.u.145 6231456230Vspnion1(σ2= 0)Pcpln=0.27 p.u.Figure 4.1: State-plane and time domain representations of start-up (Ê-Ì) and load step-up(Ì-Ï) transients of a buck converter with a CSS controller connected to a constant powerload (CPL).When the CSS controlled converter is in parallel with other slower converters in a micro-grid, its transient response can be affected by the interaction among converters. However,the circular approximation of the converter trajectories still describes the general behavior39IC3C Control avw dpervtionO hingle Converter vny bixrogriy hxenvriosCDroop LinevonVspn0tntnvon00VspniLn,2iLn,3iLn,3iLn,1iLn,2iLn,11112333442iLn,2,MaxiLn,2,Max24Start-up begins,switch ON actionConv. 2 switch OFF action124 Target point atno loadConv. 2 starts continuous switching3* iLn,X = iL,X / Iref,XFigure 4.2: State-plane and time domain representations of synchronized start-up of threeparallel converters connected to a constant power load (CPL). The middle power converter,with current tLnB2, uses a CSS controller.of the converter in the state-plane domain. As seen for a single converter with a CPL, al-though the trajectory approximation is less accurate for large signal perturbations, it stilldescribes the converter’s general behavior and succeeds in bringing the operating point closerto the target value in very few switching actions. In an area close to the target point thetrajectories estimation accuracy increases substantially guaranteeing convergence. A perfor-mance assessment of the use of a fast CSS controller in a microgrid is showed in Fig. 4.2 and40IC3C Control avw dpervtionO hingle Converter vny bixrogriy hxenvriosCFig. 4.3. The analysis shows how the use of a single CSS controller improves the performancefor start-up and load step-up transient responses in a microgrid. The microgrid comprisesthree parallel converters with similar specifications as those in Table 3.1, with the exceptionthat the middle power converter linear compensator is replaced with a CSS one. As donefor the single converter case, the transients are shown in both the state-plane and the timedomain. The inductor current of each converter (tLnBX) is normalized with respect to its ownreference current (TrlmBX) in order to simplify the state-plane plots showing a single droopline instead of multiple ones.The system is first analyzed for a synchronized start-up at no load. In Fig. 4.2, it isobserved that the three converters switch ON right after the start-up begins (Ê). However,converter 2 stays in the ON structure for a longer time, allowing tLnB2 to keep growing andthe target voltage to be reached at a faster rate. In Ë the control law command converter 2to switch off, since it estimates that the OFF trajectory for the present operating conditionswill lead the converter to the present target point. Since the operating point slides down inthe droop line while the output current tvnB2 decreases, the target point changes continuously.This can be observed from Ì to Í where the variation in the target point forces converter 2to switch continuously until it reaches steady state Í. Although this effect slows down therate at which the steady-state target point could be reached, it is evident when the systemis close to the target value and the transient is almost solved.On the other hand, Fig. 4.3 shows the system’s transient response for a CPL step-up.The state plane representation of the three converters trajectories shows how both controllersinteract with the droop line. The plot shows that converter 2, with the CSS controller, hitsthe droop line within a few switching actions (Ê). Its faster controller allows the inductorcurrent to increase at a larger rate than that of the other two converters, and to reach thetarget value before. Once on the droop line, converter 2 moves on the straight line until it41IC3C Control avw dpervtionO hingle Converter vny bixrogriy hxenvriosC000Droop LineVspn111222333Operating point at no loadMaximum Voltage drop123 Target op. pointfor Pcpln= 0.13 p.u.5vonPcplniLn,ssiLn,sstntntnVspnvoniLn,2,MaxiLn,2,MaxiLn,2iLn,2iLn,3iLn,3iLn,10.75Vspn0.75Vspnvon,Minvon,MiniLn,10.13 p.u.* iLn,X = iL,X / Iref,Xvon,ssvon,ssFigure 4.3: State-plane and time domain representations of a load step-up transient of threeparallel converters connected to a constant power load (CPL). The middle power converter,with current tLnB2, uses a CSS controller.reaches an operating point where the current is shared equally among the three converters(Ì).42ICIC hummvryIt is important to mention that although the CSS controller slides on the line as it maybe expected for a sliding mode controller, its principle of operation is different and leads todissimilar transient responses from initial to hitting point (Ê-Ë). Moreover, if the droop lineis used as a sliding line, its slope does not guarantee that the converter will enter the regionof existence right after the hitting point, which can lead to oscillations before the converterstarts sliding. Then, if a sliding mode controller is implemented, its sliding line slope shouldbe chosen adequately and will generally differ from that of the droop line.ICI hummvryIn this chapter, the Circular Switching Surface controller (CSS) was introduced for its op-eration under a current sharing scenario. The control law bases its operation in a circularapproximation of the converter trajectories in the state plane and includes a conventionaldroop law to enable its use for a converter in a microgrid. The operation of the CSS con-troller with embedded droop was first assessed for a single converter with a CPL. This simplescenario allowed to present the control operation for a simple case.Later, the CSS control operation for a converter in parallel with other two converters,ruled with linear compensators, was assessed for start-up and CPL step-up transients. Theinspection of the system transient responses revealed that the proposed control law allowsthe converter’s current to increase at a faster rate during the transient, increasing the rateat which the target operating point is reached.43Chvptzr 5himulvtions vny evrvmztrix AnvlysisWhile converters that are controlled using a CSS strategy, work very close to their physicallimit, those that use conventional linear controllers, usually PI dual-loop, are tuned for abandwidth several times lower than the plant’s. Consequently, the substitution of linearcontrollers with CSS controllers within microgrids is expected to improve the individualdynamic performance of the converters and the combined bandwidth of the whole microgrid.The level of improvement that results from using the proposed approach can be assessed bycomparing the performance of the different system configurations when subjected to largesignal transients.In this chapter, a simulation model is created to compare the performance of the tra-ditional approach, where all the converters are controlled using conventional PI dual-loopcompensators, and the proposed approach, where one compensator is substituted with a CSSone. The performance improvement of the proposed approach is assessed in terms of voltagedynamic regulation, and system resilience for a microgrid. In order to expand the results to alarger range of conditions, a parameter analysis is done in the second section of the chapter.5CF himulvtionsA microgrid comprising three different converters powering a CPL is controlled using twodifferent approaches and simulated using MATLAB/Simulink+PLECS. The converters arebuck topologies of different power ratings. In a traditional strategy (case B), converters are445CFC himulvtions048-9090 5 25 50-909i o(B) [A]i o(A) [A]v o [V]t [ms]17.8 ms2.4 ms5.3 V (10.9%)2.0 V (4.1%)BBAABProposed CSSTraditional PIAFigure 5.1: Simulation of a start-up transient for the proposed CSS (A) and traditional PI(B) approaches. Percentage of overshoot and settling time are measured for both strategiesconsidering the 5% criterion. Case A is more than 7 times faster, and exhibits less than halfof the overshoot.controlled using traditional PI current-mode compensators. On the other hand, the proposedstrategy (case A) involves replacing one of the controllers of the traditional approach systemwith a CSS one. Results of the simulations are presented for start-up, load step-up and loadstep-down transients in Fig. 5.1, 5.2 and 5.3 respectively. It is observed that the voltagedynamic regulation improves significantly for the strategy that replaces a linear controllerwith a CSS controller. It is noted how, during transients, the CSS-controlled convertersupplies most of the current, avoiding major voltage drops in the dc bus. Moreover, Fig. 5.4shows the simulation results of the two system configurations for different power steps. Theproposed approach withstands higher power steps before becoming unstable, showing it canget closer to the theoretic critical power step specified in Fig. 2.2(b). for the given initialconditions.455CFC himulvtions3040480153007.5150 10 50 10007.515t [ms]i o(B) [A]i o(A) [A]i o,CPL [A]v o [V]BBBAAA20.7 ms21.8 ms -4.5 V (-10.4%) -14.1 V (-32.4%)BProposed CSSTraditional PIAFigure 5.2: Simulation of a load step-up transient for the proposed (A) and traditional (B)approaches. Percentage of overshoot and recovery time are measured considering the 5%criterion. Cases A and B do not present a considerable difference in their recovery time.However, case B shows a dramatic voltage drop that is 3 times larger than case (A).465CFC himulvtions4048600153007.507.50 10 50 100t [ms]i o(B) [A]i o(A) [A]i o,CPL [A]v o [V]17.0 ms23.1 ms 8.6 V (17.9%)3.6 V (7.4%)ABABBProposed CSSTraditional PIAFigure 5.3: Simulation of a load step-down transient for the proposed CSS (A) and traditionalPI (B) approaches. Case A presents much smaller overshoot when compared to 17.9% of caseB, and a decrease in the transient recovery time.475CFC himulvtions02448024480 2.5 20 4009121392t [ms]5111PMax(B)PMax(A)50% increase of PMax4`4553322423v o(A) [V]v o(B) [V]Po  [V]BAB Traditional PIProposed CSSAFigure 5.4: Microgrid dc bus voltage response for different power step-up values. Simulationresults for the proposed CSS (A) and the traditional PI (B) approaches. Simulations for theproposed approach show that the microgrid can withstand larger power steps (up to 50%larger) while remaining stable.485C2C evrvmetrix Anvlysis5C2 evrvmztrix AnvlysisIn order to test the validity of the proposed approach on a wider range of system conditions,a parametric analysis is performed. The study is done with a system of three buck convertersin order to assess the impact that both the bandwidth of the linear controllers and the use ofCSS compensators have on the critical power step of the system and the voltage drop duringtransients. The study is performed in the normalized domain as it was done in previouschapters, where converter 2 base quantities were used as normalizing values. The microgrid ofthree parallel converters shares most of its parameters with the approximated model analyzedin section 3.2. In this section, Table 5.1 is expanded to Table 3.1 to include those parametersthat were not used in the linear model approximation. This time, a more comprehensivemodel of a microgrid of three parallel buck converters with normalized power 1, 2/3 and1/3 p.u. is analyzed for different system configurations. The parametric sweep requiresTable 5.1: Simulations ParametersParameter Formula Conv. 1 Conv. 2 Conv. 3ZnBX ZrlmBX=ZrlmB2 = 1=Z0nBX 3=2 1 1=2Z0nBX Z0BX=Z0B2 2=3 1 2P0nBX P0BX=P0B2 F:8 1 1:2VccnBX C 1 1 1VnBX12a0n;XM0n;X5121256CnBX121a0n;XM0n;X121516524RdnBX F:4 · Z0nBX F:27 F:4 F:8fswn0BX 8F · P0n 4 8F 96KiP0BX C 1F 1F 1FKiP0BX F:7 · fswn0BX ·KiP0BX 44:8 56 67:2KvP0BX C 1 1 1KvP0BX F:F1 · fswn0BX ·KvP0BX F:64 F:8 F:96495C2C evrvmetrix Anvlysis0.5 1 1.5 2 2.5k = fswn,X / fswn0,X(bandwidth increases with fswn,X )ΔPMaxn [p.u.]ΔPCritn3 PI2 PI + 1 CSS in conv. 32 PI + 1 CSS in conv. 22 PI + 1 CSS in conv. 1Pn,3 < Pn,2 < Pn,1AAABFigure 5.5: Maximum power step (CPL) vs. compensators’ bandwidths for a microgrid com-prising three converters, which are controlled using four different approaches. The bandwidthof each linear compensator is approximately proportional to the normalized switching fre-quency. For the proposed approach (A), the systems can withstand larger power steps atlower switching frequencies than it does for the traditional case (B). When the bandwidthof the linear compensators is increased, ∆PTaxn for all the cases tends to the theoreticalmaximum ∆PCrptn. Moreover, the improvements in stability are larger when the relativepower rate of the CSS-controlled converter is larger.varying the bandwidth of the converters with linear compensators. The bandwidth is modifiedby multiplying the converter’s nominal switching frequency fswn0BX and the compensatorsnominal gains VtP0BX , VtP0BX , VvP0BX , and VvP0BX by a constant v. The nominal parametersare those that result from tuning the compensators with fswn ≈ 80.Two different sets of simulations are performed. The analysis of the results of the first setof simulations is depicted in Fig. 5.5, and shows how the maximum stable power step increaseswhen the bandwidth of the converters with linear compensators are expanded by doing vlarger than 1. The influence of the use of CSS compensators on the maximum stable power505C2C evrvmetrix Anvlysis0.15 0.3-0.4-0.20ΔPVn,drop0.450.8tv on[p.u.]t[p.u.]2PI +1CSS 3PI0V2∆P1 ∆P1V1-V1-V2∆P1[p.u.]fswn0,XAB3 fswn0,XFigure 5.6: Voltage drop within a load step-up transient vs. power step for different com-pensators’ bandwidths. The proposed case (A) shows a smaller voltage drops than doesthe traditional case (B). The difference increases when the switching frequency of the linearcompensator is lower.step ∆PTaxn is assessed when comparing the results of the simulations across four systemconfigurations. In the first setup, the three buck converters are controlled using only twonested PI loops (3PI), while in the other three cases, one of the converters’ compensators isreplaced with a CSS one (2PI+1CSS). For a given v it is observed that the 3PI setup presentslower ∆PTaxn than do the 2PI+1CSS setups. Moreover, when the power rate of the CSSconverter increases, the system maximum power step also increases. The analysis shows thecapability of CSS-controlled converters to increase the stability margin of the system. Whenv increases so does the switching frequency and the linear compensators’ bandwidth, and the515C3C hummvrybehavior of each converter gets closer to its physical limit. In this case, when fswnBX doublewith respect to the nominal frequency fswn0BX (v = 2), the system ∆PTaxn gets very closeto the theoretical maximum ∆PCrptn (found in section 2.1). The results of this analysis showthe system’s large-signal stability improvement when the speed of the converters’ controllersis increased, and are aligned with the analysis done in section 3.2 for small signal stability.The analysis of the results of the second set of simulations is presented in Fig. 5.6. Thisfigure shows the voltage drop of the dc bus during a load power step-up transient for differentpower step values. Both the proposed and traditional approaches are simulated for twodifferent sets of switching frequencies, and the linear compensators’ bandwidth is adjustedaccordingly. It is observed that the 2PI+1CSS setup results in lower voltage drop for anypower step. As was noticed in the first set of simulations, when the switching frequency isincreased together with the PI compensators’ bandwidth, both approaches present a similarbehavior.5C3 hummvryIn this chapter, a model of a microgrid with three parallel converters was simulated for twomain different approaches. The traditional approach (B) comprised converters controlledusing conventional PI dual-loop compensators, while the proposed approach (A) replacedone of the compensators with a CSS faster controller. The results for approach A showedbetter dynamic regulation for start-up and CPL steps transients, and larger resilience underCPL step-up events. Moreover, a parametric analysis was done to expand the range ofsystem conditions and include the effect of the bandwidth of the linear compensators forboth approaches. Results showed that an increase in bandwidth in the linear controllersimproves the overall system performance, but this happens at expense of a larger switchingfrequency. The proposed approach results present lower levels of voltage drop and better525C3C hummvryresilience during CPL step-ups transients for all the conditions analyzed. If compared withthe proposed scenario, approach B would need to increase the switching frequency of theirconverters significantly, to achieve the same performance.53Chvptzr 6Expzrimzntvl gzsultsConverter 1 Converter 2 Converter 3Oscilloscopemultimeter multimeter aux. power supplyaux. power supply function generatordc load (CPL)dc power supplyvcplVccio,1 io,2 io,3Figure 6.1: Photograph of the experimental setup54Chvpter 6C Experimentvl gesultsTable 6.1: Experimental ParametersParameter lalue Conv. 1 lalue Conv. 2 lalue Conv. 3Vcc 6F V 6F V 6F VVrlm 48 V 48 V 48 VRdrvvw F:48 Ω 1:8 Ω 3:6 ΩV 1:9 mR 2:3 mR 4:F mRC 12FF P 68F P 33F Pfv 1F5 Rz 127 Rz 138 kRzfsw 8 kRz 1F kRz 12 kRzThe setup simulated in chapter 5, consisting of three parallel buck converters supplyinga CPL, was implemented in an experimental platform, which photograph can be observed inFig. 6.1, with the parameters detailed in Table 6.1.The converters were synchronous buck prototypes working at a maximum power rate of400 W each, and were controlled locally using a TI TMS320F28335 DSP. The circuit’s loadwas an NHR 4760 dc electronic load configured in constant power mode and the dc powersupply is an AMETEK Sorensen SGI 100/150. A function generator was used to synchronizethe converters turn-on and turn-off in order to be able to capture the start-up transient.Experimental results of the system for start-up and load step-up and step-down transientsare displayed in Figs. 6.2 to 6.4.Fig. 6.2 b) shows a synchronized start-up of the linear-controlled converters when no loadis present. The same transient is repeated when the compensator in converter 2 is replacedwith a CSS controller. The results for the proposed approach are shown in Fig. 6.2 a),showing a 2.8 times decrease in the overshoot of the dc bus and a 6.7 times faster settlingtime compared with the results obtained using the traditional approach.Figs. 6.3 shows the results for an 800 W CPL step-up. In this case, the dynamics ofthe droop control are responsible for the slow response and no significant difference can be55Chvpter 6C Experimentvl gesultsappreciated between both settling times. However, a decrease of almost 3 times can beobserved for the negative overshoot of the output voltage.CPL step-down transients for both approaches are shown in Figs. 6.4. Once more, areduction in the overshoot is observed when using the CSS approach (2.3 times). Moreover,the settling time is decreased from 18.4 ms to 12.4 ms.Finally, Fig. 6.5 b) shows the response of the linear-controlled system crossing the limitof stability after a 900 W step-up is applied. On the other hand, when converter 2 employs aCSS control instead of the traditional current-mode control, the whole system can withstanda CPL step-up of at least 1080 W (20% higher), as shown in Fig. 6.5 a).56Chvpter 6C Experimentvl gesultsvo20 msio,1 lineario,3 lineario,2 linear±5% 48 V6.7 VBvo3 msio,1 lineario,3 lineario,2 CSS±5% 48 V2.4 VA Proposed CSSTraditional PIa)b)Figure 6.2: Experimental results of a start-up transient in a microgrid comprising three linearcontrolled converters. a) shows the proposed approach (A) when the controller of converter 2is replaced with a CSS controller; and b) is the result for the traditional approach, wherethe three converters are controlled with linear PI controllers. (A) shows a smaller voltageovershoot (almost 3 times lower) and faster settling time (almost 7 times faster) than does(B).57Chvpter 6C Experimentvl gesultsvo16.6 ms48 Vio,1 lineario,3 lineario,2 CSSio,CPL±5% 4.5 V43.5 V18.4 A20 Avo16.5 ms48 Vio,1 lineario,3 lineario,2 lineario,CPL±5% 43.5 V18.4 A26.5 A13.2 Va)b)BA Proposed CSSTraditional PIFigure 6.3: Experimental result of a 800 W CPL step-up transient. a) proposed approach (A),(compensator in converter 2 is replaced with a CSS controller); and b) traditional approach(B) for the traditional approach. The proposed approach (A) shows a smaller voltage drop(almost 3 times lower) than does the traditional approach (B).58Chvpter 6C Experimentvl gesultsvo18.4 ms43.5 Vio,1 lineario,3 lineario,2 lineario,CPL8.75 V±5% 48 V0 A18.4 Avo12.4 ms43.5 Vio,1 lineario,3 lineario,2 CSSio,CPL±5% 48 V0 A18.4 A3.8 Va)b)BA Proposed CSSTraditional PIFigure 6.4: Experimental result of a 800 W CPL step-down transient for the traditionalapproach. a) proposed approach (A), (compensator in converter 2 is replaced with a CSScontroller); and b) traditional approach (B) for the traditional approach. (A) has a smallervoltage overshoot (less than 2 times lower), and faster recovery time (30% improvement)than does (B).59Chvpter 6C Experimentvl gesultsvoio,1 lineario,3 lineario,2 CSSio,CPLvoio,1 lineario,3 lineario,2 lineario,CPLa)b)A Proposed CSSsystem failure: vo decreases while io,CPL increases exceeding CPL operating range.B Traditional PIFigure 6.5: Experimental result showing maximum CPL step-up that the system can with-stand for both approaches. a) proposed approach (A) for a 1080 W CPL step-up Experimen-tal; b) traditional approach (B) for a 900 W CPL step-up. The use of a CSS controller inone of the converters in case (A) extends the maximum load step-up 20% if compared withthe traditional approach (B).60Chvptzr LConxlusionLCF hummvryThe presence of CPLs within microgrids leads to instability, observed in the dc bus as voltageoscillations, and as voltage drops after sudden load increments. This work introduced a CSSstate-trajectory controller with an embedded droop law, to improve three critical aspectsof the microgrid operation: 1) resiliency under large CPL’s steps; 2) load transient voltageregulation; 3) voltage transient recovery time.The negative effects of CPLs were studied in the state-plane domain and the theoreticalmaximum stable power-step (∆PCrpt) was obtained, and generalized for a microgrid withparallel converters. ∆PCrpt can be employed as a benchmark to evaluate the performance ofa given system and its source-end controllers.The implications of having a faster controller in one of the source-end converters of thedc microgrid were analyzed using a linearized model, which was analytically derived. Themodel’s root locus, obtained while varying the speed of one of the source-end controllers,showed that a single faster compensator can improve the small signal stability of the wholesystem. The tradeoff between current sharing dynamics and voltage regulation performancewas noted, an unequal current sharing during the transient enables the faster converter toincrease its load improving the transient performance. Moreover, the limitations of conven-tional nested dual-loop PI controllers were discussed, explaining the need for using a different61LCFC hummvrycontrol technique that does not require increased switching frequency in order to improvethe system dynamic performance.The CSS control law with embedded droop was derived for a buck converter, which isa structure frequently found in microgrids. Since the control law bases its operation on acircular approximation of the converter’s state-plane trajectories, its performance assessmentwas explained and analyzed geometrically in the state-variables domain.Then, a microgrid consisting of three buck converters connected in parallel to a CPLwas simulated in MATLAB/Simulink+PLECS confirming the advantages of the proposedapproach. A parametric analysis of a normalized version of the system was done to investigatethe large signal behavior of the proposed approach. The results showed for a wide range ofconditions that the proposed approach can withstand larger load step-ups with less voltagedrop, and that switching frequency would need to be increased several times in order toobtain the same stability margins using PI current-mode compensators.Finally, the system was implemented in an experimental platform comprised of threebuck converters prototypes working at a maximum power of 400 W each. The converterswere controlled using local DSPs and the system was loaded with an electronic load workingin CPL mode. The experiments, run in the proposed setup for start-up, load step-up and loadstep-down transients, showed a reduction in overshoot of two to three times and a decrease insettling and recovery time for start-up and load step-down cases, as compared to the resultsobtained using the traditional setup. Moreover, the replacement of a single controller with aCSS control with embedded droop led to a 20% increase in the power level of the maximumCPL step that the system can withstand.62LC2C Future lorkLC2 Futurz lorkThe concept developed in this work provides an original contribution to the field of stabiliza-tion of dc microgrids under constant power loads. The work could be extended to differentmicrogrid configurations with diverse converter topologies. In particular, its application inhybrid energy storage systems, where a single faster controller drives a super-capacitor bank,could be very effective.63Wiwliogrvphy[1] M. A. Bianchi, I. G. Zurbriggen, M. Ordonez, and M. Anun, “Dynamic performanceimprovement of dc microgrids during constant-power load transients,” in FDEJ IEEE7th Intyrnutionul gymposium on dowyr Elywtroniws for Distrivutyx Gynyrution gystyms(dEDG), June 2016, pp. 1–6.[2] fEbFE. FDE7. fynywuvlys FDE7 Glovul gtutus fyport, 2017.[3] R. H. Lasseter, “Microgrids,” in FDDF IEEE dowyr Enginyyring gowiyty Wintyr ayyting.Wonfyrynwy drowyyxings (Wut. bo.DFWHG7GDM), vol. 1, 2002, pp. 305–308 vol.1.[4] R. Pen˜a-Alzola, M. A. Bianchi, and M. Ordonez, “Control design of a pfc with harmonicmitigation function for small hybrid ac/dc buildings,” IEEE hrunsuwtions on dowyrElywtroniws, vol. 31, no. 9, pp. 6607–6620, Sept 2016.[5] X. Lu, K. Sun, J. M. Guerrero, J. C. Vasquez, L. Huang, and J. Wang, “Stabilityenhancement based on virtual impedance for dc microgrids with constant power loads,”IEEE hrunsuwtions on gmurt Grix, vol. 6, no. 6, pp. 2770–2783, Nov 2015.[6] M. Su, Z. Liu, Y. Sun, H. Han, and X. Hou, “Stability analysis and stabilization methodsof dc microgrid with multiple parallel-connected dc-dc converters loaded by cpls,” IEEEhrunsuwtions on gmurt Grix, vol. PP, no. 99, pp. 1–1, 2016.64Wiwliogrvphy[7] I. Kondratiev, E. Santi, and R. Dougal, “Nonlinear synergetic control for m parallel-connected dc-dc buck converters: Droop current sharing,” in FDDJ G7th IEEE dowyrElywtroniws gpywiulists Wonfyrynwy, June 2006, pp. 1–7.[8] S. Anand and B. G. Fernandes, “Reduced-order model and stability analysis of low-voltage dc microgrid,” IEEE hrunsuwtions on Inxustriul Elywtroniws, vol. 60, no. 11, pp.5040–5049, Nov 2013.[9] J. M. Guerrero, J. C. Vasquez, J. Matas, L. G. de Vicuna, and M. Castilla, “Hierarchicalcontrol of droop-controlled ac and dc microgrids - a general approach toward standard-ization,” IEEE hrunsuwtions on Inxustriul Elywtroniws, vol. 58, no. 1, pp. 158–172, Jan2011.[10] A. Paquette and D. Divan, “Transient droop for improved transient load sharing inmicrogrids,” in Enyrgy Wonvyrsion Wongryss unx Efiposition (EWWE), FDEH IEEE, Sept2014, pp. 84–91.[11] Y. Tu, Z. Liu, J. Liu, T. Liu, and Z. Liu, “An additional stabilizer for mitigating theinstability in dc/dc cascaded system with constant power loads,” in FDE7 IEEE GrxIntyrnutionul Futury Enyrgy Elywtroniws Wonfyrynwy unx EWWE Asiu (IFEEW FDE7 AEWWE Asiu), June 2017, pp. 1982–1986.[12] O. Pizniur, Z. Shan, and J. Jatskevich, “Ensuring dynamic stability of constant powerloads in dc telecom power systems and data centers using active damping,” in FDEH IEEEGJth Intyrnutionul hylywommuniwutions Enyrgy Wonfyrynwy (IbhELEW), Sept 2014, pp.1–8.[13] J. You, Z. P. Fan, Y. Hu, and M. J. Deng, “Virtual resistor based dbvc and activedamping method for dc bus stabilization of cascaded power converters system,” in FDE765WiwliogrvphyIEEE hrunsportution Elywtriwution Wonfyrynwy unx Efipo, AsiuAduwiw (IhEW AsiuAduwiw), Aug 2017, pp. 1–6.[14] X. Chang, Y. Li, X. Li, and X. Chen, “An active damping method based on a supercapac-itor energy storage system to overcome the destabilizing effect of instantaneous constantpower loads in dc microgrids,” IEEE hrunsuwtions on Enyrgy Wonvyrsion, vol. 32, no. 1,pp. 36–47, March 2017.[15] M. Cespedes, L. Xing, and J. Sun, “Constant-power load system stabilization by passivedamping,” IEEE hrunsuwtions on dowyr Elywtroniws, vol. 26, no. 7, pp. 1832–1836, July2011.[16] A. Kwasinski and C. N. Onwuchekwa, “Dynamic behavior and stabilization of dc mi-crogrids with instantaneous constant-power loads,” IEEE hrunsuwtions on dowyr ElywAtroniws, vol. 26, no. 3, pp. 822–834, March 2011.[17] A. Riccobono and E. Santi, “Comprehensive review of stability criteria for dc distributionsystems,” in FDEF IEEE Enyrgy Wonvyrsion Wongryss unx Efiposition (EWWE), Sept2012, pp. 3917–3925.[18] P. Magne, D. Marx, B. Nahid-Mobarakeh, and S. Pierfederici, “Large-signal stabilizationof a dc-link supplying a constant power load using a virtual capacitor: Impact on thedomain of attraction,” IEEE hrunsuwtions on Inxustry Appliwutions, vol. 48, no. 3, pp.878–887, May 2012.[19] P. Magne, B. Nahid-Mobarakeh, and S. Pierfederici, “Dynamic consideration of dc mi-crogrids with constant power loads and active damping system -a design method forfault-tolerant stabilizing system,” IEEE Journul of Emyrging unx gylywtyx hopiws indowyr Elywtroniws, vol. 2, no. 3, pp. 562–570, Sept 2014.66Wiwliogrvphy[20] X. Zhang, X. Ruan, and Q. C. Zhong, “Improving the stability of cascaded dc/dc con-verter systems via shaping the input impedance of the load converter with a parallel orseries virtual impedance,” IEEE hrunsuwtions on Inxustriul Elywtroniws, vol. 62, no. 12,pp. 7499–7512, Dec 2015.[21] F. Zhang and Y. Yan, “Start-up process and step response of a dc-dc converter loadedby constant power loads,” IEEE hrunsuwtions on Inxustriul Elywtroniws, vol. 58, no. 1,pp. 298–304, Jan 2011.[22] Q. Shafiee, T. Dragievi, J. C. Vasquez, and J. M. Guerrero, “Hierarchical control formultiple dc-microgrids clusters,” IEEE hrunsuwtions on Enyrgy Wonvyrsion, vol. 29,no. 4, pp. 922–933, Dec 2014.[23] D. Zammit, C. S. Staines, M. Apap, and A. Micallef, “Paralleling of buck convertersfor dc microgrid operation,” in FDEJ Intyrnutionul Wonfyrynwy on Wontrol, Dywision unxInformution hywhnologiys (WoDIh), April 2016, pp. 070–076.[24] A. M. Rahimi and A. Emadi, “Active damping in dc/dc power electronic converters: Anovel method to overcome the problems of constant power loads,” IEEE hrunsuwtionson Inxustriul Elywtroniws, vol. 56, no. 5, pp. 1428–1439, May 2009.[25] M. Wu and D. D. C. Lu, “A novel stabilization method of lc input filter with constantpower loads without load performance compromise in dc microgrids,” IEEE hrunsuwtionson Inxustriul Elywtroniws, vol. 62, no. 7, pp. 4552–4562, July 2015.[26] M. K. Zadeh, R. Gavagsaz-Ghoachani, S. Pierfederici, B. Nahid-Mobarakeh, andM. Molinas, “Stability analysis and dynamic performance evaluation of a powerelectronics-based dc distribution system with active stabilizer,” IEEE Journul of EmyrgAing unx gylywtyx hopiws in dowyr Elywtroniws, vol. 4, no. 1, pp. 93–102, March 2016.67Wiwliogrvphy[27] P. Lin, P. Wang, J. Xiao, J. Wang, C. Jin, and Y. Tang, “An integral droop for transientpower allocation and output impedance shaping of hybrid energy storage system in dcmicrogrid,” IEEE hrunsuwtions on dowyr Elywtroniws, vol. PP, no. 99, pp. 1–1, 2017.[28] Q. Xu, X. Hu, P. Wang, J. Xiao, P. Tu, C. Wen, and M. Y. Lee, “A decentralizeddynamic power sharing strategy for hybrid energy storage system in autonomous dcmicrogrid,” IEEE hrunsuwtions on Inxustriul Elywtroniws, vol. 64, no. 7, pp. 5930–5941,July 2017.[29] S. K. Kollimalla, M. K. Mishra, A. Ukil, and H. B. Gooi, “Dc grid voltage regulationusing new hess control strategy,” IEEE hrunsuwtions on gustuinuvly Enyrgy, vol. 8, no. 2,pp. 772–781, April 2017.[30] C. H. Rivetta, A. Emadi, G. A. Williamson, R. Jayabalan, and B. Fahimi, “Analysisand control of a buck dc-dc converter operating with constant power load in sea andundersea vehicles,” IEEE hrunsuwtions on Inxustry Appliwutions, vol. 42, no. 2, pp.559–572, March 2006.[31] W. Du, J. Zhang, Y. Zhang, and Z. Qian, “Stability criterion for cascaded system withconstant power load,” IEEE hrunsuwtions on dowyr Elywtroniws, vol. 28, no. 4, pp.1843–1851, April 2013.[32] I. G. Zurbriggen, M. Anun, and M. Ordonez, “Dual-loop geometric-based control ofbuck converters,” in FDEI IEEE Appliyx dowyr Elywtroniws Wonfyrynwy unx Efiposition(AdEW), March 2015, pp. 392–396.[33] I. G. Zurbriggen, M. Ordonez, and M. A. Bianchi, “Dual-loop geometric-based control ofboost converters,” in FDEI IEEE Enyrgy Wonvyrsion Wongryss unx Efiposition (EWWE),Sept 2015, pp. 2072–2077.68Wiwliogrvphy[34] M. A. Bianchi, I. G. Zurbriggen, and M. Ordonez, “Dual-loop geometric-based controlof full-bridge inverters for stand-alone distributed generation systems,” in FDEJ IEEE7th Intyrnutionul gymposium on dowyr Elywtroniws for Distrivutyx Gynyrution gystyms(dEDG), June 2016, pp. 1–5.[35] F. Paz and M. Ordonez, “High-accuracy impedance detection to improve transient sta-bility in microgrids,” IEEE hrunsuwtions on Inxustriul Elywtroniws, vol. 64, no. 10, pp.8167–8176, Oct 2017.[36] A. M. Rahimi, G. A. Williamson, and A. Emadi, “Loop-cancellation technique: A novelnonlinear feedback to overcome the destabilizing effect of constant-power loads,” IEEEhrunsuwtions on Vyhiwulur hywhnology, vol. 59, no. 2, pp. 650–661, Feb 2010.[37] G. Sulligoi, D. Bosich, G. Giadrossi, L. Zhu, M. Cupelli, and A. Monti, “Multiconvertermedium voltage dc power systems on ships: Constant-power loads instability solutionusing linearization via state feedback control,” IEEE hrunsuwtions on gmurt Grix, vol. 5,no. 5, pp. 2543–2552, Sept 2014.[38] J. A. Solsona, S. G. Jorge, and C. A. Busada, “Nonlinear control of a buck converterwhich feeds a constant power load,” IEEE hrunsuwtions on dowyr Elywtroniws, vol. 30,no. 12, pp. 7193–7201, Dec 2015.[39] Q. Xu, C. Zhang, C. Wen, and P. Wang, “A novel composite nonlinear controller forstabilization of constant power load in dc microgrid,” IEEE hrunsuwtions on gmurt Grix,vol. PP, no. 99, pp. 1–1, 2017.[40] I. G. Zurbriggen, M. Ordonez, and M. Anun, “Dynamic physical limits of buck convert-ers: The t0/4 transient benchmark rule,” in FDEG hwyntyAEighth Annuul IEEE Appliyxdowyr Elywtroniws Wonfyrynwy unx Efiposition (AdEW), March 2013, pp. 421–428.69Wiwliogrvphy[41] Y. Zhao, W. Qiao, and D. Ha, “A sliding-mode duty-ratio controller for dc/dc buckconverters with constant power loads,” IEEE hrunsuwtions on Inxustry Appliwutions,vol. 50, no. 2, pp. 1448–1458, March 2014.[42] C. N. Onwuchekwa and A. Kwasinski, “Analysis of boundary control for buck converterswith instantaneous constant-power loads,” IEEE hrunsuwtions on dowyr Elywtroniws,vol. 25, no. 8, pp. 2018–2032, Aug 2010.[43] M. Ordonez, M. Iqbal, and J. Quaicoe, “Selection of a curved switching surface for buckconverters,” IEEE hruns. dowyr Elywtron., vol. 21, no. 4, pp. 1148–1153, 2006.[44] W. Burns and T. Wilson, “A state-trajectory control law for DC-to-DC converters,”IEEE hruns. Ayrosp. Elywtron. gyst., vol. AES-14, no. 1, pp. 2–20, 1978.[45] M. Anun, M. Ordonez, I. Zurbriggen, and G. Oggier, “Circular switching surface tech-nique: High-performance constant power load stabilization for electric vehicle systems,”dowyr Elywtroniws, IEEE hrunsuwtions on, vol. 30, no. 8, pp. 4560–4572, Aug 2015.[46] R. W. Erickson and D. Maksimovic´, “Fundamentals of power electronics.” KluwerAcademic Publishers, 2004, ch. 9.[47] S. Buso and P. Mattavelli, Digitul wontrol in powyr ylywtroniws. Morgan and Claypool,2006.[48] B. T. Irving and M. M. Jovanovic, “Analysis, design, and performance evaluation ofdroop current-sharing method,” in AdEW FDDD. Fiftyynth Annuul IEEE Appliyx dowyrElywtroniws Wonfyrynwy unx Efiposition (Wut. bo.DDWHG7DIL), vol. 1, 2000, pp. 235–241vol.1.70


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